# Properties

 Label 825.2.c.g.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: 6.0.350464.1 Defining polynomial: $$x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2$$ x^6 - 2*x^5 + 2*x^4 + 2*x^3 + 4*x^2 - 4*x + 2 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 165) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 199.4 Root $$0.403032 + 0.403032i$$ of defining polynomial Character $$\chi$$ $$=$$ 825.199 Dual form 825.2.c.g.199.3

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+0.193937i q^{2} +1.00000i q^{3} +1.96239 q^{4} -0.193937 q^{6} -3.35026i q^{7} +0.768452i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q+0.193937i q^{2} +1.00000i q^{3} +1.96239 q^{4} -0.193937 q^{6} -3.35026i q^{7} +0.768452i q^{8} -1.00000 q^{9} +1.00000 q^{11} +1.96239i q^{12} +2.96239i q^{13} +0.649738 q^{14} +3.77575 q^{16} +4.57452i q^{17} -0.193937i q^{18} +4.31265 q^{19} +3.35026 q^{21} +0.193937i q^{22} -6.70052i q^{23} -0.768452 q^{24} -0.574515 q^{26} -1.00000i q^{27} -6.57452i q^{28} +3.61213 q^{29} +9.92478 q^{31} +2.26916i q^{32} +1.00000i q^{33} -0.887166 q^{34} -1.96239 q^{36} +2.00000i q^{37} +0.836381i q^{38} -2.96239 q^{39} -4.38787 q^{41} +0.649738i q^{42} -9.27504i q^{43} +1.96239 q^{44} +1.29948 q^{46} +9.92478i q^{47} +3.77575i q^{48} -4.22425 q^{49} -4.57452 q^{51} +5.81336i q^{52} +4.70052i q^{53} +0.193937 q^{54} +2.57452 q^{56} +4.31265i q^{57} +0.700523i q^{58} -10.7005 q^{59} -8.70052 q^{61} +1.92478i q^{62} +3.35026i q^{63} +7.11142 q^{64} -0.193937 q^{66} -5.92478i q^{67} +8.97698i q^{68} +6.70052 q^{69} +9.92478 q^{71} -0.768452i q^{72} -7.73813i q^{73} -0.387873 q^{74} +8.46310 q^{76} -3.35026i q^{77} -0.574515i q^{78} -11.5369 q^{79} +1.00000 q^{81} -0.850969i q^{82} +10.8872i q^{83} +6.57452 q^{84} +1.79877 q^{86} +3.61213i q^{87} +0.768452i q^{88} +2.77575 q^{89} +9.92478 q^{91} -13.1490i q^{92} +9.92478i q^{93} -1.92478 q^{94} -2.26916 q^{96} -0.0752228i q^{97} -0.819237i q^{98} -1.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 10 q^{4} - 2 q^{6} - 6 q^{9}+O(q^{10})$$ 6 * q - 10 * q^4 - 2 * q^6 - 6 * q^9 $$6 q - 10 q^{4} - 2 q^{6} - 6 q^{9} + 6 q^{11} + 24 q^{14} + 26 q^{16} - 16 q^{19} + 18 q^{24} + 20 q^{26} + 20 q^{29} + 16 q^{31} + 60 q^{34} + 10 q^{36} + 4 q^{39} - 28 q^{41} - 10 q^{44} + 48 q^{46} - 22 q^{49} - 4 q^{51} + 2 q^{54} - 8 q^{56} - 24 q^{59} - 12 q^{61} - 26 q^{64} - 2 q^{66} + 16 q^{71} - 4 q^{74} + 96 q^{76} - 24 q^{79} + 6 q^{81} + 16 q^{84} - 16 q^{86} + 20 q^{89} + 16 q^{91} + 32 q^{94} - 58 q^{96} - 6 q^{99}+O(q^{100})$$ 6 * q - 10 * q^4 - 2 * q^6 - 6 * q^9 + 6 * q^11 + 24 * q^14 + 26 * q^16 - 16 * q^19 + 18 * q^24 + 20 * q^26 + 20 * q^29 + 16 * q^31 + 60 * q^34 + 10 * q^36 + 4 * q^39 - 28 * q^41 - 10 * q^44 + 48 * q^46 - 22 * q^49 - 4 * q^51 + 2 * q^54 - 8 * q^56 - 24 * q^59 - 12 * q^61 - 26 * q^64 - 2 * q^66 + 16 * q^71 - 4 * q^74 + 96 * q^76 - 24 * q^79 + 6 * q^81 + 16 * q^84 - 16 * q^86 + 20 * q^89 + 16 * q^91 + 32 * q^94 - 58 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0.193937i 0.137134i 0.997647 + 0.0685669i $$0.0218427\pi$$
−0.997647 + 0.0685669i $$0.978157\pi$$
$$3$$ 1.00000i 0.577350i
$$4$$ 1.96239 0.981194
$$5$$ 0 0
$$6$$ −0.193937 −0.0791743
$$7$$ − 3.35026i − 1.26628i −0.774037 0.633140i $$-0.781766\pi$$
0.774037 0.633140i $$-0.218234\pi$$
$$8$$ 0.768452i 0.271689i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 1.00000 0.301511
$$12$$ 1.96239i 0.566493i
$$13$$ 2.96239i 0.821619i 0.911721 + 0.410809i $$0.134754\pi$$
−0.911721 + 0.410809i $$0.865246\pi$$
$$14$$ 0.649738 0.173650
$$15$$ 0 0
$$16$$ 3.77575 0.943937
$$17$$ 4.57452i 1.10948i 0.832023 + 0.554741i $$0.187183\pi$$
−0.832023 + 0.554741i $$0.812817\pi$$
$$18$$ − 0.193937i − 0.0457113i
$$19$$ 4.31265 0.989390 0.494695 0.869067i $$-0.335280\pi$$
0.494695 + 0.869067i $$0.335280\pi$$
$$20$$ 0 0
$$21$$ 3.35026 0.731087
$$22$$ 0.193937i 0.0413474i
$$23$$ − 6.70052i − 1.39716i −0.715534 0.698578i $$-0.753817\pi$$
0.715534 0.698578i $$-0.246183\pi$$
$$24$$ −0.768452 −0.156860
$$25$$ 0 0
$$26$$ −0.574515 −0.112672
$$27$$ − 1.00000i − 0.192450i
$$28$$ − 6.57452i − 1.24247i
$$29$$ 3.61213 0.670755 0.335378 0.942084i $$-0.391136\pi$$
0.335378 + 0.942084i $$0.391136\pi$$
$$30$$ 0 0
$$31$$ 9.92478 1.78254 0.891271 0.453470i $$-0.149814\pi$$
0.891271 + 0.453470i $$0.149814\pi$$
$$32$$ 2.26916i 0.401134i
$$33$$ 1.00000i 0.174078i
$$34$$ −0.887166 −0.152148
$$35$$ 0 0
$$36$$ −1.96239 −0.327065
$$37$$ 2.00000i 0.328798i 0.986394 + 0.164399i $$0.0525685\pi$$
−0.986394 + 0.164399i $$0.947432\pi$$
$$38$$ 0.836381i 0.135679i
$$39$$ −2.96239 −0.474362
$$40$$ 0 0
$$41$$ −4.38787 −0.685271 −0.342635 0.939468i $$-0.611320\pi$$
−0.342635 + 0.939468i $$0.611320\pi$$
$$42$$ 0.649738i 0.100257i
$$43$$ − 9.27504i − 1.41443i −0.706998 0.707215i $$-0.749951\pi$$
0.706998 0.707215i $$-0.250049\pi$$
$$44$$ 1.96239 0.295841
$$45$$ 0 0
$$46$$ 1.29948 0.191597
$$47$$ 9.92478i 1.44768i 0.689969 + 0.723839i $$0.257624\pi$$
−0.689969 + 0.723839i $$0.742376\pi$$
$$48$$ 3.77575i 0.544982i
$$49$$ −4.22425 −0.603465
$$50$$ 0 0
$$51$$ −4.57452 −0.640560
$$52$$ 5.81336i 0.806168i
