# Properties

 Label 4650.2.d.bd.3349.1 Level $4650$ Weight $2$ Character 4650.3349 Analytic conductor $37.130$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4650.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$37.1304369399$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{17})$$ Defining polynomial: $$x^{4} + 9x^{2} + 16$$ x^4 + 9*x^2 + 16 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 930) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 3349.1 Root $$-2.56155i$$ of defining polynomial Character $$\chi$$ $$=$$ 4650.3349 Dual form 4650.2.d.bd.3349.4

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -2.56155i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -2.56155i q^{7} +1.00000i q^{8} -1.00000 q^{9} +2.56155 q^{11} +1.00000i q^{12} +2.00000i q^{13} -2.56155 q^{14} +1.00000 q^{16} +3.12311i q^{17} +1.00000i q^{18} +7.68466 q^{19} -2.56155 q^{21} -2.56155i q^{22} +1.43845i q^{23} +1.00000 q^{24} +2.00000 q^{26} +1.00000i q^{27} +2.56155i q^{28} -7.12311 q^{29} +1.00000 q^{31} -1.00000i q^{32} -2.56155i q^{33} +3.12311 q^{34} +1.00000 q^{36} +3.12311i q^{37} -7.68466i q^{38} +2.00000 q^{39} +7.12311 q^{41} +2.56155i q^{42} +12.8078i q^{43} -2.56155 q^{44} +1.43845 q^{46} -5.12311i q^{47} -1.00000i q^{48} +0.438447 q^{49} +3.12311 q^{51} -2.00000i q^{52} +7.43845i q^{53} +1.00000 q^{54} +2.56155 q^{56} -7.68466i q^{57} +7.12311i q^{58} +13.1231 q^{59} +6.00000 q^{61} -1.00000i q^{62} +2.56155i q^{63} -1.00000 q^{64} -2.56155 q^{66} +15.3693i q^{67} -3.12311i q^{68} +1.43845 q^{69} -7.68466 q^{71} -1.00000i q^{72} -10.8078i q^{73} +3.12311 q^{74} -7.68466 q^{76} -6.56155i q^{77} -2.00000i q^{78} +4.31534 q^{79} +1.00000 q^{81} -7.12311i q^{82} +14.2462i q^{83} +2.56155 q^{84} +12.8078 q^{86} +7.12311i q^{87} +2.56155i q^{88} +13.6847 q^{89} +5.12311 q^{91} -1.43845i q^{92} -1.00000i q^{93} -5.12311 q^{94} -1.00000 q^{96} +6.00000i q^{97} -0.438447i q^{98} -2.56155 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4} - 4 q^{6} - 4 q^{9}+O(q^{10})$$ 4 * q - 4 * q^4 - 4 * q^6 - 4 * q^9 $$4 q - 4 q^{4} - 4 q^{6} - 4 q^{9} + 2 q^{11} - 2 q^{14} + 4 q^{16} + 6 q^{19} - 2 q^{21} + 4 q^{24} + 8 q^{26} - 12 q^{29} + 4 q^{31} - 4 q^{34} + 4 q^{36} + 8 q^{39} + 12 q^{41} - 2 q^{44} + 14 q^{46} + 10 q^{49} - 4 q^{51} + 4 q^{54} + 2 q^{56} + 36 q^{59} + 24 q^{61} - 4 q^{64} - 2 q^{66} + 14 q^{69} - 6 q^{71} - 4 q^{74} - 6 q^{76} + 42 q^{79} + 4 q^{81} + 2 q^{84} + 10 q^{86} + 30 q^{89} + 4 q^{91} - 4 q^{94} - 4 q^{96} - 2 q^{99}+O(q^{100})$$ 4 * q - 4 * q^4 - 4 * q^6 - 4 * q^9 + 2 * q^11 - 2 * q^14 + 4 * q^16 + 6 * q^19 - 2 * q^21 + 4 * q^24 + 8 * q^26 - 12 * q^29 + 4 * q^31 - 4 * q^34 + 4 * q^36 + 8 * q^39 + 12 * q^41 - 2 * q^44 + 14 * q^46 + 10 * q^49 - 4 * q^51 + 4 * q^54 + 2 * q^56 + 36 * q^59 + 24 * q^61 - 4 * q^64 - 2 * q^66 + 14 * q^69 - 6 * q^71 - 4 * q^74 - 6 * q^76 + 42 * q^79 + 4 * q^81 + 2 * q^84 + 10 * q^86 + 30 * q^89 + 4 * q^91 - 4 * q^94 - 4 * q^96 - 2 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4650\mathbb{Z}\right)^\times$$.

 $$n$$ $$1801$$ $$2977$$ $$3101$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 1.00000i − 0.707107i
$$3$$ − 1.00000i − 0.577350i
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ −1.00000 −0.408248
$$7$$ − 2.56155i − 0.968176i −0.875019 0.484088i $$-0.839151\pi$$
0.875019 0.484088i $$-0.160849\pi$$
$$8$$ 1.00000i 0.353553i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 2.56155 0.772337 0.386169 0.922428i $$-0.373798\pi$$
0.386169 + 0.922428i $$0.373798\pi$$
$$12$$ 1.00000i 0.288675i
$$13$$ 2.00000i 0.554700i 0.960769 + 0.277350i $$0.0894562\pi$$
−0.960769 + 0.277350i $$0.910544\pi$$
$$14$$ −2.56155 −0.684604
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 3.12311i 0.757464i 0.925506 + 0.378732i $$0.123640\pi$$
−0.925506 + 0.378732i $$0.876360\pi$$
$$18$$ 1.00000i 0.235702i
$$19$$ 7.68466 1.76298 0.881491 0.472201i $$-0.156540\pi$$
0.881491 + 0.472201i $$0.156540\pi$$
$$20$$ 0 0
$$21$$ −2.56155 −0.558977
$$22$$ − 2.56155i − 0.546125i
$$23$$ 1.43845i 0.299937i 0.988691 + 0.149968i $$0.0479172\pi$$
−0.988691 + 0.149968i $$0.952083\pi$$
$$24$$ 1.00000 0.204124
$$25$$ 0 0
$$26$$ 2.00000 0.392232
$$27$$ 1.00000i 0.192450i
$$28$$ 2.56155i 0.484088i
$$29$$ −7.12311 −1.32273 −0.661364 0.750065i $$-0.730022\pi$$
−0.661364 + 0.750065i $$0.730022\pi$$
$$30$$ 0 0
$$31$$ 1.00000 0.179605
$$32$$ − 1.00000i − 0.176777i
$$33$$ − 2.56155i − 0.445909i
$$34$$ 3.12311 0.535608
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 3.12311i 0.513435i 0.966486 + 0.256718i $$0.0826411\pi$$
−0.966486 + 0.256718i $$0.917359\pi$$
$$38$$ − 7.68466i − 1.24662i
$$39$$ 2.00000 0.320256
$$40$$ 0 0
$$41$$ 7.12311 1.11244 0.556221 0.831034i $$-0.312251\pi$$
0.556221 + 0.831034i $$0.312251\pi$$
$$42$$ 2.56155i 0.395256i
$$43$$ 12.8078i 1.95317i 0.215142 + 0.976583i $$0.430979\pi$$
−0.215142 + 0.976583i $$0.569021\pi$$
$$44$$ −2.56155 −0.386169
$$45$$ 0 0
$$46$$ 1.43845 0.212087
$$47$$ − 5.12311i − 0.747282i −0.927573 0.373641i $$-0.878109\pi$$
