# Properties

 Label 9408.2.a.el.1.1 Level $9408$ Weight $2$ Character 9408.1 Self dual yes Analytic conductor $75.123$ Analytic rank $1$ Dimension $4$ CM no Inner twists $1$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$9408 = 2^{6} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9408.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$75.1232582216$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.2624.1 Defining polynomial: $$x^{4} - 2x^{3} - 3x^{2} + 2x + 1$$ x^4 - 2*x^3 - 3*x^2 + 2*x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 4704) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-0.360409$$ of defining polynomial Character $$\chi$$ $$=$$ 9408.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} -2.13503 q^{5} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} -2.13503 q^{5} +1.00000 q^{9} -0.298573 q^{11} +6.43361 q^{13} +2.13503 q^{15} -1.11564 q^{17} +1.01939 q^{19} -4.29857 q^{23} -0.441637 q^{25} -1.00000 q^{27} +0.422246 q^{29} -10.6762 q^{31} +0.298573 q^{33} +5.65685 q^{37} -6.43361 q^{39} -8.32600 q^{41} +7.09849 q^{43} -2.13503 q^{45} +8.11788 q^{47} +1.11564 q^{51} +3.44164 q^{53} +0.637463 q^{55} -1.01939 q^{57} -6.46103 q^{59} +5.83646 q^{61} -13.7360 q^{65} -5.05971 q^{67} +4.29857 q^{69} +1.35828 q^{71} +2.85585 q^{73} +0.441637 q^{75} +7.69564 q^{79} +1.00000 q^{81} -6.03878 q^{83} +2.38193 q^{85} -0.422246 q^{87} -15.3696 q^{89} +10.6762 q^{93} -2.17643 q^{95} +0.512705 q^{97} -0.298573 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{3} + 4 q^{5} + 4 q^{9}+O(q^{10})$$ 4 * q - 4 * q^3 + 4 * q^5 + 4 * q^9 $$4 q - 4 q^{3} + 4 q^{5} + 4 q^{9} + 4 q^{11} + 8 q^{13} - 4 q^{15} - 4 q^{17} - 8 q^{19} - 12 q^{23} + 12 q^{25} - 4 q^{27} - 8 q^{31} - 4 q^{33} - 8 q^{39} - 20 q^{41} - 8 q^{43} + 4 q^{45} - 16 q^{47} + 4 q^{51} - 8 q^{55} + 8 q^{57} + 16 q^{61} - 8 q^{65} - 8 q^{67} + 12 q^{69} - 12 q^{71} - 8 q^{73} - 12 q^{75} - 16 q^{79} + 4 q^{81} + 8 q^{85} - 28 q^{89} + 8 q^{93} - 24 q^{95} - 40 q^{97} + 4 q^{99}+O(q^{100})$$ 4 * q - 4 * q^3 + 4 * q^5 + 4 * q^9 + 4 * q^11 + 8 * q^13 - 4 * q^15 - 4 * q^17 - 8 * q^19 - 12 * q^23 + 12 * q^25 - 4 * q^27 - 8 * q^31 - 4 * q^33 - 8 * q^39 - 20 * q^41 - 8 * q^43 + 4 * q^45 - 16 * q^47 + 4 * q^51 - 8 * q^55 + 8 * q^57 + 16 * q^61 - 8 * q^65 - 8 * q^67 + 12 * q^69 - 12 * q^71 - 8 * q^73 - 12 * q^75 - 16 * q^79 + 4 * q^81 + 8 * q^85 - 28 * q^89 + 8 * q^93 - 24 * q^95 - 40 * q^97 + 4 * q^99

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −1.00000 −0.577350
$$4$$ 0 0
$$5$$ −2.13503 −0.954815 −0.477408 0.878682i $$-0.658424\pi$$
−0.477408 + 0.878682i $$0.658424\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −0.298573 −0.0900231 −0.0450116 0.998986i $$-0.514332\pi$$
−0.0450116 + 0.998986i $$0.514332\pi$$
$$12$$ 0 0
$$13$$ 6.43361 1.78436 0.892181 0.451679i $$-0.149175\pi$$
0.892181 + 0.451679i $$0.149175\pi$$
$$14$$ 0 0
$$15$$ 2.13503 0.551263
$$16$$ 0 0
$$17$$ −1.11564 −0.270583 −0.135291 0.990806i $$-0.543197\pi$$
−0.135291 + 0.990806i $$0.543197\pi$$
$$18$$ 0 0
$$19$$ 1.01939 0.233864 0.116932 0.993140i $$-0.462694\pi$$
0.116932 + 0.993140i $$0.462694\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −4.29857 −0.896314 −0.448157 0.893955i $$-0.647920\pi$$
−0.448157 + 0.893955i $$0.647920\pi$$
$$24$$ 0 0
$$25$$ −0.441637 −0.0883275
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ 0.422246 0.0784091 0.0392045 0.999231i $$-0.487518\pi$$
0.0392045 + 0.999231i $$0.487518\pi$$
$$30$$ 0 0
$$31$$ −10.6762 −1.91751 −0.958755 0.284233i $$-0.908261\pi$$
−0.958755 + 0.284233i $$0.908261\pi$$
$$32$$ 0 0
$$33$$ 0.298573 0.0519749
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 5.65685 0.929981 0.464991 0.885316i $$-0.346058\pi$$
0.464991 + 0.885316i $$0.346058\pi$$
$$38$$ 0 0
$$39$$ −6.43361 −1.03020
$$40$$ 0 0
$$41$$ −8.32600 −1.30030 −0.650151 0.759805i $$-0.725294\pi$$
−0.650151 + 0.759805i $$0.725294\pi$$
$$42$$ 0 0
$$43$$ 7.09849 1.08251 0.541255 0.840859i $$-0.317949\pi$$
0.541255 + 0.840859i $$0.317949\pi$$
$$44$$ 0 0
$$45$$ −2.13503 −0.318272
$$46$$ 0 0
$$47$$ 8.11788 1.18411 0.592057 0.805896i $$-0.298316\pi$$
0.592057 + 0.805896i $$0.298316\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 1.11564 0.156221
$$52$$ 0 0