$$53$$ 4.70052i 0.645667i 0.946456 + 0.322833i $$0.104635\pi$$
−0.946456 + 0.322833i $$0.895365\pi$$
$$54$$ 0.193937 0.0263914
$$55$$ 0 0
$$56$$ 2.57452 0.344034
$$57$$ 4.31265i 0.571224i
$$58$$ 0.700523i 0.0919832i
$$59$$ −10.7005 −1.39309 −0.696545 0.717513i $$-0.745280\pi$$
−0.696545 + 0.717513i $$0.745280\pi$$
$$60$$ 0 0
$$61$$ −8.70052 −1.11399 −0.556994 0.830517i $$-0.688045\pi$$
−0.556994 + 0.830517i $$0.688045\pi$$
$$62$$ 1.92478i 0.244447i
$$63$$ 3.35026i 0.422093i
$$64$$ 7.11142 0.888927
$$65$$ 0 0
$$66$$ −0.193937 −0.0238719
$$67$$ − 5.92478i − 0.723827i −0.932212 0.361913i $$-0.882124\pi$$
0.932212 0.361913i $$-0.117876\pi$$
$$68$$ 8.97698i 1.08862i
$$69$$ 6.70052 0.806648
$$70$$ 0 0
$$71$$ 9.92478 1.17785 0.588927 0.808186i $$-0.299550\pi$$
0.588927 + 0.808186i $$0.299550\pi$$
$$72$$ − 0.768452i − 0.0905629i
$$73$$ − 7.73813i − 0.905680i −0.891592 0.452840i $$-0.850411\pi$$
0.891592 0.452840i $$-0.149589\pi$$
$$74$$ −0.387873 −0.0450893
$$75$$ 0 0
$$76$$ 8.46310 0.970784
$$77$$ − 3.35026i − 0.381798i
$$78$$ − 0.574515i − 0.0650511i
$$79$$ −11.5369 −1.29800 −0.649002 0.760787i $$-0.724813\pi$$
−0.649002 + 0.760787i $$0.724813\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ − 0.850969i − 0.0939738i
$$83$$ 10.8872i 1.19502i 0.801861 + 0.597511i $$0.203844\pi$$
−0.801861 + 0.597511i $$0.796156\pi$$
$$84$$ 6.57452 0.717338
$$85$$ 0 0
$$86$$ 1.79877 0.193966
$$87$$ 3.61213i 0.387261i
$$88$$ 0.768452i 0.0819173i
$$89$$ 2.77575 0.294229 0.147114 0.989120i $$-0.453001\pi$$
0.147114 + 0.989120i $$0.453001\pi$$
$$90$$ 0 0
$$91$$ 9.92478 1.04040
$$92$$ − 13.1490i − 1.37088i
$$93$$ 9.92478i 1.02915i
$$94$$ −1.92478 −0.198526
$$95$$ 0 0
$$96$$ −2.26916 −0.231595
$$97$$ − 0.0752228i − 0.00763772i −0.999993 0.00381886i $$-0.998784\pi$$
0.999993 0.00381886i $$-0.00121558\pi$$
$$98$$ − 0.819237i − 0.0827555i
$$99$$ −1.00000 −0.100504
$$100$$ 0 0
$$101$$ −15.0884 −1.50135 −0.750676 0.660671i $$-0.770272\pi$$
−0.750676 + 0.660671i $$0.770272\pi$$
$$102$$ − 0.887166i − 0.0878425i
$$103$$ − 3.22425i − 0.317695i −0.987303 0.158848i $$-0.949222\pi$$
0.987303 0.158848i $$-0.0507778\pi$$
$$104$$ −2.27645 −0.223225
$$105$$ 0 0
$$106$$ −0.911603 −0.0885427
$$107$$ 0.962389i 0.0930376i 0.998917 + 0.0465188i $$0.0148127\pi$$
−0.998917 + 0.0465188i $$0.985187\pi$$
$$108$$ − 1.96239i − 0.188831i
$$109$$ −11.4010 −1.09202 −0.546011 0.837778i $$-0.683854\pi$$
−0.546011 + 0.837778i $$0.683854\pi$$
$$110$$ 0 0
$$111$$ −2.00000 −0.189832
$$112$$ − 12.6497i − 1.19529i
$$113$$ − 6.00000i − 0.564433i −0.959351 0.282216i $$-0.908930\pi$$
0.959351 0.282216i $$-0.0910696\pi$$
$$114$$ −0.836381 −0.0783342
$$115$$ 0 0
$$116$$ 7.08840 0.658141
$$117$$ − 2.96239i − 0.273873i
$$118$$ − 2.07522i − 0.191040i
$$119$$ 15.3258 1.40492
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ − 1.68735i − 0.152765i
$$123$$ − 4.38787i − 0.395641i
$$124$$ 19.4763 1.74902
$$125$$ 0 0
$$126$$ −0.649738 −0.0578833
$$127$$ 14.5745i 1.29328i 0.762796 + 0.646640i $$0.223826\pi$$
−0.762796 + 0.646640i $$0.776174\pi$$
$$128$$ 5.91748i 0.523037i
$$129$$ 9.27504 0.816622
$$130$$ 0 0
$$131$$ −5.92478 −0.517650 −0.258825 0.965924i $$-0.583335\pi$$
−0.258825 + 0.965924i $$0.583335\pi$$
$$132$$ 1.96239i 0.170804i
$$133$$ − 14.4485i − 1.25284i
$$134$$ 1.14903 0.0992612
$$135$$ 0 0
$$136$$ −3.51530 −0.301434
$$137$$ − 13.8496i − 1.18325i −0.806214 0.591624i $$-0.798487\pi$$
0.806214 0.591624i $$-0.201513\pi$$
$$138$$ 1.29948i 0.110619i
$$139$$ −13.6121 −1.15457 −0.577283 0.816544i $$-0.695887\pi$$
−0.577283 + 0.816544i $$0.695887\pi$$
$$140$$ 0 0
$$141$$ −9.92478 −0.835817
$$142$$ 1.92478i 0.161524i
$$143$$ 2.96239i 0.247727i
$$144$$ −3.77575 −0.314646
$$145$$ 0 0
$$146$$ 1.50071 0.124199
$$147$$ − 4.22425i − 0.348411i
$$148$$ 3.92478i 0.322615i
$$149$$ −1.53690 −0.125908 −0.0629540 0.998016i $$-0.520052\pi$$
−0.0629540 + 0.998016i $$0.520052\pi$$
$$150$$ 0 0
$$151$$ −6.76116 −0.550215 −0.275108 0.961413i $$-0.588713\pi$$
−0.275108 + 0.961413i $$0.588713\pi$$
$$152$$ 3.31406i 0.268806i
$$153$$ − 4.57452i − 0.369828i
$$154$$ 0.649738 0.0523574
$$155$$ 0 0
$$156$$ −5.81336 −0.465441
$$157$$ 5.47627i 0.437054i 0.975831 + 0.218527i $$0.0701252\pi$$
−0.975831 + 0.218527i $$0.929875\pi$$
$$158$$ − 2.23743i − 0.178000i
$$159$$ −4.70052 −0.372776
$$160$$ 0 0
$$161$$ −22.4485 −1.76919
$$162$$ 0.193937i 0.0152371i
$$163$$ 12.6253i 0.988890i 0.869209 + 0.494445i $$0.164629\pi$$
−0.869209 + 0.494445i $$0.835371\pi$$
$$164$$ −8.61071 −0.672384
$$165$$ 0 0
$$166$$ −2.11142 −0.163878
$$167$$ − 18.3634i − 1.42101i −0.703695 0.710503i $$-0.748468\pi$$
0.703695 0.710503i $$-0.251532\pi$$
$$168$$ 2.57452i 0.198628i
$$169$$ 4.22425 0.324943
$$170$$ 0 0
$$171$$ −4.31265 −0.329797
$$172$$ − 18.2012i − 1.38783i
$$173$$ − 8.57452i − 0.651908i −0.945386 0.325954i $$-0.894314\pi$$
0.945386 0.325954i $$-0.105686\pi$$
$$174$$ −0.700523 −0.0531065
$$175$$ 0 0
$$176$$ 3.77575 0.284608
$$177$$ − 10.7005i − 0.804301i
$$178$$ 0.538319i 0.0403487i
$$179$$ −14.1768 −1.05962 −0.529812 0.848115i $$-0.677737\pi$$
−0.529812 + 0.848115i $$0.677737\pi$$
$$180$$ 0 0