0.927573 0.373641i $$-0.121891\pi$$
$$48$$ − 1.00000i − 0.144338i
$$49$$ 0.438447 0.0626353
$$50$$ 0 0
$$51$$ 3.12311 0.437322
$$52$$ − 2.00000i − 0.277350i
$$53$$ 7.43845i 1.02175i 0.859655 + 0.510875i $$0.170678\pi$$
−0.859655 + 0.510875i $$0.829322\pi$$
$$54$$ 1.00000 0.136083
$$55$$ 0 0
$$56$$ 2.56155 0.342302
$$57$$ − 7.68466i − 1.01786i
$$58$$ 7.12311i 0.935310i
$$59$$ 13.1231 1.70848 0.854241 0.519877i $$-0.174022\pi$$
0.854241 + 0.519877i $$0.174022\pi$$
$$60$$ 0 0
$$61$$ 6.00000 0.768221 0.384111 0.923287i $$-0.374508\pi$$
0.384111 + 0.923287i $$0.374508\pi$$
$$62$$ − 1.00000i − 0.127000i
$$63$$ 2.56155i 0.322725i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ −2.56155 −0.315305
$$67$$ 15.3693i 1.87766i 0.344380 + 0.938830i $$0.388089\pi$$
−0.344380 + 0.938830i $$0.611911\pi$$
$$68$$ − 3.12311i − 0.378732i
$$69$$ 1.43845 0.173169
$$70$$ 0 0
$$71$$ −7.68466 −0.912001 −0.456001 0.889979i $$-0.650719\pi$$
−0.456001 + 0.889979i $$0.650719\pi$$
$$72$$ − 1.00000i − 0.117851i
$$73$$ − 10.8078i − 1.26495i −0.774580 0.632477i $$-0.782038\pi$$
0.774580 0.632477i $$-0.217962\pi$$
$$74$$ 3.12311 0.363054
$$75$$ 0 0
$$76$$ −7.68466 −0.881491
$$77$$ − 6.56155i − 0.747758i
$$78$$ − 2.00000i − 0.226455i
$$79$$ 4.31534 0.485514 0.242757 0.970087i $$-0.421948\pi$$
0.242757 + 0.970087i $$0.421948\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ − 7.12311i − 0.786615i
$$83$$ 14.2462i 1.56372i 0.623451 + 0.781862i $$0.285730\pi$$
−0.623451 + 0.781862i $$0.714270\pi$$
$$84$$ 2.56155 0.279488
$$85$$ 0 0
$$86$$ 12.8078 1.38110
$$87$$ 7.12311i 0.763677i
$$88$$ 2.56155i 0.273062i
$$89$$ 13.6847 1.45057 0.725285 0.688448i $$-0.241708\pi$$
0.725285 + 0.688448i $$0.241708\pi$$
$$90$$ 0 0
$$91$$ 5.12311 0.537047
$$92$$ − 1.43845i − 0.149968i
$$93$$ − 1.00000i − 0.103695i
$$94$$ −5.12311 −0.528408
$$95$$ 0 0
$$96$$ −1.00000 −0.102062
$$97$$ 6.00000i 0.609208i 0.952479 + 0.304604i $$0.0985241\pi$$
−0.952479 + 0.304604i $$0.901476\pi$$
$$98$$ − 0.438447i − 0.0442899i
$$99$$ −2.56155 −0.257446
$$100$$ 0 0
$$101$$ −0.561553 −0.0558766 −0.0279383 0.999610i $$-0.508894\pi$$
−0.0279383 + 0.999610i $$0.508894\pi$$
$$102$$ − 3.12311i − 0.309234i
$$103$$ − 1.75379i − 0.172806i −0.996260 0.0864030i $$-0.972463\pi$$
0.996260 0.0864030i $$-0.0275373\pi$$
$$104$$ −2.00000 −0.196116
$$105$$ 0 0
$$106$$ 7.43845 0.722486
$$107$$ 2.56155i 0.247635i 0.992305 + 0.123817i $$0.0395137\pi$$
−0.992305 + 0.123817i $$0.960486\pi$$
$$108$$ − 1.00000i − 0.0962250i
$$109$$ −5.36932 −0.514287 −0.257144 0.966373i $$-0.582781\pi$$
−0.257144 + 0.966373i $$0.582781\pi$$
$$110$$ 0 0
$$111$$ 3.12311 0.296432
$$112$$ − 2.56155i − 0.242044i
$$113$$ − 1.68466i − 0.158479i −0.996856 0.0792397i $$-0.974751\pi$$
0.996856 0.0792397i $$-0.0252492\pi$$
$$114$$ −7.68466 −0.719734
$$115$$ 0 0
$$116$$ 7.12311 0.661364
$$117$$ − 2.00000i − 0.184900i
$$118$$ − 13.1231i − 1.20808i
$$119$$ 8.00000 0.733359
$$120$$ 0 0
$$121$$ −4.43845 −0.403495
$$122$$ − 6.00000i − 0.543214i
$$123$$ − 7.12311i − 0.642269i
$$124$$ −1.00000 −0.0898027
$$125$$ 0 0
$$126$$ 2.56155 0.228201
$$127$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$128$$ 1.00000i 0.0883883i
$$129$$ 12.8078 1.12766
$$130$$ 0 0
$$131$$ −15.3693 −1.34282 −0.671412 0.741085i $$-0.734312\pi$$
−0.671412 + 0.741085i $$0.734312\pi$$
$$132$$ 2.56155i 0.222955i
$$133$$ − 19.6847i − 1.70688i
$$134$$ 15.3693 1.32771
$$135$$ 0 0
$$136$$ −3.12311 −0.267804
$$137$$ − 20.2462i − 1.72975i −0.501987 0.864875i $$-0.667397\pi$$
0.501987 0.864875i $$-0.332603\pi$$
$$138$$ − 1.43845i − 0.122449i
$$139$$ 17.1231 1.45236 0.726181 0.687503i $$-0.241293\pi$$
0.726181 + 0.687503i $$0.241293\pi$$
$$140$$ 0 0
$$141$$ −5.12311 −0.431443
$$142$$ 7.68466i 0.644882i
$$143$$ 5.12311i 0.428416i
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ −10.8078 −0.894457
$$147$$ − 0.438447i − 0.0361625i
$$148$$ − 3.12311i − 0.256718i
$$149$$ −17.0540 −1.39712 −0.698558 0.715553i $$-0.746175\pi$$
−0.698558 + 0.715553i $$0.746175\pi$$
$$150$$ 0 0
$$151$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$152$$ 7.68466i 0.623308i
$$153$$ − 3.12311i − 0.252488i
$$154$$ −6.56155 −0.528745
$$155$$ 0 0
$$156$$ −2.00000 −0.160128
$$157$$ − 17.0540i − 1.36106i −0.732722 0.680528i $$-0.761751\pi$$
0.732722 0.680528i $$-0.238249\pi$$
$$158$$ − 4.31534i − 0.343310i
$$159$$ 7.43845 0.589907
$$160$$ 0 0
$$161$$ 3.68466 0.290392
$$162$$ − 1.00000i − 0.0785674i
$$163$$ − 15.3693i − 1.20382i −0.798565 0.601909i $$-0.794407\pi$$
0.798565 0.601909i $$-0.205593\pi$$
$$164$$ −7.12311 −0.556221
$$165$$ 0 0
$$166$$ 14.2462 1.10572
$$167$$ 6.56155i 0.507748i 0.967237 + 0.253874i $$0.0817049\pi$$
−0.967237 + 0.253874i $$0.918295\pi$$
$$168$$ − 2.56155i − 0.197628i
$$169$$ 9.00000 0.692308
$$170$$ 0 0
$$171$$ −7.68466 −0.587661
$$172$$ − 12.8078i − 0.976583i
$$173$$ 8.24621i 0.626948i 0.949597 + 0.313474i $$0.101493\pi$$
−0.949597 + 0.313474i $$0.898507\pi$$
$$174$$ 7.12311 0.540001
$$175$$ 0 0
$$176$$ 2.56155 0.193084
$$177$$ − 13.1231i − 0.986393i