$$53$$ 3.44164 0.472745 0.236373 0.971662i $$-0.424041\pi$$
0.236373 + 0.971662i $$0.424041\pi$$
$$54$$ 0 0
$$55$$ 0.637463 0.0859555
$$56$$ 0 0
$$57$$ −1.01939 −0.135022
$$58$$ 0 0
$$59$$ −6.46103 −0.841154 −0.420577 0.907257i $$-0.638172\pi$$
−0.420577 + 0.907257i $$0.638172\pi$$
$$60$$ 0 0
$$61$$ 5.83646 0.747282 0.373641 0.927573i $$-0.378109\pi$$
0.373641 + 0.927573i $$0.378109\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −13.7360 −1.70374
$$66$$ 0 0
$$67$$ −5.05971 −0.618142 −0.309071 0.951039i $$-0.600018\pi$$
−0.309071 + 0.951039i $$0.600018\pi$$
$$68$$ 0 0
$$69$$ 4.29857 0.517487
$$70$$ 0 0
$$71$$ 1.35828 0.161198 0.0805992 0.996747i $$-0.474317\pi$$
0.0805992 + 0.996747i $$0.474317\pi$$
$$72$$ 0 0
$$73$$ 2.85585 0.334252 0.167126 0.985936i $$-0.446551\pi$$
0.167126 + 0.985936i $$0.446551\pi$$
$$74$$ 0 0
$$75$$ 0.441637 0.0509959
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 7.69564 0.865827 0.432913 0.901436i $$-0.357486\pi$$
0.432913 + 0.901436i $$0.357486\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −6.03878 −0.662843 −0.331421 0.943483i $$-0.607528\pi$$
−0.331421 + 0.943483i $$0.607528\pi$$
$$84$$ 0 0
$$85$$ 2.38193 0.258356
$$86$$ 0 0
$$87$$ −0.422246 −0.0452695
$$88$$ 0 0
$$89$$ −15.3696 −1.62918 −0.814589 0.580038i $$-0.803038\pi$$
−0.814589 + 0.580038i $$0.803038\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 10.6762 1.10708
$$94$$ 0 0
$$95$$ −2.17643 −0.223297
$$96$$ 0 0
$$97$$ 0.512705 0.0520573 0.0260287 0.999661i $$-0.491714\pi$$
0.0260287 + 0.999661i $$0.491714\pi$$
$$98$$ 0 0
$$99$$ −0.298573 −0.0300077
$$100$$ 0 0
$$101$$ −4.96346 −0.493883 −0.246941 0.969030i $$-0.579425\pi$$
−0.246941 + 0.969030i $$0.579425\pi$$
$$102$$ 0 0
$$103$$ −9.48195 −0.934285 −0.467142 0.884182i $$-0.654716\pi$$
−0.467142 + 0.884182i $$0.654716\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 16.8387 1.62786 0.813929 0.580964i $$-0.197324\pi$$
0.813929 + 0.580964i $$0.197324\pi$$
$$108$$ 0 0
$$109$$ −3.09849 −0.296782 −0.148391 0.988929i $$-0.547409\pi$$
−0.148391 + 0.988929i $$0.547409\pi$$
$$110$$ 0 0
$$111$$ −5.65685 −0.536925
$$112$$ 0 0
$$113$$ 9.31371 0.876160 0.438080 0.898936i $$-0.355659\pi$$
0.438080 + 0.898936i $$0.355659\pi$$
$$114$$ 0 0
$$115$$ 9.17759 0.855815
$$116$$ 0 0
$$117$$ 6.43361 0.594787
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −10.9109 −0.991896
$$122$$ 0 0
$$123$$ 8.32600 0.750730
$$124$$ 0 0
$$125$$ 11.6181 1.03915
$$126$$ 0 0
$$127$$ 17.1373 1.52069 0.760344 0.649521i $$-0.225031\pi$$
0.760344 + 0.649521i $$0.225031\pi$$
$$128$$ 0 0
$$129$$ −7.09849 −0.624987
$$130$$ 0 0
$$131$$ −16.1582 −1.41175 −0.705874 0.708337i $$-0.749446\pi$$
−0.705874 + 0.708337i $$0.749446\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 2.13503 0.183754
$$136$$ 0 0
$$137$$ 21.7747 1.86034 0.930171 0.367127i $$-0.119659\pi$$
0.930171 + 0.367127i $$0.119659\pi$$
$$138$$ 0 0
$$139$$ −15.3137 −1.29889 −0.649446 0.760408i $$-0.724999\pi$$
−0.649446 + 0.760408i $$0.724999\pi$$
$$140$$ 0 0
$$141$$ −8.11788 −0.683649
$$142$$ 0 0
$$143$$ −1.92090 −0.160634
$$144$$ 0 0
$$145$$ −0.901508 −0.0748662
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 3.87207 0.317212 0.158606 0.987342i $$-0.449300\pi$$
0.158606 + 0.987342i $$0.449300\pi$$
$$150$$ 0 0
$$151$$ −4.75535 −0.386985 −0.193492 0.981102i $$-0.561981\pi$$
−0.193492 + 0.981102i $$0.561981\pi$$
$$152$$ 0 0
$$153$$ −1.11564 −0.0901942
$$154$$ 0 0
$$155$$ 22.7941 1.83087
$$156$$ 0 0
$$157$$ 11.5482 0.921644 0.460822 0.887493i $$-0.347555\pi$$
0.460822 + 0.887493i $$0.347555\pi$$
$$158$$ 0 0
$$159$$ −3.44164 −0.272940
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 11.9109 0.932930 0.466465 0.884540i $$-0.345527\pi$$
0.466465 + 0.884540i $$0.345527\pi$$
$$164$$ 0 0
$$165$$ −0.637463 −0.0496264
$$166$$ 0 0
$$167$$ −15.2734 −1.18189 −0.590945 0.806712i $$-0.701245\pi$$
−0.590945 + 0.806712i $$0.701245\pi$$
$$168$$ 0 0
$$169$$ 28.3913 2.18394
$$170$$ 0 0
$$171$$ 1.01939 0.0779548
$$172$$ 0 0
$$173$$ −4.00710 −0.304654 −0.152327 0.988330i $$-0.548677\pi$$
−0.152327 + 0.988330i $$0.548677\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 6.46103 0.485641
$$178$$ 0 0
$$179$$ −20.7496 −1.55089 −0.775447 0.631412i $$-0.782476\pi$$
