# Properties

 Label 1323.2.a.bb.1.1 Level $1323$ Weight $2$ Character 1323.1 Self dual yes Analytic conductor $10.564$ Analytic rank $1$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1323 = 3^{3} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1323.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$10.5642081874$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{5})$$ Defining polynomial: $$x^{4} - 6 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-2.28825$$ of defining polynomial Character $$\chi$$ $$=$$ 1323.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-2.28825 q^{2} +3.23607 q^{4} -4.23607 q^{5} -2.82843 q^{8} +O(q^{10})$$ $$q-2.28825 q^{2} +3.23607 q^{4} -4.23607 q^{5} -2.82843 q^{8} +9.69316 q^{10} +3.70246 q^{11} -2.28825 q^{13} -5.00000 q^{17} +5.45052 q^{19} -13.7082 q^{20} -8.47214 q^{22} -0.333851 q^{23} +12.9443 q^{25} +5.23607 q^{26} +7.19859 q^{29} -3.36861 q^{31} +5.65685 q^{32} +11.4412 q^{34} -4.70820 q^{37} -12.4721 q^{38} +11.9814 q^{40} +4.70820 q^{41} +2.23607 q^{43} +11.9814 q^{44} +0.763932 q^{46} +1.47214 q^{47} -29.6197 q^{50} -7.40492 q^{52} -3.16228 q^{53} -15.6839 q^{55} -16.4721 q^{58} +3.94427 q^{59} +3.70246 q^{61} +7.70820 q^{62} -12.9443 q^{64} +9.69316 q^{65} -11.2361 q^{67} -16.1803 q^{68} -13.0618 q^{71} +0.746512 q^{73} +10.7735 q^{74} +17.6383 q^{76} -11.4721 q^{79} -10.7735 q^{82} +0.236068 q^{83} +21.1803 q^{85} -5.11667 q^{86} -10.4721 q^{88} +1.70820 q^{89} -1.08036 q^{92} -3.36861 q^{94} -23.0888 q^{95} +1.95440 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{4} - 8q^{5} + O(q^{10})$$ $$4q + 4q^{4} - 8q^{5} - 20q^{17} - 28q^{20} - 16q^{22} + 16q^{25} + 12q^{26} + 8q^{37} - 32q^{38} - 8q^{41} + 12q^{46} - 12q^{47} - 48q^{58} - 20q^{59} + 4q^{62} - 16q^{64} - 36q^{67} - 20q^{68} - 28q^{79} - 8q^{83} + 40q^{85} - 24q^{88} - 20q^{89} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.28825 −1.61803 −0.809017 0.587785i $$-0.800000\pi$$
−0.809017 + 0.587785i $$0.800000\pi$$
$$3$$ 0 0
$$4$$ 3.23607 1.61803
$$5$$ −4.23607 −1.89443 −0.947214 0.320603i $$-0.896114\pi$$
−0.947214 + 0.320603i $$0.896114\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ −2.82843 −1.00000
$$9$$ 0 0
$$10$$ 9.69316 3.06525
$$11$$ 3.70246 1.11633 0.558167 0.829729i $$-0.311505\pi$$
0.558167 + 0.829729i $$0.311505\pi$$
$$12$$ 0 0
$$13$$ −2.28825 −0.634645 −0.317323 0.948318i $$-0.602784\pi$$
−0.317323 + 0.948318i $$0.602784\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −5.00000 −1.21268 −0.606339 0.795206i $$-0.707363\pi$$
−0.606339 + 0.795206i $$0.707363\pi$$
$$18$$ 0 0
$$19$$ 5.45052 1.25044 0.625218 0.780450i $$-0.285010\pi$$
0.625218 + 0.780450i $$0.285010\pi$$
$$20$$ −13.7082 −3.06525
$$21$$ 0 0
$$22$$ −8.47214 −1.80627
$$23$$ −0.333851 −0.0696126 −0.0348063 0.999394i $$-0.511081\pi$$
−0.0348063 + 0.999394i $$0.511081\pi$$
$$24$$ 0 0
$$25$$ 12.9443 2.58885
$$26$$ 5.23607 1.02688
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 7.19859 1.33674 0.668372 0.743827i $$-0.266991\pi$$
0.668372 + 0.743827i $$0.266991\pi$$
$$30$$ 0 0
$$31$$ −3.36861 −0.605020 −0.302510 0.953146i $$-0.597825\pi$$
−0.302510 + 0.953146i $$0.597825\pi$$
$$32$$ 5.65685 1.00000
$$33$$ 0 0
$$34$$ 11.4412 1.96215
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −4.70820 −0.774024 −0.387012 0.922075i $$-0.626493\pi$$
−0.387012 + 0.922075i $$0.626493\pi$$
$$38$$ −12.4721 −2.02325
$$39$$ 0 0
$$40$$ 11.9814 1.89443
$$41$$ 4.70820 0.735298 0.367649 0.929965i $$-0.380163\pi$$
0.367649 + 0.929965i $$0.380163\pi$$
$$42$$ 0 0
$$43$$ 2.23607 0.340997 0.170499 0.985358i $$-0.445462\pi$$
0.170499 + 0.985358i $$0.445462\pi$$
$$44$$ 11.9814 1.80627
$$45$$ 0 0
$$46$$ 0.763932 0.112636
$$47$$ 1.47214 0.214733 0.107367 0.994220i $$-0.465758\pi$$
0.107367 + 0.994220i $$0.465758\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ −29.6197 −4.18885
$$51$$ 0 0
$$52$$ −7.40492 −1.02688
$$53$$ −3.16228 −0.434372 −0.217186 0.976130i $$-0.569688\pi$$
−0.217186 + 0.976130i $$0.569688\pi$$
$$54$$ 0 0
$$55$$ −15.6839 −2.11481
$$56$$ 0 0
$$57$$ 0 0
$$58$$ −16.4721 −2.16290
$$59$$ 3.94427 0.513500 0.256750 0.966478i $$-0.417348\pi$$
0.256750 + 0.966478i $$0.417348\pi$$
$$60$$ 0 0
$$61$$ 3.70246 0.474051 0.237026 0.971503i $$-0.423827\pi$$
0.237026 + 0.971503i $$0.423827\pi$$
$$62$$ 7.70820 0.978943
$$63$$ 0 0
$$64$$ −12.9443 −1.61803
$$65$$ 9.69316 1.20229
$$66$$ 0 0
$$67$$ −11.2361 −1.37270 −0.686352 0.727269i $$-0.740789\pi$$
−0.686352 + 0.727269i $$0.740789\pi$$
$$68$$ −16.1803 −1.96215
