# Properties

 Label 825.2.c.f.199.4 Level $825$ Weight $2$ Character 825.199 Analytic conductor $6.588$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$825 = 3 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 825.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.58765816676$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.5161984.1 Defining polynomial: $$x^{6} - 4x^{3} + 25x^{2} - 20x + 8$$ x^6 - 4*x^3 + 25*x^2 - 20*x + 8 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 199.4 Root $$-1.75233 + 1.75233i$$ of defining polynomial Character $$\chi$$ $$=$$ 825.199 Dual form 825.2.c.f.199.3

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.36333i q^{2} -1.00000i q^{3} +0.141336 q^{4} +1.36333 q^{6} +2.50466i q^{7} +2.91934i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q+1.36333i q^{2} -1.00000i q^{3} +0.141336 q^{4} +1.36333 q^{6} +2.50466i q^{7} +2.91934i q^{8} -1.00000 q^{9} +1.00000 q^{11} -0.141336i q^{12} -1.14134i q^{13} -3.41468 q^{14} -3.69735 q^{16} +7.64600i q^{17} -1.36333i q^{18} -1.77801 q^{19} +2.50466 q^{21} +1.36333i q^{22} -1.41468i q^{23} +2.91934 q^{24} +1.55602 q^{26} +1.00000i q^{27} +0.354000i q^{28} +0.726656 q^{29} +2.85866 q^{31} +0.797984i q^{32} -1.00000i q^{33} -10.4240 q^{34} -0.141336 q^{36} -8.42401i q^{37} -2.42401i q^{38} -1.14134 q^{39} +0.636672 q^{41} +3.41468i q^{42} +12.6974i q^{43} +0.141336 q^{44} +1.92867 q^{46} +6.14134i q^{47} +3.69735i q^{48} +0.726656 q^{49} +7.64600 q^{51} -0.161312i q^{52} +12.0187i q^{53} -1.36333 q^{54} -7.31198 q^{56} +1.77801i q^{57} +0.990671i q^{58} +3.41468 q^{59} +4.59465 q^{61} +3.89730i q^{62} -2.50466i q^{63} -8.48262 q^{64} +1.36333 q^{66} -9.32131i q^{67} +1.08066i q^{68} -1.41468 q^{69} +5.85866 q^{71} -2.91934i q^{72} -7.55602i q^{73} +11.4847 q^{74} -0.251297 q^{76} +2.50466i q^{77} -1.55602i q^{78} -6.91934 q^{79} +1.00000 q^{81} +0.867993i q^{82} +6.17997i q^{83} +0.354000 q^{84} -17.3107 q^{86} -0.726656i q^{87} +2.91934i q^{88} -3.45331 q^{89} +2.85866 q^{91} -0.199945i q^{92} -2.85866i q^{93} -8.37266 q^{94} +0.797984 q^{96} -19.4626i q^{97} +0.990671i q^{98} -1.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 16 q^{4} + 4 q^{6} - 6 q^{9}+O(q^{10})$$ 6 * q - 16 * q^4 + 4 * q^6 - 6 * q^9 $$6 q - 16 q^{4} + 4 q^{6} - 6 q^{9} + 6 q^{11} - 12 q^{14} + 20 q^{16} + 2 q^{19} - 6 q^{21} - 12 q^{24} - 16 q^{26} - 4 q^{29} + 34 q^{31} - 12 q^{34} + 16 q^{36} + 10 q^{39} + 8 q^{41} - 16 q^{44} - 60 q^{46} - 4 q^{49} + 8 q^{51} - 4 q^{54} - 44 q^{56} + 12 q^{59} - 6 q^{61} - 68 q^{64} + 4 q^{66} + 52 q^{71} - 28 q^{74} - 48 q^{76} - 12 q^{79} + 6 q^{81} + 40 q^{84} + 56 q^{86} - 4 q^{89} + 34 q^{91} - 4 q^{94} + 68 q^{96} - 6 q^{99}+O(q^{100})$$ 6 * q - 16 * q^4 + 4 * q^6 - 6 * q^9 + 6 * q^11 - 12 * q^14 + 20 * q^16 + 2 * q^19 - 6 * q^21 - 12 * q^24 - 16 * q^26 - 4 * q^29 + 34 * q^31 - 12 * q^34 + 16 * q^36 + 10 * q^39 + 8 * q^41 - 16 * q^44 - 60 * q^46 - 4 * q^49 + 8 * q^51 - 4 * q^54 - 44 * q^56 + 12 * q^59 - 6 * q^61 - 68 * q^64 + 4 * q^66 + 52 * q^71 - 28 * q^74 - 48 * q^76 - 12 * q^79 + 6 * q^81 + 40 * q^84 + 56 * q^86 - 4 * q^89 + 34 * q^91 - 4 * q^94 + 68 * q^96 - 6 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/825\mathbb{Z}\right)^\times$$.

 $$n$$ $$376$$ $$551$$ $$727$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.36333i 0.964019i 0.876166 + 0.482009i $$0.160093\pi$$
−0.876166 + 0.482009i $$0.839907\pi$$
$$3$$ − 1.00000i − 0.577350i
$$4$$ 0.141336 0.0706681
$$5$$ 0 0
$$6$$ 1.36333 0.556576
$$7$$ 2.50466i 0.946674i 0.880881 + 0.473337i $$0.156951\pi$$
−0.880881 + 0.473337i $$0.843049\pi$$
$$8$$ 2.91934i 1.03214i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 1.00000 0.301511
$$12$$ − 0.141336i − 0.0408002i
$$13$$ − 1.14134i − 0.316550i −0.987395 0.158275i $$-0.949407\pi$$
0.987395 0.158275i $$-0.0505932\pi$$
$$14$$ −3.41468 −0.912612
$$15$$ 0 0
$$16$$ −3.69735 −0.924338
$$17$$ 7.64600i 1.85443i 0.374533 + 0.927214i $$0.377803\pi$$
−0.374533 + 0.927214i $$0.622197\pi$$
$$18$$ − 1.36333i − 0.321340i
$$19$$ −1.77801 −0.407903 −0.203951 0.978981i $$-0.565378\pi$$
−0.203951 + 0.978981i $$0.565378\pi$$
$$20$$ 0 0
$$21$$ 2.50466 0.546563
$$22$$ 1.36333i 0.290663i
$$23$$ − 1.41468i − 0.294981i −0.989063 0.147491i $$-0.952880\pi$$
0.989063 0.147491i $$-0.0471196\pi$$
$$24$$ 2.91934 0.595909
$$25$$ 0 0
$$26$$ 1.55602 0.305160
$$27$$ 1.00000i 0.192450i
$$28$$ 0.354000i 0.0668996i
$$29$$ 0.726656 0.134937 0.0674684 0.997721i $$-0.478508\pi$$
0.0674684 + 0.997721i $$0.478508\pi$$
$$30$$ 0 0
$$31$$ 2.85866 0.513431 0.256716 0.966487i $$-0.417360\pi$$
0.256716 + 0.966487i $$0.417360\pi$$
$$32$$ 0.797984i 0.141065i
$$33$$ − 1.00000i − 0.174078i
$$34$$ −10.4240 −1.78770
$$35$$ 0 0
$$36$$ −0.141336 −0.0235560
$$37$$ − 8.42401i − 1.38490i −0.721467 0.692449i $$-0.756532\pi$$
0.721467 0.692449i $$-0.243468\pi$$
$$38$$ − 2.42401i − 0.393226i
$$39$$ −1.14134 −0.182760
$$40$$ 0 0
$$41$$ 0.636672 0.0994314 0.0497157 0.998763i $$-0.484168\pi$$
0.0497157 + 0.998763i $$0.484168\pi$$
$$42$$ 3.41468i 0.526897i
$$43$$ 12.6974i 1.93633i 0.250317 + 0.968164i $$0.419465\pi$$