−0.775447 + 0.631412i $$0.782476\pi$$
$$180$$ 0 0
$$181$$ 21.7246 1.61478 0.807388 0.590021i $$-0.200880\pi$$
0.807388 + 0.590021i $$0.200880\pi$$
$$182$$ 0 0
$$183$$ −5.83646 −0.431443
$$184$$ 0 0
$$185$$ −12.0776 −0.887960
$$186$$ 0 0
$$187$$ 0.333100 0.0243587
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 20.6720 1.49577 0.747886 0.663827i $$-0.231069\pi$$
0.747886 + 0.663827i $$0.231069\pi$$
$$192$$ 0 0
$$193$$ 15.5998 1.12290 0.561450 0.827510i $$-0.310244\pi$$
0.561450 + 0.827510i $$0.310244\pi$$
$$194$$ 0 0
$$195$$ 13.7360 0.983652
$$196$$ 0 0
$$197$$ −25.6386 −1.82668 −0.913338 0.407202i $$-0.866504\pi$$
−0.913338 + 0.407202i $$0.866504\pi$$
$$198$$ 0 0
$$199$$ 2.88327 0.204390 0.102195 0.994764i $$-0.467413\pi$$
0.102195 + 0.994764i $$0.467413\pi$$
$$200$$ 0 0
$$201$$ 5.05971 0.356884
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 17.7763 1.24155
$$206$$ 0 0
$$207$$ −4.29857 −0.298771
$$208$$ 0 0
$$209$$ −0.304363 −0.0210532
$$210$$ 0 0
$$211$$ −24.5401 −1.68941 −0.844706 0.535230i $$-0.820225\pi$$
−0.844706 + 0.535230i $$0.820225\pi$$
$$212$$ 0 0
$$213$$ −1.35828 −0.0930679
$$214$$ 0 0
$$215$$ −15.1555 −1.03360
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ −2.85585 −0.192981
$$220$$ 0 0
$$221$$ −7.17759 −0.482817
$$222$$ 0 0
$$223$$ 20.5080 1.37332 0.686659 0.726980i $$-0.259077\pi$$
0.686659 + 0.726980i $$0.259077\pi$$
$$224$$ 0 0
$$225$$ −0.441637 −0.0294425
$$226$$ 0 0
$$227$$ −13.6972 −0.909113 −0.454557 0.890718i $$-0.650202\pi$$
−0.454557 + 0.890718i $$0.650202\pi$$
$$228$$ 0 0
$$229$$ 16.4175 1.08490 0.542451 0.840088i $$-0.317496\pi$$
0.542451 + 0.840088i $$0.317496\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −10.8915 −0.713523 −0.356762 0.934195i $$-0.616119\pi$$
−0.356762 + 0.934195i $$0.616119\pi$$
$$234$$ 0 0
$$235$$ −17.3319 −1.13061
$$236$$ 0 0
$$237$$ −7.69564 −0.499885
$$238$$ 0 0
$$239$$ −4.29857 −0.278052 −0.139026 0.990289i $$-0.544397\pi$$
−0.139026 + 0.990289i $$0.544397\pi$$
$$240$$ 0 0
$$241$$ −7.12808 −0.459160 −0.229580 0.973290i $$-0.573735\pi$$
−0.229580 + 0.973290i $$0.573735\pi$$
$$242$$ 0 0
$$243$$ −1.00000 −0.0641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 6.55836 0.417299
$$248$$ 0 0
$$249$$ 6.03878 0.382692
$$250$$ 0 0
$$251$$ 18.9690 1.19731 0.598657 0.801005i $$-0.295701\pi$$
0.598657 + 0.801005i $$0.295701\pi$$
$$252$$ 0 0
$$253$$ 1.28344 0.0806890
$$254$$ 0 0
$$255$$ −2.38193 −0.149162
$$256$$ 0 0
$$257$$ −20.4842 −1.27777 −0.638885 0.769303i $$-0.720604\pi$$
−0.638885 + 0.769303i $$0.720604\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 0.422246 0.0261364
$$262$$ 0 0
$$263$$ −23.5553 −1.45248 −0.726240 0.687441i $$-0.758734\pi$$
−0.726240 + 0.687441i $$0.758734\pi$$
$$264$$ 0 0
$$265$$ −7.34801 −0.451384
$$266$$ 0 0
$$267$$ 15.3696 0.940607
$$268$$ 0 0
$$269$$ 12.9086 0.787052 0.393526 0.919313i $$-0.371255\pi$$
0.393526 + 0.919313i $$0.371255\pi$$
$$270$$ 0 0
$$271$$ −19.8705 −1.20705 −0.603525 0.797344i $$-0.706237\pi$$
−0.603525 + 0.797344i $$0.706237\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0.131861 0.00795151
$$276$$ 0 0
$$277$$ −25.9884 −1.56149 −0.780746 0.624848i $$-0.785161\pi$$
−0.780746 + 0.624848i $$0.785161\pi$$
$$278$$ 0 0
$$279$$ −10.6762 −0.639170
$$280$$ 0 0
$$281$$ −19.6553 −1.17254 −0.586269 0.810116i $$-0.699404\pi$$
−0.586269 + 0.810116i $$0.699404\pi$$
$$282$$ 0 0
$$283$$ −19.0582 −1.13289 −0.566445 0.824099i $$-0.691682\pi$$
−0.566445 + 0.824099i $$0.691682\pi$$
$$284$$ 0 0
$$285$$ 2.17643 0.128921
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −15.7553 −0.926785
$$290$$ 0 0
$$291$$ −0.512705 −0.0300553
$$292$$ 0 0
$$293$$ 7.84890 0.458538 0.229269 0.973363i $$-0.426366\pi$$
0.229269 + 0.973363i $$0.426366\pi$$
$$294$$ 0 0
$$295$$ 13.7945 0.803147
$$296$$ 0 0
$$297$$ 0.298573 0.0173250
$$298$$ 0 0
$$299$$ −27.6553 −1.59935
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 4.96346 0.285143
$$304$$ 0 0
$$305$$ −12.4610 −0.713516
$$306$$ 0 0
$$307$$ −26.5301 −1.51415 −0.757076 0.653327i $$-0.773373\pi$$
−0.757076 + 0.653327i $$0.773373\pi$$
$$308$$ 0 0
$$309$$ 9.48195 0.539410
$$310$$ 0 0
$$311$$ −0.726608 −0.0412022 −0.0206011 0.999788i $$-0.506558\pi$$
−0.0206011 + 0.999788i $$0.506558\pi$$
$$312$$ 0 0
