# Properties

 Label 4600.2.a.bi.1.7 Level $4600$ Weight $2$ Character 4600.1 Self dual yes Analytic conductor $36.731$ Analytic rank $1$ Dimension $7$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4600 = 2^{3} \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4600.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$36.7311849298$$ Analytic rank: $$1$$ Dimension: $$7$$ Coefficient field: $$\mathbb{Q}[x]/(x^{7} - \cdots)$$ Defining polynomial: $$x^{7} - 3 x^{6} - 7 x^{5} + 24 x^{4} + x^{3} - 35 x^{2} + 17 x - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 920) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.7 Root $$2.98707$$ of defining polynomial Character $$\chi$$ $$=$$ 4600.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.98707 q^{3} -0.980560 q^{7} +5.92257 q^{9} +O(q^{10})$$ $$q+2.98707 q^{3} -0.980560 q^{7} +5.92257 q^{9} -6.14721 q^{11} -6.37400 q^{13} +3.36098 q^{17} +1.08276 q^{19} -2.92900 q^{21} -1.00000 q^{23} +8.72993 q^{27} -0.271042 q^{29} -8.77792 q^{31} -18.3621 q^{33} -8.84665 q^{37} -19.0396 q^{39} -4.85308 q^{41} +1.87756 q^{43} +0.196089 q^{47} -6.03850 q^{49} +10.0395 q^{51} -1.93157 q^{53} +3.23429 q^{57} +13.0036 q^{59} +6.00189 q^{61} -5.80744 q^{63} +2.26847 q^{67} -2.98707 q^{69} +10.2677 q^{71} +1.38188 q^{73} +6.02771 q^{77} -4.67459 q^{79} +8.30917 q^{81} -15.7171 q^{83} -0.809622 q^{87} -11.1548 q^{89} +6.25008 q^{91} -26.2202 q^{93} -10.2868 q^{97} -36.4073 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$7q + 3q^{3} + 4q^{7} + 2q^{9} + O(q^{10})$$ $$7q + 3q^{3} + 4q^{7} + 2q^{9} - 7q^{11} - 7q^{13} - 7q^{19} - 6q^{21} - 7q^{23} - 11q^{29} - 10q^{31} - 19q^{33} - 19q^{37} - 24q^{39} - 16q^{41} + 6q^{43} + 6q^{47} - 17q^{49} - 7q^{51} - 15q^{53} - 8q^{57} - 11q^{59} + 5q^{61} + 13q^{63} + 9q^{67} - 3q^{69} - 14q^{71} - 10q^{73} - 6q^{77} - 32q^{79} - 5q^{81} + q^{83} + 10q^{87} - 24q^{89} - 7q^{91} - 26q^{93} + 7q^{97} - 61q^{99} + O(q^{100})$$

## 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 0
$$3$$ 2.98707 1.72458 0.862292 0.506411i $$-0.169028\pi$$
0.862292 + 0.506411i $$0.169028\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −0.980560 −0.370617 −0.185308 0.982680i $$-0.559328\pi$$
−0.185308 + 0.982680i $$0.559328\pi$$
$$8$$ 0 0
$$9$$ 5.92257 1.97419
$$10$$ 0 0
$$11$$ −6.14721 −1.85345 −0.926727 0.375734i $$-0.877391\pi$$
−0.926727 + 0.375734i $$0.877391\pi$$
$$12$$ 0 0
$$13$$ −6.37400 −1.76783 −0.883914 0.467649i $$-0.845101\pi$$
−0.883914 + 0.467649i $$0.845101\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 3.36098 0.815157 0.407579 0.913170i $$-0.366373\pi$$
0.407579 + 0.913170i $$0.366373\pi$$
$$18$$ 0 0
$$19$$ 1.08276 0.248403 0.124202 0.992257i $$-0.460363\pi$$
0.124202 + 0.992257i $$0.460363\pi$$
$$20$$ 0 0
$$21$$ −2.92900 −0.639160
$$22$$ 0 0
$$23$$ −1.00000 −0.208514
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 8.72993 1.68008
$$28$$ 0 0
$$29$$ −0.271042 −0.0503313 −0.0251657 0.999683i $$-0.508011\pi$$
−0.0251657 + 0.999683i $$0.508011\pi$$
$$30$$ 0 0
$$31$$ −8.77792 −1.57656 −0.788280 0.615316i $$-0.789028\pi$$
−0.788280 + 0.615316i $$0.789028\pi$$
$$32$$ 0 0
$$33$$ −18.3621 −3.19644
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −8.84665 −1.45438 −0.727190 0.686436i $$-0.759174\pi$$
−0.727190 + 0.686436i $$0.759174\pi$$
$$38$$ 0 0
$$39$$ −19.0396 −3.04877
$$40$$ 0 0
$$41$$ −4.85308 −0.757923 −0.378962 0.925412i $$-0.623719\pi$$
−0.378962 + 0.925412i $$0.623719\pi$$
$$42$$ 0 0
$$43$$ 1.87756 0.286326 0.143163 0.989699i $$-0.454273\pi$$
0.143163 + 0.989699i $$0.454273\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 0.196089 0.0286026 0.0143013 0.999898i $$-0.495448\pi$$
0.0143013 + 0.999898i $$0.495448\pi$$
$$48$$ 0 0
$$49$$ −6.03850 −0.862643
$$50$$ 0 0
$$51$$ 10.0395 1.40581
$$52$$ 0 0
$$53$$ −1.93157 −0.265321 −0.132661 0.991162i $$-0.542352\pi$$
−0.132661 + 0.991162i $$0.542352\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 3.23429 0.428392
$$58$$ 0 0
$$59$$ 13.0036 1.69293 0.846465 0.532444i $$-0.178726\pi$$
0.846465 + 0.532444i $$0.178726\pi$$
$$60$$ 0 0
$$61$$ 6.00189 0.768464 0.384232 0.923237i $$-0.374466\pi$$
