Properties

 Label 1575.2.d.i.1324.3 Level $1575$ Weight $2$ Character 1575.1324 Analytic conductor $12.576$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$1575 = 3^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1575.d (of order $$2$$, degree $$1$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$12.5764383184$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 63) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

 Embedding label 1324.3 Root $$0.866025 + 0.500000i$$ of defining polynomial Character $$\chi$$ $$=$$ 1575.1324 Dual form 1575.2.d.i.1324.2

$q$-expansion

 $$f(q)$$ $$=$$ $$q+1.73205i q^{2} -1.00000 q^{4} -1.00000i q^{7} +1.73205i q^{8} +O(q^{10})$$ $$q+1.73205i q^{2} -1.00000 q^{4} -1.00000i q^{7} +1.73205i q^{8} -3.46410 q^{11} +2.00000i q^{13} +1.73205 q^{14} -5.00000 q^{16} +3.46410i q^{17} +4.00000 q^{19} -6.00000i q^{22} +3.46410i q^{23} -3.46410 q^{26} +1.00000i q^{28} -4.00000 q^{31} -5.19615i q^{32} -6.00000 q^{34} -2.00000i q^{37} +6.92820i q^{38} -10.3923 q^{41} -4.00000i q^{43} +3.46410 q^{44} -6.00000 q^{46} +6.92820i q^{47} -1.00000 q^{49} -2.00000i q^{52} +6.92820i q^{53} +1.73205 q^{56} -6.92820 q^{59} -10.0000 q^{61} -6.92820i q^{62} -1.00000 q^{64} +4.00000i q^{67} -3.46410i q^{68} +10.3923 q^{71} +14.0000i q^{73} +3.46410 q^{74} -4.00000 q^{76} +3.46410i q^{77} -8.00000 q^{79} -18.0000i q^{82} +6.92820 q^{86} -6.00000i q^{88} -3.46410 q^{89} +2.00000 q^{91} -3.46410i q^{92} -12.0000 q^{94} -14.0000i q^{97} -1.73205i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4} + O(q^{10})$$ $$4 q - 4 q^{4} - 20 q^{16} + 16 q^{19} - 16 q^{31} - 24 q^{34} - 24 q^{46} - 4 q^{49} - 40 q^{61} - 4 q^{64} - 16 q^{76} - 32 q^{79} + 8 q^{91} - 48 q^{94} + O(q^{100})$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1575\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$451$$ $$1226$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.73205i 1.22474i 0.790569 + 0.612372i $$0.209785\pi$$
−0.790569 + 0.612372i $$0.790215\pi$$
$$3$$ 0 0
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ − 1.00000i − 0.377964i
$$8$$ 1.73205i 0.612372i
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −3.46410 −1.04447 −0.522233 0.852803i $$-0.674901\pi$$
−0.522233 + 0.852803i $$0.674901\pi$$
$$12$$ 0 0
$$13$$ 2.00000i 0.554700i 0.960769 + 0.277350i $$0.0894562\pi$$
−0.960769 + 0.277350i $$0.910544\pi$$
$$14$$ 1.73205 0.462910
$$15$$ 0 0
$$16$$ −5.00000 −1.25000
$$17$$ 3.46410i 0.840168i 0.907485 + 0.420084i $$0.137999\pi$$
−0.907485 + 0.420084i $$0.862001\pi$$
$$18$$ 0 0
$$19$$ 4.00000 0.917663 0.458831 0.888523i $$-0.348268\pi$$
0.458831 + 0.888523i $$0.348268\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ − 6.00000i − 1.27920i
$$23$$ 3.46410i 0.722315i 0.932505 + 0.361158i $$0.117618\pi$$
−0.932505 + 0.361158i $$0.882382\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −3.46410 −0.679366
$$27$$ 0 0
$$28$$ 1.00000i 0.188982i
$$29$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$30$$ 0 0
$$31$$ −4.00000 −0.718421 −0.359211 0.933257i $$-0.616954\pi$$
−0.359211 + 0.933257i $$0.616954\pi$$
$$32$$ − 5.19615i − 0.918559i
$$33$$ 0 0
$$34$$ −6.00000 −1.02899
$$35$$ 0 0
$$36$$ 0 0
$$37$$ − 2.00000i − 0.328798i −0.986394 0.164399i $$-0.947432\pi$$
0.986394 0.164399i $$-0.0525685\pi$$
$$38$$ 6.92820i 1.12390i
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −10.3923 −1.62301 −0.811503 0.584349i $$-0.801350\pi$$
−0.811503 + 0.584349i $$0.801350\pi$$
$$42$$ 0 0
$$43$$ − 4.00000i − 0.609994i −0.952353 0.304997i $$-0.901344\pi$$
0.952353 0.304997i $$-0.0986555\pi$$
$$44$$ 3.46410 0.522233
$$45$$ 0 0
$$46$$ −6.00000 −0.884652
$$47$$ 6.92820i 1.01058i 0.862949 + 0.505291i $$0.168615\pi$$
−0.862949 + 0.505291i $$0.831385\pi$$
$$48$$ 0 0
$$49$$ −1.00000 −0.142857
$$50$$ 0 0
$$51$$ 0 0
$$52$$ − 2.00000i − 0.277350i
$$53$$ 6.92820i 0.951662i 0.879537 + 0.475831i $$0.157853\pi$$
−0.879537 + 0.475831i $$0.842147\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 1.73205 0.231455
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −6.92820 −0.901975 −0.450988 0.892530i $$-0.648928\pi$$
−0.450988 + 0.892530i $$0.648928\pi$$
$$60$$ 0 0
$$61$$ −10.0000 −1.28037 −0.640184 0.768221i $$-0.721142\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ − 6.92820i − 0.879883i