$$313$$ −15.1259 −0.854967 −0.427484 0.904023i $$-0.640600\pi$$
−0.427484 + 0.904023i $$0.640600\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 19.9496 1.12048 0.560242 0.828329i $$-0.310708\pi$$
0.560242 + 0.828329i $$0.310708\pi$$
$$318$$ 0 0
$$319$$ −0.126071 −0.00705863
$$320$$ 0 0
$$321$$ −16.8387 −0.939845
$$322$$ 0 0
$$323$$ −1.13727 −0.0632797
$$324$$ 0 0
$$325$$ −2.84132 −0.157608
$$326$$ 0 0
$$327$$ 3.09849 0.171347
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −17.1943 −0.945084 −0.472542 0.881308i $$-0.656663\pi$$
−0.472542 + 0.881308i $$0.656663\pi$$
$$332$$ 0 0
$$333$$ 5.65685 0.309994
$$334$$ 0 0
$$335$$ 10.8026 0.590211
$$336$$ 0 0
$$337$$ 10.0388 0.546847 0.273424 0.961894i $$-0.411844\pi$$
0.273424 + 0.961894i $$0.411844\pi$$
$$338$$ 0 0
$$339$$ −9.31371 −0.505851
$$340$$ 0 0
$$341$$ 3.18764 0.172620
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ −9.17759 −0.494105
$$346$$ 0 0
$$347$$ −2.41799 −0.129804 −0.0649022 0.997892i $$-0.520674\pi$$
−0.0649022 + 0.997892i $$0.520674\pi$$
$$348$$ 0 0
$$349$$ −6.26689 −0.335459 −0.167730 0.985833i $$-0.553644\pi$$
−0.167730 + 0.985833i $$0.553644\pi$$
$$350$$ 0 0
$$351$$ −6.43361 −0.343400
$$352$$ 0 0
$$353$$ 10.3099 0.548742 0.274371 0.961624i $$-0.411530\pi$$
0.274371 + 0.961624i $$0.411530\pi$$
$$354$$ 0 0
$$355$$ −2.89997 −0.153915
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −17.8984 −0.944642 −0.472321 0.881427i $$-0.656584\pi$$
−0.472321 + 0.881427i $$0.656584\pi$$
$$360$$ 0 0
$$361$$ −17.9608 −0.945307
$$362$$ 0 0
$$363$$ 10.9109 0.572671
$$364$$ 0 0
$$365$$ −6.09733 −0.319149
$$366$$ 0 0
$$367$$ −23.0415 −1.20276 −0.601378 0.798965i $$-0.705381\pi$$
−0.601378 + 0.798965i $$0.705381\pi$$
$$368$$ 0 0
$$369$$ −8.32600 −0.433434
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −3.68898 −0.191008 −0.0955042 0.995429i $$-0.530446\pi$$
−0.0955042 + 0.995429i $$0.530446\pi$$
$$374$$ 0 0
$$375$$ −11.6181 −0.599955
$$376$$ 0 0
$$377$$ 2.71656 0.139910
$$378$$ 0 0
$$379$$ −8.76386 −0.450169 −0.225085 0.974339i $$-0.572266\pi$$
−0.225085 + 0.974339i $$0.572266\pi$$
$$380$$ 0 0
$$381$$ −17.1373 −0.877969
$$382$$ 0 0
$$383$$ −21.7665 −1.11222 −0.556109 0.831109i $$-0.687706\pi$$
−0.556109 + 0.831109i $$0.687706\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 7.09849 0.360837
$$388$$ 0 0
$$389$$ −6.38346 −0.323654 −0.161827 0.986819i $$-0.551739\pi$$
−0.161827 + 0.986819i $$0.551739\pi$$
$$390$$ 0 0
$$391$$ 4.79566 0.242527
$$392$$ 0 0
$$393$$ 16.1582 0.815073
$$394$$ 0 0
$$395$$ −16.4304 −0.826705
$$396$$ 0 0
$$397$$ 5.53210 0.277648 0.138824 0.990317i $$-0.455668\pi$$
0.138824 + 0.990317i $$0.455668\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −15.5778 −0.777916 −0.388958 0.921256i $$-0.627165\pi$$
−0.388958 + 0.921256i $$0.627165\pi$$
$$402$$ 0 0
$$403$$ −68.6867 −3.42153
$$404$$ 0 0
$$405$$ −2.13503 −0.106091
$$406$$ 0 0
$$407$$ −1.68898 −0.0837198
$$408$$ 0 0
$$409$$ 16.8625 0.833797 0.416899 0.908953i $$-0.363117\pi$$
0.416899 + 0.908953i $$0.363117\pi$$
$$410$$ 0 0
$$411$$ −21.7747 −1.07407
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 12.8930 0.632892
$$416$$ 0 0
$$417$$ 15.3137 0.749916
$$418$$ 0 0
$$419$$ 27.8942 1.36272 0.681359 0.731949i $$-0.261389\pi$$
0.681359 + 0.731949i $$0.261389\pi$$
$$420$$ 0 0
$$421$$ 31.6774 1.54386 0.771931 0.635706i $$-0.219291\pi$$
0.771931 + 0.635706i $$0.219291\pi$$
$$422$$ 0 0
$$423$$ 8.11788 0.394705
$$424$$ 0 0
$$425$$ 0.492709 0.0238999
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 1.92090 0.0927419
$$430$$ 0 0
$$431$$ −24.8066 −1.19489 −0.597445 0.801910i $$-0.703817\pi$$
−0.597445 + 0.801910i $$0.703817\pi$$
$$432$$ 0 0
$$433$$ −13.4918 −0.648374 −0.324187 0.945993i $$-0.605091\pi$$
−0.324187 + 0.945993i $$0.605091\pi$$
$$434$$ 0 0
$$435$$ 0.901508 0.0432240
$$436$$ 0 0
$$437$$ −4.38193 −0.209616
$$438$$ 0 0
$$439$$ −35.3137 −1.68543 −0.842716 0.538359i $$-0.819044\pi$$
−0.842716 + 0.538359i $$0.819044\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −30.4385 −1.44618 −0.723089 0.690755i $$-0.757279\pi$$
−0.723089 + 0.690755i $$0.757279\pi$$
$$444$$ 0 0
$$445$$ 32.8147 1.55556
$$446$$ 0 0
$$447$$ −3.87207 −0.183143
$$448$$ 0 0
$$449$$ −18.0000 −0.849473 −0.424736 0.905317i $$-0.639633\pi$$