$$181$$ −5.22425 −0.388316 −0.194158 0.980970i $$-0.562197\pi$$
−0.194158 + 0.980970i $$0.562197\pi$$
$$182$$ 1.92478i 0.142674i
$$183$$ − 8.70052i − 0.643161i
$$184$$ 5.14903 0.379592
$$185$$ 0 0
$$186$$ −1.92478 −0.141132
$$187$$ 4.57452i 0.334522i
$$188$$ 19.4763i 1.42045i
$$189$$ −3.35026 −0.243696
$$190$$ 0 0
$$191$$ −16.6253 −1.20296 −0.601482 0.798886i $$-0.705423\pi$$
−0.601482 + 0.798886i $$0.705423\pi$$
$$192$$ 7.11142i 0.513222i
$$193$$ − 16.3634i − 1.17787i −0.808182 0.588933i $$-0.799548\pi$$
0.808182 0.588933i $$-0.200452\pi$$
$$194$$ 0.0145884 0.00104739
$$195$$ 0 0
$$196$$ −8.28963 −0.592116
$$197$$ 20.4241i 1.45515i 0.686026 + 0.727577i $$0.259354\pi$$
−0.686026 + 0.727577i $$0.740646\pi$$
$$198$$ − 0.193937i − 0.0137825i
$$199$$ 8.62530 0.611431 0.305716 0.952123i $$-0.401104\pi$$
0.305716 + 0.952123i $$0.401104\pi$$
$$200$$ 0 0
$$201$$ 5.92478 0.417902
$$202$$ − 2.92619i − 0.205886i
$$203$$ − 12.1016i − 0.849364i
$$204$$ −8.97698 −0.628514
$$205$$ 0 0
$$206$$ 0.625301 0.0435668
$$207$$ 6.70052i 0.465719i
$$208$$ 11.1852i 0.775556i
$$209$$ 4.31265 0.298312
$$210$$ 0 0
$$211$$ 9.08840 0.625671 0.312836 0.949807i $$-0.398721\pi$$
0.312836 + 0.949807i $$0.398721\pi$$
$$212$$ 9.22425i 0.633524i
$$213$$ 9.92478i 0.680035i
$$214$$ −0.186642 −0.0127586
$$215$$ 0 0
$$216$$ 0.768452 0.0522865
$$217$$ − 33.2506i − 2.25720i
$$218$$ − 2.21108i − 0.149753i
$$219$$ 7.73813 0.522895
$$220$$ 0 0
$$221$$ −13.5515 −0.911572
$$222$$ − 0.387873i − 0.0260323i
$$223$$ − 6.70052i − 0.448700i −0.974509 0.224350i $$-0.927974\pi$$
0.974509 0.224350i $$-0.0720259\pi$$
$$224$$ 7.60228 0.507949
$$225$$ 0 0
$$226$$ 1.16362 0.0774028
$$227$$ − 16.9624i − 1.12583i −0.826514 0.562917i $$-0.809679\pi$$
0.826514 0.562917i $$-0.190321\pi$$
$$228$$ 8.46310i 0.560482i
$$229$$ −25.8496 −1.70819 −0.854093 0.520120i $$-0.825887\pi$$
−0.854093 + 0.520120i $$0.825887\pi$$
$$230$$ 0 0
$$231$$ 3.35026 0.220431
$$232$$ 2.77575i 0.182237i
$$233$$ − 19.2750i − 1.26275i −0.775478 0.631375i $$-0.782491\pi$$
0.775478 0.631375i $$-0.217509\pi$$
$$234$$ 0.574515 0.0375573
$$235$$ 0 0
$$236$$ −20.9986 −1.36689
$$237$$ − 11.5369i − 0.749402i
$$238$$ 2.97224i 0.192662i
$$239$$ −26.5501 −1.71738 −0.858691 0.512494i $$-0.828722\pi$$
−0.858691 + 0.512494i $$0.828722\pi$$
$$240$$ 0 0
$$241$$ 28.5501 1.83907 0.919536 0.393006i $$-0.128565\pi$$
0.919536 + 0.393006i $$0.128565\pi$$
$$242$$ 0.193937i 0.0124667i
$$243$$ 1.00000i 0.0641500i
$$244$$ −17.0738 −1.09304
$$245$$ 0 0
$$246$$ 0.850969 0.0542558
$$247$$ 12.7757i 0.812901i
$$248$$ 7.62672i 0.484297i
$$249$$ −10.8872 −0.689946
$$250$$ 0 0
$$251$$ 29.9248 1.88884 0.944418 0.328748i $$-0.106627\pi$$
0.944418 + 0.328748i $$0.106627\pi$$
$$252$$ 6.57452i 0.414156i
$$253$$ − 6.70052i − 0.421258i
$$254$$ −2.82653 −0.177352
$$255$$ 0 0
$$256$$ 13.0752 0.817201
$$257$$ − 8.70052i − 0.542724i −0.962477 0.271362i $$-0.912526\pi$$
0.962477 0.271362i $$-0.0874740\pi$$
$$258$$ 1.79877i 0.111986i
$$259$$ 6.70052 0.416350
$$260$$ 0 0
$$261$$ −3.61213 −0.223585
$$262$$ − 1.14903i − 0.0709874i
$$263$$ 12.2882i 0.757724i 0.925453 + 0.378862i $$0.123684\pi$$
−0.925453 + 0.378862i $$0.876316\pi$$
$$264$$ −0.768452 −0.0472950
$$265$$ 0 0
$$266$$ 2.80209 0.171807
$$267$$ 2.77575i 0.169873i
$$268$$ − 11.6267i − 0.710215i
$$269$$ 5.84955 0.356654 0.178327 0.983971i $$-0.442932\pi$$
0.178327 + 0.983971i $$0.442932\pi$$
$$270$$ 0 0
$$271$$ −5.08840 −0.309098 −0.154549 0.987985i $$-0.549392\pi$$
−0.154549 + 0.987985i $$0.549392\pi$$
$$272$$ 17.2722i 1.04728i
$$273$$ 9.92478i 0.600675i
$$274$$ 2.68594 0.162263
$$275$$ 0 0
$$276$$ 13.1490 0.791479
$$277$$ − 1.41090i − 0.0847725i −0.999101 0.0423863i $$-0.986504\pi$$
0.999101 0.0423863i $$-0.0134960\pi$$
$$278$$ − 2.63989i − 0.158330i
$$279$$ −9.92478 −0.594181
$$280$$ 0 0
$$281$$ −4.38787 −0.261759 −0.130879 0.991398i $$-0.541780\pi$$
−0.130879 + 0.991398i $$0.541780\pi$$
$$282$$ − 1.92478i − 0.114619i
$$283$$ 26.5745i 1.57969i 0.613306 + 0.789845i $$0.289839\pi$$
−0.613306 + 0.789845i $$0.710161\pi$$
$$284$$ 19.4763 1.15570
$$285$$ 0 0
$$286$$ −0.574515 −0.0339718
$$287$$ 14.7005i 0.867744i
$$288$$ − 2.26916i − 0.133711i
$$289$$ −3.92619 −0.230952
$$290$$ 0 0
$$291$$ 0.0752228 0.00440964
$$292$$ − 15.1852i − 0.888648i
$$293$$ − 3.42548i − 0.200119i −0.994981 0.100059i $$-0.968097\pi$$
0.994981 0.100059i $$-0.0319033\pi$$
$$294$$ 0.819237 0.0477789
$$295$$ 0 0
$$296$$ −1.53690 −0.0893307
$$297$$ − 1.00000i − 0.0580259i
$$298$$ − 0.298062i − 0.0172663i
$$299$$ 19.8496 1.14793
$$300$$ 0 0
$$301$$ −31.0738 −1.79106
$$302$$ − 1.31124i − 0.0754531i
$$303$$ − 15.0884i − 0.866806i
$$304$$ 16.2835 0.933921
$$305$$ 0 0
$$306$$ 0.887166 0.0507159
$$307$$ 16.6497i 0.950251i 0.879918 + 0.475125i $$0.157597\pi$$
−0.879918 + 0.475125i $$0.842403\pi$$
$$308$$ − 6.57452i − 0.374618i
$$309$$ 3.22425 0.183421
$$310$$ 0 0
$$311$$ 32.9986 1.87118 0.935589 0.353091i $$-0.114869\pi$$
0.935589 + 0.353091i $$0.114869\pi$$
$$312$$ − 2.27645i − 0.128879i
$$313$$ 15.4010i 0.870519i 0.900305 + 0.435259i $$0.143343\pi$$
−0.900305 + 0.435259i $$0.856657\pi$$
$$314$$ −1.06205 −0.0599349
$$315$$ 0 0