−0.250317 + 0.968164i $$0.580535\pi$$
$$44$$ 0.141336 0.0213072
$$45$$ 0 0
$$46$$ 1.92867 0.284367
$$47$$ 6.14134i 0.895806i 0.894082 + 0.447903i $$0.147829\pi$$
−0.894082 + 0.447903i $$0.852171\pi$$
$$48$$ 3.69735i 0.533667i
$$49$$ 0.726656 0.103808
$$50$$ 0 0
$$51$$ 7.64600 1.07065
$$52$$ − 0.161312i − 0.0223700i
$$53$$ 12.0187i 1.65089i 0.564483 + 0.825445i $$0.309076\pi$$
−0.564483 + 0.825445i $$0.690924\pi$$
$$54$$ −1.36333 −0.185525
$$55$$ 0 0
$$56$$ −7.31198 −0.977104
$$57$$ 1.77801i 0.235503i
$$58$$ 0.990671i 0.130082i
$$59$$ 3.41468 0.444553 0.222277 0.974984i $$-0.428651\pi$$
0.222277 + 0.974984i $$0.428651\pi$$
$$60$$ 0 0
$$61$$ 4.59465 0.588285 0.294142 0.955762i $$-0.404966\pi$$
0.294142 + 0.955762i $$0.404966\pi$$
$$62$$ 3.89730i 0.494957i
$$63$$ − 2.50466i − 0.315558i
$$64$$ −8.48262 −1.06033
$$65$$ 0 0
$$66$$ 1.36333 0.167814
$$67$$ − 9.32131i − 1.13878i −0.822068 0.569389i $$-0.807180\pi$$
0.822068 0.569389i $$-0.192820\pi$$
$$68$$ 1.08066i 0.131049i
$$69$$ −1.41468 −0.170307
$$70$$ 0 0
$$71$$ 5.85866 0.695295 0.347648 0.937625i $$-0.386981\pi$$
0.347648 + 0.937625i $$0.386981\pi$$
$$72$$ − 2.91934i − 0.344048i
$$73$$ − 7.55602i − 0.884365i −0.896925 0.442182i $$-0.854204\pi$$
0.896925 0.442182i $$-0.145796\pi$$
$$74$$ 11.4847 1.33507
$$75$$ 0 0
$$76$$ −0.251297 −0.0288257
$$77$$ 2.50466i 0.285433i
$$78$$ − 1.55602i − 0.176184i
$$79$$ −6.91934 −0.778487 −0.389244 0.921135i $$-0.627264\pi$$
−0.389244 + 0.921135i $$0.627264\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0.867993i 0.0958537i
$$83$$ 6.17997i 0.678340i 0.940725 + 0.339170i $$0.110146\pi$$
−0.940725 + 0.339170i $$0.889854\pi$$
$$84$$ 0.354000 0.0386245
$$85$$ 0 0
$$86$$ −17.3107 −1.86666
$$87$$ − 0.726656i − 0.0779058i
$$88$$ 2.91934i 0.311203i
$$89$$ −3.45331 −0.366050 −0.183025 0.983108i $$-0.558589\pi$$
−0.183025 + 0.983108i $$0.558589\pi$$
$$90$$ 0 0
$$91$$ 2.85866 0.299669
$$92$$ − 0.199945i − 0.0208457i
$$93$$ − 2.85866i − 0.296430i
$$94$$ −8.37266 −0.863574
$$95$$ 0 0
$$96$$ 0.797984 0.0814439
$$97$$ − 19.4626i − 1.97613i −0.154032 0.988066i $$-0.549226\pi$$
0.154032 0.988066i $$-0.450774\pi$$
$$98$$ 0.990671i 0.100073i
$$99$$ −1.00000 −0.100504
$$100$$ 0 0
$$101$$ −8.37266 −0.833111 −0.416555 0.909110i $$-0.636763\pi$$
−0.416555 + 0.909110i $$0.636763\pi$$
$$102$$ 10.4240i 1.03213i
$$103$$ − 2.54669i − 0.250933i −0.992098 0.125466i $$-0.959957\pi$$
0.992098 0.125466i $$-0.0400427\pi$$
$$104$$ 3.33195 0.326725
$$105$$ 0 0
$$106$$ −16.3854 −1.59149
$$107$$ − 14.5653i − 1.40808i −0.710158 0.704042i $$-0.751376\pi$$
0.710158 0.704042i $$-0.248624\pi$$
$$108$$ 0.141336i 0.0136001i
$$109$$ −2.41468 −0.231284 −0.115642 0.993291i $$-0.536893\pi$$
−0.115642 + 0.993291i $$0.536893\pi$$
$$110$$ 0 0
$$111$$ −8.42401 −0.799571
$$112$$ − 9.26063i − 0.875047i
$$113$$ − 6.00000i − 0.564433i −0.959351 0.282216i $$-0.908930\pi$$
0.959351 0.282216i $$-0.0910696\pi$$
$$114$$ −2.42401 −0.227029
$$115$$ 0 0
$$116$$ 0.102703 0.00953572
$$117$$ 1.14134i 0.105517i
$$118$$ 4.65533i 0.428558i
$$119$$ −19.1507 −1.75554
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 6.26401i 0.567117i
$$123$$ − 0.636672i − 0.0574068i
$$124$$ 0.404032 0.0362832
$$125$$ 0 0
$$126$$ 3.41468 0.304204
$$127$$ − 15.2020i − 1.34896i −0.738293 0.674480i $$-0.764368\pi$$
0.738293 0.674480i $$-0.235632\pi$$
$$128$$ − 9.96862i − 0.881110i
$$129$$ 12.6974 1.11794
$$130$$ 0 0
$$131$$ 18.7453 1.63779 0.818893 0.573946i $$-0.194588\pi$$
0.818893 + 0.573946i $$0.194588\pi$$
$$132$$ − 0.141336i − 0.0123017i
$$133$$ − 4.45331i − 0.386151i
$$134$$ 12.7080 1.09780
$$135$$ 0 0
$$136$$ −22.3213 −1.91404
$$137$$ 8.74531i 0.747163i 0.927597 + 0.373581i $$0.121870\pi$$
−0.927597 + 0.373581i $$0.878130\pi$$
$$138$$ − 1.92867i − 0.164180i
$$139$$ 22.3013 1.89157 0.945787 0.324787i $$-0.105293\pi$$
0.945787 + 0.324787i $$0.105293\pi$$
$$140$$ 0 0
$$141$$ 6.14134 0.517194
$$142$$ 7.98728i 0.670278i
$$143$$ − 1.14134i − 0.0954433i
$$144$$ 3.69735 0.308113
$$145$$ 0 0
$$146$$ 10.3013 0.852544
$$147$$ − 0.726656i − 0.0599336i
$$148$$ − 1.19062i − 0.0978681i
$$149$$ 15.9287 1.30493 0.652464 0.757820i $$-0.273735\pi$$
0.652464 + 0.757820i $$0.273735\pi$$
$$150$$ 0 0
$$151$$ −9.88665 −0.804564 −0.402282 0.915516i $$-0.631783\pi$$
−0.402282 + 0.915516i $$0.631783\pi$$
$$152$$ − 5.19062i − 0.421015i
$$153$$ − 7.64600i − 0.618143i
$$154$$ −3.41468 −0.275163
$$155$$ 0 0
$$156$$ −0.161312 −0.0129153
$$157$$ − 0.132007i − 0.0105353i −0.999986 0.00526767i $$-0.998323\pi$$
0.999986 0.00526767i $$-0.00167676\pi$$
$$158$$ − 9.43334i − 0.750476i
$$159$$ 12.0187 0.953142
$$160$$ 0 0
$$161$$ 3.54330 0.279251
$$162$$ 1.36333i 0.107113i
$$163$$ 4.31198i 0.337740i 0.985638 + 0.168870i $$0.0540118\pi$$
−0.985638 + 0.168870i $$0.945988\pi$$
$$164$$ 0.0899847 0.00702663
$$165$$ 0 0
$$166$$ −8.42533 −0.653932
$$167$$ − 11.2920i − 0.873801i −0.899510 0.436901i $$-0.856076\pi$$
0.899510 0.436901i $$-0.143924\pi$$
$$168$$ 7.31198i 0.564131i
$$169$$ 11.6974 0.899796
$$170$$ 0 0
$$171$$ 1.77801 0.135968
$$172$$ 1.79459i 0.136837i
$$173$$ − 21.6460i − 1.64571i −0.568248 0.822857i $$-0.692379\pi$$
0.568248 0.822857i $$-0.307621\pi$$