0.384232 + 0.923237i $$0.374466\pi$$
$$62$$ 0 0
$$63$$ −5.80744 −0.731668
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 2.26847 0.277138 0.138569 0.990353i $$-0.455750\pi$$
0.138569 + 0.990353i $$0.455750\pi$$
$$68$$ 0 0
$$69$$ −2.98707 −0.359601
$$70$$ 0 0
$$71$$ 10.2677 1.21855 0.609277 0.792958i $$-0.291460\pi$$
0.609277 + 0.792958i $$0.291460\pi$$
$$72$$ 0 0
$$73$$ 1.38188 0.161737 0.0808685 0.996725i $$-0.474231\pi$$
0.0808685 + 0.996725i $$0.474231\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 6.02771 0.686921
$$78$$ 0 0
$$79$$ −4.67459 −0.525932 −0.262966 0.964805i $$-0.584701\pi$$
−0.262966 + 0.964805i $$0.584701\pi$$
$$80$$ 0 0
$$81$$ 8.30917 0.923241
$$82$$ 0 0
$$83$$ −15.7171 −1.72517 −0.862587 0.505909i $$-0.831157\pi$$
−0.862587 + 0.505909i $$0.831157\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ −0.809622 −0.0868006
$$88$$ 0 0
$$89$$ −11.1548 −1.18241 −0.591206 0.806521i $$-0.701348\pi$$
−0.591206 + 0.806521i $$0.701348\pi$$
$$90$$ 0 0
$$91$$ 6.25008 0.655187
$$92$$ 0 0
$$93$$ −26.2202 −2.71891
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −10.2868 −1.04447 −0.522234 0.852802i $$-0.674901\pi$$
−0.522234 + 0.852802i $$0.674901\pi$$
$$98$$ 0 0
$$99$$ −36.4073 −3.65908
$$100$$ 0 0
$$101$$ −15.2625 −1.51868 −0.759339 0.650695i $$-0.774478\pi$$
−0.759339 + 0.650695i $$0.774478\pi$$
$$102$$ 0 0
$$103$$ −7.60795 −0.749634 −0.374817 0.927099i $$-0.622294\pi$$
−0.374817 + 0.927099i $$0.622294\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 11.3081 1.09320 0.546599 0.837395i $$-0.315922\pi$$
0.546599 + 0.837395i $$0.315922\pi$$
$$108$$ 0 0
$$109$$ 3.45336 0.330772 0.165386 0.986229i $$-0.447113\pi$$
0.165386 + 0.986229i $$0.447113\pi$$
$$110$$ 0 0
$$111$$ −26.4256 −2.50820
$$112$$ 0 0
$$113$$ 14.6889 1.38181 0.690907 0.722944i $$-0.257212\pi$$
0.690907 + 0.722944i $$0.257212\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −37.7505 −3.49003
$$118$$ 0 0
$$119$$ −3.29564 −0.302111
$$120$$ 0 0
$$121$$ 26.7882 2.43530
$$122$$ 0 0
$$123$$ −14.4965 −1.30710
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −1.19312 −0.105873 −0.0529363 0.998598i $$-0.516858\pi$$
−0.0529363 + 0.998598i $$0.516858\pi$$
$$128$$ 0 0
$$129$$ 5.60841 0.493793
$$130$$ 0 0
$$131$$ −11.7326 −1.02509 −0.512543 0.858662i $$-0.671296\pi$$
−0.512543 + 0.858662i $$0.671296\pi$$
$$132$$ 0 0
$$133$$ −1.06171 −0.0920623
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −10.4665 −0.894210 −0.447105 0.894481i $$-0.647545\pi$$
−0.447105 + 0.894481i $$0.647545\pi$$
$$138$$ 0 0
$$139$$ 4.11349 0.348902 0.174451 0.984666i $$-0.444185\pi$$
0.174451 + 0.984666i $$0.444185\pi$$
$$140$$ 0 0
$$141$$ 0.585732 0.0493275
$$142$$ 0 0
$$143$$ 39.1823 3.27659
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ −18.0374 −1.48770
$$148$$ 0 0
$$149$$ 1.37793 0.112885 0.0564424 0.998406i $$-0.482024\pi$$
0.0564424 + 0.998406i $$0.482024\pi$$
$$150$$ 0 0
$$151$$ −3.49384 −0.284325 −0.142162 0.989843i $$-0.545406\pi$$
−0.142162 + 0.989843i $$0.545406\pi$$
$$152$$ 0 0
$$153$$ 19.9057 1.60928
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −3.32373 −0.265262 −0.132631 0.991165i $$-0.542343\pi$$
−0.132631 + 0.991165i $$0.542343\pi$$
$$158$$ 0 0
$$159$$ −5.76973 −0.457569
$$160$$ 0 0
$$161$$ 0.980560 0.0772789
$$162$$ 0 0
$$163$$ −8.52075 −0.667397 −0.333698 0.942680i $$-0.608297\pi$$
−0.333698 + 0.942680i $$0.608297\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 15.6190 1.20864 0.604319 0.796742i $$-0.293445\pi$$
0.604319 + 0.796742i $$0.293445\pi$$
$$168$$ 0 0
$$169$$ 27.6278 2.12522
$$170$$ 0 0
$$171$$ 6.41275 0.490395
$$172$$ 0 0
$$173$$ −8.42727 −0.640713 −0.320357 0.947297i $$-0.603803\pi$$
−0.320357 + 0.947297i $$0.603803\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 38.8428 2.91960
$$178$$ 0 0
$$179$$ 10.8047 0.807580 0.403790 0.914852i $$-0.367693\pi$$
0.403790 + 0.914852i $$0.367693\pi$$
$$180$$ 0 0
$$181$$ −15.4282 −1.14677 −0.573385 0.819286i $$-0.694370\pi$$
−0.573385 + 0.819286i $$0.694370\pi$$
$$182$$ 0 0
$$183$$ 17.9281 1.32528
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −20.6607 −1.51086
$$188$$ 0 0
$$189$$ −8.56022 −0.622664