$$316$$ −22.6399 −1.27359
$$317$$ − 2.15045i − 0.120781i −0.998175 0.0603905i $$-0.980765\pi$$
0.998175 0.0603905i $$-0.0192346\pi$$
$$318$$ − 0.911603i − 0.0511202i
$$319$$ 3.61213 0.202240
$$320$$ 0 0
$$321$$ −0.962389 −0.0537153
$$322$$ − 4.35359i − 0.242616i
$$323$$ 19.7283i 1.09771i
$$324$$ 1.96239 0.109022
$$325$$ 0 0
$$326$$ −2.44851 −0.135610
$$327$$ − 11.4010i − 0.630479i
$$328$$ − 3.37187i − 0.186180i
$$329$$ 33.2506 1.83316
$$330$$ 0 0
$$331$$ −14.5501 −0.799745 −0.399872 0.916571i $$-0.630946\pi$$
−0.399872 + 0.916571i $$0.630946\pi$$
$$332$$ 21.3649i 1.17255i
$$333$$ − 2.00000i − 0.109599i
$$334$$ 3.56134 0.194868
$$335$$ 0 0
$$336$$ 12.6497 0.690100
$$337$$ − 16.2619i − 0.885840i −0.896561 0.442920i $$-0.853943\pi$$
0.896561 0.442920i $$-0.146057\pi$$
$$338$$ 0.819237i 0.0445606i
$$339$$ 6.00000 0.325875
$$340$$ 0 0
$$341$$ 9.92478 0.537457
$$342$$ − 0.836381i − 0.0452263i
$$343$$ − 9.29948i − 0.502125i
$$344$$ 7.12742 0.384285
$$345$$ 0 0
$$346$$ 1.66291 0.0893987
$$347$$ 0.962389i 0.0516637i 0.999666 + 0.0258319i $$0.00822345\pi$$
−0.999666 + 0.0258319i $$0.991777\pi$$
$$348$$ 7.08840i 0.379978i
$$349$$ −20.7005 −1.10807 −0.554037 0.832492i $$-0.686913\pi$$
−0.554037 + 0.832492i $$0.686913\pi$$
$$350$$ 0 0
$$351$$ 2.96239 0.158121
$$352$$ 2.26916i 0.120947i
$$353$$ 20.5501i 1.09377i 0.837208 + 0.546885i $$0.184187\pi$$
−0.837208 + 0.546885i $$0.815813\pi$$
$$354$$ 2.07522 0.110297
$$355$$ 0 0
$$356$$ 5.44709 0.288695
$$357$$ 15.3258i 0.811129i
$$358$$ − 2.74940i − 0.145310i
$$359$$ −17.9248 −0.946034 −0.473017 0.881053i $$-0.656835\pi$$
−0.473017 + 0.881053i $$0.656835\pi$$
$$360$$ 0 0
$$361$$ −0.401047 −0.0211077
$$362$$ − 1.01317i − 0.0532512i
$$363$$ 1.00000i 0.0524864i
$$364$$ 19.4763 1.02083
$$365$$ 0 0
$$366$$ 1.68735 0.0881992
$$367$$ 29.6531i 1.54788i 0.633261 + 0.773939i $$0.281716\pi$$
−0.633261 + 0.773939i $$0.718284\pi$$
$$368$$ − 25.2995i − 1.31883i
$$369$$ 4.38787 0.228424
$$370$$ 0 0
$$371$$ 15.7480 0.817595
$$372$$ 19.4763i 1.00980i
$$373$$ − 9.13918i − 0.473209i −0.971606 0.236604i $$-0.923965\pi$$
0.971606 0.236604i $$-0.0760345\pi$$
$$374$$ −0.887166 −0.0458743
$$375$$ 0 0
$$376$$ −7.62672 −0.393318
$$377$$ 10.7005i 0.551105i
$$378$$ − 0.649738i − 0.0334189i
$$379$$ 20.0000 1.02733 0.513665 0.857991i $$-0.328287\pi$$
0.513665 + 0.857991i $$0.328287\pi$$
$$380$$ 0 0
$$381$$ −14.5745 −0.746675
$$382$$ − 3.22425i − 0.164967i
$$383$$ − 34.9234i − 1.78450i −0.451541 0.892250i $$-0.649126\pi$$
0.451541 0.892250i $$-0.350874\pi$$
$$384$$ −5.91748 −0.301975
$$385$$ 0 0
$$386$$ 3.17347 0.161525
$$387$$ 9.27504i 0.471477i
$$388$$ − 0.147616i − 0.00749408i
$$389$$ −2.77575 −0.140736 −0.0703680 0.997521i $$-0.522417\pi$$
−0.0703680 + 0.997521i $$0.522417\pi$$
$$390$$ 0 0
$$391$$ 30.6516 1.55012
$$392$$ − 3.24614i − 0.163955i
$$393$$ − 5.92478i − 0.298865i
$$394$$ −3.96097 −0.199551
$$395$$ 0 0
$$396$$ −1.96239 −0.0986137
$$397$$ 19.9248i 0.999996i 0.866027 + 0.499998i $$0.166666\pi$$
−0.866027 + 0.499998i $$0.833334\pi$$
$$398$$ 1.67276i 0.0838479i
$$399$$ 14.4485 0.723330
$$400$$ 0 0
$$401$$ 2.00000 0.0998752 0.0499376 0.998752i $$-0.484098\pi$$
0.0499376 + 0.998752i $$0.484098\pi$$
$$402$$ 1.14903i 0.0573085i
$$403$$ 29.4010i 1.46457i
$$404$$ −29.6093 −1.47312
$$405$$ 0 0
$$406$$ 2.34694 0.116477
$$407$$ 2.00000i 0.0991363i
$$408$$ − 3.51530i − 0.174033i
$$409$$ 13.0738 0.646458 0.323229 0.946321i $$-0.395232\pi$$
0.323229 + 0.946321i $$0.395232\pi$$
$$410$$ 0 0
$$411$$ 13.8496 0.683148
$$412$$ − 6.32724i − 0.311721i
$$413$$ 35.8496i 1.76404i
$$414$$ −1.29948 −0.0638658
$$415$$ 0 0
$$416$$ −6.72213 −0.329580
$$417$$ − 13.6121i − 0.666589i
$$418$$ 0.836381i 0.0409087i
$$419$$ −7.22425 −0.352928 −0.176464 0.984307i $$-0.556466\pi$$
−0.176464 + 0.984307i $$0.556466\pi$$
$$420$$ 0 0
$$421$$ 30.6253 1.49259 0.746293 0.665618i $$-0.231832\pi$$
0.746293 + 0.665618i $$0.231832\pi$$
$$422$$ 1.76257i 0.0858007i
$$423$$ − 9.92478i − 0.482559i
$$424$$ −3.61213 −0.175420
$$425$$ 0 0
$$426$$ −1.92478 −0.0932558
$$427$$ 29.1490i 1.41062i
$$428$$ 1.88858i 0.0912880i
$$429$$ −2.96239 −0.143025
$$430$$ 0 0
$$431$$ −33.8759 −1.63174 −0.815872 0.578232i $$-0.803743\pi$$
−0.815872 + 0.578232i $$0.803743\pi$$
$$432$$ − 3.77575i − 0.181661i
$$433$$ − 9.47627i − 0.455400i −0.973731 0.227700i $$-0.926879\pi$$
0.973731 0.227700i $$-0.0731206\pi$$
$$434$$ 6.44851 0.309538
$$435$$ 0 0
$$436$$ −22.3733 −1.07149
$$437$$ − 28.8970i − 1.38233i
$$438$$ 1.50071i 0.0717066i
$$439$$ 29.4617 1.40613 0.703065 0.711126i $$-0.251814\pi$$
0.703065 + 0.711126i $$0.251814\pi$$
$$440$$ 0 0
$$441$$ 4.22425 0.201155
$$442$$ − 2.62813i − 0.125007i
$$443$$ − 19.0738i − 0.906224i −0.891454 0.453112i $$-0.850314\pi$$
0.891454 0.453112i $$-0.149686\pi$$
$$444$$ −3.92478 −0.186262
$$445$$ 0 0
$$446$$ 1.29948 0.0615320
$$447$$ − 1.53690i − 0.0726931i
$$448$$ − 23.8251i − 1.12563i
$$449$$ −35.8759 −1.69309 −0.846544 0.532318i $$-0.821321\pi$$
−0.846544 + 0.532318i $$0.821321\pi$$
$$450$$ 0 0
$$451$$ −4.38787 −0.206617
$$452$$ − 11.7743i − 0.553818i
$$453$$ − 6.76116i − 0.317667i
$$454$$ 3.28963 0.154390
$$455$$ 0 0