$$174$$ 0.990671 0.0751026
$$175$$ 0 0
$$176$$ −3.69735 −0.278698
$$177$$ − 3.41468i − 0.256663i
$$178$$ − 4.70800i − 0.352879i
$$179$$ −16.1413 −1.20646 −0.603230 0.797567i $$-0.706120\pi$$
−0.603230 + 0.797567i $$0.706120\pi$$
$$180$$ 0 0
$$181$$ 20.7546 1.54268 0.771340 0.636423i $$-0.219587\pi$$
0.771340 + 0.636423i $$0.219587\pi$$
$$182$$ 3.89730i 0.288887i
$$183$$ − 4.59465i − 0.339646i
$$184$$ 4.12994 0.304463
$$185$$ 0 0
$$186$$ 3.89730 0.285764
$$187$$ 7.64600i 0.559131i
$$188$$ 0.867993i 0.0633049i
$$189$$ −2.50466 −0.182188
$$190$$ 0 0
$$191$$ 3.31198 0.239646 0.119823 0.992795i $$-0.461767\pi$$
0.119823 + 0.992795i $$0.461767\pi$$
$$192$$ 8.48262i 0.612180i
$$193$$ 2.13201i 0.153465i 0.997052 + 0.0767326i $$0.0244488\pi$$
−0.997052 + 0.0767326i $$0.975551\pi$$
$$194$$ 26.5340 1.90503
$$195$$ 0 0
$$196$$ 0.102703 0.00733591
$$197$$ 17.9287i 1.27737i 0.769470 + 0.638683i $$0.220520\pi$$
−0.769470 + 0.638683i $$0.779480\pi$$
$$198$$ − 1.36333i − 0.0968875i
$$199$$ −15.1600 −1.07466 −0.537332 0.843371i $$-0.680568\pi$$
−0.537332 + 0.843371i $$0.680568\pi$$
$$200$$ 0 0
$$201$$ −9.32131 −0.657474
$$202$$ − 11.4147i − 0.803134i
$$203$$ 1.82003i 0.127741i
$$204$$ 1.08066 0.0756611
$$205$$ 0 0
$$206$$ 3.47197 0.241904
$$207$$ 1.41468i 0.0983270i
$$208$$ 4.21992i 0.292599i
$$209$$ −1.77801 −0.122987
$$210$$ 0 0
$$211$$ −13.3213 −0.917076 −0.458538 0.888675i $$-0.651627\pi$$
−0.458538 + 0.888675i $$0.651627\pi$$
$$212$$ 1.69867i 0.116665i
$$213$$ − 5.85866i − 0.401429i
$$214$$ 19.8573 1.35742
$$215$$ 0 0
$$216$$ −2.91934 −0.198636
$$217$$ 7.15999i 0.486052i
$$218$$ − 3.29200i − 0.222962i
$$219$$ −7.55602 −0.510588
$$220$$ 0 0
$$221$$ 8.72666 0.587018
$$222$$ − 11.4847i − 0.770802i
$$223$$ 12.2534i 0.820546i 0.911963 + 0.410273i $$0.134567\pi$$
−0.911963 + 0.410273i $$0.865433\pi$$
$$224$$ −1.99868 −0.133543
$$225$$ 0 0
$$226$$ 8.17997 0.544123
$$227$$ 8.74531i 0.580447i 0.956959 + 0.290223i $$0.0937296\pi$$
−0.956959 + 0.290223i $$0.906270\pi$$
$$228$$ 0.251297i 0.0166425i
$$229$$ 11.4626 0.757473 0.378736 0.925505i $$-0.376359\pi$$
0.378736 + 0.925505i $$0.376359\pi$$
$$230$$ 0 0
$$231$$ 2.50466 0.164795
$$232$$ 2.12136i 0.139274i
$$233$$ − 7.48469i − 0.490338i −0.969480 0.245169i $$-0.921157\pi$$
0.969480 0.245169i $$-0.0788435\pi$$
$$234$$ −1.55602 −0.101720
$$235$$ 0 0
$$236$$ 0.482618 0.0314157
$$237$$ 6.91934i 0.449460i
$$238$$ − 26.1086i − 1.69237i
$$239$$ −12.9066 −0.834860 −0.417430 0.908709i $$-0.637069\pi$$
−0.417430 + 0.908709i $$0.637069\pi$$
$$240$$ 0 0
$$241$$ −16.8773 −1.08716 −0.543582 0.839356i $$-0.682932\pi$$
−0.543582 + 0.839356i $$0.682932\pi$$
$$242$$ 1.36333i 0.0876381i
$$243$$ − 1.00000i − 0.0641500i
$$244$$ 0.649390 0.0415729
$$245$$ 0 0
$$246$$ 0.867993 0.0553412
$$247$$ 2.02930i 0.129122i
$$248$$ 8.34542i 0.529935i
$$249$$ 6.17997 0.391640
$$250$$ 0 0
$$251$$ 2.28267 0.144081 0.0720405 0.997402i $$-0.477049\pi$$
0.0720405 + 0.997402i $$0.477049\pi$$
$$252$$ − 0.354000i − 0.0222999i
$$253$$ − 1.41468i − 0.0889401i
$$254$$ 20.7253 1.30042
$$255$$ 0 0
$$256$$ −3.37473 −0.210920
$$257$$ − 26.8667i − 1.67590i −0.545749 0.837949i $$-0.683755\pi$$
0.545749 0.837949i $$-0.316245\pi$$
$$258$$ 17.3107i 1.07771i
$$259$$ 21.0993 1.31105
$$260$$ 0 0
$$261$$ −0.726656 −0.0449789
$$262$$ 25.5560i 1.57886i
$$263$$ − 12.0187i − 0.741102i −0.928812 0.370551i $$-0.879169\pi$$
0.928812 0.370551i $$-0.120831\pi$$
$$264$$ 2.91934 0.179673
$$265$$ 0 0
$$266$$ 6.07133 0.372257
$$267$$ 3.45331i 0.211339i
$$268$$ − 1.31744i − 0.0804753i
$$269$$ 14.3854 0.877092 0.438546 0.898709i $$-0.355494\pi$$
0.438546 + 0.898709i $$0.355494\pi$$
$$270$$ 0 0
$$271$$ 28.6553 1.74069 0.870344 0.492445i $$-0.163897\pi$$
0.870344 + 0.492445i $$0.163897\pi$$
$$272$$ − 28.2700i − 1.71412i
$$273$$ − 2.85866i − 0.173014i
$$274$$ −11.9227 −0.720279
$$275$$ 0 0
$$276$$ −0.199945 −0.0120353
$$277$$ − 29.1600i − 1.75205i −0.482262 0.876027i $$-0.660185\pi$$
0.482262 0.876027i $$-0.339815\pi$$
$$278$$ 30.4040i 1.82351i
$$279$$ −2.85866 −0.171144
$$280$$ 0 0
$$281$$ −5.36333 −0.319949 −0.159975 0.987121i $$-0.551141\pi$$
−0.159975 + 0.987121i $$0.551141\pi$$
$$282$$ 8.37266i 0.498584i
$$283$$ 10.0420i 0.596936i 0.954420 + 0.298468i $$0.0964757\pi$$
−0.954420 + 0.298468i $$0.903524\pi$$
$$284$$ 0.828041 0.0491352
$$285$$ 0 0
$$286$$ 1.55602 0.0920091
$$287$$ 1.59465i 0.0941292i
$$288$$ − 0.797984i − 0.0470216i
$$289$$ −41.4613 −2.43890
$$290$$ 0 0
$$291$$ −19.4626 −1.14092
$$292$$ − 1.06794i − 0.0624963i
$$293$$ − 20.3540i − 1.18909i −0.804061 0.594547i $$-0.797332\pi$$
0.804061 0.594547i $$-0.202668\pi$$
$$294$$ 0.990671 0.0577771
$$295$$ 0 0
$$296$$ 24.5926 1.42941
$$297$$ 1.00000i 0.0580259i
$$298$$ 21.7160i 1.25797i
$$299$$ −1.61462 −0.0933762
$$300$$ 0 0
$$301$$ −31.8026 −1.83307
$$302$$ − 13.4787i − 0.775615i
$$303$$ 8.37266i 0.480997i
$$304$$ 6.57392 0.377040
$$305$$ 0 0
$$306$$ 10.4240 0.595901
$$307$$ 25.6226i 1.46236i 0.682184 + 0.731181i $$0.261030\pi$$
−0.682184 + 0.731181i $$0.738970\pi$$
$$308$$ 0.354000i 0.0201710i
$$309$$ −2.54669 −0.144876
$$310$$ 0 0
$$311$$ 11.9414 0.677134 0.338567 0.940942i $$-0.390058\pi$$