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −13.0618 −1.55015 −0.775074 0.631871i $$-0.782287\pi$$
−0.775074 + 0.631871i $$0.782287\pi$$
$$72$$ 0 0
$$73$$ 0.746512 0.0873727 0.0436863 0.999045i $$-0.486090\pi$$
0.0436863 + 0.999045i $$0.486090\pi$$
$$74$$ 10.7735 1.25240
$$75$$ 0 0
$$76$$ 17.6383 2.02325
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −11.4721 −1.29072 −0.645358 0.763880i $$-0.723292\pi$$
−0.645358 + 0.763880i $$0.723292\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ −10.7735 −1.18974
$$83$$ 0.236068 0.0259118 0.0129559 0.999916i $$-0.495876\pi$$
0.0129559 + 0.999916i $$0.495876\pi$$
$$84$$ 0 0
$$85$$ 21.1803 2.29733
$$86$$ −5.11667 −0.551745
$$87$$ 0 0
$$88$$ −10.4721 −1.11633
$$89$$ 1.70820 0.181069 0.0905346 0.995893i $$-0.471142\pi$$
0.0905346 + 0.995893i $$0.471142\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −1.08036 −0.112636
$$93$$ 0 0
$$94$$ −3.36861 −0.347445
$$95$$ −23.0888 −2.36886
$$96$$ 0 0
$$97$$ 1.95440 0.198439 0.0992194 0.995066i $$-0.468365\pi$$
0.0992194 + 0.995066i $$0.468365\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 41.8885 4.18885
$$101$$ −9.23607 −0.919023 −0.459512 0.888172i $$-0.651976\pi$$
−0.459512 + 0.888172i $$0.651976\pi$$
$$102$$ 0 0
$$103$$ −8.40647 −0.828314 −0.414157 0.910205i $$-0.635924\pi$$
−0.414157 + 0.910205i $$0.635924\pi$$
$$104$$ 6.47214 0.634645
$$105$$ 0 0
$$106$$ 7.23607 0.702829
$$107$$ 9.89949 0.957020 0.478510 0.878082i $$-0.341177\pi$$
0.478510 + 0.878082i $$0.341177\pi$$
$$108$$ 0 0
$$109$$ 9.76393 0.935215 0.467608 0.883936i $$-0.345116\pi$$
0.467608 + 0.883936i $$0.345116\pi$$
$$110$$ 35.8885 3.42184
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −1.20788 −0.113628 −0.0568140 0.998385i $$-0.518094\pi$$
−0.0568140 + 0.998385i $$0.518094\pi$$
$$114$$ 0 0
$$115$$ 1.41421 0.131876
$$116$$ 23.2951 2.16290
$$117$$ 0 0
$$118$$ −9.02546 −0.830861
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 2.70820 0.246200
$$122$$ −8.47214 −0.767031
$$123$$ 0 0
$$124$$ −10.9010 −0.978943
$$125$$ −33.6525 −3.00997
$$126$$ 0 0
$$127$$ 5.47214 0.485574 0.242787 0.970080i $$-0.421938\pi$$
0.242787 + 0.970080i $$0.421938\pi$$
$$128$$ 18.3060 1.61803
$$129$$ 0 0
$$130$$ −22.1803 −1.94534
$$131$$ −10.9443 −0.956205 −0.478103 0.878304i $$-0.658675\pi$$
−0.478103 + 0.878304i $$0.658675\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 25.7109 2.22108
$$135$$ 0 0
$$136$$ 14.1421 1.21268
$$137$$ −4.44897 −0.380101 −0.190051 0.981774i $$-0.560865\pi$$
−0.190051 + 0.981774i $$0.560865\pi$$
$$138$$ 0 0
$$139$$ −19.1800 −1.62683 −0.813413 0.581687i $$-0.802393\pi$$
−0.813413 + 0.581687i $$0.802393\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 29.8885 2.50819
$$143$$ −8.47214 −0.708476
$$144$$ 0 0
$$145$$ −30.4937 −2.53236
$$146$$ −1.70820 −0.141372
$$147$$ 0 0
$$148$$ −15.2361 −1.25240
$$149$$ −22.0084 −1.80300 −0.901500 0.432779i $$-0.857533\pi$$
−0.901500 + 0.432779i $$0.857533\pi$$
$$150$$ 0 0
$$151$$ −16.4164 −1.33595 −0.667974 0.744184i $$-0.732838\pi$$
−0.667974 + 0.744184i $$0.732838\pi$$
$$152$$ −15.4164 −1.25044
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 14.2697 1.14617
$$156$$ 0 0
$$157$$ −20.6730 −1.64989 −0.824943 0.565215i $$-0.808793\pi$$
−0.824943 + 0.565215i $$0.808793\pi$$
$$158$$ 26.2511 2.08842
$$159$$ 0 0
$$160$$ −23.9628 −1.89443
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −5.18034 −0.405756 −0.202878 0.979204i $$-0.565029\pi$$
−0.202878 + 0.979204i $$0.565029\pi$$
$$164$$ 15.2361 1.18974
$$165$$ 0 0
$$166$$ −0.540182 −0.0419262
$$167$$ 3.76393 0.291262 0.145631 0.989339i $$-0.453479\pi$$
0.145631 + 0.989339i $$0.453479\pi$$
$$168$$ 0 0
$$169$$ −7.76393 −0.597226
$$170$$ −48.4658 −3.71716
$$171$$ 0 0
$$172$$ 7.23607 0.551745
$$173$$ −18.4721 −1.40441 −0.702205 0.711975i $$-0.747801\pi$$
−0.702205 + 0.711975i $$0.747801\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 0 0
$$178$$ −3.90879 −0.292976
$$179$$ −14.6823 −1.09741 −0.548704 0.836017i $$-0.684879\pi$$
−0.548704 + 0.836017i $$0.684879\pi$$
$$180$$ 0 0
$$181$$ 22.8825 1.70084 0.850420 0.526105i $$-0.176348\pi$$
0.850420 + 0.526105i $$0.176348\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0.944272 0.0696126
$$185$$ 19.9443 1.46633
$$186$$ 0 0
$$187$$ −18.5123 −1.35375
$$188$$ 4.76393 0.347445
$$189$$ 0 0
$$190$$ 52.8328 3.83290
$$191$$ 4.24264 0.306987 0.153493 0.988150i $$-0.450948\pi$$
0.153493 + 0.988150i $$0.450948\pi$$
$$192$$ 0 0
$$193$$ 17.1803 1.23667 0.618334 0.785915i $$-0.287808\pi$$