$$178$$ − 13.6847i − 1.02571i
$$179$$ 12.0000 0.896922 0.448461 0.893802i $$-0.351972\pi$$
0.448461 + 0.893802i $$0.351972\pi$$
$$180$$ 0 0
$$181$$ −13.0540 −0.970294 −0.485147 0.874433i $$-0.661234\pi$$
−0.485147 + 0.874433i $$0.661234\pi$$
$$182$$ − 5.12311i − 0.379750i
$$183$$ − 6.00000i − 0.443533i
$$184$$ −1.43845 −0.106044
$$185$$ 0 0
$$186$$ −1.00000 −0.0733236
$$187$$ 8.00000i 0.585018i
$$188$$ 5.12311i 0.373641i
$$189$$ 2.56155 0.186326
$$190$$ 0 0
$$191$$ −14.2462 −1.03082 −0.515410 0.856944i $$-0.672360\pi$$
−0.515410 + 0.856944i $$0.672360\pi$$
$$192$$ 1.00000i 0.0721688i
$$193$$ 14.4924i 1.04319i 0.853194 + 0.521594i $$0.174662\pi$$
−0.853194 + 0.521594i $$0.825338\pi$$
$$194$$ 6.00000 0.430775
$$195$$ 0 0
$$196$$ −0.438447 −0.0313177
$$197$$ − 6.00000i − 0.427482i −0.976890 0.213741i $$-0.931435\pi$$
0.976890 0.213741i $$-0.0685649\pi$$
$$198$$ 2.56155i 0.182042i
$$199$$ 16.8078 1.19147 0.595735 0.803181i $$-0.296861\pi$$
0.595735 + 0.803181i $$0.296861\pi$$
$$200$$ 0 0
$$201$$ 15.3693 1.08407
$$202$$ 0.561553i 0.0395107i
$$203$$ 18.2462i 1.28063i
$$204$$ −3.12311 −0.218661
$$205$$ 0 0
$$206$$ −1.75379 −0.122192
$$207$$ − 1.43845i − 0.0999790i
$$208$$ 2.00000i 0.138675i
$$209$$ 19.6847 1.36162
$$210$$ 0 0
$$211$$ −23.6847 −1.63052 −0.815260 0.579096i $$-0.803406\pi$$
−0.815260 + 0.579096i $$0.803406\pi$$
$$212$$ − 7.43845i − 0.510875i
$$213$$ 7.68466i 0.526544i
$$214$$ 2.56155 0.175104
$$215$$ 0 0
$$216$$ −1.00000 −0.0680414
$$217$$ − 2.56155i − 0.173890i
$$218$$ 5.36932i 0.363656i
$$219$$ −10.8078 −0.730321
$$220$$ 0 0
$$221$$ −6.24621 −0.420166
$$222$$ − 3.12311i − 0.209609i
$$223$$ − 21.1231i − 1.41451i −0.706960 0.707254i $$-0.749934\pi$$
0.706960 0.707254i $$-0.250066\pi$$
$$224$$ −2.56155 −0.171151
$$225$$ 0 0
$$226$$ −1.68466 −0.112062
$$227$$ − 7.68466i − 0.510049i −0.966935 0.255024i $$-0.917917\pi$$
0.966935 0.255024i $$-0.0820835\pi$$
$$228$$ 7.68466i 0.508929i
$$229$$ 16.5616 1.09442 0.547209 0.836996i $$-0.315690\pi$$
0.547209 + 0.836996i $$0.315690\pi$$
$$230$$ 0 0
$$231$$ −6.56155 −0.431718
$$232$$ − 7.12311i − 0.467655i
$$233$$ − 17.0540i − 1.11724i −0.829423 0.558622i $$-0.811330\pi$$
0.829423 0.558622i $$-0.188670\pi$$
$$234$$ −2.00000 −0.130744
$$235$$ 0 0
$$236$$ −13.1231 −0.854241
$$237$$ − 4.31534i − 0.280312i
$$238$$ − 8.00000i − 0.518563i
$$239$$ −24.0000 −1.55243 −0.776215 0.630468i $$-0.782863\pi$$
−0.776215 + 0.630468i $$0.782863\pi$$
$$240$$ 0 0
$$241$$ 12.2462 0.788848 0.394424 0.918929i $$-0.370944\pi$$
0.394424 + 0.918929i $$0.370944\pi$$
$$242$$ 4.43845i 0.285314i
$$243$$ − 1.00000i − 0.0641500i
$$244$$ −6.00000 −0.384111
$$245$$ 0 0
$$246$$ −7.12311 −0.454153
$$247$$ 15.3693i 0.977926i
$$248$$ 1.00000i 0.0635001i
$$249$$ 14.2462 0.902817
$$250$$ 0 0
$$251$$ 16.4924 1.04099 0.520496 0.853864i $$-0.325747\pi$$
0.520496 + 0.853864i $$0.325747\pi$$
$$252$$ − 2.56155i − 0.161363i
$$253$$ 3.68466i 0.231652i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 1.68466i 0.105086i 0.998619 + 0.0525431i $$0.0167327\pi$$
−0.998619 + 0.0525431i $$0.983267\pi$$
$$258$$ − 12.8078i − 0.797377i
$$259$$ 8.00000 0.497096
$$260$$ 0 0
$$261$$ 7.12311 0.440909
$$262$$ 15.3693i 0.949520i
$$263$$ − 20.4924i − 1.26362i −0.775125 0.631808i $$-0.782313\pi$$
0.775125 0.631808i $$-0.217687\pi$$
$$264$$ 2.56155 0.157653
$$265$$ 0 0
$$266$$ −19.6847 −1.20694
$$267$$ − 13.6847i − 0.837487i
$$268$$ − 15.3693i − 0.938830i
$$269$$ 0.246211 0.0150118 0.00750588 0.999972i $$-0.497611\pi$$
0.00750588 + 0.999972i $$0.497611\pi$$
$$270$$ 0 0
$$271$$ 27.0540 1.64341 0.821706 0.569912i $$-0.193023\pi$$
0.821706 + 0.569912i $$0.193023\pi$$
$$272$$ 3.12311i 0.189366i
$$273$$ − 5.12311i − 0.310064i
$$274$$ −20.2462 −1.22312
$$275$$ 0 0
$$276$$ −1.43845 −0.0865843
$$277$$ 5.36932i 0.322611i 0.986905 + 0.161305i $$0.0515704\pi$$
−0.986905 + 0.161305i $$0.948430\pi$$
$$278$$ − 17.1231i − 1.02698i
$$279$$ −1.00000 −0.0598684
$$280$$ 0 0
$$281$$ 4.87689 0.290931 0.145466 0.989363i $$-0.453532\pi$$
0.145466 + 0.989363i $$0.453532\pi$$
$$282$$ 5.12311i 0.305077i
$$283$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$284$$ 7.68466 0.456001
$$285$$ 0 0
$$286$$ 5.12311 0.302936
$$287$$ − 18.2462i − 1.07704i
$$288$$ 1.00000i 0.0589256i
$$289$$ 7.24621 0.426248
$$290$$ 0 0
$$291$$ 6.00000 0.351726
$$292$$ 10.8078i 0.632477i
$$293$$ 26.4924i 1.54770i 0.633367 + 0.773852i $$0.281673\pi$$
−0.633367 + 0.773852i $$0.718327\pi$$
$$294$$ −0.438447 −0.0255708
$$295$$ 0 0
$$296$$ −3.12311 −0.181527
$$297$$ 2.56155i 0.148636i
$$298$$ 17.0540i 0.987910i
$$299$$ −2.87689 −0.166375
$$300$$ 0 0
$$301$$ 32.8078 1.89101
$$302$$ 0 0
$$303$$ 0.561553i 0.0322604i
$$304$$ 7.68466 0.440745
$$305$$ 0 0
$$306$$ −3.12311 −0.178536
$$307$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$308$$ 6.56155i 0.373879i
$$309$$ −1.75379 −0.0997696
$$310$$ 0 0
$$311$$ −1.75379 −0.0994482 −0.0497241 0.998763i $$-0.515834\pi$$
−0.0497241 + 0.998763i $$0.515834\pi$$
$$312$$ 2.00000i 0.113228i