0.338567 + 0.940942i $$0.390058\pi$$
$$312$$ − 3.33195i − 0.188635i
$$313$$ − 17.1986i − 0.972124i −0.873924 0.486062i $$-0.838433\pi$$
0.873924 0.486062i $$-0.161567\pi$$
$$314$$ 0.179969 0.0101563
$$315$$ 0 0
$$316$$ −0.977953 −0.0550142
$$317$$ − 22.7453i − 1.27750i −0.769413 0.638752i $$-0.779451\pi$$
0.769413 0.638752i $$-0.220549\pi$$
$$318$$ 16.3854i 0.918846i
$$319$$ 0.726656 0.0406850
$$320$$ 0 0
$$321$$ −14.5653 −0.812958
$$322$$ 4.83068i 0.269203i
$$323$$ − 13.5946i − 0.756427i
$$324$$ 0.141336 0.00785201
$$325$$ 0 0
$$326$$ −5.87864 −0.325588
$$327$$ 2.41468i 0.133532i
$$328$$ 1.85866i 0.102628i
$$329$$ −15.3820 −0.848036
$$330$$ 0 0
$$331$$ 25.2733 1.38915 0.694574 0.719421i $$-0.255593\pi$$
0.694574 + 0.719421i $$0.255593\pi$$
$$332$$ 0.873453i 0.0479370i
$$333$$ 8.42401i 0.461633i
$$334$$ 15.3947 0.842361
$$335$$ 0 0
$$336$$ −9.26063 −0.505209
$$337$$ − 17.6880i − 0.963528i −0.876301 0.481764i $$-0.839996\pi$$
0.876301 0.481764i $$-0.160004\pi$$
$$338$$ 15.9473i 0.867420i
$$339$$ −6.00000 −0.325875
$$340$$ 0 0
$$341$$ 2.85866 0.154805
$$342$$ 2.42401i 0.131075i
$$343$$ 19.3527i 1.04495i
$$344$$ −37.0679 −1.99857
$$345$$ 0 0
$$346$$ 29.5106 1.58650
$$347$$ 20.1027i 1.07917i 0.841931 + 0.539585i $$0.181419\pi$$
−0.841931 + 0.539585i $$0.818581\pi$$
$$348$$ − 0.102703i − 0.00550545i
$$349$$ 10.9907 0.588317 0.294159 0.955757i $$-0.404961\pi$$
0.294159 + 0.955757i $$0.404961\pi$$
$$350$$ 0 0
$$351$$ 1.14134 0.0609200
$$352$$ 0.797984i 0.0425327i
$$353$$ − 16.7267i − 0.890270i −0.895463 0.445135i $$-0.853156\pi$$
0.895463 0.445135i $$-0.146844\pi$$
$$354$$ 4.65533 0.247428
$$355$$ 0 0
$$356$$ −0.488078 −0.0258681
$$357$$ 19.1507i 1.01356i
$$358$$ − 22.0059i − 1.16305i
$$359$$ 13.5933 0.717429 0.358714 0.933447i $$-0.383215\pi$$
0.358714 + 0.933447i $$0.383215\pi$$
$$360$$ 0 0
$$361$$ −15.8387 −0.833615
$$362$$ 28.2954i 1.48717i
$$363$$ − 1.00000i − 0.0524864i
$$364$$ 0.404032 0.0211771
$$365$$ 0 0
$$366$$ 6.26401 0.327425
$$367$$ 28.9987i 1.51372i 0.653578 + 0.756859i $$0.273267\pi$$
−0.653578 + 0.756859i $$0.726733\pi$$
$$368$$ 5.23057i 0.272662i
$$369$$ −0.636672 −0.0331438
$$370$$ 0 0
$$371$$ −30.1027 −1.56285
$$372$$ − 0.404032i − 0.0209481i
$$373$$ 17.8867i 0.926136i 0.886323 + 0.463068i $$0.153251\pi$$
−0.886323 + 0.463068i $$0.846749\pi$$
$$374$$ −10.4240 −0.539013
$$375$$ 0 0
$$376$$ −17.9287 −0.924601
$$377$$ − 0.829359i − 0.0427142i
$$378$$ − 3.41468i − 0.175632i
$$379$$ −23.7839 −1.22170 −0.610850 0.791747i $$-0.709172\pi$$
−0.610850 + 0.791747i $$0.709172\pi$$
$$380$$ 0 0
$$381$$ −15.2020 −0.778823
$$382$$ 4.51531i 0.231023i
$$383$$ 31.7360i 1.62163i 0.585300 + 0.810817i $$0.300977\pi$$
−0.585300 + 0.810817i $$0.699023\pi$$
$$384$$ −9.96862 −0.508709
$$385$$ 0 0
$$386$$ −2.90663 −0.147943
$$387$$ − 12.6974i − 0.645443i
$$388$$ − 2.75077i − 0.139649i
$$389$$ −25.7546 −1.30581 −0.652906 0.757439i $$-0.726450\pi$$
−0.652906 + 0.757439i $$0.726450\pi$$
$$390$$ 0 0
$$391$$ 10.8166 0.547021
$$392$$ 2.12136i 0.107145i
$$393$$ − 18.7453i − 0.945576i
$$394$$ −24.4427 −1.23140
$$395$$ 0 0
$$396$$ −0.141336 −0.00710241
$$397$$ − 3.03863i − 0.152505i −0.997089 0.0762523i $$-0.975705\pi$$
0.997089 0.0762523i $$-0.0242954\pi$$
$$398$$ − 20.6680i − 1.03600i
$$399$$ −4.45331 −0.222945
$$400$$ 0 0
$$401$$ −18.1800 −0.907864 −0.453932 0.891036i $$-0.649979\pi$$
−0.453932 + 0.891036i $$0.649979\pi$$
$$402$$ − 12.7080i − 0.633817i
$$403$$ − 3.26270i − 0.162526i
$$404$$ −1.18336 −0.0588743
$$405$$ 0 0
$$406$$ −2.48130 −0.123145
$$407$$ − 8.42401i − 0.417563i
$$408$$ 22.3213i 1.10507i
$$409$$ 20.5946 1.01834 0.509170 0.860666i $$-0.329952\pi$$
0.509170 + 0.860666i $$0.329952\pi$$
$$410$$ 0 0
$$411$$ 8.74531 0.431375
$$412$$ − 0.359939i − 0.0177329i
$$413$$ 8.55263i 0.420847i
$$414$$ −1.92867 −0.0947891
$$415$$ 0 0
$$416$$ 0.910768 0.0446541
$$417$$ − 22.3013i − 1.09210i
$$418$$ − 2.42401i − 0.118562i
$$419$$ −32.7253 −1.59874 −0.799369 0.600841i $$-0.794833\pi$$
−0.799369 + 0.600841i $$0.794833\pi$$
$$420$$ 0 0
$$421$$ 32.5640 1.58707 0.793537 0.608522i $$-0.208237\pi$$
0.793537 + 0.608522i $$0.208237\pi$$
$$422$$ − 18.1613i − 0.884079i
$$423$$ − 6.14134i − 0.298602i
$$424$$ −35.0866 −1.70396
$$425$$ 0 0
$$426$$ 7.98728 0.386985
$$427$$ 11.5081i 0.556914i
$$428$$ − 2.05861i − 0.0995066i
$$429$$ −1.14134 −0.0551042
$$430$$ 0 0
$$431$$ 1.82003 0.0876678 0.0438339 0.999039i $$-0.486043\pi$$
0.0438339 + 0.999039i $$0.486043\pi$$
$$432$$ − 3.69735i − 0.177889i
$$433$$ 29.8280i 1.43344i 0.697359 + 0.716722i $$0.254358\pi$$
−0.697359 + 0.716722i $$0.745642\pi$$
$$434$$ −9.76142 −0.468563
$$435$$ 0 0
$$436$$ −0.341281 −0.0163444
$$437$$ 2.51531i 0.120324i
$$438$$ − 10.3013i − 0.492217i
$$439$$ −18.2220 −0.869688 −0.434844 0.900506i $$-0.643197\pi$$
−0.434844 + 0.900506i $$0.643197\pi$$
$$440$$ 0 0
$$441$$ −0.726656 −0.0346027
$$442$$ 11.8973i 0.565897i
$$443$$ 22.2627i 1.05773i 0.848705 + 0.528866i $$0.177383\pi$$
−0.848705 + 0.528866i $$0.822617\pi$$
$$444$$ −1.19062 −0.0565042
$$445$$ 0 0
$$446$$ −16.7054 −0.791022
$$447$$ − 15.9287i − 0.753400i
$$448$$ − 21.2461i − 1.00378i