0.618334 + 0.785915i $$0.287808\pi$$
$$194$$ −4.47214 −0.321081
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −0.461370 −0.0328713 −0.0164356 0.999865i $$-0.505232\pi$$
−0.0164356 + 0.999865i $$0.505232\pi$$
$$198$$ 0 0
$$199$$ 13.4744 0.955177 0.477589 0.878584i $$-0.341511\pi$$
0.477589 + 0.878584i $$0.341511\pi$$
$$200$$ −36.6119 −2.58885
$$201$$ 0 0
$$202$$ 21.1344 1.48701
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −19.9443 −1.39297
$$206$$ 19.2361 1.34024
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 20.1803 1.39590
$$210$$ 0 0
$$211$$ −17.4164 −1.19899 −0.599497 0.800377i $$-0.704633\pi$$
−0.599497 + 0.800377i $$0.704633\pi$$
$$212$$ −10.2333 −0.702829
$$213$$ 0 0
$$214$$ −22.6525 −1.54849
$$215$$ −9.47214 −0.645994
$$216$$ 0 0
$$217$$ 0 0
$$218$$ −22.3423 −1.51321
$$219$$ 0 0
$$220$$ −50.7541 −3.42184
$$221$$ 11.4412 0.769620
$$222$$ 0 0
$$223$$ −19.5138 −1.30674 −0.653372 0.757037i $$-0.726646\pi$$
−0.653372 + 0.757037i $$0.726646\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 2.76393 0.183854
$$227$$ −12.1803 −0.808438 −0.404219 0.914662i $$-0.632457\pi$$
−0.404219 + 0.914662i $$0.632457\pi$$
$$228$$ 0 0
$$229$$ 10.6460 0.703508 0.351754 0.936092i $$-0.385585\pi$$
0.351754 + 0.936092i $$0.385585\pi$$
$$230$$ −3.23607 −0.213380
$$231$$ 0 0
$$232$$ −20.3607 −1.33674
$$233$$ 0.127520 0.00835408 0.00417704 0.999991i $$-0.498670\pi$$
0.00417704 + 0.999991i $$0.498670\pi$$
$$234$$ 0 0
$$235$$ −6.23607 −0.406796
$$236$$ 12.7639 0.830861
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 25.5834 1.65485 0.827425 0.561576i $$-0.189805\pi$$
0.827425 + 0.561576i $$0.189805\pi$$
$$240$$ 0 0
$$241$$ 6.86474 0.442197 0.221098 0.975252i $$-0.429036\pi$$
0.221098 + 0.975252i $$0.429036\pi$$
$$242$$ −6.19704 −0.398361
$$243$$ 0 0
$$244$$ 11.9814 0.767031
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −12.4721 −0.793583
$$248$$ 9.52786 0.605020
$$249$$ 0 0
$$250$$ 77.0051 4.87023
$$251$$ 9.29180 0.586493 0.293246 0.956037i $$-0.405264\pi$$
0.293246 + 0.956037i $$0.405264\pi$$
$$252$$ 0 0
$$253$$ −1.23607 −0.0777109
$$254$$ −12.5216 −0.785675
$$255$$ 0 0
$$256$$ −16.0000 −1.00000
$$257$$ 5.41641 0.337866 0.168933 0.985628i $$-0.445968\pi$$
0.168933 + 0.985628i $$0.445968\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 31.3677 1.94534
$$261$$ 0 0
$$262$$ 25.0432 1.54717
$$263$$ 20.1815 1.24445 0.622224 0.782839i $$-0.286229\pi$$
0.622224 + 0.782839i $$0.286229\pi$$
$$264$$ 0 0
$$265$$ 13.3956 0.822887
$$266$$ 0 0
$$267$$ 0 0
$$268$$ −36.3607 −2.22108
$$269$$ −7.00000 −0.426798 −0.213399 0.976965i $$-0.568453\pi$$
−0.213399 + 0.976965i $$0.568453\pi$$
$$270$$ 0 0
$$271$$ 5.57804 0.338842 0.169421 0.985544i $$-0.445810\pi$$
0.169421 + 0.985544i $$0.445810\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 10.1803 0.615017
$$275$$ 47.9256 2.89002
$$276$$ 0 0
$$277$$ −13.7639 −0.826995 −0.413497 0.910505i $$-0.635693\pi$$
−0.413497 + 0.910505i $$0.635693\pi$$
$$278$$ 43.8885 2.63226
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 22.2936 1.32992 0.664961 0.746878i $$-0.268448\pi$$
0.664961 + 0.746878i $$0.268448\pi$$
$$282$$ 0 0
$$283$$ 16.3516 0.972000 0.486000 0.873959i $$-0.338456\pi$$
0.486000 + 0.873959i $$0.338456\pi$$
$$284$$ −42.2688 −2.50819
$$285$$ 0 0
$$286$$ 19.3863 1.14634
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 8.00000 0.470588
$$290$$ 69.7771 4.09745
$$291$$ 0 0
$$292$$ 2.41577 0.141372
$$293$$ −7.47214 −0.436527 −0.218263 0.975890i $$-0.570039\pi$$
−0.218263 + 0.975890i $$0.570039\pi$$
$$294$$ 0 0
$$295$$ −16.7082 −0.972789
$$296$$ 13.3168 0.774024
$$297$$ 0 0
$$298$$ 50.3607 2.91732
$$299$$ 0.763932 0.0441793
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 37.5648 2.16161
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −15.6839 −0.898056
$$306$$ 0 0
$$307$$ 2.74962 0.156929 0.0784644 0.996917i $$-0.474998\pi$$
0.0784644 + 0.996917i $$0.474998\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ −32.6525 −1.85454
$$311$$ −27.0689 −1.53494 −0.767468 0.641088i $$-0.778484\pi$$
−0.767468 + 0.641088i $$0.778484\pi$$
$$312$$ 0 0
$$313$$ −28.9520 −1.63646 −0.818231 0.574889i $$-0.805045\pi$$
−0.818231 + 0.574889i $$0.805045\pi$$
$$314$$ 47.3050 2.66957
$$315$$ 0 0
$$316$$ −37.1246 −2.08842
$$317$$ 2.08191 0.116932 0.0584660 0.998289i $$-0.481379\pi$$
0.0584660 + 0.998289i $$0.481379\pi$$
$$318$$ 0 0
$$319$$ 26.6525 1.49225
$$320$$ 54.8328 3.06525
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −27.2526 −1.51638