−0.424736 + 0.905317i $$0.639633\pi$$
$$450$$ 0 0
$$451$$ 2.48592 0.117057
$$452$$ 0 0
$$453$$ 4.75535 0.223426
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 15.1943 0.710759 0.355379 0.934722i $$-0.384352\pi$$
0.355379 + 0.934722i $$0.384352\pi$$
$$458$$ 0 0
$$459$$ 1.11564 0.0520736
$$460$$ 0 0
$$461$$ 38.3935 1.78816 0.894082 0.447903i $$-0.147829\pi$$
0.894082 + 0.447903i $$0.147829\pi$$
$$462$$ 0 0
$$463$$ −5.59984 −0.260247 −0.130123 0.991498i $$-0.541537\pi$$
−0.130123 + 0.991498i $$0.541537\pi$$
$$464$$ 0 0
$$465$$ −22.7941 −1.05705
$$466$$ 0 0
$$467$$ 23.9717 1.10928 0.554639 0.832091i $$-0.312856\pi$$
0.554639 + 0.832091i $$0.312856\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −11.5482 −0.532111
$$472$$ 0 0
$$473$$ −2.11942 −0.0974509
$$474$$ 0 0
$$475$$ −0.450201 −0.0206567
$$476$$ 0 0
$$477$$ 3.44164 0.157582
$$478$$ 0 0
$$479$$ 5.23461 0.239175 0.119588 0.992824i $$-0.461843\pi$$
0.119588 + 0.992824i $$0.461843\pi$$
$$480$$ 0 0
$$481$$ 36.3940 1.65942
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −1.09464 −0.0497051
$$486$$ 0 0
$$487$$ 11.2447 0.509544 0.254772 0.967001i $$-0.418000\pi$$
0.254772 + 0.967001i $$0.418000\pi$$
$$488$$ 0 0
$$489$$ −11.9109 −0.538627
$$490$$ 0 0
$$491$$ −13.5250 −0.610374 −0.305187 0.952292i $$-0.598719\pi$$
−0.305187 + 0.952292i $$0.598719\pi$$
$$492$$ 0 0
$$493$$ −0.471075 −0.0212161
$$494$$ 0 0
$$495$$ 0.637463 0.0286518
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 9.87207 0.441935 0.220967 0.975281i $$-0.429079\pi$$
0.220967 + 0.975281i $$0.429079\pi$$
$$500$$ 0 0
$$501$$ 15.2734 0.682365
$$502$$ 0 0
$$503$$ −39.0415 −1.74077 −0.870387 0.492369i $$-0.836131\pi$$
−0.870387 + 0.492369i $$0.836131\pi$$
$$504$$ 0 0
$$505$$ 10.5971 0.471567
$$506$$ 0 0
$$507$$ −28.3913 −1.26090
$$508$$ 0 0
$$509$$ −24.2202 −1.07354 −0.536770 0.843729i $$-0.680356\pi$$
−0.536770 + 0.843729i $$0.680356\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ −1.01939 −0.0450072
$$514$$ 0 0
$$515$$ 20.2443 0.892069
$$516$$ 0 0
$$517$$ −2.42378 −0.106598
$$518$$ 0 0
$$519$$ 4.00710 0.175892
$$520$$ 0 0
$$521$$ 22.1638 0.971012 0.485506 0.874233i $$-0.338635\pi$$
0.485506 + 0.874233i $$0.338635\pi$$
$$522$$ 0 0
$$523$$ −35.8053 −1.56566 −0.782829 0.622237i $$-0.786224\pi$$
−0.782829 + 0.622237i $$0.786224\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 11.9109 0.518845
$$528$$ 0 0
$$529$$ −4.52227 −0.196620
$$530$$ 0 0
$$531$$ −6.46103 −0.280385
$$532$$ 0 0
$$533$$ −53.5662 −2.32021
$$534$$ 0 0
$$535$$ −35.9512 −1.55430
$$536$$ 0 0
$$537$$ 20.7496 0.895409
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −12.7941 −0.550063 −0.275031 0.961435i $$-0.588688\pi$$
−0.275031 + 0.961435i $$0.588688\pi$$
$$542$$ 0 0
$$543$$ −21.7246 −0.932292
$$544$$ 0 0
$$545$$ 6.61538 0.283372
$$546$$ 0 0
$$547$$ 28.9675 1.23856 0.619280 0.785170i $$-0.287424\pi$$
0.619280 + 0.785170i $$0.287424\pi$$
$$548$$ 0 0
$$549$$ 5.83646 0.249094
$$550$$ 0 0
$$551$$ 0.430434 0.0183371
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 12.0776 0.512664
$$556$$ 0 0
$$557$$ −9.75265 −0.413233 −0.206617 0.978422i $$-0.566245\pi$$
−0.206617 + 0.978422i $$0.566245\pi$$
$$558$$ 0 0
$$559$$ 45.6689 1.93159
$$560$$ 0 0
$$561$$ −0.333100 −0.0140635
$$562$$ 0 0
$$563$$ 10.6192 0.447547 0.223774 0.974641i $$-0.428162\pi$$
0.223774 + 0.974641i $$0.428162\pi$$
$$564$$ 0 0
$$565$$ −19.8851 −0.836571
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 37.1660 1.55808 0.779040 0.626974i $$-0.215707\pi$$
0.779040 + 0.626974i $$0.215707\pi$$
$$570$$ 0 0
$$571$$ −23.6956 −0.991632 −0.495816 0.868428i $$-0.665131\pi$$
−0.495816 + 0.868428i $$0.665131\pi$$
$$572$$ 0 0
$$573$$ −20.6720 −0.863585
$$574$$ 0 0
$$575$$ 1.89841 0.0791692
$$576$$ 0 0
$$577$$ −9.84465 −0.409838 −0.204919 0.978779i $$-0.565693\pi$$
−0.204919 + 0.978779i $$0.565693\pi$$
$$578$$ 0 0
$$579$$ −15.5998 −0.648307
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −1.02758 −0.0425580
$$584$$ 0 0
$$585$$ −13.7360 −0.567912
$$586$$ 0 0
$$587$$ −30.4610 −1.25726 −0.628631 0.777704i $$-0.716384\pi$$
−0.628631 + 0.777704i $$0.716384\pi$$
$$588$$ 0 0
$$589$$ −10.8833 −0.448438
$$590$$ 0 0
$$591$$ 25.6386 1.05463
$$592$$ 0 0