$$313$$ − 10.0000i − 0.565233i −0.959233 0.282617i $$-0.908798\pi$$
0.959233 0.282617i $$-0.0912024\pi$$
$$314$$ −17.0540 −0.962412
$$315$$ 0 0
$$316$$ −4.31534 −0.242757
$$317$$ − 16.8769i − 0.947901i −0.880552 0.473950i $$-0.842828\pi$$
0.880552 0.473950i $$-0.157172\pi$$
$$318$$ − 7.43845i − 0.417127i
$$319$$ −18.2462 −1.02159
$$320$$ 0 0
$$321$$ 2.56155 0.142972
$$322$$ − 3.68466i − 0.205338i
$$323$$ 24.0000i 1.33540i
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ −15.3693 −0.851228
$$327$$ 5.36932i 0.296924i
$$328$$ 7.12311i 0.393308i
$$329$$ −13.1231 −0.723500
$$330$$ 0 0
$$331$$ −6.24621 −0.343323 −0.171661 0.985156i $$-0.554914\pi$$
−0.171661 + 0.985156i $$0.554914\pi$$
$$332$$ − 14.2462i − 0.781862i
$$333$$ − 3.12311i − 0.171145i
$$334$$ 6.56155 0.359032
$$335$$ 0 0
$$336$$ −2.56155 −0.139744
$$337$$ 10.0000i 0.544735i 0.962193 + 0.272367i $$0.0878066\pi$$
−0.962193 + 0.272367i $$0.912193\pi$$
$$338$$ − 9.00000i − 0.489535i
$$339$$ −1.68466 −0.0914981
$$340$$ 0 0
$$341$$ 2.56155 0.138716
$$342$$ 7.68466i 0.415539i
$$343$$ − 19.0540i − 1.02882i
$$344$$ −12.8078 −0.690548
$$345$$ 0 0
$$346$$ 8.24621 0.443319
$$347$$ − 14.2462i − 0.764777i −0.924002 0.382388i $$-0.875102\pi$$
0.924002 0.382388i $$-0.124898\pi$$
$$348$$ − 7.12311i − 0.381839i
$$349$$ −5.36932 −0.287413 −0.143706 0.989620i $$-0.545902\pi$$
−0.143706 + 0.989620i $$0.545902\pi$$
$$350$$ 0 0
$$351$$ −2.00000 −0.106752
$$352$$ − 2.56155i − 0.136531i
$$353$$ − 0.246211i − 0.0131045i −0.999979 0.00655225i $$-0.997914\pi$$
0.999979 0.00655225i $$-0.00208566\pi$$
$$354$$ −13.1231 −0.697485
$$355$$ 0 0
$$356$$ −13.6847 −0.725285
$$357$$ − 8.00000i − 0.423405i
$$358$$ − 12.0000i − 0.634220i
$$359$$ 31.6847 1.67225 0.836126 0.548537i $$-0.184815\pi$$
0.836126 + 0.548537i $$0.184815\pi$$
$$360$$ 0 0
$$361$$ 40.0540 2.10810
$$362$$ 13.0540i 0.686102i
$$363$$ 4.43845i 0.232958i
$$364$$ −5.12311 −0.268524
$$365$$ 0 0
$$366$$ −6.00000 −0.313625
$$367$$ − 15.3693i − 0.802272i −0.916019 0.401136i $$-0.868616\pi$$
0.916019 0.401136i $$-0.131384\pi$$
$$368$$ 1.43845i 0.0749842i
$$369$$ −7.12311 −0.370814
$$370$$ 0 0
$$371$$ 19.0540 0.989233
$$372$$ 1.00000i 0.0518476i
$$373$$ − 5.68466i − 0.294340i −0.989111 0.147170i $$-0.952983\pi$$
0.989111 0.147170i $$-0.0470165\pi$$
$$374$$ 8.00000 0.413670
$$375$$ 0 0
$$376$$ 5.12311 0.264204
$$377$$ − 14.2462i − 0.733717i
$$378$$ − 2.56155i − 0.131752i
$$379$$ 7.05398 0.362338 0.181169 0.983452i $$-0.442012\pi$$
0.181169 + 0.983452i $$0.442012\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 14.2462i 0.728900i
$$383$$ − 10.2462i − 0.523557i −0.965128 0.261778i $$-0.915691\pi$$
0.965128 0.261778i $$-0.0843090\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 0 0
$$386$$ 14.4924 0.737645
$$387$$ − 12.8078i − 0.651055i
$$388$$ − 6.00000i − 0.304604i
$$389$$ −7.75379 −0.393133 −0.196566 0.980491i $$-0.562979\pi$$
−0.196566 + 0.980491i $$0.562979\pi$$
$$390$$ 0 0
$$391$$ −4.49242 −0.227192
$$392$$ 0.438447i 0.0221449i
$$393$$ 15.3693i 0.775279i
$$394$$ −6.00000 −0.302276
$$395$$ 0 0
$$396$$ 2.56155 0.128723
$$397$$ − 9.05398i − 0.454406i −0.973847 0.227203i $$-0.927042\pi$$
0.973847 0.227203i $$-0.0729581\pi$$
$$398$$ − 16.8078i − 0.842497i
$$399$$ −19.6847 −0.985466
$$400$$ 0 0
$$401$$ −13.6847 −0.683379 −0.341690 0.939813i $$-0.610999\pi$$
−0.341690 + 0.939813i $$0.610999\pi$$
$$402$$ − 15.3693i − 0.766552i
$$403$$ 2.00000i 0.0996271i
$$404$$ 0.561553 0.0279383
$$405$$ 0 0
$$406$$ 18.2462 0.905544
$$407$$ 8.00000i 0.396545i
$$408$$ 3.12311i 0.154617i
$$409$$ 37.3693 1.84779 0.923897 0.382642i $$-0.124986\pi$$
0.923897 + 0.382642i $$0.124986\pi$$
$$410$$ 0 0
$$411$$ −20.2462 −0.998672
$$412$$ 1.75379i 0.0864030i
$$413$$ − 33.6155i − 1.65411i
$$414$$ −1.43845 −0.0706958
$$415$$ 0 0
$$416$$ 2.00000 0.0980581
$$417$$ − 17.1231i − 0.838522i
$$418$$ − 19.6847i − 0.962808i
$$419$$ −5.75379 −0.281091 −0.140545 0.990074i $$-0.544886\pi$$
−0.140545 + 0.990074i $$0.544886\pi$$
$$420$$ 0 0
$$421$$ −14.4924 −0.706317 −0.353159 0.935563i $$-0.614892\pi$$
−0.353159 + 0.935563i $$0.614892\pi$$
$$422$$ 23.6847i 1.15295i
$$423$$ 5.12311i 0.249094i
$$424$$ −7.43845 −0.361243
$$425$$ 0 0
$$426$$ 7.68466 0.372323
$$427$$ − 15.3693i − 0.743773i
$$428$$ − 2.56155i − 0.123817i
$$429$$ 5.12311 0.247346
$$430$$ 0 0
$$431$$ −30.2462 −1.45691 −0.728454 0.685094i $$-0.759761\pi$$
−0.728454 + 0.685094i $$0.759761\pi$$
$$432$$ 1.00000i 0.0481125i
$$433$$ 35.3002i 1.69642i 0.529661 + 0.848209i $$0.322319\pi$$
−0.529661 + 0.848209i $$0.677681\pi$$
$$434$$ −2.56155 −0.122958
$$435$$ 0 0
$$436$$ 5.36932 0.257144
$$437$$ 11.0540i 0.528783i
$$438$$ 10.8078i 0.516415i
$$439$$ −9.61553 −0.458924 −0.229462 0.973318i $$-0.573697\pi$$
−0.229462 + 0.973318i $$0.573697\pi$$
$$440$$ 0 0
$$441$$ −0.438447 −0.0208784
$$442$$ 6.24621i 0.297102i
$$443$$ 23.0540i 1.09533i 0.836699 + 0.547664i $$0.184483\pi$$
−0.836699 + 0.547664i $$0.815517\pi$$
$$444$$ −3.12311 −0.148216
$$445$$ 0 0
$$446$$ −21.1231 −1.00021
$$447$$ 17.0540i 0.806625i
$$448$$ 2.56155i 0.121022i