$$190$$ 0 0
$$191$$ −22.2634 −1.61092 −0.805462 0.592648i $$-0.798083\pi$$
−0.805462 + 0.592648i $$0.798083\pi$$
$$192$$ 0 0
$$193$$ 16.1399 1.16178 0.580889 0.813983i $$-0.302705\pi$$
0.580889 + 0.813983i $$0.302705\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 13.9936 0.997004 0.498502 0.866889i $$-0.333884\pi$$
0.498502 + 0.866889i $$0.333884\pi$$
$$198$$ 0 0
$$199$$ −6.91926 −0.490493 −0.245246 0.969461i $$-0.578869\pi$$
−0.245246 + 0.969461i $$0.578869\pi$$
$$200$$ 0 0
$$201$$ 6.77608 0.477948
$$202$$ 0 0
$$203$$ 0.265773 0.0186536
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ −5.92257 −0.411647
$$208$$ 0 0
$$209$$ −6.65598 −0.460404
$$210$$ 0 0
$$211$$ 12.9170 0.889246 0.444623 0.895718i $$-0.353338\pi$$
0.444623 + 0.895718i $$0.353338\pi$$
$$212$$ 0 0
$$213$$ 30.6704 2.10150
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 8.60728 0.584300
$$218$$ 0 0
$$219$$ 4.12777 0.278929
$$220$$ 0 0
$$221$$ −21.4229 −1.44106
$$222$$ 0 0
$$223$$ 26.6579 1.78514 0.892572 0.450905i $$-0.148899\pi$$
0.892572 + 0.450905i $$0.148899\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −9.44407 −0.626825 −0.313412 0.949617i $$-0.601472\pi$$
−0.313412 + 0.949617i $$0.601472\pi$$
$$228$$ 0 0
$$229$$ 14.1994 0.938322 0.469161 0.883113i $$-0.344556\pi$$
0.469161 + 0.883113i $$0.344556\pi$$
$$230$$ 0 0
$$231$$ 18.0052 1.18465
$$232$$ 0 0
$$233$$ −9.41134 −0.616557 −0.308279 0.951296i $$-0.599753\pi$$
−0.308279 + 0.951296i $$0.599753\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −13.9633 −0.907014
$$238$$ 0 0
$$239$$ −15.4781 −1.00119 −0.500597 0.865680i $$-0.666886\pi$$
−0.500597 + 0.865680i $$0.666886\pi$$
$$240$$ 0 0
$$241$$ −3.51052 −0.226133 −0.113066 0.993587i $$-0.536067\pi$$
−0.113066 + 0.993587i $$0.536067\pi$$
$$242$$ 0 0
$$243$$ −1.36974 −0.0878689
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −6.90153 −0.439134
$$248$$ 0 0
$$249$$ −46.9480 −2.97521
$$250$$ 0 0
$$251$$ −23.6649 −1.49372 −0.746859 0.664982i $$-0.768439\pi$$
−0.746859 + 0.664982i $$0.768439\pi$$
$$252$$ 0 0
$$253$$ 6.14721 0.386472
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 30.8326 1.92328 0.961641 0.274311i $$-0.0884497\pi$$
0.961641 + 0.274311i $$0.0884497\pi$$
$$258$$ 0 0
$$259$$ 8.67467 0.539018
$$260$$ 0 0
$$261$$ −1.60527 −0.0993636
$$262$$ 0 0
$$263$$ 20.5880 1.26951 0.634755 0.772714i $$-0.281101\pi$$
0.634755 + 0.772714i $$0.281101\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −33.3203 −2.03917
$$268$$ 0 0
$$269$$ −17.0277 −1.03820 −0.519099 0.854714i $$-0.673732\pi$$
−0.519099 + 0.854714i $$0.673732\pi$$
$$270$$ 0 0
$$271$$ 13.6709 0.830450 0.415225 0.909719i $$-0.363703\pi$$
0.415225 + 0.909719i $$0.363703\pi$$
$$272$$ 0 0
$$273$$ 18.6694 1.12993
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 16.2380 0.975647 0.487824 0.872942i $$-0.337791\pi$$
0.487824 + 0.872942i $$0.337791\pi$$
$$278$$ 0 0
$$279$$ −51.9879 −3.11243
$$280$$ 0 0
$$281$$ −0.362407 −0.0216194 −0.0108097 0.999942i $$-0.503441\pi$$
−0.0108097 + 0.999942i $$0.503441\pi$$
$$282$$ 0 0
$$283$$ 6.80039 0.404241 0.202120 0.979361i $$-0.435217\pi$$
0.202120 + 0.979361i $$0.435217\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 4.75873 0.280899
$$288$$ 0 0
$$289$$ −5.70382 −0.335519
$$290$$ 0 0
$$291$$ −30.7274 −1.80127
$$292$$ 0 0
$$293$$ −26.2172 −1.53162 −0.765812 0.643065i $$-0.777663\pi$$
−0.765812 + 0.643065i $$0.777663\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −53.6647 −3.11394
$$298$$ 0 0
$$299$$ 6.37400 0.368618
$$300$$ 0 0
$$301$$ −1.84106 −0.106117
$$302$$ 0 0
$$303$$ −45.5902 −2.61909
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 19.7146 1.12517 0.562587 0.826738i $$-0.309806\pi$$
0.562587 + 0.826738i $$0.309806\pi$$
$$308$$ 0 0
$$309$$ −22.7255 −1.29281
$$310$$ 0 0
$$311$$ 15.1003 0.856258 0.428129 0.903718i $$-0.359173\pi$$
0.428129 + 0.903718i $$0.359173\pi$$
$$312$$ 0 0
$$313$$ 12.9813 0.733747 0.366874 0.930271i $$-0.380428\pi$$
0.366874 + 0.930271i $$0.380428\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 17.3503 0.974489 0.487244 0.873266i $$-0.338002\pi$$
0.487244 + 0.873266i $$0.338002\pi$$
$$318$$ 0 0
$$319$$ 1.66616 0.0932868