$$63$$ 0 0
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 4.00000i 0.488678i 0.969690 + 0.244339i $$0.0785709\pi$$
−0.969690 + 0.244339i $$0.921429\pi$$
$$68$$ − 3.46410i − 0.420084i
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 10.3923 1.23334 0.616670 0.787222i $$-0.288481\pi$$
0.616670 + 0.787222i $$0.288481\pi$$
$$72$$ 0 0
$$73$$ 14.0000i 1.63858i 0.573382 + 0.819288i $$0.305631\pi$$
−0.573382 + 0.819288i $$0.694369\pi$$
$$74$$ 3.46410 0.402694
$$75$$ 0 0
$$76$$ −4.00000 −0.458831
$$77$$ 3.46410i 0.394771i
$$78$$ 0 0
$$79$$ −8.00000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ − 18.0000i − 1.98777i
$$83$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 6.92820 0.747087
$$87$$ 0 0
$$88$$ − 6.00000i − 0.639602i
$$89$$ −3.46410 −0.367194 −0.183597 0.983002i $$-0.558774\pi$$
−0.183597 + 0.983002i $$0.558774\pi$$
$$90$$ 0 0
$$91$$ 2.00000 0.209657
$$92$$ − 3.46410i − 0.361158i
$$93$$ 0 0
$$94$$ −12.0000 −1.23771
$$95$$ 0 0
$$96$$ 0 0
$$97$$ − 14.0000i − 1.42148i −0.703452 0.710742i $$-0.748359\pi$$
0.703452 0.710742i $$-0.251641\pi$$
$$98$$ − 1.73205i − 0.174964i
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −3.46410 −0.344691 −0.172345 0.985037i $$-0.555135\pi$$
−0.172345 + 0.985037i $$0.555135\pi$$
$$102$$ 0 0
$$103$$ − 4.00000i − 0.394132i −0.980390 0.197066i $$-0.936859\pi$$
0.980390 0.197066i $$-0.0631413\pi$$
$$104$$ −3.46410 −0.339683
$$105$$ 0 0
$$106$$ −12.0000 −1.16554
$$107$$ 17.3205i 1.67444i 0.546869 + 0.837218i $$0.315820\pi$$
−0.546869 + 0.837218i $$0.684180\pi$$
$$108$$ 0 0
$$109$$ −2.00000 −0.191565 −0.0957826 0.995402i $$-0.530535\pi$$
−0.0957826 + 0.995402i $$0.530535\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 5.00000i 0.472456i
$$113$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 0 0
$$118$$ − 12.0000i − 1.10469i
$$119$$ 3.46410 0.317554
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ − 17.3205i − 1.56813i
$$123$$ 0 0
$$124$$ 4.00000 0.359211
$$125$$ 0 0
$$126$$ 0 0
$$127$$ − 8.00000i − 0.709885i −0.934888 0.354943i $$-0.884500\pi$$
0.934888 0.354943i $$-0.115500\pi$$
$$128$$ − 12.1244i − 1.07165i
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 13.8564 1.21064 0.605320 0.795982i $$-0.293045\pi$$
0.605320 + 0.795982i $$0.293045\pi$$
$$132$$ 0 0
$$133$$ − 4.00000i − 0.346844i
$$134$$ −6.92820 −0.598506
$$135$$ 0 0
$$136$$ −6.00000 −0.514496
$$137$$ − 6.92820i − 0.591916i −0.955201 0.295958i $$-0.904361\pi$$
0.955201 0.295958i $$-0.0956389\pi$$
$$138$$ 0 0
$$139$$ 16.0000 1.35710 0.678551 0.734553i $$-0.262608\pi$$
0.678551 + 0.734553i $$0.262608\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 18.0000i 1.51053i
$$143$$ − 6.92820i − 0.579365i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ −24.2487 −2.00684
$$147$$ 0 0
$$148$$ 2.00000i 0.164399i
$$149$$ 6.92820 0.567581 0.283790 0.958886i $$-0.408408\pi$$
0.283790 + 0.958886i $$0.408408\pi$$
$$150$$ 0 0
$$151$$ 8.00000 0.651031 0.325515 0.945537i $$-0.394462\pi$$
0.325515 + 0.945537i $$0.394462\pi$$
$$152$$ 6.92820i 0.561951i
$$153$$ 0 0
$$154$$ −6.00000 −0.483494
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 10.0000i 0.798087i 0.916932 + 0.399043i $$0.130658\pi$$
−0.916932 + 0.399043i $$0.869342\pi$$
$$158$$ − 13.8564i − 1.10236i
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 3.46410 0.273009
$$162$$ 0 0
$$163$$ 20.0000i 1.56652i 0.621694 + 0.783260i $$0.286445\pi$$
−0.621694 + 0.783260i $$0.713555\pi$$
$$164$$ 10.3923 0.811503
$$165$$ 0 0
$$166$$ 0 0
$$167$$ − 20.7846i − 1.60836i −0.594385 0.804181i $$-0.702604\pi$$
0.594385 0.804181i $$-0.297396\pi$$
$$168$$ 0 0
$$169$$ 9.00000 0.692308
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 4.00000i 0.304997i
$$173$$ − 17.3205i − 1.31685i −0.752645 0.658427i $$-0.771222\pi$$
0.752645 0.658427i $$-0.228778\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 17.3205 1.30558
$$177$$ 0 0
$$178$$ − 6.00000i − 0.449719i
$$179$$ −17.3205 −1.29460 −0.647298 0.762237i $$-0.724101\pi$$
−0.647298 + 0.762237i $$0.724101\pi$$
$$180$$ 0 0
$$181$$ 2.00000 0.148659 0.0743294 0.997234i $$-0.476318\pi$$
0.0743294 + 0.997234i $$0.476318\pi$$
$$182$$ 3.46410i 0.256776i
$$183$$ 0 0
$$184$$ −6.00000 −0.442326
$$185$$ 0 0
$$186$$ 0 0
$$187$$ − 12.0000i − 0.877527i