$$449$$ −10.4924 −0.495168 −0.247584 0.968866i $$-0.579637\pi$$
−0.247584 + 0.968866i $$0.579637\pi$$
$$450$$ 0 0
$$451$$ 18.2462 0.859181
$$452$$ 1.68466i 0.0792397i
$$453$$ 0 0
$$454$$ −7.68466 −0.360659
$$455$$ 0 0
$$456$$ 7.68466 0.359867
$$457$$ 24.7386i 1.15722i 0.815603 + 0.578612i $$0.196406\pi$$
−0.815603 + 0.578612i $$0.803594\pi$$
$$458$$ − 16.5616i − 0.773871i
$$459$$ −3.12311 −0.145774
$$460$$ 0 0
$$461$$ 9.36932 0.436373 0.218186 0.975907i $$-0.429986\pi$$
0.218186 + 0.975907i $$0.429986\pi$$
$$462$$ 6.56155i 0.305271i
$$463$$ 30.7386i 1.42855i 0.699867 + 0.714273i $$0.253242\pi$$
−0.699867 + 0.714273i $$0.746758\pi$$
$$464$$ −7.12311 −0.330682
$$465$$ 0 0
$$466$$ −17.0540 −0.790010
$$467$$ 9.75379i 0.451352i 0.974202 + 0.225676i $$0.0724590\pi$$
−0.974202 + 0.225676i $$0.927541\pi$$
$$468$$ 2.00000i 0.0924500i
$$469$$ 39.3693 1.81791
$$470$$ 0 0
$$471$$ −17.0540 −0.785806
$$472$$ 13.1231i 0.604040i
$$473$$ 32.8078i 1.50850i
$$474$$ −4.31534 −0.198210
$$475$$ 0 0
$$476$$ −8.00000 −0.366679
$$477$$ − 7.43845i − 0.340583i
$$478$$ 24.0000i 1.09773i
$$479$$ −10.5616 −0.482570 −0.241285 0.970454i $$-0.577569\pi$$
−0.241285 + 0.970454i $$0.577569\pi$$
$$480$$ 0 0
$$481$$ −6.24621 −0.284803
$$482$$ − 12.2462i − 0.557800i
$$483$$ − 3.68466i − 0.167658i
$$484$$ 4.43845 0.201748
$$485$$ 0 0
$$486$$ −1.00000 −0.0453609
$$487$$ − 24.0000i − 1.08754i −0.839233 0.543772i $$-0.816996\pi$$
0.839233 0.543772i $$-0.183004\pi$$
$$488$$ 6.00000i 0.271607i
$$489$$ −15.3693 −0.695025
$$490$$ 0 0
$$491$$ −25.9309 −1.17024 −0.585122 0.810945i $$-0.698953\pi$$
−0.585122 + 0.810945i $$0.698953\pi$$
$$492$$ 7.12311i 0.321134i
$$493$$ − 22.2462i − 1.00192i
$$494$$ 15.3693 0.691498
$$495$$ 0 0
$$496$$ 1.00000 0.0449013
$$497$$ 19.6847i 0.882978i
$$498$$ − 14.2462i − 0.638388i
$$499$$ −20.0000 −0.895323 −0.447661 0.894203i $$-0.647743\pi$$
−0.447661 + 0.894203i $$0.647743\pi$$
$$500$$ 0 0
$$501$$ 6.56155 0.293149
$$502$$ − 16.4924i − 0.736093i
$$503$$ 26.2462i 1.17026i 0.810939 + 0.585130i $$0.198957\pi$$
−0.810939 + 0.585130i $$0.801043\pi$$
$$504$$ −2.56155 −0.114101
$$505$$ 0 0
$$506$$ 3.68466 0.163803
$$507$$ − 9.00000i − 0.399704i
$$508$$ 0 0
$$509$$ −19.6155 −0.869443 −0.434721 0.900565i $$-0.643153\pi$$
−0.434721 + 0.900565i $$0.643153\pi$$
$$510$$ 0 0
$$511$$ −27.6847 −1.22470
$$512$$ − 1.00000i − 0.0441942i
$$513$$ 7.68466i 0.339286i
$$514$$ 1.68466 0.0743071
$$515$$ 0 0
$$516$$ −12.8078 −0.563830
$$517$$ − 13.1231i − 0.577154i
$$518$$ − 8.00000i − 0.351500i
$$519$$ 8.24621 0.361968
$$520$$ 0 0
$$521$$ −32.2462 −1.41273 −0.706366 0.707847i $$-0.749667\pi$$
−0.706366 + 0.707847i $$0.749667\pi$$
$$522$$ − 7.12311i − 0.311770i
$$523$$ − 22.4233i − 0.980502i −0.871581 0.490251i $$-0.836905\pi$$
0.871581 0.490251i $$-0.163095\pi$$
$$524$$ 15.3693 0.671412
$$525$$ 0 0
$$526$$ −20.4924 −0.893512
$$527$$ 3.12311i 0.136045i
$$528$$ − 2.56155i − 0.111477i
$$529$$ 20.9309 0.910038
$$530$$ 0 0
$$531$$ −13.1231 −0.569494
$$532$$ 19.6847i 0.853438i
$$533$$ 14.2462i 0.617072i
$$534$$ −13.6847 −0.592193
$$535$$ 0 0
$$536$$ −15.3693 −0.663853
$$537$$ − 12.0000i − 0.517838i
$$538$$ − 0.246211i − 0.0106149i
$$539$$ 1.12311 0.0483756
$$540$$ 0 0
$$541$$ 19.7538 0.849282 0.424641 0.905362i $$-0.360400\pi$$
0.424641 + 0.905362i $$0.360400\pi$$
$$542$$ − 27.0540i − 1.16207i
$$543$$ 13.0540i 0.560200i
$$544$$ 3.12311 0.133902
$$545$$ 0 0
$$546$$ −5.12311 −0.219249
$$547$$ − 26.8769i − 1.14917i −0.818444 0.574587i $$-0.805163\pi$$
0.818444 0.574587i $$-0.194837\pi$$
$$548$$ 20.2462i 0.864875i
$$549$$ −6.00000 −0.256074
$$550$$ 0 0
$$551$$ −54.7386 −2.33194
$$552$$ 1.43845i 0.0612244i
$$553$$ − 11.0540i − 0.470063i
$$554$$ 5.36932 0.228120
$$555$$ 0 0
$$556$$ −17.1231 −0.726181
$$557$$ 26.8078i 1.13588i 0.823069 + 0.567941i $$0.192260\pi$$
−0.823069 + 0.567941i $$0.807740\pi$$
$$558$$ 1.00000i 0.0423334i
$$559$$ −25.6155 −1.08342
$$560$$ 0 0
$$561$$ 8.00000 0.337760
$$562$$ − 4.87689i − 0.205719i
$$563$$ 16.4924i 0.695073i 0.937667 + 0.347536i $$0.112982\pi$$
−0.937667 + 0.347536i $$0.887018\pi$$
$$564$$ 5.12311 0.215722
$$565$$ 0 0
$$566$$ 0 0
$$567$$ − 2.56155i − 0.107575i
$$568$$ − 7.68466i − 0.322441i
$$569$$ −30.8078 −1.29153 −0.645764 0.763537i $$-0.723461\pi$$
−0.645764 + 0.763537i $$0.723461\pi$$
$$570$$ 0 0
$$571$$ 41.1231 1.72095 0.860474 0.509494i $$-0.170167\pi$$
0.860474 + 0.509494i $$0.170167\pi$$
$$572$$ − 5.12311i − 0.214208i
$$573$$ 14.2462i 0.595144i
$$574$$ −18.2462 −0.761582
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ − 15.1231i − 0.629583i −0.949161 0.314792i $$-0.898065\pi$$
0.949161 0.314792i $$-0.101935\pi$$
$$578$$ − 7.24621i − 0.301403i
$$579$$ 14.4924 0.602285
$$580$$ 0 0
$$581$$ 36.4924 1.51396
$$582$$ − 6.00000i − 0.248708i
$$583$$ 19.0540i 0.789135i
$$584$$ 10.8078 0.447228
$$585$$ 0 0
$$586$$ 26.4924 1.09439
$$587$$ 0.492423i 0.0203245i 0.999948 + 0.0101622i $$0.00323479\pi$$
−0.999948 + 0.0101622i $$0.996765\pi$$
$$588$$ 0.438447i 0.0180813i
$$589$$ 7.68466 0.316641
$$590$$ 0 0