$$593$$ 10.0720 0.413607 0.206804 0.978382i $$-0.433694\pi$$
0.206804 + 0.978382i $$0.433694\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −2.88327 −0.118005
$$598$$ 0 0
$$599$$ −31.7220 −1.29612 −0.648062 0.761587i $$-0.724420\pi$$
−0.648062 + 0.761587i $$0.724420\pi$$
$$600$$ 0 0
$$601$$ −29.7533 −1.21366 −0.606832 0.794830i $$-0.707560\pi$$
−0.606832 + 0.794830i $$0.707560\pi$$
$$602$$ 0 0
$$603$$ −5.05971 −0.206047
$$604$$ 0 0
$$605$$ 23.2950 0.947077
$$606$$ 0 0
$$607$$ −17.6247 −0.715366 −0.357683 0.933843i $$-0.616433\pi$$
−0.357683 + 0.933843i $$0.616433\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 52.2273 2.11289
$$612$$ 0 0
$$613$$ −12.7068 −0.513224 −0.256612 0.966514i $$-0.582606\pi$$
−0.256612 + 0.966514i $$0.582606\pi$$
$$614$$ 0 0
$$615$$ −17.7763 −0.716808
$$616$$ 0 0
$$617$$ −40.7386 −1.64008 −0.820038 0.572309i $$-0.806048\pi$$
−0.820038 + 0.572309i $$0.806048\pi$$
$$618$$ 0 0
$$619$$ 46.9407 1.88671 0.943354 0.331788i $$-0.107652\pi$$
0.943354 + 0.331788i $$0.107652\pi$$
$$620$$ 0 0
$$621$$ 4.29857 0.172496
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −22.5968 −0.903871
$$626$$ 0 0
$$627$$ 0.304363 0.0121551
$$628$$ 0 0
$$629$$ −6.31102 −0.251637
$$630$$ 0 0
$$631$$ 27.4804 1.09398 0.546989 0.837140i $$-0.315774\pi$$
0.546989 + 0.837140i $$0.315774\pi$$
$$632$$ 0 0
$$633$$ 24.5401 0.975383
$$634$$ 0 0
$$635$$ −36.5886 −1.45198
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 1.35828 0.0537328
$$640$$ 0 0
$$641$$ −48.6580 −1.92188 −0.960938 0.276764i $$-0.910738\pi$$
−0.960938 + 0.276764i $$0.910738\pi$$
$$642$$ 0 0
$$643$$ −17.6856 −0.697452 −0.348726 0.937225i $$-0.613386\pi$$
−0.348726 + 0.937225i $$0.613386\pi$$
$$644$$ 0 0
$$645$$ 15.1555 0.596748
$$646$$ 0 0
$$647$$ 4.04032 0.158841 0.0794206 0.996841i $$-0.474693\pi$$
0.0794206 + 0.996841i $$0.474693\pi$$
$$648$$ 0 0
$$649$$ 1.92909 0.0757233
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −30.9690 −1.21191 −0.605956 0.795498i $$-0.707209\pi$$
−0.605956 + 0.795498i $$0.707209\pi$$
$$654$$ 0 0
$$655$$ 34.4983 1.34796
$$656$$ 0 0
$$657$$ 2.85585 0.111417
$$658$$ 0 0
$$659$$ 4.81815 0.187689 0.0938443 0.995587i $$-0.470084\pi$$
0.0938443 + 0.995587i $$0.470084\pi$$
$$660$$ 0 0
$$661$$ −4.38324 −0.170488 −0.0852442 0.996360i $$-0.527167\pi$$
−0.0852442 + 0.996360i $$0.527167\pi$$
$$662$$ 0 0
$$663$$ 7.17759 0.278755
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −1.81505 −0.0702792
$$668$$ 0 0
$$669$$ −20.5080 −0.792885
$$670$$ 0 0
$$671$$ −1.74261 −0.0672727
$$672$$ 0 0
$$673$$ −8.67471 −0.334386 −0.167193 0.985924i $$-0.553470\pi$$
−0.167193 + 0.985924i $$0.553470\pi$$
$$674$$ 0 0
$$675$$ 0.441637 0.0169986
$$676$$ 0 0
$$677$$ 1.04103 0.0400099 0.0200049 0.999800i $$-0.493632\pi$$
0.0200049 + 0.999800i $$0.493632\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 13.6972 0.524877
$$682$$ 0 0
$$683$$ 35.7220 1.36686 0.683432 0.730014i $$-0.260487\pi$$
0.683432 + 0.730014i $$0.260487\pi$$
$$684$$ 0 0
$$685$$ −46.4898 −1.77628
$$686$$ 0 0
$$687$$ −16.4175 −0.626368
$$688$$ 0 0
$$689$$ 22.1421 0.843548
$$690$$ 0 0
$$691$$ 16.9221 0.643745 0.321873 0.946783i $$-0.395688\pi$$
0.321873 + 0.946783i $$0.395688\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 32.6953 1.24020
$$696$$ 0 0
$$697$$ 9.28882 0.351839
$$698$$ 0 0
$$699$$ 10.8915 0.411953
$$700$$ 0 0
$$701$$ 22.2052 0.838678 0.419339 0.907830i $$-0.362262\pi$$
0.419339 + 0.907830i $$0.362262\pi$$
$$702$$ 0 0
$$703$$ 5.76655 0.217490
$$704$$ 0 0
$$705$$ 17.3319 0.652759
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 7.35435 0.276198 0.138099 0.990418i $$-0.455901\pi$$
0.138099 + 0.990418i $$0.455901\pi$$
$$710$$ 0 0
$$711$$ 7.69564 0.288609
$$712$$ 0 0
$$713$$ 45.8926 1.71869
$$714$$ 0 0
$$715$$ 4.10118 0.153376
$$716$$ 0 0
$$717$$ 4.29857 0.160533
$$718$$ 0 0
$$719$$ −27.5495 −1.02742 −0.513711 0.857963i $$-0.671730\pi$$
−0.513711 + 0.857963i $$0.671730\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 7.12808 0.265096
$$724$$ 0 0
$$725$$ −0.186480 −0.00692568
$$726$$ 0 0
$$727$$ −14.9321 −0.553801 −0.276901 0.960899i $$-0.589307\pi$$
−0.276901 + 0.960899i $$0.589307\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −7.91937 −0.292908
$$732$$ 0 0