$$188$$ − 6.92820i − 0.505291i
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 24.2487 1.75458 0.877288 0.479965i $$-0.159351\pi$$
0.877288 + 0.479965i $$0.159351\pi$$
$$192$$ 0 0
$$193$$ 14.0000i 1.00774i 0.863779 + 0.503871i $$0.168091\pi$$
−0.863779 + 0.503871i $$0.831909\pi$$
$$194$$ 24.2487 1.74096
$$195$$ 0 0
$$196$$ 1.00000 0.0714286
$$197$$ 20.7846i 1.48084i 0.672143 + 0.740421i $$0.265374\pi$$
−0.672143 + 0.740421i $$0.734626\pi$$
$$198$$ 0 0
$$199$$ 16.0000 1.13421 0.567105 0.823646i $$-0.308063\pi$$
0.567105 + 0.823646i $$0.308063\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ − 6.00000i − 0.422159i
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 6.92820 0.482711
$$207$$ 0 0
$$208$$ − 10.0000i − 0.693375i
$$209$$ −13.8564 −0.958468
$$210$$ 0 0
$$211$$ 20.0000 1.37686 0.688428 0.725304i $$-0.258301\pi$$
0.688428 + 0.725304i $$0.258301\pi$$
$$212$$ − 6.92820i − 0.475831i
$$213$$ 0 0
$$214$$ −30.0000 −2.05076
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 4.00000i 0.271538i
$$218$$ − 3.46410i − 0.234619i
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −6.92820 −0.466041
$$222$$ 0 0
$$223$$ 8.00000i 0.535720i 0.963458 + 0.267860i $$0.0863164\pi$$
−0.963458 + 0.267860i $$0.913684\pi$$
$$224$$ −5.19615 −0.347183
$$225$$ 0 0
$$226$$ 0 0
$$227$$ − 6.92820i − 0.459841i −0.973209 0.229920i $$-0.926153\pi$$
0.973209 0.229920i $$-0.0738466\pi$$
$$228$$ 0 0
$$229$$ 22.0000 1.45380 0.726900 0.686743i $$-0.240960\pi$$
0.726900 + 0.686743i $$0.240960\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ − 6.92820i − 0.453882i −0.973909 0.226941i $$-0.927128\pi$$
0.973909 0.226941i $$-0.0728724\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 6.92820 0.450988
$$237$$ 0 0
$$238$$ 6.00000i 0.388922i
$$239$$ 10.3923 0.672222 0.336111 0.941822i $$-0.390888\pi$$
0.336111 + 0.941822i $$0.390888\pi$$
$$240$$ 0 0
$$241$$ −10.0000 −0.644157 −0.322078 0.946713i $$-0.604381\pi$$
−0.322078 + 0.946713i $$0.604381\pi$$
$$242$$ 1.73205i 0.111340i
$$243$$ 0 0
$$244$$ 10.0000 0.640184
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 8.00000i 0.509028i
$$248$$ − 6.92820i − 0.439941i
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −20.7846 −1.31191 −0.655956 0.754799i $$-0.727735\pi$$
−0.655956 + 0.754799i $$0.727735\pi$$
$$252$$ 0 0
$$253$$ − 12.0000i − 0.754434i
$$254$$ 13.8564 0.869428
$$255$$ 0 0
$$256$$ 19.0000 1.18750
$$257$$ − 3.46410i − 0.216085i −0.994146 0.108042i $$-0.965542\pi$$
0.994146 0.108042i $$-0.0344582\pi$$
$$258$$ 0 0
$$259$$ −2.00000 −0.124274
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 24.0000i 1.48272i
$$263$$ 17.3205i 1.06803i 0.845476 + 0.534014i $$0.179317\pi$$
−0.845476 + 0.534014i $$0.820683\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 6.92820 0.424795
$$267$$ 0 0
$$268$$ − 4.00000i − 0.244339i
$$269$$ −17.3205 −1.05605 −0.528025 0.849229i $$-0.677067\pi$$
−0.528025 + 0.849229i $$0.677067\pi$$
$$270$$ 0 0
$$271$$ 20.0000 1.21491 0.607457 0.794353i $$-0.292190\pi$$
0.607457 + 0.794353i $$0.292190\pi$$
$$272$$ − 17.3205i − 1.05021i
$$273$$ 0 0
$$274$$ 12.0000 0.724947
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 10.0000i 0.600842i 0.953807 + 0.300421i $$0.0971271\pi$$
−0.953807 + 0.300421i $$0.902873\pi$$
$$278$$ 27.7128i 1.66210i
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 20.7846 1.23991 0.619953 0.784639i $$-0.287152\pi$$
0.619953 + 0.784639i $$0.287152\pi$$
$$282$$ 0 0
$$283$$ − 4.00000i − 0.237775i −0.992908 0.118888i $$-0.962067\pi$$
0.992908 0.118888i $$-0.0379328\pi$$
$$284$$ −10.3923 −0.616670
$$285$$ 0 0
$$286$$ 12.0000 0.709575
$$287$$ 10.3923i 0.613438i
$$288$$ 0 0
$$289$$ 5.00000 0.294118
$$290$$ 0 0
$$291$$ 0 0
$$292$$ − 14.0000i − 0.819288i
$$293$$ 10.3923i 0.607125i 0.952812 + 0.303562i $$0.0981761\pi$$
−0.952812 + 0.303562i $$0.901824\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 3.46410 0.201347
$$297$$ 0 0
$$298$$ 12.0000i 0.695141i
$$299$$ −6.92820 −0.400668
$$300$$ 0 0
$$301$$ −4.00000 −0.230556
$$302$$ 13.8564i 0.797347i
$$303$$ 0 0
$$304$$ −20.0000 −1.14708
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 28.0000i 1.59804i 0.601302 + 0.799022i $$0.294649\pi$$
−0.601302 + 0.799022i $$0.705351\pi$$
$$308$$ − 3.46410i − 0.197386i