$$320$$ 0 0
$$321$$ 33.7781 1.88531
$$322$$ 0 0
$$323$$ 3.63915 0.202488
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 10.3154 0.570444
$$328$$ 0 0
$$329$$ −0.192277 −0.0106006
$$330$$ 0 0
$$331$$ 0.984756 0.0541271 0.0270635 0.999634i $$-0.491384\pi$$
0.0270635 + 0.999634i $$0.491384\pi$$
$$332$$ 0 0
$$333$$ −52.3950 −2.87123
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 23.5575 1.28326 0.641629 0.767015i $$-0.278259\pi$$
0.641629 + 0.767015i $$0.278259\pi$$
$$338$$ 0 0
$$339$$ 43.8767 2.38305
$$340$$ 0 0
$$341$$ 53.9598 2.92208
$$342$$ 0 0
$$343$$ 12.7850 0.690327
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 26.0916 1.40067 0.700337 0.713813i $$-0.253033\pi$$
0.700337 + 0.713813i $$0.253033\pi$$
$$348$$ 0 0
$$349$$ −17.8609 −0.956070 −0.478035 0.878341i $$-0.658651\pi$$
−0.478035 + 0.878341i $$0.658651\pi$$
$$350$$ 0 0
$$351$$ −55.6445 −2.97009
$$352$$ 0 0
$$353$$ −13.1080 −0.697667 −0.348833 0.937185i $$-0.613422\pi$$
−0.348833 + 0.937185i $$0.613422\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ −9.84430 −0.521016
$$358$$ 0 0
$$359$$ 22.1119 1.16702 0.583510 0.812106i $$-0.301679\pi$$
0.583510 + 0.812106i $$0.301679\pi$$
$$360$$ 0 0
$$361$$ −17.8276 −0.938296
$$362$$ 0 0
$$363$$ 80.0183 4.19987
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 7.98495 0.416811 0.208406 0.978043i $$-0.433173\pi$$
0.208406 + 0.978043i $$0.433173\pi$$
$$368$$ 0 0
$$369$$ −28.7427 −1.49629
$$370$$ 0 0
$$371$$ 1.89402 0.0983326
$$372$$ 0 0
$$373$$ 23.8712 1.23600 0.618002 0.786177i $$-0.287942\pi$$
0.618002 + 0.786177i $$0.287942\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 1.72762 0.0889771
$$378$$ 0 0
$$379$$ −34.5647 −1.77547 −0.887735 0.460355i $$-0.847722\pi$$
−0.887735 + 0.460355i $$0.847722\pi$$
$$380$$ 0 0
$$381$$ −3.56394 −0.182586
$$382$$ 0 0
$$383$$ 12.3139 0.629211 0.314606 0.949223i $$-0.398128\pi$$
0.314606 + 0.949223i $$0.398128\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 11.1200 0.565262
$$388$$ 0 0
$$389$$ −27.0447 −1.37122 −0.685611 0.727968i $$-0.740465\pi$$
−0.685611 + 0.727968i $$0.740465\pi$$
$$390$$ 0 0
$$391$$ −3.36098 −0.169972
$$392$$ 0 0
$$393$$ −35.0462 −1.76785
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −9.07631 −0.455527 −0.227763 0.973717i $$-0.573141\pi$$
−0.227763 + 0.973717i $$0.573141\pi$$
$$398$$ 0 0
$$399$$ −3.17141 −0.158769
$$400$$ 0 0
$$401$$ 6.73762 0.336461 0.168230 0.985748i $$-0.446195\pi$$
0.168230 + 0.985748i $$0.446195\pi$$
$$402$$ 0 0
$$403$$ 55.9504 2.78709
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 54.3823 2.69563
$$408$$ 0 0
$$409$$ 25.7205 1.27180 0.635898 0.771773i $$-0.280630\pi$$
0.635898 + 0.771773i $$0.280630\pi$$
$$410$$ 0 0
$$411$$ −31.2640 −1.54214
$$412$$ 0 0
$$413$$ −12.7509 −0.627428
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 12.2873 0.601711
$$418$$ 0 0
$$419$$ −16.2666 −0.794676 −0.397338 0.917672i $$-0.630066\pi$$
−0.397338 + 0.917672i $$0.630066\pi$$
$$420$$ 0 0
$$421$$ −18.9106 −0.921645 −0.460822 0.887492i $$-0.652445\pi$$
−0.460822 + 0.887492i $$0.652445\pi$$
$$422$$ 0 0
$$423$$ 1.16135 0.0564669
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −5.88522 −0.284806
$$428$$ 0 0
$$429$$ 117.040 5.65076
$$430$$ 0 0
$$431$$ −11.4086 −0.549535 −0.274767 0.961511i $$-0.588601\pi$$
−0.274767 + 0.961511i $$0.588601\pi$$
$$432$$ 0 0
$$433$$ −23.2756 −1.11856 −0.559278 0.828980i $$-0.688922\pi$$
−0.559278 + 0.828980i $$0.688922\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −1.08276 −0.0517956
$$438$$ 0 0
$$439$$ −13.0581 −0.623231 −0.311615 0.950208i $$-0.600870\pi$$
−0.311615 + 0.950208i $$0.600870\pi$$
$$440$$ 0 0
$$441$$ −35.7635 −1.70302
$$442$$ 0 0
$$443$$ −0.113844 −0.00540891 −0.00270446 0.999996i $$-0.500861\pi$$
−0.00270446 + 0.999996i $$0.500861\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 4.11598 0.194679
$$448$$ 0 0
$$449$$ −11.0371 −0.520874 −0.260437 0.965491i $$-0.583867\pi$$
−0.260437 + 0.965491i $$0.583867\pi$$
$$450$$ 0 0
$$451$$ 29.8329 1.40478
$$452$$ 0 0
$$453$$ −10.4363 −0.490342
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −30.1638 −1.41100 −0.705501 0.708709i $$-0.749278\pi$$