$$324$$ 0 0
$$325$$ −29.6197 −1.64300
$$326$$ 11.8539 0.656526
$$327$$ 0 0
$$328$$ −13.3168 −0.735298
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 18.4164 1.01226 0.506129 0.862458i $$-0.331076\pi$$
0.506129 + 0.862458i $$0.331076\pi$$
$$332$$ 0.763932 0.0419262
$$333$$ 0 0
$$334$$ −8.61280 −0.471271
$$335$$ 47.5967 2.60049
$$336$$ 0 0
$$337$$ 3.76393 0.205034 0.102517 0.994731i $$-0.467310\pi$$
0.102517 + 0.994731i $$0.467310\pi$$
$$338$$ 17.7658 0.966331
$$339$$ 0 0
$$340$$ 68.5410 3.71716
$$341$$ −12.4721 −0.675404
$$342$$ 0 0
$$343$$ 0 0
$$344$$ −6.32456 −0.340997
$$345$$ 0 0
$$346$$ 42.2688 2.27238
$$347$$ −23.1676 −1.24370 −0.621851 0.783136i $$-0.713619\pi$$
−0.621851 + 0.783136i $$0.713619\pi$$
$$348$$ 0 0
$$349$$ 10.9010 0.583520 0.291760 0.956492i $$-0.405759\pi$$
0.291760 + 0.956492i $$0.405759\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 20.9443 1.11633
$$353$$ −30.1246 −1.60337 −0.801686 0.597746i $$-0.796063\pi$$
−0.801686 + 0.597746i $$0.796063\pi$$
$$354$$ 0 0
$$355$$ 55.3306 2.93664
$$356$$ 5.52786 0.292976
$$357$$ 0 0
$$358$$ 33.5967 1.77564
$$359$$ 4.65530 0.245697 0.122849 0.992425i $$-0.460797\pi$$
0.122849 + 0.992425i $$0.460797\pi$$
$$360$$ 0 0
$$361$$ 10.7082 0.563590
$$362$$ −52.3607 −2.75202
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −3.16228 −0.165521
$$366$$ 0 0
$$367$$ 18.6398 0.972990 0.486495 0.873683i $$-0.338275\pi$$
0.486495 + 0.873683i $$0.338275\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ −45.6374 −2.37258
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −13.6525 −0.706898 −0.353449 0.935454i $$-0.614991\pi$$
−0.353449 + 0.935454i $$0.614991\pi$$
$$374$$ 42.3607 2.19042
$$375$$ 0 0
$$376$$ −4.16383 −0.214733
$$377$$ −16.4721 −0.848358
$$378$$ 0 0
$$379$$ −30.8885 −1.58664 −0.793319 0.608806i $$-0.791649\pi$$
−0.793319 + 0.608806i $$0.791649\pi$$
$$380$$ −74.7169 −3.83290
$$381$$ 0 0
$$382$$ −9.70820 −0.496715
$$383$$ 11.1803 0.571289 0.285644 0.958336i $$-0.407792\pi$$
0.285644 + 0.958336i $$0.407792\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −39.3128 −2.00097
$$387$$ 0 0
$$388$$ 6.32456 0.321081
$$389$$ −23.0888 −1.17065 −0.585324 0.810800i $$-0.699033\pi$$
−0.585324 + 0.810800i $$0.699033\pi$$
$$390$$ 0 0
$$391$$ 1.66925 0.0844177
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 1.05573 0.0531868
$$395$$ 48.5967 2.44517
$$396$$ 0 0
$$397$$ 27.7928 1.39488 0.697440 0.716643i $$-0.254322\pi$$
0.697440 + 0.716643i $$0.254322\pi$$
$$398$$ −30.8328 −1.54551
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −23.7078 −1.18391 −0.591955 0.805971i $$-0.701644\pi$$
−0.591955 + 0.805971i $$0.701644\pi$$
$$402$$ 0 0
$$403$$ 7.70820 0.383973
$$404$$ −29.8885 −1.48701
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −17.4319 −0.864069
$$408$$ 0 0
$$409$$ 32.0354 1.58405 0.792025 0.610488i $$-0.209027\pi$$
0.792025 + 0.610488i $$0.209027\pi$$
$$410$$ 45.6374 2.25387
$$411$$ 0 0
$$412$$ −27.2039 −1.34024
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −1.00000 −0.0490881
$$416$$ −12.9443 −0.634645
$$417$$ 0 0
$$418$$ −46.1776 −2.25862
$$419$$ −15.6525 −0.764673 −0.382337 0.924023i $$-0.624881\pi$$
−0.382337 + 0.924023i $$0.624881\pi$$
$$420$$ 0 0
$$421$$ −4.47214 −0.217959 −0.108979 0.994044i $$-0.534758\pi$$
−0.108979 + 0.994044i $$0.534758\pi$$
$$422$$ 39.8530 1.94001
$$423$$ 0 0
$$424$$ 8.94427 0.434372
$$425$$ −64.7214 −3.13945
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 32.0354 1.54849
$$429$$ 0 0
$$430$$ 21.6746 1.04524
$$431$$ −20.0540 −0.965969 −0.482984 0.875629i $$-0.660447\pi$$
−0.482984 + 0.875629i $$0.660447\pi$$
$$432$$ 0 0
$$433$$ −16.2728 −0.782019 −0.391009 0.920387i $$-0.627874\pi$$
−0.391009 + 0.920387i $$0.627874\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 31.5967 1.51321
$$437$$ −1.81966 −0.0870461
$$438$$ 0 0
$$439$$ −0.412662 −0.0196953 −0.00984764 0.999952i $$-0.503135\pi$$
−0.00984764 + 0.999952i $$0.503135\pi$$
$$440$$ 44.3607 2.11481
$$441$$ 0 0
$$442$$ −26.1803 −1.24527
$$443$$ −21.5958 −1.02605 −0.513023 0.858375i $$-0.671474\pi$$
−0.513023 + 0.858375i $$0.671474\pi$$
$$444$$ 0 0
$$445$$ −7.23607 −0.343023
$$446$$ 44.6525 2.11436
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 34.7363 1.63931 0.819655 0.572858i $$-0.194165\pi$$
0.819655 + 0.572858i $$0.194165\pi$$
$$450$$ 0 0
$$451$$ 17.4319 0.820838
$$452$$ −3.90879 −0.183854
$$453$$ 0 0
$$454$$ 27.8716 1.30808
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 32.8328 1.53585 0.767927 0.640537i $$-0.221288\pi$$