$$733$$ 42.4684 1.56860 0.784302 0.620379i $$-0.213021\pi$$
0.784302 + 0.620379i $$0.213021\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 1.51069 0.0556470
$$738$$ 0 0
$$739$$ −7.83329 −0.288152 −0.144076 0.989567i $$-0.546021\pi$$
−0.144076 + 0.989567i $$0.546021\pi$$
$$740$$ 0 0
$$741$$ −6.55836 −0.240927
$$742$$ 0 0
$$743$$ 30.4064 1.11550 0.557752 0.830008i $$-0.311664\pi$$
0.557752 + 0.830008i $$0.311664\pi$$
$$744$$ 0 0
$$745$$ −8.26700 −0.302879
$$746$$ 0 0
$$747$$ −6.03878 −0.220948
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 28.2843 1.03211 0.516054 0.856556i $$-0.327400\pi$$
0.516054 + 0.856556i $$0.327400\pi$$
$$752$$ 0 0
$$753$$ −18.9690 −0.691270
$$754$$ 0 0
$$755$$ 10.1528 0.369499
$$756$$ 0 0
$$757$$ 19.5289 0.709791 0.354895 0.934906i $$-0.384516\pi$$
0.354895 + 0.934906i $$0.384516\pi$$
$$758$$ 0 0
$$759$$ −1.28344 −0.0465858
$$760$$ 0 0
$$761$$ −34.0746 −1.23520 −0.617602 0.786491i $$-0.711896\pi$$
−0.617602 + 0.786491i $$0.711896\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 2.38193 0.0861188
$$766$$ 0 0
$$767$$ −41.5677 −1.50092
$$768$$ 0 0
$$769$$ 28.3523 1.02241 0.511206 0.859458i $$-0.329199\pi$$
0.511206 + 0.859458i $$0.329199\pi$$
$$770$$ 0 0
$$771$$ 20.4842 0.737720
$$772$$ 0 0
$$773$$ 1.96615 0.0707175 0.0353588 0.999375i $$-0.488743\pi$$
0.0353588 + 0.999375i $$0.488743\pi$$
$$774$$ 0 0
$$775$$ 4.71503 0.169369
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −8.48745 −0.304094
$$780$$ 0 0
$$781$$ −0.405546 −0.0145116
$$782$$ 0 0
$$783$$ −0.422246 −0.0150898
$$784$$ 0 0
$$785$$ −24.6557 −0.880000
$$786$$ 0 0
$$787$$ 2.39165 0.0852531 0.0426266 0.999091i $$-0.486427\pi$$
0.0426266 + 0.999091i $$0.486427\pi$$
$$788$$ 0 0
$$789$$ 23.5553 0.838590
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 37.5495 1.33342
$$794$$ 0 0
$$795$$ 7.34801 0.260607
$$796$$ 0 0
$$797$$ −30.2674 −1.07213 −0.536064 0.844177i $$-0.680089\pi$$
−0.536064 + 0.844177i $$0.680089\pi$$
$$798$$ 0 0
$$799$$ −9.05664 −0.320401
$$800$$ 0 0
$$801$$ −15.3696 −0.543060
$$802$$ 0 0
$$803$$ −0.852680 −0.0300904
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −12.9086 −0.454405
$$808$$ 0 0
$$809$$ 28.7692 1.01147 0.505736 0.862688i $$-0.331221\pi$$
0.505736 + 0.862688i $$0.331221\pi$$
$$810$$ 0 0
$$811$$ −51.6270 −1.81287 −0.906435 0.422345i $$-0.861207\pi$$
−0.906435 + 0.422345i $$0.861207\pi$$
$$812$$ 0 0
$$813$$ 19.8705 0.696890
$$814$$ 0 0
$$815$$ −25.4301 −0.890776
$$816$$ 0 0
$$817$$ 7.23614 0.253161
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −44.1885 −1.54219 −0.771094 0.636721i $$-0.780290\pi$$
−0.771094 + 0.636721i $$0.780290\pi$$
$$822$$ 0 0
$$823$$ 33.8151 1.17872 0.589359 0.807871i $$-0.299380\pi$$
0.589359 + 0.807871i $$0.299380\pi$$
$$824$$ 0 0
$$825$$ −0.131861 −0.00459081
$$826$$ 0 0
$$827$$ 48.0748 1.67173 0.835863 0.548938i $$-0.184968\pi$$
0.835863 + 0.548938i $$0.184968\pi$$
$$828$$ 0 0
$$829$$ −9.26203 −0.321684 −0.160842 0.986980i $$-0.551421\pi$$
−0.160842 + 0.986980i $$0.551421\pi$$
$$830$$ 0 0
$$831$$ 25.9884 0.901528
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 32.6092 1.12849
$$836$$ 0 0
$$837$$ 10.6762 0.369025
$$838$$ 0 0
$$839$$ −18.1567 −0.626838 −0.313419 0.949615i $$-0.601474\pi$$
−0.313419 + 0.949615i $$0.601474\pi$$
$$840$$ 0 0
$$841$$ −28.8217 −0.993852
$$842$$ 0 0
$$843$$ 19.6553 0.676965
$$844$$ 0 0
$$845$$ −60.6163 −2.08526
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 19.0582 0.654075
$$850$$ 0 0
$$851$$ −24.3164 −0.833555
$$852$$ 0 0
$$853$$ −40.7652 −1.39577 −0.697886 0.716208i $$-0.745876\pi$$
−0.697886 + 0.716208i $$0.745876\pi$$
$$854$$ 0 0
$$855$$ −2.17643 −0.0744325
$$856$$ 0 0
$$857$$ −12.0559 −0.411823 −0.205911 0.978571i $$-0.566016\pi$$
−0.205911 + 0.978571i $$0.566016\pi$$
$$858$$ 0 0
$$859$$ 4.25247 0.145092 0.0725461 0.997365i $$-0.476888\pi$$
0.0725461 + 0.997365i $$0.476888\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −11.3280 −0.385610 −0.192805 0.981237i $$-0.561758\pi$$
−0.192805 + 0.981237i $$0.561758\pi$$
$$864$$ 0 0
$$865$$ 8.55529 0.290889
$$866$$ 0 0
$$867$$ 15.7553 0.535080
$$868$$ 0 0
$$869$$ −2.29771 −0.0779444
$$870$$ 0 0
$$871$$ −32.5522 −1.10299
$$872$$ 0 0
$$873$$ 0.512705 0.0173524