−0.705501 + 0.708709i $$0.749278\pi$$
$$458$$ 0 0
$$459$$ 29.3411 1.36953
$$460$$ 0 0
$$461$$ −37.3908 −1.74146 −0.870732 0.491759i $$-0.836354\pi$$
−0.870732 + 0.491759i $$0.836354\pi$$
$$462$$ 0 0
$$463$$ −16.7365 −0.777810 −0.388905 0.921278i $$-0.627147\pi$$
−0.388905 + 0.921278i $$0.627147\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −18.2260 −0.843399 −0.421699 0.906736i $$-0.638566\pi$$
−0.421699 + 0.906736i $$0.638566\pi$$
$$468$$ 0 0
$$469$$ −2.22437 −0.102712
$$470$$ 0 0
$$471$$ −9.92820 −0.457467
$$472$$ 0 0
$$473$$ −11.5418 −0.530692
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −11.4399 −0.523795
$$478$$ 0 0
$$479$$ 3.31948 0.151671 0.0758355 0.997120i $$-0.475838\pi$$
0.0758355 + 0.997120i $$0.475838\pi$$
$$480$$ 0 0
$$481$$ 56.3885 2.57110
$$482$$ 0 0
$$483$$ 2.92900 0.133274
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −19.7412 −0.894558 −0.447279 0.894395i $$-0.647607\pi$$
−0.447279 + 0.894395i $$0.647607\pi$$
$$488$$ 0 0
$$489$$ −25.4521 −1.15098
$$490$$ 0 0
$$491$$ 18.1353 0.818432 0.409216 0.912437i $$-0.365802\pi$$
0.409216 + 0.912437i $$0.365802\pi$$
$$492$$ 0 0
$$493$$ −0.910968 −0.0410279
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −10.0681 −0.451616
$$498$$ 0 0
$$499$$ 14.9797 0.670582 0.335291 0.942115i $$-0.391165\pi$$
0.335291 + 0.942115i $$0.391165\pi$$
$$500$$ 0 0
$$501$$ 46.6552 2.08440
$$502$$ 0 0
$$503$$ 7.90238 0.352350 0.176175 0.984359i $$-0.443628\pi$$
0.176175 + 0.984359i $$0.443628\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 82.5262 3.66512
$$508$$ 0 0
$$509$$ −21.1882 −0.939152 −0.469576 0.882892i $$-0.655593\pi$$
−0.469576 + 0.882892i $$0.655593\pi$$
$$510$$ 0 0
$$511$$ −1.35502 −0.0599425
$$512$$ 0 0
$$513$$ 9.45245 0.417336
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −1.20540 −0.0530136
$$518$$ 0 0
$$519$$ −25.1728 −1.10496
$$520$$ 0 0
$$521$$ −11.1210 −0.487219 −0.243610 0.969873i $$-0.578332\pi$$
−0.243610 + 0.969873i $$0.578332\pi$$
$$522$$ 0 0
$$523$$ 16.2670 0.711304 0.355652 0.934618i $$-0.384259\pi$$
0.355652 + 0.934618i $$0.384259\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −29.5024 −1.28515
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ 77.0151 3.34217
$$532$$ 0 0
$$533$$ 30.9335 1.33988
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 32.2743 1.39274
$$538$$ 0 0
$$539$$ 37.1200 1.59887
$$540$$ 0 0
$$541$$ −13.8797 −0.596733 −0.298367 0.954451i $$-0.596442\pi$$
−0.298367 + 0.954451i $$0.596442\pi$$
$$542$$ 0 0
$$543$$ −46.0851 −1.97770
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 5.91647 0.252970 0.126485 0.991969i $$-0.459630\pi$$
0.126485 + 0.991969i $$0.459630\pi$$
$$548$$ 0 0
$$549$$ 35.5467 1.51709
$$550$$ 0 0
$$551$$ −0.293475 −0.0125025
$$552$$ 0 0
$$553$$ 4.58371 0.194919
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −28.2857 −1.19850 −0.599252 0.800560i $$-0.704535\pi$$
−0.599252 + 0.800560i $$0.704535\pi$$
$$558$$ 0 0
$$559$$ −11.9676 −0.506175
$$560$$ 0 0
$$561$$ −61.7148 −2.60560
$$562$$ 0 0
$$563$$ −31.7025 −1.33610 −0.668050 0.744117i $$-0.732871\pi$$
−0.668050 + 0.744117i $$0.732871\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ −8.14763 −0.342169
$$568$$ 0 0
$$569$$ −24.0702 −1.00907 −0.504537 0.863390i $$-0.668337\pi$$
−0.504537 + 0.863390i $$0.668337\pi$$
$$570$$ 0 0
$$571$$ 14.3935 0.602349 0.301175 0.953569i $$-0.402621\pi$$
0.301175 + 0.953569i $$0.402621\pi$$
$$572$$ 0 0
$$573$$ −66.5023 −2.77817
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −5.77998 −0.240624 −0.120312 0.992736i $$-0.538389\pi$$
−0.120312 + 0.992736i $$0.538389\pi$$
$$578$$ 0 0
$$579$$ 48.2111 2.00358
$$580$$ 0 0
$$581$$ 15.4115 0.639378
$$582$$ 0 0
$$583$$ 11.8738 0.491761
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 27.7897 1.14700 0.573502 0.819204i $$-0.305584\pi$$
0.573502 + 0.819204i $$0.305584\pi$$
$$588$$ 0 0
$$589$$ −9.50441 −0.391623
$$590$$ 0 0
$$591$$ 41.7999 1.71942
$$592$$ 0 0
$$593$$ −22.1872 −0.911118 −0.455559 0.890205i $$-0.650561\pi$$
−0.455559 + 0.890205i $$0.650561\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −20.6683 −0.845897
$$598$$ 0 0