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −34.6410 −1.96431 −0.982156 0.188069i $$-0.939777\pi$$
−0.982156 + 0.188069i $$0.939777\pi$$
$$312$$ 0 0
$$313$$ 2.00000i 0.113047i 0.998401 + 0.0565233i $$0.0180015\pi$$
−0.998401 + 0.0565233i $$0.981998\pi$$
$$314$$ −17.3205 −0.977453
$$315$$ 0 0
$$316$$ 8.00000 0.450035
$$317$$ 6.92820i 0.389127i 0.980890 + 0.194563i $$0.0623290\pi$$
−0.980890 + 0.194563i $$0.937671\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 6.00000i 0.334367i
$$323$$ 13.8564i 0.770991i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −34.6410 −1.91859
$$327$$ 0 0
$$328$$ − 18.0000i − 0.993884i
$$329$$ 6.92820 0.381964
$$330$$ 0 0
$$331$$ 20.0000 1.09930 0.549650 0.835395i $$-0.314761\pi$$
0.549650 + 0.835395i $$0.314761\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 36.0000 1.96983
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 14.0000i − 0.762629i −0.924445 0.381314i $$-0.875472\pi$$
0.924445 0.381314i $$-0.124528\pi$$
$$338$$ 15.5885i 0.847900i
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 13.8564 0.750366
$$342$$ 0 0
$$343$$ 1.00000i 0.0539949i
$$344$$ 6.92820 0.373544
$$345$$ 0 0
$$346$$ 30.0000 1.61281
$$347$$ − 17.3205i − 0.929814i −0.885360 0.464907i $$-0.846088\pi$$
0.885360 0.464907i $$-0.153912\pi$$
$$348$$ 0 0
$$349$$ −14.0000 −0.749403 −0.374701 0.927146i $$-0.622255\pi$$
−0.374701 + 0.927146i $$0.622255\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 18.0000i 0.959403i
$$353$$ − 3.46410i − 0.184376i −0.995742 0.0921878i $$-0.970614\pi$$
0.995742 0.0921878i $$-0.0293860\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 3.46410 0.183597
$$357$$ 0 0
$$358$$ − 30.0000i − 1.58555i
$$359$$ −24.2487 −1.27980 −0.639899 0.768459i $$-0.721024\pi$$
−0.639899 + 0.768459i $$0.721024\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ 3.46410i 0.182069i
$$363$$ 0 0
$$364$$ −2.00000 −0.104828
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 16.0000i 0.835193i 0.908633 + 0.417597i $$0.137127\pi$$
−0.908633 + 0.417597i $$0.862873\pi$$
$$368$$ − 17.3205i − 0.902894i
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 6.92820 0.359694
$$372$$ 0 0
$$373$$ − 10.0000i − 0.517780i −0.965907 0.258890i $$-0.916643\pi$$
0.965907 0.258890i $$-0.0833568\pi$$
$$374$$ 20.7846 1.07475
$$375$$ 0 0
$$376$$ −12.0000 −0.618853
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 28.0000 1.43826 0.719132 0.694874i $$-0.244540\pi$$
0.719132 + 0.694874i $$0.244540\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 42.0000i 2.14891i
$$383$$ 13.8564i 0.708029i 0.935240 + 0.354015i $$0.115184\pi$$
−0.935240 + 0.354015i $$0.884816\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −24.2487 −1.23423
$$387$$ 0 0
$$388$$ 14.0000i 0.710742i
$$389$$ 13.8564 0.702548 0.351274 0.936273i $$-0.385749\pi$$
0.351274 + 0.936273i $$0.385749\pi$$
$$390$$ 0 0
$$391$$ −12.0000 −0.606866
$$392$$ − 1.73205i − 0.0874818i
$$393$$ 0 0
$$394$$ −36.0000 −1.81365
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 38.0000i − 1.90717i −0.301131 0.953583i $$-0.597364\pi$$
0.301131 0.953583i $$-0.402636\pi$$
$$398$$ 27.7128i 1.38912i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −6.92820 −0.345978 −0.172989 0.984924i $$-0.555343\pi$$
−0.172989 + 0.984924i $$0.555343\pi$$
$$402$$ 0 0
$$403$$ − 8.00000i − 0.398508i
$$404$$ 3.46410 0.172345
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 6.92820i 0.343418i
$$408$$ 0 0
$$409$$ −14.0000 −0.692255 −0.346128 0.938187i $$-0.612504\pi$$
−0.346128 + 0.938187i $$0.612504\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 4.00000i 0.197066i
$$413$$ 6.92820i 0.340915i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 10.3923 0.509525
$$417$$ 0 0
$$418$$ − 24.0000i − 1.17388i
$$419$$ 20.7846 1.01539 0.507697 0.861536i $$-0.330497\pi$$
0.507697 + 0.861536i $$0.330497\pi$$
$$420$$ 0 0
$$421$$ −10.0000 −0.487370 −0.243685 0.969854i $$-0.578356\pi$$
−0.243685 + 0.969854i $$0.578356\pi$$
$$422$$ 34.6410i 1.68630i
$$423$$ 0 0
$$424$$ −12.0000 −0.582772
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 10.0000i 0.483934i
$$428$$ − 17.3205i − 0.837218i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −3.46410 −0.166860 −0.0834300 0.996514i $$-0.526587\pi$$
−0.0834300 + 0.996514i $$0.526587\pi$$