$$591$$ −6.00000 −0.246807
$$592$$ 3.12311i 0.128359i
$$593$$ − 30.0000i − 1.23195i −0.787765 0.615976i $$-0.788762\pi$$
0.787765 0.615976i $$-0.211238\pi$$
$$594$$ 2.56155 0.105102
$$595$$ 0 0
$$596$$ 17.0540 0.698558
$$597$$ − 16.8078i − 0.687896i
$$598$$ 2.87689i 0.117645i
$$599$$ −38.4233 −1.56993 −0.784967 0.619538i $$-0.787320\pi$$
−0.784967 + 0.619538i $$0.787320\pi$$
$$600$$ 0 0
$$601$$ 2.00000 0.0815817 0.0407909 0.999168i $$-0.487012\pi$$
0.0407909 + 0.999168i $$0.487012\pi$$
$$602$$ − 32.8078i − 1.33714i
$$603$$ − 15.3693i − 0.625887i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0.561553 0.0228115
$$607$$ − 7.05398i − 0.286312i −0.989700 0.143156i $$-0.954275\pi$$
0.989700 0.143156i $$-0.0457251\pi$$
$$608$$ − 7.68466i − 0.311654i
$$609$$ 18.2462 0.739374
$$610$$ 0 0
$$611$$ 10.2462 0.414517
$$612$$ 3.12311i 0.126244i
$$613$$ 2.00000i 0.0807792i 0.999184 + 0.0403896i $$0.0128599\pi$$
−0.999184 + 0.0403896i $$0.987140\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 6.56155 0.264372
$$617$$ 15.4384i 0.621528i 0.950487 + 0.310764i $$0.100585\pi$$
−0.950487 + 0.310764i $$0.899415\pi$$
$$618$$ 1.75379i 0.0705477i
$$619$$ −11.3693 −0.456971 −0.228486 0.973547i $$-0.573377\pi$$
−0.228486 + 0.973547i $$0.573377\pi$$
$$620$$ 0 0
$$621$$ −1.43845 −0.0577229
$$622$$ 1.75379i 0.0703205i
$$623$$ − 35.0540i − 1.40441i
$$624$$ 2.00000 0.0800641
$$625$$ 0 0
$$626$$ −10.0000 −0.399680
$$627$$ − 19.6847i − 0.786130i
$$628$$ 17.0540i 0.680528i
$$629$$ −9.75379 −0.388909
$$630$$ 0 0
$$631$$ 29.3002 1.16642 0.583211 0.812321i $$-0.301796\pi$$
0.583211 + 0.812321i $$0.301796\pi$$
$$632$$ 4.31534i 0.171655i
$$633$$ 23.6847i 0.941381i
$$634$$ −16.8769 −0.670267
$$635$$ 0 0
$$636$$ −7.43845 −0.294954
$$637$$ 0.876894i 0.0347438i
$$638$$ 18.2462i 0.722374i
$$639$$ 7.68466 0.304000
$$640$$ 0 0
$$641$$ −5.50758 −0.217536 −0.108768 0.994067i $$-0.534691\pi$$
−0.108768 + 0.994067i $$0.534691\pi$$
$$642$$ − 2.56155i − 0.101096i
$$643$$ − 7.05398i − 0.278182i −0.990280 0.139091i $$-0.955582\pi$$
0.990280 0.139091i $$-0.0444180\pi$$
$$644$$ −3.68466 −0.145196
$$645$$ 0 0
$$646$$ 24.0000 0.944267
$$647$$ 45.9309i 1.80573i 0.429925 + 0.902864i $$0.358540\pi$$
−0.429925 + 0.902864i $$0.641460\pi$$
$$648$$ 1.00000i 0.0392837i
$$649$$ 33.6155 1.31952
$$650$$ 0 0
$$651$$ −2.56155 −0.100395
$$652$$ 15.3693i 0.601909i
$$653$$ 40.2462i 1.57496i 0.616343 + 0.787478i $$0.288614\pi$$
−0.616343 + 0.787478i $$0.711386\pi$$
$$654$$ 5.36932 0.209957
$$655$$ 0 0
$$656$$ 7.12311 0.278111
$$657$$ 10.8078i 0.421651i
$$658$$ 13.1231i 0.511592i
$$659$$ 30.7386 1.19741 0.598704 0.800971i $$-0.295683\pi$$
0.598704 + 0.800971i $$0.295683\pi$$
$$660$$ 0 0
$$661$$ 26.4924 1.03044 0.515218 0.857059i $$-0.327711\pi$$
0.515218 + 0.857059i $$0.327711\pi$$
$$662$$ 6.24621i 0.242766i
$$663$$ 6.24621i 0.242583i
$$664$$ −14.2462 −0.552860
$$665$$ 0 0
$$666$$ −3.12311 −0.121018
$$667$$ − 10.2462i − 0.396735i
$$668$$ − 6.56155i − 0.253874i
$$669$$ −21.1231 −0.816666
$$670$$ 0 0
$$671$$ 15.3693 0.593326
$$672$$ 2.56155i 0.0988140i
$$673$$ 11.7538i 0.453075i 0.974002 + 0.226538i $$0.0727406\pi$$
−0.974002 + 0.226538i $$0.927259\pi$$
$$674$$ 10.0000 0.385186
$$675$$ 0 0
$$676$$ −9.00000 −0.346154
$$677$$ − 32.4233i − 1.24613i −0.782171 0.623064i $$-0.785888\pi$$
0.782171 0.623064i $$-0.214112\pi$$
$$678$$ 1.68466i 0.0646989i
$$679$$ 15.3693 0.589820
$$680$$ 0 0
$$681$$ −7.68466 −0.294477
$$682$$ − 2.56155i − 0.0980869i
$$683$$ 31.6847i 1.21238i 0.795320 + 0.606190i $$0.207303\pi$$
−0.795320 + 0.606190i $$0.792697\pi$$
$$684$$ 7.68466 0.293830
$$685$$ 0 0
$$686$$ −19.0540 −0.727484
$$687$$ − 16.5616i − 0.631863i
$$688$$ 12.8078i 0.488291i
$$689$$ −14.8769 −0.566765
$$690$$ 0 0
$$691$$ 15.0540 0.572680 0.286340 0.958128i $$-0.407561\pi$$
0.286340 + 0.958128i $$0.407561\pi$$
$$692$$ − 8.24621i − 0.313474i
$$693$$ 6.56155i 0.249253i
$$694$$ −14.2462 −0.540779
$$695$$ 0 0
$$696$$ −7.12311 −0.270001
$$697$$ 22.2462i 0.842635i
$$698$$ 5.36932i 0.203232i
$$699$$ −17.0540 −0.645041
$$700$$ 0 0
$$701$$ 5.19224 0.196108 0.0980540 0.995181i $$-0.468738\pi$$
0.0980540 + 0.995181i $$0.468738\pi$$
$$702$$ 2.00000i 0.0754851i
$$703$$ 24.0000i 0.905177i
$$704$$ −2.56155 −0.0965422
$$705$$ 0 0
$$706$$ −0.246211 −0.00926628
$$707$$ 1.43845i 0.0540984i
$$708$$ 13.1231i 0.493197i
$$709$$ −17.0540 −0.640475 −0.320238 0.947337i $$-0.603763\pi$$
−0.320238 + 0.947337i $$0.603763\pi$$
$$710$$ 0 0
$$711$$ −4.31534 −0.161838
$$712$$ 13.6847i 0.512854i
$$713$$ 1.43845i 0.0538703i
$$714$$ −8.00000 −0.299392
$$715$$ 0 0
$$716$$ −12.0000 −0.448461
$$717$$ 24.0000i 0.896296i
$$718$$ − 31.6847i − 1.18246i
$$719$$ 34.8769 1.30069 0.650344 0.759640i $$-0.274625\pi$$
0.650344 + 0.759640i $$0.274625\pi$$
$$720$$ 0 0
$$721$$ −4.49242 −0.167307
$$722$$ − 40.0540i − 1.49065i
$$723$$ − 12.2462i − 0.455441i
$$724$$ 13.0540 0.485147
$$725$$ 0 0
$$726$$ 4.43845 0.164726
$$727$$ − 23.0540i − 0.855025i −0.904010 0.427512i $$-0.859390\pi$$
0.904010 0.427512i $$-0.140610\pi$$