$$599$$ 15.1380 0.618521 0.309260 0.950977i $$-0.399919\pi$$
0.309260 + 0.950977i $$0.399919\pi$$
$$600$$ 0 0
$$601$$ 20.5353 0.837652 0.418826 0.908067i $$-0.362442\pi$$
0.418826 + 0.908067i $$0.362442\pi$$
$$602$$ 0 0
$$603$$ 13.4352 0.547123
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 28.9906 1.17669 0.588345 0.808610i $$-0.299780\pi$$
0.588345 + 0.808610i $$0.299780\pi$$
$$608$$ 0 0
$$609$$ 0.793883 0.0321698
$$610$$ 0 0
$$611$$ −1.24987 −0.0505644
$$612$$ 0 0
$$613$$ −9.03621 −0.364969 −0.182485 0.983209i $$-0.558414\pi$$
−0.182485 + 0.983209i $$0.558414\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −4.42741 −0.178241 −0.0891203 0.996021i $$-0.528406\pi$$
−0.0891203 + 0.996021i $$0.528406\pi$$
$$618$$ 0 0
$$619$$ 14.8403 0.596483 0.298241 0.954490i $$-0.403600\pi$$
0.298241 + 0.954490i $$0.403600\pi$$
$$620$$ 0 0
$$621$$ −8.72993 −0.350320
$$622$$ 0 0
$$623$$ 10.9380 0.438222
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ −19.8819 −0.794005
$$628$$ 0 0
$$629$$ −29.7334 −1.18555
$$630$$ 0 0
$$631$$ 41.8369 1.66550 0.832751 0.553648i $$-0.186765\pi$$
0.832751 + 0.553648i $$0.186765\pi$$
$$632$$ 0 0
$$633$$ 38.5841 1.53358
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 38.4894 1.52501
$$638$$ 0 0
$$639$$ 60.8113 2.40566
$$640$$ 0 0
$$641$$ 5.86478 0.231645 0.115822 0.993270i $$-0.463050\pi$$
0.115822 + 0.993270i $$0.463050\pi$$
$$642$$ 0 0
$$643$$ −0.460745 −0.0181700 −0.00908500 0.999959i $$-0.502892\pi$$
−0.00908500 + 0.999959i $$0.502892\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 15.5690 0.612079 0.306040 0.952019i $$-0.400996\pi$$
0.306040 + 0.952019i $$0.400996\pi$$
$$648$$ 0 0
$$649$$ −79.9362 −3.13777
$$650$$ 0 0
$$651$$ 25.7105 1.00767
$$652$$ 0 0
$$653$$ 29.4538 1.15262 0.576308 0.817233i $$-0.304493\pi$$
0.576308 + 0.817233i $$0.304493\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 8.18430 0.319300
$$658$$ 0 0
$$659$$ 22.5604 0.878829 0.439415 0.898284i $$-0.355186\pi$$
0.439415 + 0.898284i $$0.355186\pi$$
$$660$$ 0 0
$$661$$ −39.7530 −1.54621 −0.773106 0.634277i $$-0.781298\pi$$
−0.773106 + 0.634277i $$0.781298\pi$$
$$662$$ 0 0
$$663$$ −63.9916 −2.48523
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0.271042 0.0104948
$$668$$ 0 0
$$669$$ 79.6289 3.07863
$$670$$ 0 0
$$671$$ −36.8949 −1.42431
$$672$$ 0 0
$$673$$ −11.5047 −0.443475 −0.221737 0.975106i $$-0.571173\pi$$
−0.221737 + 0.975106i $$0.571173\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −31.7703 −1.22103 −0.610515 0.792005i $$-0.709038\pi$$
−0.610515 + 0.792005i $$0.709038\pi$$
$$678$$ 0 0
$$679$$ 10.0868 0.387098
$$680$$ 0 0
$$681$$ −28.2101 −1.08101
$$682$$ 0 0
$$683$$ −7.22683 −0.276527 −0.138264 0.990395i $$-0.544152\pi$$
−0.138264 + 0.990395i $$0.544152\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 42.4145 1.61822
$$688$$ 0 0
$$689$$ 12.3118 0.469043
$$690$$ 0 0
$$691$$ 3.29788 0.125457 0.0627286 0.998031i $$-0.480020\pi$$
0.0627286 + 0.998031i $$0.480020\pi$$
$$692$$ 0 0
$$693$$ 35.6996 1.35611
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −16.3111 −0.617827
$$698$$ 0 0
$$699$$ −28.1123 −1.06331
$$700$$ 0 0
$$701$$ 18.7936 0.709824 0.354912 0.934900i $$-0.384511\pi$$
0.354912 + 0.934900i $$0.384511\pi$$
$$702$$ 0 0
$$703$$ −9.57884 −0.361273
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 14.9658 0.562848
$$708$$ 0 0
$$709$$ 14.1039 0.529685 0.264842 0.964292i $$-0.414680\pi$$
0.264842 + 0.964292i $$0.414680\pi$$
$$710$$ 0 0
$$711$$ −27.6856 −1.03829
$$712$$ 0 0
$$713$$ 8.77792 0.328736
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −46.2341 −1.72664
$$718$$ 0 0
$$719$$ −23.5631 −0.878754 −0.439377 0.898303i $$-0.644801\pi$$
−0.439377 + 0.898303i $$0.644801\pi$$
$$720$$ 0 0
$$721$$ 7.46005 0.277827
$$722$$ 0 0
$$723$$ −10.4862 −0.389985
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 3.57021 0.132412 0.0662059 0.997806i $$-0.478911\pi$$
0.0662059 + 0.997806i $$0.478911\pi$$
$$728$$ 0 0
$$729$$ −29.0190 −1.07478
$$730$$ 0 0
$$731$$ 6.31045 0.233401
$$732$$ 0 0
$$733$$ −10.3158 −0.381022 −0.190511 0.981685i $$-0.561015\pi$$
−0.190511 + 0.981685i $$0.561015\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −13.9448 −0.513663