$$432$$ 0 0
$$433$$ 26.0000i 1.24948i 0.780833 + 0.624740i $$0.214795\pi$$
−0.780833 + 0.624740i $$0.785205\pi$$
$$434$$ −6.92820 −0.332564
$$435$$ 0 0
$$436$$ 2.00000 0.0957826
$$437$$ 13.8564i 0.662842i
$$438$$ 0 0
$$439$$ 16.0000 0.763638 0.381819 0.924237i $$-0.375298\pi$$
0.381819 + 0.924237i $$0.375298\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ − 12.0000i − 0.570782i
$$443$$ 3.46410i 0.164584i 0.996608 + 0.0822922i $$0.0262241\pi$$
−0.996608 + 0.0822922i $$0.973776\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −13.8564 −0.656120
$$447$$ 0 0
$$448$$ 1.00000i 0.0472456i
$$449$$ 41.5692 1.96177 0.980886 0.194581i $$-0.0623348\pi$$
0.980886 + 0.194581i $$0.0623348\pi$$
$$450$$ 0 0
$$451$$ 36.0000 1.69517
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 12.0000 0.563188
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 10.0000i 0.467780i 0.972263 + 0.233890i $$0.0751456\pi$$
−0.972263 + 0.233890i $$0.924854\pi$$
$$458$$ 38.1051i 1.78054i
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −31.1769 −1.45205 −0.726027 0.687666i $$-0.758635\pi$$
−0.726027 + 0.687666i $$0.758635\pi$$
$$462$$ 0 0
$$463$$ 32.0000i 1.48717i 0.668644 + 0.743583i $$0.266875\pi$$
−0.668644 + 0.743583i $$0.733125\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 12.0000 0.555889
$$467$$ 6.92820i 0.320599i 0.987068 + 0.160300i $$0.0512460\pi$$
−0.987068 + 0.160300i $$0.948754\pi$$
$$468$$ 0 0
$$469$$ 4.00000 0.184703
$$470$$ 0 0
$$471$$ 0 0
$$472$$ − 12.0000i − 0.552345i
$$473$$ 13.8564i 0.637118i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −3.46410 −0.158777
$$477$$ 0 0
$$478$$ 18.0000i 0.823301i
$$479$$ −6.92820 −0.316558 −0.158279 0.987394i $$-0.550594\pi$$
−0.158279 + 0.987394i $$0.550594\pi$$
$$480$$ 0 0
$$481$$ 4.00000 0.182384
$$482$$ − 17.3205i − 0.788928i
$$483$$ 0 0
$$484$$ −1.00000 −0.0454545
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 40.0000i 1.81257i 0.422664 + 0.906287i $$0.361095\pi$$
−0.422664 + 0.906287i $$0.638905\pi$$
$$488$$ − 17.3205i − 0.784063i
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −10.3923 −0.468998 −0.234499 0.972116i $$-0.575345\pi$$
−0.234499 + 0.972116i $$0.575345\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ −13.8564 −0.623429
$$495$$ 0 0
$$496$$ 20.0000 0.898027
$$497$$ − 10.3923i − 0.466159i
$$498$$ 0 0
$$499$$ 4.00000 0.179065 0.0895323 0.995984i $$-0.471463\pi$$
0.0895323 + 0.995984i $$0.471463\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ − 36.0000i − 1.60676i
$$503$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 20.7846 0.923989
$$507$$ 0 0
$$508$$ 8.00000i 0.354943i
$$509$$ −3.46410 −0.153544 −0.0767718 0.997049i $$-0.524461\pi$$
−0.0767718 + 0.997049i $$0.524461\pi$$
$$510$$ 0 0
$$511$$ 14.0000 0.619324
$$512$$ 8.66025i 0.382733i
$$513$$ 0 0
$$514$$ 6.00000 0.264649
$$515$$ 0 0
$$516$$ 0 0
$$517$$ − 24.0000i − 1.05552i
$$518$$ − 3.46410i − 0.152204i
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −3.46410 −0.151765 −0.0758825 0.997117i $$-0.524177\pi$$
−0.0758825 + 0.997117i $$0.524177\pi$$
$$522$$ 0 0
$$523$$ − 16.0000i − 0.699631i −0.936819 0.349816i $$-0.886244\pi$$
0.936819 0.349816i $$-0.113756\pi$$
$$524$$ −13.8564 −0.605320
$$525$$ 0 0
$$526$$ −30.0000 −1.30806
$$527$$ − 13.8564i − 0.603595i
$$528$$ 0 0
$$529$$ 11.0000 0.478261
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 4.00000i 0.173422i
$$533$$ − 20.7846i − 0.900281i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −6.92820 −0.299253
$$537$$ 0 0
$$538$$ − 30.0000i − 1.29339i
$$539$$ 3.46410 0.149209
$$540$$ 0 0
$$541$$ 14.0000 0.601907 0.300954 0.953639i $$-0.402695\pi$$
0.300954 + 0.953639i $$0.402695\pi$$
$$542$$ 34.6410i 1.48796i
$$543$$ 0 0
$$544$$ 18.0000 0.771744
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 4.00000i 0.171028i 0.996337 + 0.0855138i $$0.0272532\pi$$
−0.996337 + 0.0855138i $$0.972747\pi$$
$$548$$ 6.92820i 0.295958i
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 8.00000i 0.340195i
$$554$$ −17.3205 −0.735878
$$555$$ 0 0
$$556$$ −16.0000 −0.678551
$$557$$ − 6.92820i − 0.293557i −0.989169 0.146779i $$-0.953109\pi$$
0.989169 0.146779i $$-0.0468905\pi$$
$$558$$ 0 0
$$559$$ 8.00000 0.338364
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 36.0000i 1.51857i