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −31.6871 −1.07000 −0.534999 0.844853i $$-0.679688\pi$$
−0.534999 + 0.844853i $$0.679688\pi$$
$$878$$ 0 0
$$879$$ −7.84890 −0.264737
$$880$$ 0 0
$$881$$ 24.2690 0.817643 0.408821 0.912614i $$-0.365940\pi$$
0.408821 + 0.912614i $$0.365940\pi$$
$$882$$ 0 0
$$883$$ −47.1439 −1.58652 −0.793260 0.608883i $$-0.791618\pi$$
−0.793260 + 0.608883i $$0.791618\pi$$
$$884$$ 0 0
$$885$$ −13.7945 −0.463697
$$886$$ 0 0
$$887$$ 14.6647 0.492391 0.246196 0.969220i $$-0.420819\pi$$
0.246196 + 0.969220i $$0.420819\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −0.298573 −0.0100026
$$892$$ 0 0
$$893$$ 8.27530 0.276922
$$894$$ 0 0
$$895$$ 44.3010 1.48082
$$896$$ 0 0
$$897$$ 27.6553 0.923384
$$898$$ 0 0
$$899$$ −4.50800 −0.150350
$$900$$ 0 0
$$901$$ −3.83963 −0.127917
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −46.3827 −1.54181
$$906$$ 0 0
$$907$$ 3.99149 0.132535 0.0662676 0.997802i $$-0.478891\pi$$
0.0662676 + 0.997802i $$0.478891\pi$$
$$908$$ 0 0
$$909$$ −4.96346 −0.164628
$$910$$ 0 0
$$911$$ −18.9260 −0.627046 −0.313523 0.949581i $$-0.601509\pi$$
−0.313523 + 0.949581i $$0.601509\pi$$
$$912$$ 0 0
$$913$$ 1.80302 0.0596711
$$914$$ 0 0
$$915$$ 12.4610 0.411949
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −46.4898 −1.53356 −0.766778 0.641912i $$-0.778141\pi$$
−0.766778 + 0.641912i $$0.778141\pi$$
$$920$$ 0 0
$$921$$ 26.5301 0.874196
$$922$$ 0 0
$$923$$ 8.73865 0.287636
$$924$$ 0 0
$$925$$ −2.49828 −0.0821429
$$926$$ 0 0
$$927$$ −9.48195 −0.311428
$$928$$ 0 0
$$929$$ −28.7221 −0.942343 −0.471171 0.882042i $$-0.656169\pi$$
−0.471171 + 0.882042i $$0.656169\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ 0.726608 0.0237881
$$934$$ 0 0
$$935$$ −0.711179 −0.0232581
$$936$$ 0 0
$$937$$ 21.2560 0.694404 0.347202 0.937790i $$-0.387132\pi$$
0.347202 + 0.937790i $$0.387132\pi$$
$$938$$ 0 0
$$939$$ 15.1259 0.493616
$$940$$ 0 0
$$941$$ 19.4100 0.632747 0.316373 0.948635i $$-0.397535\pi$$
0.316373 + 0.948635i $$0.397535\pi$$
$$942$$ 0 0
$$943$$ 35.7899 1.16548
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 27.6123 0.897279 0.448639 0.893713i $$-0.351909\pi$$
0.448639 + 0.893713i $$0.351909\pi$$
$$948$$ 0 0
$$949$$ 18.3734 0.596426
$$950$$ 0 0
$$951$$ −19.9496 −0.646911
$$952$$ 0 0
$$953$$ 14.5080 0.469960 0.234980 0.972000i $$-0.424497\pi$$
0.234980 + 0.972000i $$0.424497\pi$$
$$954$$ 0 0
$$955$$ −44.1354 −1.42819
$$956$$ 0 0
$$957$$ 0.126071 0.00407530
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 82.9822 2.67685
$$962$$ 0 0
$$963$$ 16.8387 0.542620
$$964$$ 0 0
$$965$$ −33.3062 −1.07216
$$966$$ 0 0
$$967$$ 56.8449 1.82801 0.914005 0.405703i $$-0.132973\pi$$
0.914005 + 0.405703i $$0.132973\pi$$
$$968$$ 0 0
$$969$$ 1.13727 0.0365345
$$970$$ 0 0
$$971$$ 10.1970 0.327237 0.163618 0.986524i $$-0.447683\pi$$
0.163618 + 0.986524i $$0.447683\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 2.84132 0.0909951
$$976$$ 0 0
$$977$$ −8.53628 −0.273100 −0.136550 0.990633i $$-0.543601\pi$$
−0.136550 + 0.990633i $$0.543601\pi$$
$$978$$ 0 0
$$979$$ 4.58896 0.146664
$$980$$ 0 0
$$981$$ −3.09849 −0.0989272
$$982$$ 0 0
$$983$$ −38.1970 −1.21829 −0.609147 0.793057i $$-0.708488\pi$$
−0.609147 + 0.793057i $$0.708488\pi$$
$$984$$ 0 0
$$985$$ 54.7393 1.74414
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −30.5134 −0.970269
$$990$$ 0 0
$$991$$ −45.8810 −1.45746 −0.728730 0.684802i $$-0.759889\pi$$
−0.728730 + 0.684802i $$0.759889\pi$$
$$992$$ 0 0
$$993$$ 17.1943 0.545644
$$994$$ 0 0
$$995$$ −6.15588 −0.195155
$$996$$ 0 0
$$997$$ −35.0749 −1.11083 −0.555417 0.831572i $$-0.687441\pi$$
−0.555417 + 0.831572i $$0.687441\pi$$
$$998$$ 0 0
$$999$$ −5.65685 −0.178975
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 9408.2.a.el.1.1 4
4.3 odd 2 9408.2.a.en.1.1 4
7.6 odd 2 9408.2.a.em.1.4 4
8.3 odd 2 4704.2.a.bw.1.4 4
8.5 even 2 4704.2.a.by.1.4 yes 4
28.27 even 2 9408.2.a.ek.1.4 4
56.13 odd 2 4704.2.a.bx.1.1 yes 4
56.27 even 2 4704.2.a.bz.1.1 yes 4

By twisted newform
Twist Min Dim Char Parity Ord Type
4704.2.a.bw.1.4 4 8.3 odd 2
4704.2.a.bx.1.1 yes 4 56.13 odd 2
4704.2.a.by.1.4 yes 4 8.5 even 2
4704.2.a.bz.1.1 yes 4 56.27 even 2
9408.2.a.ek.1.4 4 28.27 even 2
9408.2.a.el.1.1 4 1.1 even 1 trivial
9408.2.a.em.1.4 4 7.6 odd 2
9408.2.a.en.1.1 4 4.3 odd 2