$$449$$ −25.8760 −1.22116 −0.610582 0.791953i $$-0.709064\pi$$
−0.610582 + 0.791953i $$0.709064\pi$$
$$450$$ 0 0
$$451$$ 0.636672 0.0299797
$$452$$ − 0.848017i − 0.0398874i
$$453$$ 9.88665i 0.464515i
$$454$$ −11.9227 −0.559562
$$455$$ 0 0
$$456$$ −5.19062 −0.243073
$$457$$ − 10.4626i − 0.489422i −0.969596 0.244711i $$-0.921307\pi$$
0.969596 0.244711i $$-0.0786930\pi$$
$$458$$ 15.6273i 0.730218i
$$459$$ −7.64600 −0.356885
$$460$$ 0 0
$$461$$ 11.8387 0.551383 0.275691 0.961246i $$-0.411093\pi$$
0.275691 + 0.961246i $$0.411093\pi$$
$$462$$ 3.41468i 0.158865i
$$463$$ − 27.7546i − 1.28987i −0.764238 0.644934i $$-0.776885\pi$$
0.764238 0.644934i $$-0.223115\pi$$
$$464$$ −2.68670 −0.124727
$$465$$ 0 0
$$466$$ 10.2041 0.472695
$$467$$ 20.8294i 0.963868i 0.876208 + 0.481934i $$0.160065\pi$$
−0.876208 + 0.481934i $$0.839935\pi$$
$$468$$ 0.161312i 0.00745665i
$$469$$ 23.3467 1.07805
$$470$$ 0 0
$$471$$ −0.132007 −0.00608258
$$472$$ 9.96862i 0.458843i
$$473$$ 12.6974i 0.583825i
$$474$$ −9.43334 −0.433288
$$475$$ 0 0
$$476$$ −2.70668 −0.124061
$$477$$ − 12.0187i − 0.550297i
$$478$$ − 17.5960i − 0.804821i
$$479$$ −2.64939 −0.121054 −0.0605269 0.998167i $$-0.519278\pi$$
−0.0605269 + 0.998167i $$0.519278\pi$$
$$480$$ 0 0
$$481$$ −9.61462 −0.438389
$$482$$ − 23.0093i − 1.04805i
$$483$$ − 3.54330i − 0.161226i
$$484$$ 0.141336 0.00642437
$$485$$ 0 0
$$486$$ 1.36333 0.0618418
$$487$$ 40.5360i 1.83686i 0.395580 + 0.918432i $$0.370544\pi$$
−0.395580 + 0.918432i $$0.629456\pi$$
$$488$$ 13.4134i 0.607194i
$$489$$ 4.31198 0.194994
$$490$$ 0 0
$$491$$ −16.1214 −0.727547 −0.363773 0.931487i $$-0.618512\pi$$
−0.363773 + 0.931487i $$0.618512\pi$$
$$492$$ − 0.0899847i − 0.00405682i
$$493$$ 5.55602i 0.250230i
$$494$$ −2.76661 −0.124476
$$495$$ 0 0
$$496$$ −10.5695 −0.474584
$$497$$ 14.6740i 0.658218i
$$498$$ 8.42533i 0.377548i
$$499$$ 28.0666 1.25643 0.628217 0.778038i $$-0.283785\pi$$
0.628217 + 0.778038i $$0.283785\pi$$
$$500$$ 0 0
$$501$$ −11.2920 −0.504489
$$502$$ 3.11203i 0.138897i
$$503$$ − 0.906626i − 0.0404245i −0.999796 0.0202122i $$-0.993566\pi$$
0.999796 0.0202122i $$-0.00643419\pi$$
$$504$$ 7.31198 0.325701
$$505$$ 0 0
$$506$$ 1.92867 0.0857400
$$507$$ − 11.6974i − 0.519498i
$$508$$ − 2.14859i − 0.0953284i
$$509$$ 14.3013 0.633895 0.316948 0.948443i $$-0.397342\pi$$
0.316948 + 0.948443i $$0.397342\pi$$
$$510$$ 0 0
$$511$$ 18.9253 0.837205
$$512$$ − 24.5381i − 1.08444i
$$513$$ − 1.77801i − 0.0785010i
$$514$$ 36.6281 1.61560
$$515$$ 0 0
$$516$$ 1.79459 0.0790026
$$517$$ 6.14134i 0.270096i
$$518$$ 28.7653i 1.26387i
$$519$$ −21.6460 −0.950154
$$520$$ 0 0
$$521$$ 20.1214 0.881533 0.440766 0.897622i $$-0.354707\pi$$
0.440766 + 0.897622i $$0.354707\pi$$
$$522$$ − 0.990671i − 0.0433605i
$$523$$ 12.8459i 0.561714i 0.959750 + 0.280857i $$0.0906187\pi$$
−0.959750 + 0.280857i $$0.909381\pi$$
$$524$$ 2.64939 0.115739
$$525$$ 0 0
$$526$$ 16.3854 0.714436
$$527$$ 21.8573i 0.952121i
$$528$$ 3.69735i 0.160907i
$$529$$ 20.9987 0.912986
$$530$$ 0 0
$$531$$ −3.41468 −0.148184
$$532$$ − 0.629414i − 0.0272886i
$$533$$ − 0.726656i − 0.0314750i
$$534$$ −4.70800 −0.203735
$$535$$ 0 0
$$536$$ 27.2121 1.17538
$$537$$ 16.1413i 0.696550i
$$538$$ 19.6120i 0.845533i
$$539$$ 0.726656 0.0312993
$$540$$ 0 0
$$541$$ 17.6040 0.756854 0.378427 0.925631i $$-0.376465\pi$$
0.378427 + 0.925631i $$0.376465\pi$$
$$542$$ 39.0666i 1.67805i
$$543$$ − 20.7546i − 0.890667i
$$544$$ −6.10138 −0.261595
$$545$$ 0 0
$$546$$ 3.89730 0.166789
$$547$$ 19.9873i 0.854594i 0.904111 + 0.427297i $$0.140534\pi$$
−0.904111 + 0.427297i $$0.859466\pi$$
$$548$$ 1.23603i 0.0528005i
$$549$$ −4.59465 −0.196095
$$550$$ 0 0
$$551$$ −1.29200 −0.0550411
$$552$$ − 4.12994i − 0.175782i
$$553$$ − 17.3306i − 0.736974i
$$554$$ 39.7546 1.68901
$$555$$ 0 0
$$556$$ 3.15198 0.133674
$$557$$ − 41.5933i − 1.76237i −0.472775 0.881183i $$-0.656748\pi$$
0.472775 0.881183i $$-0.343252\pi$$
$$558$$ − 3.89730i − 0.164986i
$$559$$ 14.4919 0.612944
$$560$$ 0 0
$$561$$ 7.64600 0.322814
$$562$$ − 7.31198i − 0.308437i
$$563$$ − 2.28267i − 0.0962032i −0.998842 0.0481016i $$-0.984683\pi$$
0.998842 0.0481016i $$-0.0153171\pi$$
$$564$$ 0.867993 0.0365491
$$565$$ 0 0
$$566$$ −13.6906 −0.575458
$$567$$ 2.50466i 0.105186i
$$568$$ 17.1035i 0.717645i
$$569$$ 2.37266 0.0994670 0.0497335 0.998763i $$-0.484163\pi$$
0.0497335 + 0.998763i $$0.484163\pi$$
$$570$$ 0 0
$$571$$ −31.9053 −1.33520 −0.667598 0.744522i $$-0.732677\pi$$
−0.667598 + 0.744522i $$0.732677\pi$$
$$572$$ − 0.161312i − 0.00674479i
$$573$$ − 3.31198i − 0.138360i
$$574$$ −2.17403 −0.0907423
$$575$$ 0 0
$$576$$ 8.48262 0.353442
$$577$$ 3.28267i 0.136659i 0.997663 + 0.0683297i $$0.0217670\pi$$
−0.997663 + 0.0683297i $$0.978233\pi$$
$$578$$ − 56.5254i − 2.35115i
$$579$$ 2.13201 0.0886032
$$580$$ 0 0
$$581$$ −15.4787 −0.642167
$$582$$ − 26.5340i − 1.09987i
$$583$$ 12.0187i 0.497762i
$$584$$ 22.0586 0.912792
$$585$$ 0 0
$$586$$ 27.7492 1.14631
$$587$$ 26.3400i 1.08717i 0.839355 + 0.543583i $$0.182933\pi$$
−0.839355 + 0.543583i $$0.817067\pi$$
$$588$$ − 0.102703i − 0.00423539i
$$589$$ −5.08273 −0.209430
$$590$$ 0 0
$$591$$ 17.9287 0.737487
$$592$$ 31.1465i 1.28011i