$$728$$ 5.12311i 0.189875i
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ −40.0000 −1.47945
$$732$$ 6.00000i 0.221766i
$$733$$ − 6.49242i − 0.239803i −0.992786 0.119902i $$-0.961742\pi$$
0.992786 0.119902i $$-0.0382579\pi$$
$$734$$ −15.3693 −0.567292
$$735$$ 0 0
$$736$$ 1.43845 0.0530219
$$737$$ 39.3693i 1.45019i
$$738$$ 7.12311i 0.262205i
$$739$$ −52.9848 −1.94908 −0.974540 0.224216i $$-0.928018\pi$$
−0.974540 + 0.224216i $$0.928018\pi$$
$$740$$ 0 0
$$741$$ 15.3693 0.564606
$$742$$ − 19.0540i − 0.699493i
$$743$$ − 50.4233i − 1.84985i −0.380148 0.924926i $$-0.624127\pi$$
0.380148 0.924926i $$-0.375873\pi$$
$$744$$ 1.00000 0.0366618
$$745$$ 0 0
$$746$$ −5.68466 −0.208130
$$747$$ − 14.2462i − 0.521242i
$$748$$ − 8.00000i − 0.292509i
$$749$$ 6.56155 0.239754
$$750$$ 0 0
$$751$$ 45.1231 1.64657 0.823283 0.567631i $$-0.192140\pi$$
0.823283 + 0.567631i $$0.192140\pi$$
$$752$$ − 5.12311i − 0.186820i
$$753$$ − 16.4924i − 0.601017i
$$754$$ −14.2462 −0.518816
$$755$$ 0 0
$$756$$ −2.56155 −0.0931628
$$757$$ − 10.0000i − 0.363456i −0.983349 0.181728i $$-0.941831\pi$$
0.983349 0.181728i $$-0.0581691\pi$$
$$758$$ − 7.05398i − 0.256212i
$$759$$ 3.68466 0.133745
$$760$$ 0 0
$$761$$ −20.4233 −0.740344 −0.370172 0.928963i $$-0.620701\pi$$
−0.370172 + 0.928963i $$0.620701\pi$$
$$762$$ 0 0
$$763$$ 13.7538i 0.497921i
$$764$$ 14.2462 0.515410
$$765$$ 0 0
$$766$$ −10.2462 −0.370211
$$767$$ 26.2462i 0.947696i
$$768$$ − 1.00000i − 0.0360844i
$$769$$ 20.5616 0.741469 0.370734 0.928739i $$-0.379106\pi$$
0.370734 + 0.928739i $$0.379106\pi$$
$$770$$ 0 0
$$771$$ 1.68466 0.0606715
$$772$$ − 14.4924i − 0.521594i
$$773$$ − 11.4384i − 0.411412i −0.978614 0.205706i $$-0.934051\pi$$
0.978614 0.205706i $$-0.0659491\pi$$
$$774$$ −12.8078 −0.460366
$$775$$ 0 0
$$776$$ −6.00000 −0.215387
$$777$$ − 8.00000i − 0.286998i
$$778$$ 7.75379i 0.277987i
$$779$$ 54.7386 1.96122
$$780$$ 0 0
$$781$$ −19.6847 −0.704372
$$782$$ 4.49242i 0.160649i
$$783$$ − 7.12311i − 0.254559i
$$784$$ 0.438447 0.0156588
$$785$$ 0 0
$$786$$ 15.3693 0.548205
$$787$$ 17.9309i 0.639166i 0.947558 + 0.319583i $$0.103543\pi$$
−0.947558 + 0.319583i $$0.896457\pi$$
$$788$$ 6.00000i 0.213741i
$$789$$ −20.4924 −0.729550
$$790$$ 0 0
$$791$$ −4.31534 −0.153436
$$792$$ − 2.56155i − 0.0910208i
$$793$$ 12.0000i 0.426132i
$$794$$ −9.05398 −0.321314
$$795$$ 0 0
$$796$$ −16.8078 −0.595735
$$797$$ 26.9848i 0.955852i 0.878400 + 0.477926i $$0.158611\pi$$
−0.878400 + 0.477926i $$0.841389\pi$$
$$798$$ 19.6847i 0.696829i
$$799$$ 16.0000 0.566039
$$800$$ 0 0
$$801$$ −13.6847 −0.483524
$$802$$ 13.6847i 0.483222i
$$803$$ − 27.6847i − 0.976970i
$$804$$ −15.3693 −0.542034
$$805$$ 0 0
$$806$$ 2.00000 0.0704470
$$807$$ − 0.246211i − 0.00866705i
$$808$$ − 0.561553i − 0.0197554i
$$809$$ 37.6847 1.32492 0.662461 0.749096i $$-0.269512\pi$$
0.662461 + 0.749096i $$0.269512\pi$$
$$810$$ 0 0
$$811$$ −24.6695 −0.866263 −0.433132 0.901331i $$-0.642592\pi$$
−0.433132 + 0.901331i $$0.642592\pi$$
$$812$$ − 18.2462i − 0.640316i
$$813$$ − 27.0540i − 0.948824i
$$814$$ 8.00000 0.280400
$$815$$ 0 0
$$816$$ 3.12311 0.109331
$$817$$ 98.4233i 3.44340i
$$818$$ − 37.3693i − 1.30659i
$$819$$ −5.12311 −0.179016
$$820$$ 0 0
$$821$$ 19.6155 0.684587 0.342293 0.939593i $$-0.388796\pi$$
0.342293 + 0.939593i $$0.388796\pi$$
$$822$$ 20.2462i 0.706168i
$$823$$ − 25.6155i − 0.892901i −0.894808 0.446451i $$-0.852688\pi$$
0.894808 0.446451i $$-0.147312\pi$$
$$824$$ 1.75379 0.0610961
$$825$$ 0 0
$$826$$ −33.6155 −1.16963
$$827$$ − 1.75379i − 0.0609852i −0.999535 0.0304926i $$-0.990292\pi$$
0.999535 0.0304926i $$-0.00970760\pi$$
$$828$$ 1.43845i 0.0499895i
$$829$$ −24.4233 −0.848256 −0.424128 0.905602i $$-0.639419\pi$$
−0.424128 + 0.905602i $$0.639419\pi$$
$$830$$ 0 0
$$831$$ 5.36932 0.186260
$$832$$ − 2.00000i − 0.0693375i
$$833$$ 1.36932i 0.0474440i
$$834$$ −17.1231 −0.592925
$$835$$ 0 0
$$836$$ −19.6847 −0.680808
$$837$$ 1.00000i 0.0345651i
$$838$$ 5.75379i 0.198761i
$$839$$ −6.06913 −0.209530 −0.104765 0.994497i $$-0.533409\pi$$
−0.104765 + 0.994497i $$0.533409\pi$$
$$840$$ 0 0
$$841$$ 21.7386 0.749608
$$842$$ 14.4924i 0.499442i
$$843$$ − 4.87689i − 0.167969i
$$844$$ 23.6847 0.815260
$$845$$ 0 0
$$846$$ 5.12311 0.176136
$$847$$ 11.3693i 0.390654i
$$848$$ 7.43845i 0.255437i
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −4.49242 −0.153998
$$852$$ − 7.68466i − 0.263272i
$$853$$ 47.7926i 1.63639i 0.574942 + 0.818194i $$0.305024\pi$$
−0.574942 + 0.818194i $$0.694976\pi$$
$$854$$ −15.3693 −0.525927
$$855$$ 0 0
$$856$$ −2.56155 −0.0875521
$$857$$ − 29.2311i − 0.998514i −0.866454 0.499257i $$-0.833606\pi$$
0.866454 0.499257i $$-0.166394\pi$$
$$858$$ − 5.12311i − 0.174900i
$$859$$ −26.7386 −0.912310 −0.456155 0.889900i $$-0.650774\pi$$
−0.456155 + 0.889900i $$0.650774\pi$$
$$860$$ 0 0
$$861$$ −18.2462 −0.621829
$$862$$ 30.2462i 1.03019i
$$863$$ 39.5464i 1.34618i 0.739563 + 0.673088i $$0.235032\pi$$
−0.739563 + 0.673088i $$0.764968\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ 0 0
$$866$$ 35.3002 1.19955
$$867$$ − 7.24621i − 0.246094i