$$456$$ −3.31406 −0.155195
$$457$$ − 5.28963i − 0.247438i −0.992317 0.123719i $$-0.960518\pi$$
0.992317 0.123719i $$-0.0394822\pi$$
$$458$$ − 5.01317i − 0.234250i
$$459$$ 4.57452 0.213520
$$460$$ 0 0
$$461$$ 36.3390 1.69248 0.846238 0.532805i $$-0.178862\pi$$
0.846238 + 0.532805i $$0.178862\pi$$
$$462$$ 0.649738i 0.0302286i
$$463$$ 10.5501i 0.490304i 0.969485 + 0.245152i $$0.0788378\pi$$
−0.969485 + 0.245152i $$0.921162\pi$$
$$464$$ 13.6385 0.633150
$$465$$ 0 0
$$466$$ 3.73813 0.173166
$$467$$ − 18.7005i − 0.865357i −0.901548 0.432679i $$-0.857569\pi$$
0.901548 0.432679i $$-0.142431\pi$$
$$468$$ − 5.81336i − 0.268723i
$$469$$ −19.8496 −0.916567
$$470$$ 0 0
$$471$$ −5.47627 −0.252333
$$472$$ − 8.22284i − 0.378487i
$$473$$ − 9.27504i − 0.426467i
$$474$$ 2.23743 0.102768
$$475$$ 0 0
$$476$$ 30.0752 1.37850
$$477$$ − 4.70052i − 0.215222i
$$478$$ − 5.14903i − 0.235511i
$$479$$ 9.29948 0.424904 0.212452 0.977172i $$-0.431855\pi$$
0.212452 + 0.977172i $$0.431855\pi$$
$$480$$ 0 0
$$481$$ −5.92478 −0.270147
$$482$$ 5.53690i 0.252199i
$$483$$ − 22.4485i − 1.02144i
$$484$$ 1.96239 0.0891995
$$485$$ 0 0
$$486$$ −0.193937 −0.00879714
$$487$$ 35.4763i 1.60758i 0.594911 + 0.803792i $$0.297187\pi$$
−0.594911 + 0.803792i $$0.702813\pi$$
$$488$$ − 6.68594i − 0.302658i
$$489$$ −12.6253 −0.570936
$$490$$ 0 0
$$491$$ 24.7757 1.11811 0.559057 0.829129i $$-0.311163\pi$$
0.559057 + 0.829129i $$0.311163\pi$$
$$492$$ − 8.61071i − 0.388201i
$$493$$ 16.5237i 0.744191i
$$494$$ −2.47768 −0.111476
$$495$$ 0 0
$$496$$ 37.4734 1.68261
$$497$$ − 33.2506i − 1.49149i
$$498$$ − 2.11142i − 0.0946150i
$$499$$ −14.1768 −0.634640 −0.317320 0.948318i $$-0.602783\pi$$
−0.317320 + 0.948318i $$0.602783\pi$$
$$500$$ 0 0
$$501$$ 18.3634 0.820418
$$502$$ 5.80351i 0.259023i
$$503$$ − 8.43866i − 0.376261i −0.982144 0.188131i $$-0.939757\pi$$
0.982144 0.188131i $$-0.0602428\pi$$
$$504$$ −2.57452 −0.114678
$$505$$ 0 0
$$506$$ 1.29948 0.0577688
$$507$$ 4.22425i 0.187606i
$$508$$ 28.6009i 1.26896i
$$509$$ −1.10299 −0.0488890 −0.0244445 0.999701i $$-0.507782\pi$$
−0.0244445 + 0.999701i $$0.507782\pi$$
$$510$$ 0 0
$$511$$ −25.9248 −1.14684
$$512$$ 14.3707i 0.635103i
$$513$$ − 4.31265i − 0.190408i
$$514$$ 1.68735 0.0744258
$$515$$ 0 0
$$516$$ 18.2012 0.801265
$$517$$ 9.92478i 0.436491i
$$518$$ 1.29948i 0.0570957i
$$519$$ 8.57452 0.376379
$$520$$ 0 0
$$521$$ −12.4485 −0.545379 −0.272690 0.962102i $$-0.587913\pi$$
−0.272690 + 0.962102i $$0.587913\pi$$
$$522$$ − 0.700523i − 0.0306611i
$$523$$ 30.0508i 1.31403i 0.753878 + 0.657015i $$0.228181\pi$$
−0.753878 + 0.657015i $$0.771819\pi$$
$$524$$ −11.6267 −0.507915
$$525$$ 0 0
$$526$$ −2.38313 −0.103910
$$527$$ 45.4010i 1.97770i
$$528$$ 3.77575i 0.164318i
$$529$$ −21.8970 −0.952044
$$530$$ 0 0
$$531$$ 10.7005 0.464363
$$532$$ − 28.3536i − 1.22928i
$$533$$ − 12.9986i − 0.563031i
$$534$$ −0.538319 −0.0232953
$$535$$ 0 0
$$536$$ 4.55291 0.196656
$$537$$ − 14.1768i − 0.611774i
$$538$$ 1.13444i 0.0489093i
$$539$$ −4.22425 −0.181951
$$540$$ 0 0
$$541$$ −18.0000 −0.773880 −0.386940 0.922105i $$-0.626468\pi$$
−0.386940 + 0.922105i $$0.626468\pi$$
$$542$$ − 0.986826i − 0.0423878i
$$543$$ − 5.22425i − 0.224194i
$$544$$ −10.3803 −0.445052
$$545$$ 0 0
$$546$$ −1.92478 −0.0823729
$$547$$ 14.3028i 0.611544i 0.952105 + 0.305772i $$0.0989145\pi$$
−0.952105 + 0.305772i $$0.901086\pi$$
$$548$$ − 27.1782i − 1.16100i
$$549$$ 8.70052 0.371329
$$550$$ 0 0
$$551$$ 15.5778 0.663638
$$552$$ 5.14903i 0.219157i
$$553$$ 38.6516i 1.64364i
$$554$$ 0.273624 0.0116252
$$555$$ 0 0
$$556$$ −26.7123 −1.13285
$$557$$ 11.7988i 0.499930i 0.968255 + 0.249965i $$0.0804191\pi$$
−0.968255 + 0.249965i $$0.919581\pi$$
$$558$$ − 1.92478i − 0.0814823i
$$559$$ 27.4763 1.16212
$$560$$ 0 0
$$561$$ −4.57452 −0.193136
$$562$$ − 0.850969i − 0.0358960i
$$563$$ 30.4847i 1.28478i 0.766379 + 0.642389i $$0.222056\pi$$
−0.766379 + 0.642389i $$0.777944\pi$$
$$564$$ −19.4763 −0.820099
$$565$$ 0 0
$$566$$ −5.15377 −0.216629
$$567$$ − 3.35026i − 0.140698i
$$568$$ 7.62672i 0.320010i
$$569$$ 27.0884 1.13560 0.567802 0.823165i $$-0.307794\pi$$
0.567802 + 0.823165i $$0.307794\pi$$
$$570$$ 0 0
$$571$$ 7.28489 0.304863 0.152432 0.988314i $$-0.451290\pi$$
0.152432 + 0.988314i $$0.451290\pi$$
$$572$$ 5.81336i 0.243069i
$$573$$ − 16.6253i − 0.694532i
$$574$$ −2.85097 −0.118997
$$575$$ 0 0
$$576$$ −7.11142 −0.296309
$$577$$ 31.6239i 1.31652i 0.752791 + 0.658260i $$0.228707\pi$$
−0.752791 + 0.658260i $$0.771293\pi$$
$$578$$ − 0.761432i − 0.0316714i
$$579$$ 16.3634 0.680041
$$580$$ 0 0
$$581$$ 36.4749 1.51323
$$582$$ 0.0145884i 0 0.000604711i
$$583$$ 4.70052i 0.194676i
$$584$$ 5.94639 0.246063
$$585$$ 0 0
$$586$$ 0.664327 0.0274431
$$587$$ − 33.1490i − 1.36821i −0.729385 0.684103i $$-0.760194\pi$$
0.729385 0.684103i $$-0.239806\pi$$
$$588$$ − 8.28963i − 0.341858i
$$589$$ 42.8021 1.76363
$$590$$ 0 0
$$591$$ −20.4241 −0.840134
$$592$$ 7.55149i 0.310364i
$$593$$ 34.4993i 1.41672i 0.705853 + 0.708358i $$0.250564\pi$$
−0.705853 + 0.708358i $$0.749436\pi$$
$$594$$ 0.193937 0.00795731
$$595$$ 0 0
$$596$$ −3.01600 −0.123540
$$597$$ 8.62530i 0.353010i
$$598$$ 3.84955i 0.157420i