$$593$$ 32.9694i 1.35389i 0.736034 + 0.676945i $$0.236697\pi$$
−0.736034 + 0.676945i $$0.763303\pi$$
$$594$$ −1.36333 −0.0559380
$$595$$ 0 0
$$596$$ 2.25130 0.0922167
$$597$$ 15.1600i 0.620457i
$$598$$ − 2.20126i − 0.0900164i
$$599$$ −24.5454 −1.00290 −0.501448 0.865188i $$-0.667199\pi$$
−0.501448 + 0.865188i $$0.667199\pi$$
$$600$$ 0 0
$$601$$ −9.70668 −0.395944 −0.197972 0.980208i $$-0.563435\pi$$
−0.197972 + 0.980208i $$0.563435\pi$$
$$602$$ − 43.3574i − 1.76712i
$$603$$ 9.32131i 0.379593i
$$604$$ −1.39734 −0.0568570
$$605$$ 0 0
$$606$$ −11.4147 −0.463690
$$607$$ 1.73599i 0.0704615i 0.999379 + 0.0352307i $$0.0112166\pi$$
−0.999379 + 0.0352307i $$0.988783\pi$$
$$608$$ − 1.41882i − 0.0575408i
$$609$$ 1.82003 0.0737514
$$610$$ 0 0
$$611$$ 7.00933 0.283567
$$612$$ − 1.08066i − 0.0436829i
$$613$$ − 12.0187i − 0.485429i −0.970098 0.242715i $$-0.921962\pi$$
0.970098 0.242715i $$-0.0780378\pi$$
$$614$$ −34.9321 −1.40974
$$615$$ 0 0
$$616$$ −7.31198 −0.294608
$$617$$ 42.1587i 1.69724i 0.528999 + 0.848622i $$0.322567\pi$$
−0.528999 + 0.848622i $$0.677433\pi$$
$$618$$ − 3.47197i − 0.139663i
$$619$$ −9.03863 −0.363293 −0.181647 0.983364i $$-0.558143\pi$$
−0.181647 + 0.983364i $$0.558143\pi$$
$$620$$ 0 0
$$621$$ 1.41468 0.0567691
$$622$$ 16.2800i 0.652770i
$$623$$ − 8.64939i − 0.346530i
$$624$$ 4.21992 0.168932
$$625$$ 0 0
$$626$$ 23.4474 0.937146
$$627$$ 1.77801i 0.0710068i
$$628$$ − 0.0186574i 0 0.000744512i
$$629$$ 64.4100 2.56819
$$630$$ 0 0
$$631$$ 10.9614 0.436365 0.218183 0.975908i $$-0.429987\pi$$
0.218183 + 0.975908i $$0.429987\pi$$
$$632$$ − 20.1999i − 0.803511i
$$633$$ 13.3213i 0.529474i
$$634$$ 31.0093 1.23154
$$635$$ 0 0
$$636$$ 1.69867 0.0673567
$$637$$ − 0.829359i − 0.0328604i
$$638$$ 0.990671i 0.0392211i
$$639$$ −5.85866 −0.231765
$$640$$ 0 0
$$641$$ −44.1587 −1.74416 −0.872081 0.489361i $$-0.837230\pi$$
−0.872081 + 0.489361i $$0.837230\pi$$
$$642$$ − 19.8573i − 0.783707i
$$643$$ − 22.5840i − 0.890626i −0.895375 0.445313i $$-0.853092\pi$$
0.895375 0.445313i $$-0.146908\pi$$
$$644$$ 0.500796 0.0197341
$$645$$ 0 0
$$646$$ 18.5340 0.729209
$$647$$ − 13.3561i − 0.525081i −0.964921 0.262541i $$-0.915440\pi$$
0.964921 0.262541i $$-0.0845604\pi$$
$$648$$ 2.91934i 0.114683i
$$649$$ 3.41468 0.134038
$$650$$ 0 0
$$651$$ 7.15999 0.280622
$$652$$ 0.609438i 0.0238674i
$$653$$ − 37.5933i − 1.47114i −0.677448 0.735570i $$-0.736914\pi$$
0.677448 0.735570i $$-0.263086\pi$$
$$654$$ −3.29200 −0.128727
$$655$$ 0 0
$$656$$ −2.35400 −0.0919082
$$657$$ 7.55602i 0.294788i
$$658$$ − 20.9707i − 0.817523i
$$659$$ −7.00933 −0.273045 −0.136522 0.990637i $$-0.543593\pi$$
−0.136522 + 0.990637i $$0.543593\pi$$
$$660$$ 0 0
$$661$$ 26.6226 1.03550 0.517750 0.855532i $$-0.326770\pi$$
0.517750 + 0.855532i $$0.326770\pi$$
$$662$$ 34.4559i 1.33917i
$$663$$ − 8.72666i − 0.338915i
$$664$$ −18.0415 −0.700144
$$665$$ 0 0
$$666$$ −11.4847 −0.445023
$$667$$ − 1.02799i − 0.0398038i
$$668$$ − 1.59597i − 0.0617498i
$$669$$ 12.2534 0.473743
$$670$$ 0 0
$$671$$ 4.59465 0.177374
$$672$$ 1.99868i 0.0771008i
$$673$$ − 35.6960i − 1.37598i −0.725720 0.687990i $$-0.758493\pi$$
0.725720 0.687990i $$-0.241507\pi$$
$$674$$ 24.1146 0.928859
$$675$$ 0 0
$$676$$ 1.65326 0.0635869
$$677$$ 33.2920i 1.27952i 0.768577 + 0.639758i $$0.220965\pi$$
−0.768577 + 0.639758i $$0.779035\pi$$
$$678$$ − 8.17997i − 0.314150i
$$679$$ 48.7474 1.87075
$$680$$ 0 0
$$681$$ 8.74531 0.335121
$$682$$ 3.89730i 0.149235i
$$683$$ 36.1787i 1.38434i 0.721736 + 0.692169i $$0.243345\pi$$
−0.721736 + 0.692169i $$0.756655\pi$$
$$684$$ 0.251297 0.00960857
$$685$$ 0 0
$$686$$ −26.3841 −1.00735
$$687$$ − 11.4626i − 0.437327i
$$688$$ − 46.9466i − 1.78982i
$$689$$ 13.7173 0.522589
$$690$$ 0 0
$$691$$ 35.3293 1.34399 0.671995 0.740555i $$-0.265438\pi$$
0.671995 + 0.740555i $$0.265438\pi$$
$$692$$ − 3.05936i − 0.116299i
$$693$$ − 2.50466i − 0.0951443i
$$694$$ −27.4066 −1.04034
$$695$$ 0 0
$$696$$ 2.12136 0.0804100
$$697$$ 4.86799i 0.184388i
$$698$$ 14.9839i 0.567149i
$$699$$ −7.48469 −0.283097
$$700$$ 0 0
$$701$$ −49.4006 −1.86584 −0.932918 0.360088i $$-0.882747\pi$$
−0.932918 + 0.360088i $$0.882747\pi$$
$$702$$ 1.55602i 0.0587280i
$$703$$ 14.9780i 0.564904i
$$704$$ −8.48262 −0.319701
$$705$$ 0 0
$$706$$ 22.8039 0.858237
$$707$$ − 20.9707i − 0.788684i
$$708$$ − 0.482618i − 0.0181379i
$$709$$ 11.1027 0.416971 0.208485 0.978025i $$-0.433147\pi$$
0.208485 + 0.978025i $$0.433147\pi$$
$$710$$ 0 0
$$711$$ 6.91934 0.259496
$$712$$ − 10.0814i − 0.377817i
$$713$$ − 4.04409i − 0.151452i
$$714$$ −26.1086 −0.977091
$$715$$ 0 0
$$716$$ −2.28135 −0.0852582
$$717$$ 12.9066i 0.482007i
$$718$$ 18.5322i 0.691614i
$$719$$ 43.9787 1.64013 0.820064 0.572271i $$-0.193938\pi$$
0.820064 + 0.572271i $$0.193938\pi$$
$$720$$ 0 0
$$721$$ 6.37860 0.237551
$$722$$ − 21.5933i − 0.803621i
$$723$$ 16.8773i 0.627674i
$$724$$ 2.93338 0.109018
$$725$$ 0 0
$$726$$ 1.36333 0.0505979
$$727$$ − 44.1880i − 1.63884i −0.573193 0.819421i $$-0.694295\pi$$
0.573193 0.819421i $$-0.305705\pi$$
$$728$$ 8.34542i 0.309302i
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ −97.0840 −3.59078
$$732$$ − 0.649390i − 0.0240021i