0.767927 + 0.640537i $$0.221288\pi$$
$$458$$ −24.3607 −1.13830
$$459$$ 0 0
$$460$$ 4.57649 0.213380
$$461$$ −22.5279 −1.04923 −0.524614 0.851340i $$-0.675790\pi$$
−0.524614 + 0.851340i $$0.675790\pi$$
$$462$$ 0 0
$$463$$ 37.1803 1.72792 0.863958 0.503563i $$-0.167978\pi$$
0.863958 + 0.503563i $$0.167978\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ −0.291796 −0.0135172
$$467$$ 37.5967 1.73977 0.869885 0.493255i $$-0.164193\pi$$
0.869885 + 0.493255i $$0.164193\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 14.2697 0.658210
$$471$$ 0 0
$$472$$ −11.1561 −0.513500
$$473$$ 8.27895 0.380667
$$474$$ 0 0
$$475$$ 70.5531 3.23720
$$476$$ 0 0
$$477$$ 0 0
$$478$$ −58.5410 −2.67760
$$479$$ 31.0689 1.41957 0.709787 0.704417i $$-0.248791\pi$$
0.709787 + 0.704417i $$0.248791\pi$$
$$480$$ 0 0
$$481$$ 10.7735 0.491231
$$482$$ −15.7082 −0.715489
$$483$$ 0 0
$$484$$ 8.76393 0.398361
$$485$$ −8.27895 −0.375928
$$486$$ 0 0
$$487$$ −25.8885 −1.17312 −0.586561 0.809905i $$-0.699519\pi$$
−0.586561 + 0.809905i $$0.699519\pi$$
$$488$$ −10.4721 −0.474051
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 1.46292 0.0660207 0.0330104 0.999455i $$-0.489491\pi$$
0.0330104 + 0.999455i $$0.489491\pi$$
$$492$$ 0 0
$$493$$ −35.9929 −1.62104
$$494$$ 28.5393 1.28404
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 36.8885 1.65136 0.825679 0.564140i $$-0.190792\pi$$
0.825679 + 0.564140i $$0.190792\pi$$
$$500$$ −108.902 −4.87023
$$501$$ 0 0
$$502$$ −21.2619 −0.948966
$$503$$ −35.8328 −1.59771 −0.798853 0.601526i $$-0.794560\pi$$
−0.798853 + 0.601526i $$0.794560\pi$$
$$504$$ 0 0
$$505$$ 39.1246 1.74102
$$506$$ 2.82843 0.125739
$$507$$ 0 0
$$508$$ 17.7082 0.785675
$$509$$ 17.3607 0.769499 0.384749 0.923021i $$-0.374288\pi$$
0.384749 + 0.923021i $$0.374288\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 0 0
$$514$$ −12.3941 −0.546679
$$515$$ 35.6104 1.56918
$$516$$ 0 0
$$517$$ 5.45052 0.239714
$$518$$ 0 0
$$519$$ 0 0
$$520$$ −27.4164 −1.20229
$$521$$ 8.52786 0.373613 0.186806 0.982397i $$-0.440186\pi$$
0.186806 + 0.982397i $$0.440186\pi$$
$$522$$ 0 0
$$523$$ −4.29135 −0.187648 −0.0938238 0.995589i $$-0.529909\pi$$
−0.0938238 + 0.995589i $$0.529909\pi$$
$$524$$ −35.4164 −1.54717
$$525$$ 0 0
$$526$$ −46.1803 −2.01356
$$527$$ 16.8430 0.733694
$$528$$ 0 0
$$529$$ −22.8885 −0.995154
$$530$$ −30.6525 −1.33146
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −10.7735 −0.466653
$$534$$ 0 0
$$535$$ −41.9349 −1.81301
$$536$$ 31.7804 1.37270
$$537$$ 0 0
$$538$$ 16.0177 0.690573
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 28.3050 1.21692 0.608462 0.793583i $$-0.291787\pi$$
0.608462 + 0.793583i $$0.291787\pi$$
$$542$$ −12.7639 −0.548258
$$543$$ 0 0
$$544$$ −28.2843 −1.21268
$$545$$ −41.3607 −1.77170
$$546$$ 0 0
$$547$$ 18.1246 0.774952 0.387476 0.921880i $$-0.373347\pi$$
0.387476 + 0.921880i $$0.373347\pi$$
$$548$$ −14.3972 −0.615017
$$549$$ 0 0
$$550$$ −109.666 −4.67616
$$551$$ 39.2361 1.67151
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 31.4953 1.33811
$$555$$ 0 0
$$556$$ −62.0678 −2.63226
$$557$$ −6.32456 −0.267980 −0.133990 0.990983i $$-0.542779\pi$$
−0.133990 + 0.990983i $$0.542779\pi$$
$$558$$ 0 0
$$559$$ −5.11667 −0.216412
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −51.0132 −2.15186
$$563$$ −34.1803 −1.44053 −0.720265 0.693699i $$-0.755980\pi$$
−0.720265 + 0.693699i $$0.755980\pi$$
$$564$$ 0 0
$$565$$ 5.11667 0.215260
$$566$$ −37.4164 −1.57273
$$567$$ 0 0
$$568$$ 36.9443 1.55015
$$569$$ −25.3770 −1.06386 −0.531930 0.846788i $$-0.678533\pi$$
−0.531930 + 0.846788i $$0.678533\pi$$
$$570$$ 0 0
$$571$$ −19.0689 −0.798008 −0.399004 0.916949i $$-0.630644\pi$$
−0.399004 + 0.916949i $$0.630644\pi$$
$$572$$ −27.4164 −1.14634
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −4.32145 −0.180217
$$576$$ 0 0
$$577$$ 38.3600 1.59695 0.798474 0.602030i $$-0.205641\pi$$
0.798474 + 0.602030i $$0.205641\pi$$
$$578$$ −18.3060 −0.761428
$$579$$ 0 0
$$580$$ −98.6797 −4.09745
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −11.7082 −0.484904
$$584$$ −2.11146 −0.0873727
$$585$$ 0 0
$$586$$ 17.0981 0.706315
$$587$$ 18.4721 0.762427 0.381213 0.924487i $$-0.375506\pi$$
0.381213 + 0.924487i $$0.375506\pi$$
$$588$$ 0 0
$$589$$ −18.3607 −0.756539
$$590$$ 38.2325 1.57401
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −11.4721 −0.471104 −0.235552 0.971862i $$-0.575690\pi$$
−0.235552 + 0.971862i $$0.575690\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −71.2208 −2.91732
$$597$$ 0 0