$$599$$ 14.4485 0.590350 0.295175 0.955443i $$-0.404622\pi$$
0.295175 + 0.955443i $$0.404622\pi$$
$$600$$ 0 0
$$601$$ −15.9248 −0.649585 −0.324793 0.945785i $$-0.605295\pi$$
−0.324793 + 0.945785i $$0.605295\pi$$
$$602$$ − 6.02635i − 0.245616i
$$603$$ 5.92478i 0.241276i
$$604$$ −13.2680 −0.539868
$$605$$ 0 0
$$606$$ 2.92619 0.118868
$$607$$ 14.5745i 0.591561i 0.955256 + 0.295781i $$0.0955798\pi$$
−0.955256 + 0.295781i $$0.904420\pi$$
$$608$$ 9.78609i 0.396878i
$$609$$ 12.1016 0.490380
$$610$$ 0 0
$$611$$ −29.4010 −1.18944
$$612$$ − 8.97698i − 0.362873i
$$613$$ 16.4123i 0.662887i 0.943475 + 0.331443i $$0.107536\pi$$
−0.943475 + 0.331443i $$0.892464\pi$$
$$614$$ −3.22899 −0.130312
$$615$$ 0 0
$$616$$ 2.57452 0.103730
$$617$$ 17.8496i 0.718596i 0.933223 + 0.359298i $$0.116984\pi$$
−0.933223 + 0.359298i $$0.883016\pi$$
$$618$$ 0.625301i 0.0251533i
$$619$$ 0.402462 0.0161763 0.00808815 0.999967i $$-0.497425\pi$$
0.00808815 + 0.999967i $$0.497425\pi$$
$$620$$ 0 0
$$621$$ −6.70052 −0.268883
$$622$$ 6.39963i 0.256602i
$$623$$ − 9.29948i − 0.372576i
$$624$$ −11.1852 −0.447767
$$625$$ 0 0
$$626$$ −2.98683 −0.119378
$$627$$ 4.31265i 0.172231i
$$628$$ 10.7466i 0.428835i
$$629$$ −9.14903 −0.364796
$$630$$ 0 0
$$631$$ −38.0263 −1.51380 −0.756902 0.653528i $$-0.773288\pi$$
−0.756902 + 0.653528i $$0.773288\pi$$
$$632$$ − 8.86556i − 0.352653i
$$633$$ 9.08840i 0.361231i
$$634$$ 0.417050 0.0165632
$$635$$ 0 0
$$636$$ −9.22425 −0.365765
$$637$$ − 12.5139i − 0.495818i
$$638$$ 0.700523i 0.0277340i
$$639$$ −9.92478 −0.392618
$$640$$ 0 0
$$641$$ −28.0263 −1.10697 −0.553487 0.832858i $$-0.686703\pi$$
−0.553487 + 0.832858i $$0.686703\pi$$
$$642$$ − 0.186642i − 0.00736619i
$$643$$ − 4.62530i − 0.182404i −0.995832 0.0912020i $$-0.970929\pi$$
0.995832 0.0912020i $$-0.0290709\pi$$
$$644$$ −44.0527 −1.73592
$$645$$ 0 0
$$646$$ −3.82604 −0.150533
$$647$$ − 23.5778i − 0.926941i −0.886112 0.463470i $$-0.846604\pi$$
0.886112 0.463470i $$-0.153396\pi$$
$$648$$ 0.768452i 0.0301876i
$$649$$ −10.7005 −0.420032
$$650$$ 0 0
$$651$$ 33.2506 1.30319
$$652$$ 24.7757i 0.970293i
$$653$$ − 2.25202i − 0.0881282i −0.999029 0.0440641i $$-0.985969\pi$$
0.999029 0.0440641i $$-0.0140306\pi$$
$$654$$ 2.21108 0.0864601
$$655$$ 0 0
$$656$$ −16.5675 −0.646852
$$657$$ 7.73813i 0.301893i
$$658$$ 6.44851i 0.251389i
$$659$$ 41.4010 1.61276 0.806378 0.591401i $$-0.201425\pi$$
0.806378 + 0.591401i $$0.201425\pi$$
$$660$$ 0 0
$$661$$ 3.40105 0.132285 0.0661427 0.997810i $$-0.478931\pi$$
0.0661427 + 0.997810i $$0.478931\pi$$
$$662$$ − 2.82179i − 0.109672i
$$663$$ − 13.5515i − 0.526296i
$$664$$ −8.36626 −0.324674
$$665$$ 0 0
$$666$$ 0.387873 0.0150298
$$667$$ − 24.2031i − 0.937149i
$$668$$ − 36.0362i − 1.39428i
$$669$$ 6.70052 0.259057
$$670$$ 0 0
$$671$$ −8.70052 −0.335880
$$672$$ 7.60228i 0.293264i
$$673$$ 0.887166i 0.0341977i 0.999854 + 0.0170989i $$0.00544300\pi$$
−0.999854 + 0.0170989i $$0.994557\pi$$
$$674$$ 3.15377 0.121479
$$675$$ 0 0
$$676$$ 8.28963 0.318832
$$677$$ − 18.9018i − 0.726453i −0.931701 0.363227i $$-0.881675\pi$$
0.931701 0.363227i $$-0.118325\pi$$
$$678$$ 1.16362i 0.0446885i
$$679$$ −0.252016 −0.00967149
$$680$$ 0 0
$$681$$ 16.9624 0.650000
$$682$$ 1.92478i 0.0737035i
$$683$$ − 20.8773i − 0.798848i −0.916766 0.399424i $$-0.869210\pi$$
0.916766 0.399424i $$-0.130790\pi$$
$$684$$ −8.46310 −0.323595
$$685$$ 0 0
$$686$$ 1.80351 0.0688583
$$687$$ − 25.8496i − 0.986222i
$$688$$ − 35.0202i − 1.33513i
$$689$$ −13.9248 −0.530492
$$690$$ 0 0
$$691$$ −2.44851 −0.0931456 −0.0465728 0.998915i $$-0.514830\pi$$
−0.0465728 + 0.998915i $$0.514830\pi$$
$$692$$ − 16.8265i − 0.639649i
$$693$$ 3.35026i 0.127266i
$$694$$ −0.186642 −0.00708485
$$695$$ 0 0
$$696$$ −2.77575 −0.105214
$$697$$ − 20.0724i − 0.760296i
$$698$$ − 4.01459i − 0.151954i
$$699$$ 19.2750 0.729049
$$700$$ 0 0
$$701$$ −2.98683 −0.112811 −0.0564054 0.998408i $$-0.517964\pi$$
−0.0564054 + 0.998408i $$0.517964\pi$$
$$702$$ 0.574515i 0.0216837i
$$703$$ 8.62530i 0.325309i
$$704$$ 7.11142 0.268022
$$705$$ 0 0
$$706$$ −3.98541 −0.149993
$$707$$ 50.5501i 1.90113i
$$708$$ − 20.9986i − 0.789175i
$$709$$ −24.1768 −0.907979 −0.453989 0.891007i $$-0.650000\pi$$
−0.453989 + 0.891007i $$0.650000\pi$$
$$710$$ 0 0
$$711$$ 11.5369 0.432668
$$712$$ 2.13303i 0.0799386i
$$713$$ − 66.5012i − 2.49049i
$$714$$ −2.97224 −0.111233
$$715$$ 0 0
$$716$$ −27.8204 −1.03970
$$717$$ − 26.5501i − 0.991531i
$$718$$ − 3.47627i − 0.129733i
$$719$$ 30.0263 1.11979 0.559897 0.828562i $$-0.310841\pi$$
0.559897 + 0.828562i $$0.310841\pi$$
$$720$$ 0 0
$$721$$ −10.8021 −0.402291
$$722$$ − 0.0777777i − 0.00289459i
$$723$$ 28.5501i 1.06179i
$$724$$ −10.2520 −0.381013
$$725$$ 0 0
$$726$$ −0.193937 −0.00719766
$$727$$ − 14.9525i − 0.554559i −0.960789 0.277279i $$-0.910567\pi$$
0.960789 0.277279i $$-0.0894328\pi$$
$$728$$ 7.62672i 0.282665i
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 42.4288 1.56929
$$732$$ − 17.0738i − 0.631066i
$$733$$ − 19.1128i − 0.705949i −0.935633 0.352974i $$-0.885170\pi$$
0.935633 0.352974i $$-0.114830\pi$$
$$734$$ −5.75081 −0.212266
$$735$$ 0 0
$$736$$ 15.2046 0.560447
$$737$$ − 5.92478i − 0.218242i