$$733$$ 47.3879i 1.75031i 0.483840 + 0.875156i $$0.339242\pi$$
−0.483840 + 0.875156i $$0.660758\pi$$
$$734$$ −39.5347 −1.45925
$$735$$ 0 0
$$736$$ 1.12889 0.0416115
$$737$$ − 9.32131i − 0.343355i
$$738$$ − 0.867993i − 0.0319512i
$$739$$ 13.0407 0.479710 0.239855 0.970809i $$-0.422900\pi$$
0.239855 + 0.970809i $$0.422900\pi$$
$$740$$ 0 0
$$741$$ 2.02930 0.0745484
$$742$$ − 41.0399i − 1.50662i
$$743$$ 14.1800i 0.520213i 0.965580 + 0.260106i $$0.0837576\pi$$
−0.965580 + 0.260106i $$0.916242\pi$$
$$744$$ 8.34542 0.305958
$$745$$ 0 0
$$746$$ −24.3854 −0.892812
$$747$$ − 6.17997i − 0.226113i
$$748$$ 1.08066i 0.0395127i
$$749$$ 36.4813 1.33300
$$750$$ 0 0
$$751$$ 19.1120 0.697408 0.348704 0.937233i $$-0.386622\pi$$
0.348704 + 0.937233i $$0.386622\pi$$
$$752$$ − 22.7067i − 0.828027i
$$753$$ − 2.28267i − 0.0831852i
$$754$$ 1.13069 0.0411773
$$755$$ 0 0
$$756$$ −0.354000 −0.0128748
$$757$$ 24.7347i 0.898997i 0.893281 + 0.449498i $$0.148397\pi$$
−0.893281 + 0.449498i $$0.851603\pi$$
$$758$$ − 32.4253i − 1.17774i
$$759$$ −1.41468 −0.0513496
$$760$$ 0 0
$$761$$ −9.82003 −0.355976 −0.177988 0.984033i $$-0.556959\pi$$
−0.177988 + 0.984033i $$0.556959\pi$$
$$762$$ − 20.7253i − 0.750800i
$$763$$ − 6.04796i − 0.218951i
$$764$$ 0.468102 0.0169353
$$765$$ 0 0
$$766$$ −43.2666 −1.56328
$$767$$ − 3.89730i − 0.140723i
$$768$$ 3.37473i 0.121775i
$$769$$ 15.3026 0.551828 0.275914 0.961182i $$-0.411020\pi$$
0.275914 + 0.961182i $$0.411020\pi$$
$$770$$ 0 0
$$771$$ −26.8667 −0.967580
$$772$$ 0.301330i 0.0108451i
$$773$$ − 5.36927i − 0.193119i −0.995327 0.0965596i $$-0.969216\pi$$
0.995327 0.0965596i $$-0.0307838\pi$$
$$774$$ 17.3107 0.622219
$$775$$ 0 0
$$776$$ 56.8181 2.03965
$$777$$ − 21.0993i − 0.756934i
$$778$$ − 35.1120i − 1.25883i
$$779$$ −1.13201 −0.0405584
$$780$$ 0 0
$$781$$ 5.85866 0.209639
$$782$$ 14.7466i 0.527338i
$$783$$ 0.726656i 0.0259686i
$$784$$ −2.68670 −0.0959537
$$785$$ 0 0
$$786$$ 25.5560 0.911553
$$787$$ 15.4767i 0.551684i 0.961203 + 0.275842i $$0.0889567\pi$$
−0.961203 + 0.275842i $$0.911043\pi$$
$$788$$ 2.53397i 0.0902689i
$$789$$ −12.0187 −0.427876
$$790$$ 0 0
$$791$$ 15.0280 0.534334
$$792$$ − 2.91934i − 0.103734i
$$793$$ − 5.24404i − 0.186221i
$$794$$ 4.14265 0.147017
$$795$$ 0 0
$$796$$ −2.14265 −0.0759444
$$797$$ − 43.6774i − 1.54713i −0.633716 0.773566i $$-0.718471\pi$$
0.633716 0.773566i $$-0.281529\pi$$
$$798$$ − 6.07133i − 0.214923i
$$799$$ −46.9567 −1.66121
$$800$$ 0 0
$$801$$ 3.45331 0.122017
$$802$$ − 24.7853i − 0.875198i
$$803$$ − 7.55602i − 0.266646i
$$804$$ −1.31744 −0.0464624
$$805$$ 0 0
$$806$$ 4.44813 0.156679
$$807$$ − 14.3854i − 0.506389i
$$808$$ − 24.4427i − 0.859890i
$$809$$ 26.9966 0.949150 0.474575 0.880215i $$-0.342602\pi$$
0.474575 + 0.880215i $$0.342602\pi$$
$$810$$ 0 0
$$811$$ −14.2220 −0.499402 −0.249701 0.968323i $$-0.580332\pi$$
−0.249701 + 0.968323i $$0.580332\pi$$
$$812$$ 0.257236i 0.00902722i
$$813$$ − 28.6553i − 1.00499i
$$814$$ 11.4847 0.402538
$$815$$ 0 0
$$816$$ −28.2700 −0.989646
$$817$$ − 22.5760i − 0.789834i
$$818$$ 28.0773i 0.981699i
$$819$$ −2.85866 −0.0998898
$$820$$ 0 0
$$821$$ 23.7801 0.829930 0.414965 0.909837i $$-0.363794\pi$$
0.414965 + 0.909837i $$0.363794\pi$$
$$822$$ 11.9227i 0.415853i
$$823$$ 1.84934i 0.0644638i 0.999480 + 0.0322319i $$0.0102615\pi$$
−0.999480 + 0.0322319i $$0.989738\pi$$
$$824$$ 7.43466 0.258998
$$825$$ 0 0
$$826$$ −11.6600 −0.405705
$$827$$ 15.6987i 0.545896i 0.962029 + 0.272948i $$0.0879987\pi$$
−0.962029 + 0.272948i $$0.912001\pi$$
$$828$$ 0.199945i 0.00694858i
$$829$$ 4.34128 0.150779 0.0753895 0.997154i $$-0.475980\pi$$
0.0753895 + 0.997154i $$0.475980\pi$$
$$830$$ 0 0
$$831$$ −29.1600 −1.01155
$$832$$ 9.68152i 0.335646i
$$833$$ 5.55602i 0.192505i
$$834$$ 30.4040 1.05281
$$835$$ 0 0
$$836$$ −0.251297 −0.00869128
$$837$$ 2.85866i 0.0988099i
$$838$$ − 44.6154i − 1.54121i
$$839$$ −28.3013 −0.977070 −0.488535 0.872544i $$-0.662469\pi$$
−0.488535 + 0.872544i $$0.662469\pi$$
$$840$$ 0 0
$$841$$ −28.4720 −0.981792
$$842$$ 44.3955i 1.52997i
$$843$$ 5.36333i 0.184723i
$$844$$ −1.88278 −0.0648080
$$845$$ 0 0
$$846$$ 8.37266 0.287858
$$847$$ 2.50466i 0.0860613i
$$848$$ − 44.4372i − 1.52598i
$$849$$ 10.0420 0.344641
$$850$$ 0 0
$$851$$ −11.9173 −0.408519
$$852$$ − 0.828041i − 0.0283682i
$$853$$ 35.7653i 1.22458i 0.790633 + 0.612290i $$0.209752\pi$$
−0.790633 + 0.612290i $$0.790248\pi$$
$$854$$ −15.6893 −0.536875
$$855$$ 0 0
$$856$$ 42.5213 1.45335
$$857$$ − 24.7513i − 0.845487i −0.906249 0.422743i $$-0.861067\pi$$
0.906249 0.422743i $$-0.138933\pi$$
$$858$$ − 1.55602i − 0.0531215i
$$859$$ 10.8039 0.368625 0.184313 0.982868i $$-0.440994\pi$$
0.184313 + 0.982868i $$0.440994\pi$$
$$860$$ 0 0
$$861$$ 1.59465 0.0543455
$$862$$ 2.48130i 0.0845134i
$$863$$ 16.6027i 0.565161i 0.959244 + 0.282581i $$0.0911904\pi$$
−0.959244 + 0.282581i $$0.908810\pi$$
$$864$$ −0.797984 −0.0271480
$$865$$ 0 0
$$866$$ −40.6654 −1.38187
$$867$$ 41.4613i 1.40810i
$$868$$ 1.01197i 0.0343484i
$$869$$ −6.91934 −0.234723
$$870$$ 0 0
$$871$$ −10.6387 −0.360480
$$872$$ − 7.04928i − 0.238719i
$$873$$ 19.4626i 0.658711i
$$874$$ −3.42920 −0.115994
$$875$$ 0 0
$$876$$ −1.06794 −0.0360823