$$598$$ −1.74806 −0.0714837
$$599$$ −32.2418 −1.31736 −0.658681 0.752422i $$-0.728886\pi$$
−0.658681 + 0.752422i $$0.728886\pi$$
$$600$$ 0 0
$$601$$ −25.3283 −1.03316 −0.516582 0.856238i $$-0.672796\pi$$
−0.516582 + 0.856238i $$0.672796\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ −53.1246 −2.16161
$$605$$ −11.4721 −0.466409
$$606$$ 0 0
$$607$$ 26.6637 1.08225 0.541124 0.840943i $$-0.317999\pi$$
0.541124 + 0.840943i $$0.317999\pi$$
$$608$$ 30.8328 1.25044
$$609$$ 0 0
$$610$$ 35.8885 1.45308
$$611$$ −3.36861 −0.136279
$$612$$ 0 0
$$613$$ −31.8885 −1.28797 −0.643983 0.765040i $$-0.722719\pi$$
−0.643983 + 0.765040i $$0.722719\pi$$
$$614$$ −6.29180 −0.253916
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −10.5672 −0.425419 −0.212710 0.977115i $$-0.568229\pi$$
−0.212710 + 0.977115i $$0.568229\pi$$
$$618$$ 0 0
$$619$$ −38.5663 −1.55011 −0.775056 0.631893i $$-0.782278\pi$$
−0.775056 + 0.631893i $$0.782278\pi$$
$$620$$ 46.1776 1.85454
$$621$$ 0 0
$$622$$ 61.9403 2.48358
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 77.8328 3.11331
$$626$$ 66.2492 2.64785
$$627$$ 0 0
$$628$$ −66.8993 −2.66957
$$629$$ 23.5410 0.938642
$$630$$ 0 0
$$631$$ −26.2361 −1.04444 −0.522221 0.852810i $$-0.674896\pi$$
−0.522221 + 0.852810i $$0.674896\pi$$
$$632$$ 32.4481 1.29072
$$633$$ 0 0
$$634$$ −4.76393 −0.189200
$$635$$ −23.1803 −0.919884
$$636$$ 0 0
$$637$$ 0 0
$$638$$ −60.9874 −2.41451
$$639$$ 0 0
$$640$$ −77.5453 −3.06525
$$641$$ 34.8152 1.37512 0.687558 0.726129i $$-0.258683\pi$$
0.687558 + 0.726129i $$0.258683\pi$$
$$642$$ 0 0
$$643$$ −33.8623 −1.33540 −0.667700 0.744431i $$-0.732721\pi$$
−0.667700 + 0.744431i $$0.732721\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 62.3607 2.45355
$$647$$ 4.36068 0.171436 0.0857180 0.996319i $$-0.472682\pi$$
0.0857180 + 0.996319i $$0.472682\pi$$
$$648$$ 0 0
$$649$$ 14.6035 0.573238
$$650$$ 67.7771 2.65844
$$651$$ 0 0
$$652$$ −16.7639 −0.656526
$$653$$ 18.4335 0.721358 0.360679 0.932690i $$-0.382545\pi$$
0.360679 + 0.932690i $$0.382545\pi$$
$$654$$ 0 0
$$655$$ 46.3607 1.81146
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −38.5176 −1.50043 −0.750217 0.661192i $$-0.770051\pi$$
−0.750217 + 0.661192i $$0.770051\pi$$
$$660$$ 0 0
$$661$$ −29.3345 −1.14098 −0.570491 0.821304i $$-0.693247\pi$$
−0.570491 + 0.821304i $$0.693247\pi$$
$$662$$ −42.1413 −1.63787
$$663$$ 0 0
$$664$$ −0.667701 −0.0259118
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −2.40325 −0.0930543
$$668$$ 12.1803 0.471271
$$669$$ 0 0
$$670$$ −108.913 −4.20768
$$671$$ 13.7082 0.529199
$$672$$ 0 0
$$673$$ −41.2361 −1.58953 −0.794767 0.606915i $$-0.792407\pi$$
−0.794767 + 0.606915i $$0.792407\pi$$
$$674$$ −8.61280 −0.331753
$$675$$ 0 0
$$676$$ −25.1246 −0.966331
$$677$$ −12.1115 −0.465481 −0.232741 0.972539i $$-0.574769\pi$$
−0.232741 + 0.972539i $$0.574769\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ −59.9070 −2.29733
$$681$$ 0 0
$$682$$ 28.5393 1.09283
$$683$$ −38.9002 −1.48847 −0.744237 0.667916i $$-0.767187\pi$$
−0.744237 + 0.667916i $$0.767187\pi$$
$$684$$ 0 0
$$685$$ 18.8461 0.720074
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 7.23607 0.275672
$$690$$ 0 0
$$691$$ 0.795221 0.0302516 0.0151258 0.999886i $$-0.495185\pi$$
0.0151258 + 0.999886i $$0.495185\pi$$
$$692$$ −59.7771 −2.27238
$$693$$ 0 0
$$694$$ 53.0132 2.01235
$$695$$ 81.2478 3.08190
$$696$$ 0 0
$$697$$ −23.5410 −0.891680
$$698$$ −24.9443 −0.944155
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −9.43812 −0.356473 −0.178237 0.983988i $$-0.557039\pi$$
−0.178237 + 0.983988i $$0.557039\pi$$
$$702$$ 0 0
$$703$$ −25.6622 −0.967867
$$704$$ −47.9256 −1.80627
$$705$$ 0 0
$$706$$ 68.9325 2.59431
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −21.4721 −0.806403 −0.403201 0.915111i $$-0.632103\pi$$
−0.403201 + 0.915111i $$0.632103\pi$$
$$710$$ −126.610 −4.75159
$$711$$ 0 0
$$712$$ −4.83153 −0.181069
$$713$$ 1.12461 0.0421170
$$714$$ 0 0
$$715$$ 35.8885 1.34216
$$716$$ −47.5130 −1.77564
$$717$$ 0 0
$$718$$ −10.6525 −0.397547
$$719$$ −0.416408 −0.0155294 −0.00776470 0.999970i $$-0.502472\pi$$
−0.00776470 + 0.999970i $$0.502472\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ −24.5030 −0.911907
$$723$$ 0 0
$$724$$ 74.0492 2.75202
$$725$$ 93.1805 3.46064
$$726$$ 0 0
$$727$$ 35.4829 1.31599 0.657993 0.753024i $$-0.271406\pi$$
0.657993 + 0.753024i $$0.271406\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 7.23607 0.267819
$$731$$ −11.1803 −0.413520
$$732$$ 0 0
$$733$$ 6.32456 0.233603 0.116801 0.993155i $$-0.462736\pi$$