$$868$$ 2.56155i 0.0869448i
$$869$$ 11.0540 0.374980
$$870$$ 0 0
$$871$$ −30.7386 −1.04154
$$872$$ − 5.36932i − 0.181828i
$$873$$ − 6.00000i − 0.203069i
$$874$$ 11.0540 0.373906
$$875$$ 0 0
$$876$$ 10.8078 0.365161
$$877$$ 50.0000i 1.68838i 0.536044 + 0.844190i $$0.319918\pi$$
−0.536044 + 0.844190i $$0.680082\pi$$
$$878$$ 9.61553i 0.324508i
$$879$$ 26.4924 0.893567
$$880$$ 0 0
$$881$$ 9.50758 0.320318 0.160159 0.987091i $$-0.448799\pi$$
0.160159 + 0.987091i $$0.448799\pi$$
$$882$$ 0.438447i 0.0147633i
$$883$$ 30.4233i 1.02383i 0.859038 + 0.511913i $$0.171063\pi$$
−0.859038 + 0.511913i $$0.828937\pi$$
$$884$$ 6.24621 0.210083
$$885$$ 0 0
$$886$$ 23.0540 0.774513
$$887$$ 32.6307i 1.09563i 0.836599 + 0.547816i $$0.184540\pi$$
−0.836599 + 0.547816i $$0.815460\pi$$
$$888$$ 3.12311i 0.104805i
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 2.56155 0.0858152
$$892$$ 21.1231i 0.707254i
$$893$$ − 39.3693i − 1.31744i
$$894$$ 17.0540 0.570370
$$895$$ 0 0
$$896$$ 2.56155 0.0855755
$$897$$ 2.87689i 0.0960567i
$$898$$ 10.4924i 0.350137i
$$899$$ −7.12311 −0.237569
$$900$$ 0 0
$$901$$ −23.2311 −0.773939
$$902$$ − 18.2462i − 0.607532i
$$903$$ − 32.8078i − 1.09177i
$$904$$ 1.68466 0.0560309
$$905$$ 0 0
$$906$$ 0 0
$$907$$ − 24.0000i − 0.796907i −0.917189 0.398453i $$-0.869547\pi$$
0.917189 0.398453i $$-0.130453\pi$$
$$908$$ 7.68466i 0.255024i
$$909$$ 0.561553 0.0186255
$$910$$ 0 0
$$911$$ −11.5076 −0.381263 −0.190632 0.981662i $$-0.561054\pi$$
−0.190632 + 0.981662i $$0.561054\pi$$
$$912$$ − 7.68466i − 0.254464i
$$913$$ 36.4924i 1.20772i
$$914$$ 24.7386 0.818281
$$915$$ 0 0
$$916$$ −16.5616 −0.547209
$$917$$ 39.3693i 1.30009i
$$918$$ 3.12311i 0.103078i
$$919$$ 1.61553 0.0532914 0.0266457 0.999645i $$-0.491517\pi$$
0.0266457 + 0.999645i $$0.491517\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ − 9.36932i − 0.308562i
$$923$$ − 15.3693i − 0.505887i
$$924$$ 6.56155 0.215859
$$925$$ 0 0
$$926$$ 30.7386 1.01013
$$927$$ 1.75379i 0.0576020i
$$928$$ 7.12311i 0.233827i
$$929$$ −7.43845 −0.244048 −0.122024 0.992527i $$-0.538938\pi$$
−0.122024 + 0.992527i $$0.538938\pi$$
$$930$$ 0 0
$$931$$ 3.36932 0.110425
$$932$$ 17.0540i 0.558622i
$$933$$ 1.75379i 0.0574165i
$$934$$ 9.75379 0.319154
$$935$$ 0 0
$$936$$ 2.00000 0.0653720
$$937$$ − 41.3693i − 1.35148i −0.737142 0.675738i $$-0.763825\pi$$
0.737142 0.675738i $$-0.236175\pi$$
$$938$$ − 39.3693i − 1.28545i
$$939$$ −10.0000 −0.326338
$$940$$ 0 0
$$941$$ 2.63068 0.0857578 0.0428789 0.999080i $$-0.486347\pi$$
0.0428789 + 0.999080i $$0.486347\pi$$
$$942$$ 17.0540i 0.555649i
$$943$$ 10.2462i 0.333663i
$$944$$ 13.1231 0.427121
$$945$$ 0 0
$$946$$ 32.8078 1.06667
$$947$$ 9.12311i 0.296461i 0.988953 + 0.148231i $$0.0473578\pi$$
−0.988953 + 0.148231i $$0.952642\pi$$
$$948$$ 4.31534i 0.140156i
$$949$$ 21.6155 0.701670
$$950$$ 0 0
$$951$$ −16.8769 −0.547271
$$952$$ 8.00000i 0.259281i
$$953$$ 16.1080i 0.521788i 0.965367 + 0.260894i $$0.0840173\pi$$
−0.965367 + 0.260894i $$0.915983\pi$$
$$954$$ −7.43845 −0.240829
$$955$$ 0 0
$$956$$ 24.0000 0.776215
$$957$$ 18.2462i 0.589816i
$$958$$ 10.5616i 0.341228i
$$959$$ −51.8617 −1.67470
$$960$$ 0 0
$$961$$ 1.00000 0.0322581
$$962$$ 6.24621i 0.201386i
$$963$$ − 2.56155i − 0.0825449i
$$964$$ −12.2462 −0.394424
$$965$$ 0 0
$$966$$ −3.68466 −0.118552
$$967$$ 42.2462i 1.35855i 0.733885 + 0.679273i $$0.237705\pi$$
−0.733885 + 0.679273i $$0.762295\pi$$
$$968$$ − 4.43845i − 0.142657i
$$969$$ 24.0000 0.770991
$$970$$ 0 0
$$971$$ −37.1231 −1.19134 −0.595669 0.803230i $$-0.703113\pi$$
−0.595669 + 0.803230i $$0.703113\pi$$
$$972$$ 1.00000i 0.0320750i
$$973$$ − 43.8617i − 1.40614i
$$974$$ −24.0000 −0.769010
$$975$$ 0 0
$$976$$ 6.00000 0.192055
$$977$$ − 42.9848i − 1.37521i −0.726086 0.687604i $$-0.758663\pi$$
0.726086 0.687604i $$-0.241337\pi$$
$$978$$ 15.3693i 0.491457i
$$979$$ 35.0540 1.12033
$$980$$ 0 0
$$981$$ 5.36932 0.171429
$$982$$ 25.9309i 0.827487i
$$983$$ 40.9848i 1.30721i 0.756834 + 0.653607i $$0.226745\pi$$
−0.756834 + 0.653607i $$0.773255\pi$$
$$984$$ 7.12311 0.227076
$$985$$ 0 0
$$986$$ −22.2462 −0.708464
$$987$$ 13.1231i 0.417713i
$$988$$ − 15.3693i − 0.488963i
$$989$$ −18.4233 −0.585827
$$990$$ 0 0
$$991$$ 35.6847 1.13356 0.566780 0.823869i $$-0.308189\pi$$
0.566780 + 0.823869i $$0.308189\pi$$
$$992$$ − 1.00000i − 0.0317500i
$$993$$ 6.24621i 0.198218i
$$994$$ 19.6847 0.624359
$$995$$ 0 0
$$996$$ −14.2462 −0.451408
$$997$$ 29.5076i 0.934514i 0.884121 + 0.467257i $$0.154758\pi$$
−0.884121 + 0.467257i $$0.845242\pi$$
$$998$$ 20.0000i 0.633089i
$$999$$ −3.12311 −0.0988107
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 4650.2.d.bd.3349.1 4
5.2 odd 4 930.2.a.p.1.2 2
5.3 odd 4 4650.2.a.ce.1.1 2
5.4 even 2 inner 4650.2.d.bd.3349.4 4
15.2 even 4 2790.2.a.be.1.2 2
20.7 even 4 7440.2.a.bl.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.a.p.1.2 2 5.2 odd 4
2790.2.a.be.1.2 2 15.2 even 4
4650.2.a.ce.1.1 2 5.3 odd 4
4650.2.d.bd.3349.1 4 1.1 even 1 trivial
4650.2.d.bd.3349.4 4 5.4 even 2 inner
7440.2.a.bl.1.1 2 20.7 even 4