$$877$$ − 11.1600i − 0.376846i −0.982088 0.188423i $$-0.939662\pi$$
0.982088 0.188423i $$-0.0603376\pi$$
$$878$$ − 24.8426i − 0.838396i
$$879$$ −20.3540 −0.686523
$$880$$ 0 0
$$881$$ 54.5254 1.83701 0.918504 0.395413i $$-0.129398\pi$$
0.918504 + 0.395413i $$0.129398\pi$$
$$882$$ − 0.990671i − 0.0333576i
$$883$$ − 12.7746i − 0.429900i −0.976625 0.214950i $$-0.931041\pi$$
0.976625 0.214950i $$-0.0689589\pi$$
$$884$$ 1.23339 0.0414835
$$885$$ 0 0
$$886$$ −30.3514 −1.01967
$$887$$ − 47.1053i − 1.58164i −0.612049 0.790820i $$-0.709655\pi$$
0.612049 0.790820i $$-0.290345\pi$$
$$888$$ − 24.5926i − 0.825273i
$$889$$ 38.0759 1.27703
$$890$$ 0 0
$$891$$ 1.00000 0.0335013
$$892$$ 1.73184i 0.0579864i
$$893$$ − 10.9193i − 0.365402i
$$894$$ 21.7160 0.726292
$$895$$ 0 0
$$896$$ 24.9681 0.834124
$$897$$ 1.61462i 0.0539108i
$$898$$ − 35.2775i − 1.17722i
$$899$$ 2.07727 0.0692807
$$900$$ 0 0
$$901$$ −91.8947 −3.06146
$$902$$ 0.867993i 0.0289010i
$$903$$ 31.8026i 1.05832i
$$904$$ 17.5161 0.582576
$$905$$ 0 0
$$906$$ −13.4787 −0.447801
$$907$$ − 24.7826i − 0.822894i −0.911434 0.411447i $$-0.865024\pi$$
0.911434 0.411447i $$-0.134976\pi$$
$$908$$ 1.23603i 0.0410191i
$$909$$ 8.37266 0.277704
$$910$$ 0 0
$$911$$ −6.08273 −0.201530 −0.100765 0.994910i $$-0.532129\pi$$
−0.100765 + 0.994910i $$0.532129\pi$$
$$912$$ − 6.57392i − 0.217684i
$$913$$ 6.17997i 0.204527i
$$914$$ 14.2640 0.471812
$$915$$ 0 0
$$916$$ 1.62009 0.0535291
$$917$$ 46.9507i 1.55045i
$$918$$ − 10.4240i − 0.344044i
$$919$$ −17.9953 −0.593610 −0.296805 0.954938i $$-0.595921\pi$$
−0.296805 + 0.954938i $$0.595921\pi$$
$$920$$ 0 0
$$921$$ 25.6226 0.844295
$$922$$ 16.1400i 0.531543i
$$923$$ − 6.68670i − 0.220096i
$$924$$ 0.354000 0.0116457
$$925$$ 0 0
$$926$$ 37.8387 1.24346
$$927$$ 2.54669i 0.0836442i
$$928$$ 0.579860i 0.0190348i
$$929$$ 16.7453 0.549396 0.274698 0.961531i $$-0.411422\pi$$
0.274698 + 0.961531i $$0.411422\pi$$
$$930$$ 0 0
$$931$$ −1.29200 −0.0423436
$$932$$ − 1.05786i − 0.0346513i
$$933$$ − 11.9414i − 0.390944i
$$934$$ −28.3973 −0.929187
$$935$$ 0 0
$$936$$ −3.33195 −0.108908
$$937$$ − 31.5853i − 1.03185i −0.856635 0.515924i $$-0.827449\pi$$
0.856635 0.515924i $$-0.172551\pi$$
$$938$$ 31.8293i 1.03926i
$$939$$ −17.1986 −0.561256
$$940$$ 0 0
$$941$$ −14.8421 −0.483838 −0.241919 0.970296i $$-0.577777\pi$$
−0.241919 + 0.970296i $$0.577777\pi$$
$$942$$ − 0.179969i − 0.00586372i
$$943$$ − 0.900687i − 0.0293304i
$$944$$ −12.6253 −0.410918
$$945$$ 0 0
$$946$$ −17.3107 −0.562818
$$947$$ 2.68802i 0.0873490i 0.999046 + 0.0436745i $$0.0139065\pi$$
−0.999046 + 0.0436745i $$0.986094\pi$$
$$948$$ 0.977953i 0.0317624i
$$949$$ −8.62395 −0.279945
$$950$$ 0 0
$$951$$ −22.7453 −0.737567
$$952$$ − 55.9074i − 1.81197i
$$953$$ − 14.2700i − 0.462249i −0.972924 0.231125i $$-0.925760\pi$$
0.972924 0.231125i $$-0.0742405\pi$$
$$954$$ 16.3854 0.530496
$$955$$ 0 0
$$956$$ −1.82417 −0.0589980
$$957$$ − 0.726656i − 0.0234895i
$$958$$ − 3.61199i − 0.116698i
$$959$$ −21.9041 −0.707320
$$960$$ 0 0
$$961$$ −22.8280 −0.736388
$$962$$ − 13.1079i − 0.422615i
$$963$$ 14.5653i 0.469362i
$$964$$ −2.38538 −0.0768278
$$965$$ 0 0
$$966$$ 4.83068 0.155425
$$967$$ 10.0586i 0.323463i 0.986835 + 0.161732i $$0.0517079\pi$$
−0.986835 + 0.161732i $$0.948292\pi$$
$$968$$ 2.91934i 0.0938313i
$$969$$ −13.5946 −0.436723
$$970$$ 0 0
$$971$$ −13.4520 −0.431695 −0.215848 0.976427i $$-0.569251\pi$$
−0.215848 + 0.976427i $$0.569251\pi$$
$$972$$ − 0.141336i − 0.00453336i
$$973$$ 55.8573i 1.79070i
$$974$$ −55.2639 −1.77077
$$975$$ 0 0
$$976$$ −16.9880 −0.543774
$$977$$ − 32.8294i − 1.05030i −0.851008 0.525152i $$-0.824008\pi$$
0.851008 0.525152i $$-0.175992\pi$$
$$978$$ 5.87864i 0.187978i
$$979$$ −3.45331 −0.110368
$$980$$ 0 0
$$981$$ 2.41468 0.0770948
$$982$$ − 21.9787i − 0.701369i
$$983$$ − 43.7746i − 1.39619i −0.716003 0.698097i $$-0.754031\pi$$
0.716003 0.698097i $$-0.245969\pi$$
$$984$$ 1.85866 0.0592520
$$985$$ 0 0
$$986$$ −7.57467 −0.241227
$$987$$ 15.3820i 0.489614i
$$988$$ 0.286814i 0.00912477i
$$989$$ 17.9627 0.571180
$$990$$ 0 0
$$991$$ 42.1507 1.33896 0.669480 0.742830i $$-0.266517\pi$$
0.669480 + 0.742830i $$0.266517\pi$$
$$992$$ 2.28117i 0.0724271i
$$993$$ − 25.2733i − 0.802025i
$$994$$ −20.0055 −0.634535
$$995$$ 0 0
$$996$$ 0.873453 0.0276764
$$997$$ − 17.4347i − 0.552161i −0.961135 0.276081i $$-0.910964\pi$$
0.961135 0.276081i $$-0.0890357\pi$$
$$998$$ 38.2640i 1.21123i
$$999$$ 8.42401 0.266524
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 825.2.c.f.199.4 6
3.2 odd 2 2475.2.c.q.199.3 6
5.2 odd 4 825.2.a.i.1.2 3
5.3 odd 4 825.2.a.m.1.2 yes 3
5.4 even 2 inner 825.2.c.f.199.3 6
15.2 even 4 2475.2.a.bd.1.2 3
15.8 even 4 2475.2.a.z.1.2 3
15.14 odd 2 2475.2.c.q.199.4 6
55.32 even 4 9075.2.a.cj.1.2 3
55.43 even 4 9075.2.a.cd.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
825.2.a.i.1.2 3 5.2 odd 4
825.2.a.m.1.2 yes 3 5.3 odd 4
825.2.c.f.199.3 6 5.4 even 2 inner
825.2.c.f.199.4 6 1.1 even 1 trivial
2475.2.a.z.1.2 3 15.8 even 4
2475.2.a.bd.1.2 3 15.2 even 4
2475.2.c.q.199.3 6 3.2 odd 2
2475.2.c.q.199.4 6 15.14 odd 2
9075.2.a.cd.1.2 3 55.43 even 4
9075.2.a.cj.1.2 3 55.32 even 4