$$563$$ − 34.6410i − 1.45994i −0.683477 0.729972i $$-0.739533\pi$$
0.683477 0.729972i $$-0.260467\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 6.92820 0.291214
$$567$$ 0 0
$$568$$ 18.0000i 0.755263i
$$569$$ 6.92820 0.290445 0.145223 0.989399i $$-0.453610\pi$$
0.145223 + 0.989399i $$0.453610\pi$$
$$570$$ 0 0
$$571$$ −28.0000 −1.17176 −0.585882 0.810397i $$-0.699252\pi$$
−0.585882 + 0.810397i $$0.699252\pi$$
$$572$$ 6.92820i 0.289683i
$$573$$ 0 0
$$574$$ −18.0000 −0.751305
$$575$$ 0 0
$$576$$ 0 0
$$577$$ − 2.00000i − 0.0832611i −0.999133 0.0416305i $$-0.986745\pi$$
0.999133 0.0416305i $$-0.0132552\pi$$
$$578$$ 8.66025i 0.360219i
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ − 24.0000i − 0.993978i
$$584$$ −24.2487 −1.00342
$$585$$ 0 0
$$586$$ −18.0000 −0.743573
$$587$$ − 20.7846i − 0.857873i −0.903335 0.428936i $$-0.858888\pi$$
0.903335 0.428936i $$-0.141112\pi$$
$$588$$ 0 0
$$589$$ −16.0000 −0.659269
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 10.0000i 0.410997i
$$593$$ 24.2487i 0.995775i 0.867242 + 0.497888i $$0.165891\pi$$
−0.867242 + 0.497888i $$0.834109\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −6.92820 −0.283790
$$597$$ 0 0
$$598$$ − 12.0000i − 0.490716i
$$599$$ 45.0333 1.84001 0.920006 0.391905i $$-0.128184\pi$$
0.920006 + 0.391905i $$0.128184\pi$$
$$600$$ 0 0
$$601$$ −22.0000 −0.897399 −0.448699 0.893683i $$-0.648113\pi$$
−0.448699 + 0.893683i $$0.648113\pi$$
$$602$$ − 6.92820i − 0.282372i
$$603$$ 0 0
$$604$$ −8.00000 −0.325515
$$605$$ 0 0
$$606$$ 0 0
$$607$$ − 8.00000i − 0.324710i −0.986732 0.162355i $$-0.948091\pi$$
0.986732 0.162355i $$-0.0519090\pi$$
$$608$$ − 20.7846i − 0.842927i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −13.8564 −0.560570
$$612$$ 0 0
$$613$$ 38.0000i 1.53481i 0.641165 + 0.767403i $$0.278451\pi$$
−0.641165 + 0.767403i $$0.721549\pi$$
$$614$$ −48.4974 −1.95720
$$615$$ 0 0
$$616$$ −6.00000 −0.241747
$$617$$ − 20.7846i − 0.836757i −0.908273 0.418378i $$-0.862599\pi$$
0.908273 0.418378i $$-0.137401\pi$$
$$618$$ 0 0
$$619$$ −8.00000 −0.321547 −0.160774 0.986991i $$-0.551399\pi$$
−0.160774 + 0.986991i $$0.551399\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ − 60.0000i − 2.40578i
$$623$$ 3.46410i 0.138786i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −3.46410 −0.138453
$$627$$ 0 0
$$628$$ − 10.0000i − 0.399043i
$$629$$ 6.92820 0.276246
$$630$$ 0 0
$$631$$ 8.00000 0.318475 0.159237 0.987240i $$-0.449096\pi$$
0.159237 + 0.987240i $$0.449096\pi$$
$$632$$ − 13.8564i − 0.551178i
$$633$$ 0 0
$$634$$ −12.0000 −0.476581
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 2.00000i − 0.0792429i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 48.4974 1.91553 0.957767 0.287547i $$-0.0928398\pi$$
0.957767 + 0.287547i $$0.0928398\pi$$
$$642$$ 0 0
$$643$$ 20.0000i 0.788723i 0.918955 + 0.394362i $$0.129034\pi$$
−0.918955 + 0.394362i $$0.870966\pi$$
$$644$$ −3.46410 −0.136505
$$645$$ 0 0
$$646$$ −24.0000 −0.944267
$$647$$ − 6.92820i − 0.272376i −0.990683 0.136188i $$-0.956515\pi$$
0.990683 0.136188i $$-0.0434851\pi$$
$$648$$ 0 0
$$649$$ 24.0000 0.942082
$$650$$ 0 0
$$651$$ 0 0
$$652$$ − 20.0000i − 0.783260i
$$653$$ − 27.7128i − 1.08449i −0.840222 0.542243i $$-0.817575\pi$$
0.840222 0.542243i $$-0.182425\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 51.9615 2.02876
$$657$$ 0 0
$$658$$ 12.0000i 0.467809i
$$659$$ −10.3923 −0.404827 −0.202413 0.979300i $$-0.564878\pi$$
−0.202413 + 0.979300i $$0.564878\pi$$
$$660$$ 0 0
$$661$$ −10.0000 −0.388955 −0.194477 0.980907i $$-0.562301\pi$$
−0.194477 + 0.980907i $$0.562301\pi$$
$$662$$ 34.6410i 1.34636i
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 20.7846i 0.804181i
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 34.6410 1.33730
$$672$$ 0 0
$$673$$ − 10.0000i − 0.385472i −0.981251 0.192736i $$-0.938264\pi$$
0.981251 0.192736i $$-0.0617360\pi$$
$$674$$ 24.2487 0.934025
$$675$$ 0 0
$$676$$ −9.00000 −0.346154
$$677$$ − 24.2487i − 0.931954i −0.884797 0.465977i $$-0.845703\pi$$
0.884797 0.465977i $$-0.154297\pi$$
$$678$$ 0 0
$$679$$ −14.0000 −0.537271
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 24.0000i 0.919007i
$$683$$ − 24.2487i − 0.927851i −0.885874 0.463926i $$-0.846441\pi$$
0.885874 0.463926i $$-0.153559\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −1.73205 −0.0661300