$$738$$ 0.850969i 0.0313246i
$$739$$ 3.31406 0.121910 0.0609549 0.998141i $$-0.480585\pi$$
0.0609549 + 0.998141i $$0.480585\pi$$
$$740$$ 0 0
$$741$$ −12.7757 −0.469329
$$742$$ 3.05411i 0.112120i
$$743$$ − 34.9887i − 1.28361i −0.766867 0.641806i $$-0.778185\pi$$
0.766867 0.641806i $$-0.221815\pi$$
$$744$$ −7.62672 −0.279609
$$745$$ 0 0
$$746$$ 1.77242 0.0648930
$$747$$ − 10.8872i − 0.398341i
$$748$$ 8.97698i 0.328231i
$$749$$ 3.22425 0.117812
$$750$$ 0 0
$$751$$ −26.9234 −0.982447 −0.491224 0.871033i $$-0.663450\pi$$
−0.491224 + 0.871033i $$0.663450\pi$$
$$752$$ 37.4734i 1.36652i
$$753$$ 29.9248i 1.09052i
$$754$$ −2.07522 −0.0755752
$$755$$ 0 0
$$756$$ −6.57452 −0.239113
$$757$$ − 15.9248i − 0.578796i −0.957209 0.289398i $$-0.906545\pi$$
0.957209 0.289398i $$-0.0934551\pi$$
$$758$$ 3.87873i 0.140882i
$$759$$ 6.70052 0.243214
$$760$$ 0 0
$$761$$ −30.9380 −1.12150 −0.560750 0.827985i $$-0.689487\pi$$
−0.560750 + 0.827985i $$0.689487\pi$$
$$762$$ − 2.82653i − 0.102394i
$$763$$ 38.1965i 1.38281i
$$764$$ −32.6253 −1.18034
$$765$$ 0 0
$$766$$ 6.77292 0.244715
$$767$$ − 31.6991i − 1.14459i
$$768$$ 13.0752i 0.471811i
$$769$$ −9.32582 −0.336298 −0.168149 0.985762i $$-0.553779\pi$$
−0.168149 + 0.985762i $$0.553779\pi$$
$$770$$ 0 0
$$771$$ 8.70052 0.313342
$$772$$ − 32.1114i − 1.15572i
$$773$$ 44.7005i 1.60777i 0.594787 + 0.803883i $$0.297236\pi$$
−0.594787 + 0.803883i $$0.702764\pi$$
$$774$$ −1.79877 −0.0646554
$$775$$ 0 0
$$776$$ 0.0578051 0.00207508
$$777$$ 6.70052i 0.240380i
$$778$$ − 0.538319i − 0.0192997i
$$779$$ −18.9234 −0.678000
$$780$$ 0 0
$$781$$ 9.92478 0.355136
$$782$$ 5.94448i 0.212574i
$$783$$ − 3.61213i − 0.129087i
$$784$$ −15.9497 −0.569633
$$785$$ 0 0
$$786$$ 1.14903 0.0409846
$$787$$ − 21.6775i − 0.772719i −0.922348 0.386360i $$-0.873732\pi$$
0.922348 0.386360i $$-0.126268\pi$$
$$788$$ 40.0800i 1.42779i
$$789$$ −12.2882 −0.437472
$$790$$ 0 0
$$791$$ −20.1016 −0.714730
$$792$$ − 0.768452i − 0.0273058i
$$793$$ − 25.7743i − 0.915273i
$$794$$ −3.86414 −0.137133
$$795$$ 0 0
$$796$$ 16.9262 0.599933
$$797$$ − 22.7466i − 0.805725i −0.915261 0.402862i $$-0.868015\pi$$
0.915261 0.402862i $$-0.131985\pi$$
$$798$$ 2.80209i 0.0991930i
$$799$$ −45.4010 −1.60617
$$800$$ 0 0
$$801$$ −2.77575 −0.0980762
$$802$$ 0.387873i 0.0136963i
$$803$$ − 7.73813i − 0.273073i
$$804$$ 11.6267 0.410043
$$805$$ 0 0
$$806$$ −5.70194 −0.200842
$$807$$ 5.84955i 0.205914i
$$808$$ − 11.5947i − 0.407900i
$$809$$ 23.6121 0.830158 0.415079 0.909785i $$-0.363754\pi$$
0.415079 + 0.909785i $$0.363754\pi$$
$$810$$ 0 0
$$811$$ −26.0870 −0.916038 −0.458019 0.888942i $$-0.651441\pi$$
−0.458019 + 0.888942i $$0.651441\pi$$
$$812$$ − 23.7480i − 0.833391i
$$813$$ − 5.08840i − 0.178458i
$$814$$ −0.387873 −0.0135949
$$815$$ 0 0
$$816$$ −17.2722 −0.604648
$$817$$ − 40.0000i − 1.39942i
$$818$$ 2.53549i 0.0886513i
$$819$$ −9.92478 −0.346800
$$820$$ 0 0
$$821$$ −54.4142 −1.89907 −0.949535 0.313662i $$-0.898444\pi$$
−0.949535 + 0.313662i $$0.898444\pi$$
$$822$$ 2.68594i 0.0936827i
$$823$$ 0.121269i 0.00422716i 0.999998 + 0.00211358i $$0.000672774\pi$$
−0.999998 + 0.00211358i $$0.999327\pi$$
$$824$$ 2.47768 0.0863142
$$825$$ 0 0
$$826$$ −6.95254 −0.241910
$$827$$ 18.2130i 0.633328i 0.948538 + 0.316664i $$0.102563\pi$$
−0.948538 + 0.316664i $$0.897437\pi$$
$$828$$ 13.1490i 0.456960i
$$829$$ −13.0738 −0.454072 −0.227036 0.973886i $$-0.572904\pi$$
−0.227036 + 0.973886i $$0.572904\pi$$
$$830$$ 0 0
$$831$$ 1.41090 0.0489434
$$832$$ 21.0668i 0.730359i
$$833$$ − 19.3239i − 0.669534i
$$834$$ 2.63989 0.0914119
$$835$$ 0 0
$$836$$ 8.46310 0.292702
$$837$$ − 9.92478i − 0.343050i
$$838$$ − 1.40105i − 0.0483984i
$$839$$ 26.5501 0.916610 0.458305 0.888795i $$-0.348457\pi$$
0.458305 + 0.888795i $$0.348457\pi$$
$$840$$ 0 0
$$841$$ −15.9525 −0.550088
$$842$$ 5.93937i 0.204684i
$$843$$ − 4.38787i − 0.151126i
$$844$$ 17.8350 0.613905
$$845$$ 0 0
$$846$$ 1.92478 0.0661752
$$847$$ − 3.35026i − 0.115116i
$$848$$ 17.7480i 0.609468i
$$849$$ −26.5745 −0.912035
$$850$$ 0 0
$$851$$ 13.4010 0.459382
$$852$$ 19.4763i 0.667246i
$$853$$ 40.6155i 1.39065i 0.718697 + 0.695323i $$0.244739\pi$$
−0.718697 + 0.695323i $$0.755261\pi$$
$$854$$ −5.65306 −0.193444
$$855$$ 0 0
$$856$$ −0.739549 −0.0252773
$$857$$ − 20.1721i − 0.689064i −0.938775 0.344532i $$-0.888038\pi$$
0.938775 0.344532i $$-0.111962\pi$$
$$858$$ − 0.574515i − 0.0196136i
$$859$$ −21.8035 −0.743926 −0.371963 0.928248i $$-0.621315\pi$$
−0.371963 + 0.928248i $$0.621315\pi$$
$$860$$ 0 0
$$861$$ −14.7005 −0.500993
$$862$$ − 6.56978i − 0.223767i
$$863$$ 35.4274i 1.20596i 0.797755 + 0.602981i $$0.206021\pi$$
−0.797755 + 0.602981i $$0.793979\pi$$
$$864$$ 2.26916 0.0771984
$$865$$ 0 0
$$866$$ 1.83780 0.0624508
$$867$$ − 3.92619i − 0.133340i
$$868$$ − 65.2506i − 2.21475i
$$869$$ −11.5369 −0.391363
$$870$$ 0 0
$$871$$ 17.5515 0.594710
$$872$$ − 8.76116i − 0.296690i
$$873$$ 0.0752228i 0.00254591i
$$874$$ 5.60419 0.189564
$$875$$ 0 0
$$876$$ 15.1852 0.513061
$$877$$ 14.0362i 0.473969i 0.971513 + 0.236984i $$0.0761590\pi$$
−0.971513 + 0.236984i $$0.923841\pi$$
$$878$$ 5.71370i 0.192828i
$$879$$ 3.42548 0.115539