$$738$$ 0 0
$$739$$ −36.2056 −1.33185 −0.665923 0.746021i $$-0.731962\pi$$
−0.665923 + 0.746021i $$0.731962\pi$$
$$740$$ 0 0
$$741$$ −20.6153 −0.757324
$$742$$ 0 0
$$743$$ −3.24466 −0.119035 −0.0595174 0.998227i $$-0.518956\pi$$
−0.0595174 + 0.998227i $$0.518956\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ −93.0856 −3.40582
$$748$$ 0 0
$$749$$ −11.0883 −0.405157
$$750$$ 0 0
$$751$$ 26.9796 0.984499 0.492250 0.870454i $$-0.336175\pi$$
0.492250 + 0.870454i $$0.336175\pi$$
$$752$$ 0 0
$$753$$ −70.6888 −2.57604
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 11.7249 0.426148 0.213074 0.977036i $$-0.431652\pi$$
0.213074 + 0.977036i $$0.431652\pi$$
$$758$$ 0 0
$$759$$ 18.3621 0.666504
$$760$$ 0 0
$$761$$ −24.4303 −0.885599 −0.442800 0.896621i $$-0.646015\pi$$
−0.442800 + 0.896621i $$0.646015\pi$$
$$762$$ 0 0
$$763$$ −3.38623 −0.122590
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −82.8852 −2.99281
$$768$$ 0 0
$$769$$ 0.804588 0.0290142 0.0145071 0.999895i $$-0.495382\pi$$
0.0145071 + 0.999895i $$0.495382\pi$$
$$770$$ 0 0
$$771$$ 92.0989 3.31686
$$772$$ 0 0
$$773$$ −6.02477 −0.216696 −0.108348 0.994113i $$-0.534556\pi$$
−0.108348 + 0.994113i $$0.534556\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 25.9118 0.929582
$$778$$ 0 0
$$779$$ −5.25474 −0.188270
$$780$$ 0 0
$$781$$ −63.1179 −2.25853
$$782$$ 0 0
$$783$$ −2.36618 −0.0845604
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 12.9326 0.460996 0.230498 0.973073i $$-0.425964\pi$$
0.230498 + 0.973073i $$0.425964\pi$$
$$788$$ 0 0
$$789$$ 61.4977 2.18938
$$790$$ 0 0
$$791$$ −14.4033 −0.512123
$$792$$ 0 0
$$793$$ −38.2561 −1.35851
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −36.3660 −1.28815 −0.644075 0.764962i $$-0.722758\pi$$
−0.644075 + 0.764962i $$0.722758\pi$$
$$798$$ 0 0
$$799$$ 0.659052 0.0233156
$$800$$ 0 0
$$801$$ −66.0654 −2.33431
$$802$$ 0 0
$$803$$ −8.49472 −0.299772
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −50.8629 −1.79046
$$808$$ 0 0
$$809$$ 26.5840 0.934642 0.467321 0.884088i $$-0.345219\pi$$
0.467321 + 0.884088i $$0.345219\pi$$
$$810$$ 0 0
$$811$$ −44.7712 −1.57213 −0.786064 0.618145i $$-0.787884\pi$$
−0.786064 + 0.618145i $$0.787884\pi$$
$$812$$ 0 0
$$813$$ 40.8360 1.43218
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 2.03296 0.0711242
$$818$$ 0 0
$$819$$ 37.0166 1.29346
$$820$$ 0 0
$$821$$ 22.3949 0.781587 0.390794 0.920478i $$-0.372201\pi$$
0.390794 + 0.920478i $$0.372201\pi$$
$$822$$ 0 0
$$823$$ −2.18913 −0.0763081 −0.0381541 0.999272i $$-0.512148\pi$$
−0.0381541 + 0.999272i $$0.512148\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −39.4367 −1.37135 −0.685674 0.727909i $$-0.740492\pi$$
−0.685674 + 0.727909i $$0.740492\pi$$
$$828$$ 0 0
$$829$$ −11.0629 −0.384229 −0.192115 0.981373i $$-0.561535\pi$$
−0.192115 + 0.981373i $$0.561535\pi$$
$$830$$ 0 0
$$831$$ 48.5040 1.68259
$$832$$ 0 0
$$833$$ −20.2953 −0.703190
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −76.6306 −2.64874
$$838$$ 0 0
$$839$$ −37.8510 −1.30676 −0.653381 0.757030i $$-0.726650\pi$$
−0.653381 + 0.757030i $$0.726650\pi$$
$$840$$ 0 0
$$841$$ −28.9265 −0.997467
$$842$$ 0 0
$$843$$ −1.08254 −0.0372845
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −26.2675 −0.902561
$$848$$ 0 0
$$849$$ 20.3132 0.697148
$$850$$ 0 0
$$851$$ 8.84665 0.303259
$$852$$ 0 0
$$853$$ −28.9770 −0.992155 −0.496078 0.868278i $$-0.665227\pi$$
−0.496078 + 0.868278i $$0.665227\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 56.0462 1.91450 0.957250 0.289261i $$-0.0934095\pi$$
0.957250 + 0.289261i $$0.0934095\pi$$
$$858$$ 0 0
$$859$$ 43.8866 1.49739 0.748695 0.662914i $$-0.230681\pi$$
0.748695 + 0.662914i $$0.230681\pi$$
$$860$$ 0 0
$$861$$ 14.2147 0.484434
$$862$$ 0 0
$$863$$ −22.1955 −0.755542 −0.377771 0.925899i $$-0.623309\pi$$
−0.377771 + 0.925899i $$0.623309\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ −17.0377 −0.578630
$$868$$ 0 0
$$869$$ 28.7357 0.974791
$$870$$ 0 0
$$871$$ −14.4592 −0.489932
$$872$$ 0 0
$$873$$ −60.9245 −2.06198
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 5.44979 0.184026 0.0920131 0.995758i $$-0.470670\pi$$