$$687$$ 0 0
$$688$$ 20.0000i 0.762493i
$$689$$ −13.8564 −0.527887
$$690$$ 0 0
$$691$$ 8.00000 0.304334 0.152167 0.988355i $$-0.451375\pi$$
0.152167 + 0.988355i $$0.451375\pi$$
$$692$$ 17.3205i 0.658427i
$$693$$ 0 0
$$694$$ 30.0000 1.13878
$$695$$ 0 0
$$696$$ 0 0
$$697$$ − 36.0000i − 1.36360i
$$698$$ − 24.2487i − 0.917827i
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$702$$ 0 0
$$703$$ − 8.00000i − 0.301726i
$$704$$ 3.46410 0.130558
$$705$$ 0 0
$$706$$ 6.00000 0.225813
$$707$$ 3.46410i 0.130281i
$$708$$ 0 0
$$709$$ −26.0000 −0.976450 −0.488225 0.872718i $$-0.662356\pi$$
−0.488225 + 0.872718i $$0.662356\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ − 6.00000i − 0.224860i
$$713$$ − 13.8564i − 0.518927i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 17.3205 0.647298
$$717$$ 0 0
$$718$$ − 42.0000i − 1.56743i
$$719$$ 27.7128 1.03351 0.516757 0.856132i $$-0.327139\pi$$
0.516757 + 0.856132i $$0.327139\pi$$
$$720$$ 0 0
$$721$$ −4.00000 −0.148968
$$722$$ − 5.19615i − 0.193381i
$$723$$ 0 0
$$724$$ −2.00000 −0.0743294
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 4.00000i 0.148352i 0.997245 + 0.0741759i $$0.0236326\pi$$
−0.997245 + 0.0741759i $$0.976367\pi$$
$$728$$ 3.46410i 0.128388i
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 13.8564 0.512498
$$732$$ 0 0
$$733$$ − 22.0000i − 0.812589i −0.913742 0.406294i $$-0.866821\pi$$
0.913742 0.406294i $$-0.133179\pi$$
$$734$$ −27.7128 −1.02290
$$735$$ 0 0
$$736$$ 18.0000 0.663489
$$737$$ − 13.8564i − 0.510407i
$$738$$ 0 0
$$739$$ −20.0000 −0.735712 −0.367856 0.929883i $$-0.619908\pi$$
−0.367856 + 0.929883i $$0.619908\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 12.0000i 0.440534i
$$743$$ 10.3923i 0.381257i 0.981662 + 0.190628i $$0.0610525\pi$$
−0.981662 + 0.190628i $$0.938947\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 17.3205 0.634149
$$747$$ 0 0
$$748$$ 12.0000i 0.438763i
$$749$$ 17.3205 0.632878
$$750$$ 0 0
$$751$$ 8.00000 0.291924 0.145962 0.989290i $$-0.453372\pi$$
0.145962 + 0.989290i $$0.453372\pi$$
$$752$$ − 34.6410i − 1.26323i
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 26.0000i − 0.944986i −0.881334 0.472493i $$-0.843354\pi$$
0.881334 0.472493i $$-0.156646\pi$$
$$758$$ 48.4974i 1.76151i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −38.1051 −1.38131 −0.690655 0.723185i $$-0.742678\pi$$
−0.690655 + 0.723185i $$0.742678\pi$$
$$762$$ 0 0
$$763$$ 2.00000i 0.0724049i
$$764$$ −24.2487 −0.877288
$$765$$ 0 0
$$766$$ −24.0000 −0.867155
$$767$$ − 13.8564i − 0.500326i
$$768$$ 0 0
$$769$$ 22.0000 0.793340 0.396670 0.917961i $$-0.370166\pi$$
0.396670 + 0.917961i $$0.370166\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ − 14.0000i − 0.503871i
$$773$$ − 45.0333i − 1.61974i −0.586612 0.809868i $$-0.699539\pi$$
0.586612 0.809868i $$-0.300461\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 24.2487 0.870478
$$777$$ 0 0
$$778$$ 24.0000i 0.860442i
$$779$$ −41.5692 −1.48937
$$780$$ 0 0
$$781$$ −36.0000 −1.28818
$$782$$ − 20.7846i − 0.743256i
$$783$$ 0 0
$$784$$ 5.00000 0.178571
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 32.0000i − 1.14068i −0.821410 0.570338i $$-0.806812\pi$$
0.821410 0.570338i $$-0.193188\pi$$
$$788$$ − 20.7846i − 0.740421i
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ − 20.0000i − 0.710221i
$$794$$ 65.8179 2.33579
$$795$$ 0 0
$$796$$ −16.0000 −0.567105
$$797$$ 31.1769i 1.10434i 0.833731 + 0.552171i $$0.186201\pi$$
−0.833731 + 0.552171i $$0.813799\pi$$
$$798$$ 0 0
$$799$$ −24.0000 −0.849059
$$800$$ 0 0
$$801$$ 0 0
$$802$$ − 12.0000i − 0.423735i
$$803$$ − 48.4974i − 1.71144i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 13.8564 0.488071
$$807$$ 0 0
$$808$$ − 6.00000i − 0.211079i
$$809$$ −27.7128 −0.974331 −0.487165 0.873310i $$-0.661969\pi$$
−0.487165 + 0.873310i $$0.661969\pi$$
$$810$$ 0 0
$$811$$ −16.0000 −0.561836 −0.280918 0.959732i $$-0.590639\pi$$
−0.280918 + 0.959732i $$0.590639\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ −12.0000 −0.420600
$$815$$ 0 0
$$816$$ 0 0
$$817$$ − 16.0000i − 0.559769i
$$818$$ − 24.2487i − 0.847836i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −6.92820 −0.241796 −0.120898 0.992665i $$-0.538577\pi$$
−0.120898 + 0.992665i $$0.538577\pi$$