$$880$$ 0 0
$$881$$ −21.0738 −0.709995 −0.354997 0.934867i $$-0.615518\pi$$
−0.354997 + 0.934867i $$0.615518\pi$$
$$882$$ 0.819237i 0.0275852i
$$883$$ − 42.1476i − 1.41838i −0.705017 0.709190i $$-0.749061\pi$$
0.705017 0.709190i $$-0.250939\pi$$
$$884$$ −26.5933 −0.894429
$$885$$ 0 0
$$886$$ 3.69911 0.124274
$$887$$ − 6.93604i − 0.232889i −0.993197 0.116445i $$-0.962850\pi$$
0.993197 0.116445i $$-0.0371498\pi$$
$$888$$ − 1.53690i − 0.0515751i
$$889$$ 48.8284 1.63765
$$890$$ 0 0
$$891$$ 1.00000 0.0335013
$$892$$ − 13.1490i − 0.440262i
$$893$$ 42.8021i 1.43232i
$$894$$ 0.298062 0.00996868
$$895$$ 0 0
$$896$$ 19.8251 0.662311
$$897$$ 19.8496i 0.662757i
$$898$$ − 6.95765i − 0.232180i
$$899$$ 35.8496 1.19565
$$900$$ 0 0
$$901$$ −21.5026 −0.716356
$$902$$ − 0.850969i − 0.0283342i
$$903$$ − 31.0738i − 1.03407i
$$904$$ 4.61071 0.153350
$$905$$ 0 0
$$906$$ 1.31124 0.0435629
$$907$$ − 53.2017i − 1.76653i −0.468870 0.883267i $$-0.655339\pi$$
0.468870 0.883267i $$-0.344661\pi$$
$$908$$ − 33.2868i − 1.10466i
$$909$$ 15.0884 0.500451
$$910$$ 0 0
$$911$$ 36.4749 1.20847 0.604233 0.796808i $$-0.293480\pi$$
0.604233 + 0.796808i $$0.293480\pi$$
$$912$$ 16.2835i 0.539200i
$$913$$ 10.8872i 0.360313i
$$914$$ 1.02585 0.0339322
$$915$$ 0 0
$$916$$ −50.7269 −1.67606
$$917$$ 19.8496i 0.655490i
$$918$$ 0.887166i 0.0292808i
$$919$$ −9.73340 −0.321075 −0.160538 0.987030i $$-0.551323\pi$$
−0.160538 + 0.987030i $$0.551323\pi$$
$$920$$ 0 0
$$921$$ −16.6497 −0.548628
$$922$$ 7.04746i 0.232096i
$$923$$ 29.4010i 0.967747i
$$924$$ 6.57452 0.216286
$$925$$ 0 0
$$926$$ −2.04605 −0.0672372
$$927$$ 3.22425i 0.105898i
$$928$$ 8.19649i 0.269063i
$$929$$ 24.1768 0.793215 0.396607 0.917988i $$-0.370187\pi$$
0.396607 + 0.917988i $$0.370187\pi$$
$$930$$ 0 0
$$931$$ −18.2177 −0.597062
$$932$$ − 37.8251i − 1.23900i
$$933$$ 32.9986i 1.08033i
$$934$$ 3.62672 0.118670
$$935$$ 0 0
$$936$$ 2.27645 0.0744082
$$937$$ 7.48612i 0.244561i 0.992496 + 0.122280i $$0.0390207\pi$$
−0.992496 + 0.122280i $$0.960979\pi$$
$$938$$ − 3.84955i − 0.125692i
$$939$$ −15.4010 −0.502594
$$940$$ 0 0
$$941$$ −21.2360 −0.692274 −0.346137 0.938184i $$-0.612507\pi$$
−0.346137 + 0.938184i $$0.612507\pi$$
$$942$$ − 1.06205i − 0.0346034i
$$943$$ 29.4010i 0.957430i
$$944$$ −40.4025 −1.31499
$$945$$ 0 0
$$946$$ 1.79877 0.0584830
$$947$$ 15.4763i 0.502911i 0.967869 + 0.251456i $$0.0809092\pi$$
−0.967869 + 0.251456i $$0.919091\pi$$
$$948$$ − 22.6399i − 0.735309i
$$949$$ 22.9234 0.744124
$$950$$ 0 0
$$951$$ 2.15045 0.0697330
$$952$$ 11.7772i 0.381700i
$$953$$ − 32.0508i − 1.03823i −0.854705 0.519113i $$-0.826262\pi$$
0.854705 0.519113i $$-0.173738\pi$$
$$954$$ 0.911603 0.0295142
$$955$$ 0 0
$$956$$ −52.1016 −1.68509
$$957$$ 3.61213i 0.116763i
$$958$$ 1.80351i 0.0582687i
$$959$$ −46.3996 −1.49832
$$960$$ 0 0
$$961$$ 67.5012 2.17746
$$962$$ − 1.14903i − 0.0370462i
$$963$$ − 0.962389i − 0.0310125i
$$964$$ 56.0263 1.80449
$$965$$ 0 0
$$966$$ 4.35359 0.140074
$$967$$ 17.3766i 0.558794i 0.960176 + 0.279397i $$0.0901346\pi$$
−0.960176 + 0.279397i $$0.909865\pi$$
$$968$$ 0.768452i 0.0246990i
$$969$$ −19.7283 −0.633764
$$970$$ 0 0
$$971$$ 36.2031 1.16181 0.580907 0.813970i $$-0.302698\pi$$
0.580907 + 0.813970i $$0.302698\pi$$
$$972$$ 1.96239i 0.0629436i
$$973$$ 45.6042i 1.46200i
$$974$$ −6.88015 −0.220454
$$975$$ 0 0
$$976$$ −32.8510 −1.05153
$$977$$ 28.1476i 0.900522i 0.892897 + 0.450261i $$0.148669\pi$$
−0.892897 + 0.450261i $$0.851331\pi$$
$$978$$ − 2.44851i − 0.0782946i
$$979$$ 2.77575 0.0887132
$$980$$ 0 0
$$981$$ 11.4010 0.364007
$$982$$ 4.80492i 0.153331i
$$983$$ 7.07381i 0.225619i 0.993617 + 0.112810i $$0.0359851\pi$$
−0.993617 + 0.112810i $$0.964015\pi$$
$$984$$ 3.37187 0.107491
$$985$$ 0 0
$$986$$ −3.20456 −0.102054
$$987$$ 33.2506i 1.05838i
$$988$$ 25.0710i 0.797614i
$$989$$ −62.1476 −1.97618
$$990$$ 0 0
$$991$$ 44.4260 1.41124 0.705619 0.708592i $$-0.250669\pi$$
0.705619 + 0.708592i $$0.250669\pi$$
$$992$$ 22.5209i 0.715039i
$$993$$ − 14.5501i − 0.461733i
$$994$$ 6.44851 0.204534
$$995$$ 0 0
$$996$$ −21.3649 −0.676971
$$997$$ 28.4847i 0.902120i 0.892494 + 0.451060i $$0.148954\pi$$
−0.892494 + 0.451060i $$0.851046\pi$$
$$998$$ − 2.74940i − 0.0870307i
$$999$$ 2.00000 0.0632772
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.g.199.4 6
3.2 odd 2 2475.2.c.r.199.3 6
5.2 odd 4 165.2.a.c.1.2 3
5.3 odd 4 825.2.a.k.1.2 3
5.4 even 2 inner 825.2.c.g.199.3 6
15.2 even 4 495.2.a.e.1.2 3
15.8 even 4 2475.2.a.bb.1.2 3
15.14 odd 2 2475.2.c.r.199.4 6
20.7 even 4 2640.2.a.be.1.1 3
35.27 even 4 8085.2.a.bk.1.2 3
55.32 even 4 1815.2.a.m.1.2 3
55.43 even 4 9075.2.a.cf.1.2 3
60.47 odd 4 7920.2.a.cj.1.1 3
165.32 odd 4 5445.2.a.z.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.a.c.1.2 3 5.2 odd 4
495.2.a.e.1.2 3 15.2 even 4
825.2.a.k.1.2 3 5.3 odd 4
825.2.c.g.199.3 6 5.4 even 2 inner
825.2.c.g.199.4 6 1.1 even 1 trivial
1815.2.a.m.1.2 3 55.32 even 4
2475.2.a.bb.1.2 3 15.8 even 4
2475.2.c.r.199.3 6 3.2 odd 2
2475.2.c.r.199.4 6 15.14 odd 2
2640.2.a.be.1.1 3 20.7 even 4
5445.2.a.z.1.2 3 165.32 odd 4
7920.2.a.cj.1.1 3 60.47 odd 4
8085.2.a.bk.1.2 3 35.27 even 4
9075.2.a.cf.1.2 3 55.43 even 4