0.0920131 + 0.995758i $$0.470670\pi$$
$$878$$ 0 0
$$879$$ −78.3125 −2.64141
$$880$$ 0 0
$$881$$ −11.7854 −0.397060 −0.198530 0.980095i $$-0.563617\pi$$
−0.198530 + 0.980095i $$0.563617\pi$$
$$882$$ 0 0
$$883$$ −23.7945 −0.800748 −0.400374 0.916352i $$-0.631120\pi$$
−0.400374 + 0.916352i $$0.631120\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 44.4300 1.49181 0.745907 0.666050i $$-0.232016\pi$$
0.745907 + 0.666050i $$0.232016\pi$$
$$888$$ 0 0
$$889$$ 1.16993 0.0392381
$$890$$ 0 0
$$891$$ −51.0782 −1.71119
$$892$$ 0 0
$$893$$ 0.212318 0.00710496
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 19.0396 0.635712
$$898$$ 0 0
$$899$$ 2.37919 0.0793504
$$900$$ 0 0
$$901$$ −6.49196 −0.216279
$$902$$ 0 0
$$903$$ −5.49938 −0.183008
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 40.7542 1.35322 0.676611 0.736341i $$-0.263448\pi$$
0.676611 + 0.736341i $$0.263448\pi$$
$$908$$ 0 0
$$909$$ −90.3934 −2.99816
$$910$$ 0 0
$$911$$ −2.71520 −0.0899587 −0.0449794 0.998988i $$-0.514322\pi$$
−0.0449794 + 0.998988i $$0.514322\pi$$
$$912$$ 0 0
$$913$$ 96.6163 3.19753
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 11.5046 0.379914
$$918$$ 0 0
$$919$$ 10.7227 0.353708 0.176854 0.984237i $$-0.443408\pi$$
0.176854 + 0.984237i $$0.443408\pi$$
$$920$$ 0 0
$$921$$ 58.8890 1.94046
$$922$$ 0 0
$$923$$ −65.4464 −2.15419
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ −45.0587 −1.47992
$$928$$ 0 0
$$929$$ 38.8670 1.27518 0.637592 0.770374i $$-0.279930\pi$$
0.637592 + 0.770374i $$0.279930\pi$$
$$930$$ 0 0
$$931$$ −6.53827 −0.214283
$$932$$ 0 0
$$933$$ 45.1055 1.47669
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 38.6117 1.26139 0.630694 0.776032i $$-0.282770\pi$$
0.630694 + 0.776032i $$0.282770\pi$$
$$938$$ 0 0
$$939$$ 38.7761 1.26541
$$940$$ 0 0
$$941$$ 26.3928 0.860381 0.430191 0.902738i $$-0.358446\pi$$
0.430191 + 0.902738i $$0.358446\pi$$
$$942$$ 0 0
$$943$$ 4.85308 0.158038
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −22.3777 −0.727179 −0.363589 0.931559i $$-0.618449\pi$$
−0.363589 + 0.931559i $$0.618449\pi$$
$$948$$ 0 0
$$949$$ −8.80811 −0.285923
$$950$$ 0 0
$$951$$ 51.8265 1.68059
$$952$$ 0 0
$$953$$ −17.4362 −0.564815 −0.282407 0.959295i $$-0.591133\pi$$
−0.282407 + 0.959295i $$0.591133\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 4.97692 0.160881
$$958$$ 0 0
$$959$$ 10.2630 0.331409
$$960$$ 0 0
$$961$$ 46.0519 1.48554
$$962$$ 0 0
$$963$$ 66.9732 2.15818
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −45.1587 −1.45221 −0.726103 0.687586i $$-0.758670\pi$$
−0.726103 + 0.687586i $$0.758670\pi$$
$$968$$ 0 0
$$969$$ 10.8704 0.349207
$$970$$ 0 0
$$971$$ −34.3123 −1.10113 −0.550567 0.834791i $$-0.685588\pi$$
−0.550567 + 0.834791i $$0.685588\pi$$
$$972$$ 0 0
$$973$$ −4.03353 −0.129309
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −1.51320 −0.0484115 −0.0242058 0.999707i $$-0.507706\pi$$
−0.0242058 + 0.999707i $$0.507706\pi$$
$$978$$ 0 0
$$979$$ 68.5713 2.19155
$$980$$ 0 0
$$981$$ 20.4528 0.653007
$$982$$ 0 0
$$983$$ −15.1715 −0.483895 −0.241947 0.970289i $$-0.577786\pi$$
−0.241947 + 0.970289i $$0.577786\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ −0.574345 −0.0182816
$$988$$ 0 0
$$989$$ −1.87756 −0.0597031
$$990$$ 0 0
$$991$$ −39.4470 −1.25307 −0.626537 0.779392i $$-0.715528\pi$$
−0.626537 + 0.779392i $$0.715528\pi$$
$$992$$ 0 0
$$993$$ 2.94153 0.0933467
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 2.33319 0.0738928 0.0369464 0.999317i $$-0.488237\pi$$
0.0369464 + 0.999317i $$0.488237\pi$$
$$998$$ 0 0
$$999$$ −77.2307 −2.44347
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 4600.2.a.bi.1.7 7
4.3 odd 2 9200.2.a.cz.1.1 7
5.2 odd 4 920.2.e.b.369.1 14
5.3 odd 4 920.2.e.b.369.14 yes 14
5.4 even 2 4600.2.a.bh.1.1 7
20.3 even 4 1840.2.e.g.369.1 14
20.7 even 4 1840.2.e.g.369.14 14
20.19 odd 2 9200.2.a.dc.1.7 7

By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.e.b.369.1 14 5.2 odd 4
920.2.e.b.369.14 yes 14 5.3 odd 4
1840.2.e.g.369.1 14 20.3 even 4
1840.2.e.g.369.14 14 20.7 even 4
4600.2.a.bh.1.1 7 5.4 even 2
4600.2.a.bi.1.7 7 1.1 even 1 trivial
9200.2.a.cz.1.1 7 4.3 odd 2
9200.2.a.dc.1.7 7 20.19 odd 2