0.116801 + 0.993155i $$0.462736\pi$$
$$734$$ −42.6525 −1.57433
$$735$$ 0 0
$$736$$ −1.88854 −0.0696126
$$737$$ −41.6011 −1.53240
$$738$$ 0 0
$$739$$ −3.63932 −0.133875 −0.0669373 0.997757i $$-0.521323\pi$$
−0.0669373 + 0.997757i $$0.521323\pi$$
$$740$$ 64.5410 2.37258
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −25.9172 −0.950810 −0.475405 0.879767i $$-0.657699\pi$$
−0.475405 + 0.879767i $$0.657699\pi$$
$$744$$ 0 0
$$745$$ 93.2292 3.41565
$$746$$ 31.2402 1.14379
$$747$$ 0 0
$$748$$ −59.9070 −2.19042
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 44.6525 1.62939 0.814696 0.579888i $$-0.196904\pi$$
0.814696 + 0.579888i $$0.196904\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 37.6923 1.37267
$$755$$ 69.5410 2.53086
$$756$$ 0 0
$$757$$ −35.3607 −1.28521 −0.642603 0.766199i $$-0.722145\pi$$
−0.642603 + 0.766199i $$0.722145\pi$$
$$758$$ 70.6806 2.56723
$$759$$ 0 0
$$760$$ 65.3050 2.36886
$$761$$ 35.4721 1.28586 0.642932 0.765923i $$-0.277718\pi$$
0.642932 + 0.765923i $$0.277718\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 13.7295 0.496715
$$765$$ 0 0
$$766$$ −25.5834 −0.924365
$$767$$ −9.02546 −0.325891
$$768$$ 0 0
$$769$$ −28.6969 −1.03484 −0.517419 0.855732i $$-0.673107\pi$$
−0.517419 + 0.855732i $$0.673107\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 55.5967 2.00097
$$773$$ 0.596748 0.0214635 0.0107318 0.999942i $$-0.496584\pi$$
0.0107318 + 0.999942i $$0.496584\pi$$
$$774$$ 0 0
$$775$$ −43.6042 −1.56631
$$776$$ −5.52786 −0.198439
$$777$$ 0 0
$$778$$ 52.8328 1.89415
$$779$$ 25.6622 0.919443
$$780$$ 0 0
$$781$$ −48.3607 −1.73048
$$782$$ −3.81966 −0.136591
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 87.5723 3.12559
$$786$$ 0 0
$$787$$ −3.41732 −0.121814 −0.0609071 0.998143i $$-0.519399\pi$$
−0.0609071 + 0.998143i $$0.519399\pi$$
$$788$$ −1.49302 −0.0531868
$$789$$ 0 0
$$790$$ −111.201 −3.95636
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −8.47214 −0.300854
$$794$$ −63.5967 −2.25696
$$795$$ 0 0
$$796$$ 43.6042 1.54551
$$797$$ 16.0689 0.569189 0.284595 0.958648i $$-0.408141\pi$$
0.284595 + 0.958648i $$0.408141\pi$$
$$798$$ 0 0
$$799$$ −7.36068 −0.260402
$$800$$ 73.2239 2.58885
$$801$$ 0 0
$$802$$ 54.2492 1.91561
$$803$$ 2.76393 0.0975370
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −17.6383 −0.621281
$$807$$ 0 0
$$808$$ 26.1235 0.919023
$$809$$ 10.5672 0.371523 0.185761 0.982595i $$-0.440525\pi$$
0.185761 + 0.982595i $$0.440525\pi$$
$$810$$ 0 0
$$811$$ 37.9774 1.33357 0.666784 0.745251i $$-0.267670\pi$$
0.666784 + 0.745251i $$0.267670\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 39.8885 1.39809
$$815$$ 21.9443 0.768674
$$816$$ 0 0
$$817$$ 12.1877 0.426395
$$818$$ −73.3050 −2.56305
$$819$$ 0 0
$$820$$ −64.5410 −2.25387
$$821$$ 12.2364 0.427055 0.213528 0.976937i $$-0.431505\pi$$
0.213528 + 0.976937i $$0.431505\pi$$
$$822$$ 0 0
$$823$$ −9.58359 −0.334063 −0.167032 0.985952i $$-0.553418\pi$$
−0.167032 + 0.985952i $$0.553418\pi$$
$$824$$ 23.7771 0.828314
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 39.9805 1.39026 0.695130 0.718884i $$-0.255347\pi$$
0.695130 + 0.718884i $$0.255347\pi$$
$$828$$ 0 0
$$829$$ −22.8337 −0.793049 −0.396524 0.918024i $$-0.629784\pi$$
−0.396524 + 0.918024i $$0.629784\pi$$
$$830$$ 2.28825 0.0794262
$$831$$ 0 0
$$832$$ 29.6197 1.02688
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −15.9443 −0.551774
$$836$$ 65.3050 2.25862
$$837$$ 0 0
$$838$$ 35.8167 1.23727
$$839$$ 24.8197 0.856870 0.428435 0.903573i $$-0.359065\pi$$
0.428435 + 0.903573i $$0.359065\pi$$
$$840$$ 0 0
$$841$$ 22.8197 0.786885
$$842$$ 10.2333 0.352664
$$843$$ 0 0
$$844$$ −56.3607 −1.94001
$$845$$ 32.8885 1.13140
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 148.098 5.07973
$$851$$ 1.57184 0.0538819
$$852$$ 0 0
$$853$$ −21.3894 −0.732360 −0.366180 0.930544i $$-0.619335\pi$$
−0.366180 + 0.930544i $$0.619335\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −28.0000 −0.957020
$$857$$ 3.18034 0.108638 0.0543192 0.998524i $$-0.482701\pi$$
0.0543192 + 0.998524i $$0.482701\pi$$
$$858$$ 0 0
$$859$$ −47.5918 −1.62381 −0.811905 0.583789i $$-0.801570\pi$$
−0.811905 + 0.583789i $$0.801570\pi$$
$$860$$ −30.6525 −1.04524
$$861$$ 0 0
$$862$$ 45.8885 1.56297
$$863$$ −28.8732 −0.982854 −0.491427 0.870919i $$-0.663525\pi$$
−0.491427 + 0.870919i $$0.663525\pi$$
$$864$$ 0 0
$$865$$ 78.2492 2.66055
$$866$$ 37.2361 1.26533
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −42.4751 −1.44087
$$870$$ 0 0
$$871$$ 25.7109 0.871180
$$872$$ −27.6166 −0.935215
$$873$$ 0 0
$$874$$ 4.16383 0.140844
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −50.0132 −1.68882 −0.844412 0.535694i $$-0.820050\pi$$
−0.844412 + 0.535694i $$0.820050\pi$$
$$878$$ 0.944272 0.0318676
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −16.8328 −0.567112 −0.283556 0.958956i $$-0.591514\pi$$
−0.283556 + 0.958956i $$0.591514\pi$$
$$882$$ 0 0
$$883$$ 16.8885 0.568345 0.284172 0.958773i $$-0.408281\pi$$
0.284172 + 0.958773i $$0.408281\pi$$
$$884$$ 37.0246 1.24527
$$885$$ 0 0
$$886$$ 49.4164 1.66018
$$887$$ −52.1935 −1.75249 −0.876243 0.481869i $$-0.839958\pi$$
−0.876243 + 0.481869i $$0.839958\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 16.5579 0.555022
$$891$$ 0 0
$$892$$ −63.1481 −2.11436
$$893$$ 8.02391 0.268510
$$894$$ 0 0
$$895$$ 62.1953 2.07896
$$896$$ 0 0
$$897$$ 0 0
$$898$$ −79.4853 −2.65246
$$899$$ −24.2492 −0.808757
$$900$$ 0 0
$$901$$ 15.8114 0.526754
$$902$$ −39.8885 −1.32814
$$903$$ 0 0
$$904$$ 3.41641 0.113628
$$905$$ −96.9316 −3.22212
$$906$$ 0 0
$$907$$ 47.0000 1.56061 0.780305 0.625400i $$-0.215064\pi$$
0.780305 + 0.625400i $$0.215064\pi$$
$$908$$ −39.4164 −1.30808
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −22.8337 −0.756516 −0.378258 0.925700i $$-0.623477\pi$$
−0.378258 + 0.925700i $$0.623477\pi$$
$$912$$ 0 0
$$913$$ 0.874032 0.0289262
$$914$$ −75.1295 −2.48506
$$915$$ 0 0
$$916$$ 34.4512 1.13830
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 1.94427 0.0641356 0.0320678 0.999486i $$-0.489791\pi$$
0.0320678 + 0.999486i $$0.489791\pi$$
$$920$$ −4.00000 −0.131876
$$921$$ 0 0
$$922$$ 51.5493 1.69769
$$923$$ 29.8885 0.983793
$$924$$ 0 0
$$925$$ −60.9443 −2.00384
$$926$$ −85.0777 −2.79583
$$927$$ 0 0
$$928$$ 40.7214 1.33674
$$929$$ 17.8328 0.585076 0.292538 0.956254i $$-0.405500\pi$$
0.292538 + 0.956254i $$0.405500\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0.412662 0.0135172
$$933$$ 0 0
$$934$$ −86.0306 −2.81501
$$935$$ 78.4193 2.56459
$$936$$ 0 0
$$937$$ 17.8446 0.582958 0.291479 0.956577i $$-0.405853\pi$$
0.291479 + 0.956577i $$0.405853\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ −20.1803 −0.658210
$$941$$ −20.7082 −0.675068 −0.337534 0.941313i $$-0.609593\pi$$
−0.337534 + 0.941313i $$0.609593\pi$$
$$942$$ 0 0
$$943$$ −1.57184 −0.0511860
$$944$$ 0 0
$$945$$ 0 0
$$946$$ −18.9443 −0.615931
$$947$$ −40.7758 −1.32503 −0.662517 0.749047i $$-0.730512\pi$$
−0.662517 + 0.749047i $$0.730512\pi$$
$$948$$ 0 0
$$949$$ −1.70820 −0.0554506
$$950$$ −161.443 −5.23789
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 14.8886 0.482291 0.241145 0.970489i $$-0.422477\pi$$
0.241145 + 0.970489i $$0.422477\pi$$
$$954$$ 0 0
$$955$$ −17.9721 −0.581564
$$956$$ 82.7895 2.67760
$$957$$ 0 0
$$958$$ −71.0932 −2.29692
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −19.6525 −0.633951
$$962$$ −24.6525 −0.794828
$$963$$ 0 0
$$964$$ 22.2148 0.715489
$$965$$ −72.7771 −2.34278
$$966$$ 0 0
$$967$$ 5.52786 0.177764 0.0888821 0.996042i $$-0.471671\pi$$
0.0888821 + 0.996042i $$0.471671\pi$$
$$968$$ −7.65996 −0.246200
$$969$$ 0 0
$$970$$ 18.9443 0.608264
$$971$$ 20.5279 0.658771 0.329385 0.944196i $$-0.393159\pi$$
0.329385 + 0.944196i $$0.393159\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 59.2393 1.89815
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 4.83153 0.154574 0.0772872 0.997009i $$-0.475374\pi$$
0.0772872 + 0.997009i $$0.475374\pi$$
$$978$$ 0 0
$$979$$ 6.32456 0.202134
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −3.34752 −0.106824
$$983$$ 26.0557 0.831049 0.415524 0.909582i $$-0.363598\pi$$
0.415524 + 0.909582i $$0.363598\pi$$
$$984$$ 0 0
$$985$$ 1.95440 0.0622722
$$986$$ 82.3607 2.62290
$$987$$ 0 0
$$988$$ −40.3607 −1.28404
$$989$$ −0.746512 −0.0237377
$$990$$ 0 0
$$991$$ 0.124612 0.00395842 0.00197921 0.999998i $$-0.499370\pi$$
0.00197921 + 0.999998i $$0.499370\pi$$
$$992$$ −19.0557 −0.605020
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −57.0786 −1.80951
$$996$$ 0 0
$$997$$ 61.4488 1.94610 0.973051 0.230589i $$-0.0740653\pi$$
0.973051 + 0.230589i $$0.0740653\pi$$
$$998$$ −84.4100 −2.67195
$$999$$ 0 0
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 1323.2.a.bb.1.1 4
3.2 odd 2 1323.2.a.be.1.4 yes 4
7.6 odd 2 1323.2.a.be.1.1 yes 4
21.20 even 2 inner 1323.2.a.bb.1.4 yes 4

By twisted newform
Twist Min Dim Char Parity Ord Type
1323.2.a.bb.1.1 4 1.1 even 1 trivial
1323.2.a.bb.1.4 yes 4 21.20 even 2 inner
1323.2.a.be.1.1 yes 4 7.6 odd 2
1323.2.a.be.1.4 yes 4 3.2 odd 2