$$822$$ 0 0
$$823$$ 32.0000i 1.11545i 0.830026 + 0.557725i $$0.188326\pi$$
−0.830026 + 0.557725i $$0.811674\pi$$
$$824$$ 6.92820 0.241355
$$825$$ 0 0
$$826$$ −12.0000 −0.417533
$$827$$ − 10.3923i − 0.361376i −0.983540 0.180688i $$-0.942168\pi$$
0.983540 0.180688i $$-0.0578324\pi$$
$$828$$ 0 0
$$829$$ −50.0000 −1.73657 −0.868286 0.496064i $$-0.834778\pi$$
−0.868286 + 0.496064i $$0.834778\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ − 2.00000i − 0.0693375i
$$833$$ − 3.46410i − 0.120024i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 13.8564 0.479234
$$837$$ 0 0
$$838$$ 36.0000i 1.24360i
$$839$$ 20.7846 0.717564 0.358782 0.933421i $$-0.383192\pi$$
0.358782 + 0.933421i $$0.383192\pi$$
$$840$$ 0 0
$$841$$ −29.0000 −1.00000
$$842$$ − 17.3205i − 0.596904i
$$843$$ 0 0
$$844$$ −20.0000 −0.688428
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 1.00000i − 0.0343604i
$$848$$ − 34.6410i − 1.18958i
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 6.92820 0.237496
$$852$$ 0 0
$$853$$ − 10.0000i − 0.342393i −0.985237 0.171197i $$-0.945237\pi$$
0.985237 0.171197i $$-0.0547634\pi$$
$$854$$ −17.3205 −0.592696
$$855$$ 0 0
$$856$$ −30.0000 −1.02538
$$857$$ − 17.3205i − 0.591657i −0.955241 0.295829i $$-0.904404\pi$$
0.955241 0.295829i $$-0.0955957\pi$$
$$858$$ 0 0
$$859$$ 4.00000 0.136478 0.0682391 0.997669i $$-0.478262\pi$$
0.0682391 + 0.997669i $$0.478262\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ − 6.00000i − 0.204361i
$$863$$ − 38.1051i − 1.29711i −0.761166 0.648557i $$-0.775373\pi$$
0.761166 0.648557i $$-0.224627\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −45.0333 −1.53029
$$867$$ 0 0
$$868$$ − 4.00000i − 0.135769i
$$869$$ 27.7128 0.940093
$$870$$ 0 0
$$871$$ −8.00000 −0.271070
$$872$$ − 3.46410i − 0.117309i
$$873$$ 0 0
$$874$$ −24.0000 −0.811812
$$875$$ 0 0
$$876$$ 0 0
$$877$$ − 14.0000i − 0.472746i −0.971662 0.236373i $$-0.924041\pi$$
0.971662 0.236373i $$-0.0759588\pi$$
$$878$$ 27.7128i 0.935262i
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 51.9615 1.75063 0.875314 0.483555i $$-0.160655\pi$$
0.875314 + 0.483555i $$0.160655\pi$$
$$882$$ 0 0
$$883$$ − 52.0000i − 1.74994i −0.484178 0.874970i $$-0.660881\pi$$
0.484178 0.874970i $$-0.339119\pi$$
$$884$$ 6.92820 0.233021
$$885$$ 0 0
$$886$$ −6.00000 −0.201574
$$887$$ 6.92820i 0.232626i 0.993213 + 0.116313i $$0.0371076\pi$$
−0.993213 + 0.116313i $$0.962892\pi$$
$$888$$ 0 0
$$889$$ −8.00000 −0.268311
$$890$$ 0 0
$$891$$ 0 0
$$892$$ − 8.00000i − 0.267860i
$$893$$ 27.7128i 0.927374i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −12.1244 −0.405046
$$897$$ 0 0
$$898$$ 72.0000i 2.40267i
$$899$$ 0 0
$$900$$ 0 0
$$901$$ −24.0000 −0.799556
$$902$$ 62.3538i 2.07616i
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 28.0000i 0.929725i 0.885383 + 0.464862i $$0.153896\pi$$
−0.885383 + 0.464862i $$0.846104\pi$$
$$908$$ 6.92820i 0.229920i
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −31.1769 −1.03294 −0.516469 0.856306i $$-0.672754\pi$$
−0.516469 + 0.856306i $$0.672754\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ −17.3205 −0.572911
$$915$$ 0 0
$$916$$ −22.0000 −0.726900
$$917$$ − 13.8564i − 0.457579i
$$918$$ 0 0
$$919$$ 16.0000 0.527791 0.263896 0.964551i $$-0.414993\pi$$
0.263896 + 0.964551i $$0.414993\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ − 54.0000i − 1.77840i
$$923$$ 20.7846i 0.684134i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −55.4256 −1.82140
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −45.0333 −1.47750 −0.738748 0.673982i $$-0.764582\pi$$
−0.738748 + 0.673982i $$0.764582\pi$$
$$930$$ 0 0
$$931$$ −4.00000 −0.131095
$$932$$ 6.92820i 0.226941i
$$933$$ 0 0
$$934$$ −12.0000 −0.392652
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 46.0000i 1.50275i 0.659873 + 0.751377i $$0.270610\pi$$
−0.659873 + 0.751377i $$0.729390\pi$$
$$938$$ 6.92820i 0.226214i
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 17.3205 0.564632 0.282316 0.959321i $$-0.408897\pi$$
0.282316 + 0.959321i $$0.408897\pi$$
$$942$$ 0 0
$$943$$ − 36.0000i − 1.17232i
$$944$$ 34.6410 1.12747
$$945$$ 0 0
$$946$$ −24.0000 −0.780307
$$947$$ − 24.2487i − 0.787977i −0.919115 0.393989i $$-0.871095\pi$$
0.919115 0.393989i $$-0.128905\pi$$
$$948$$ 0 0
$$949$$ −28.0000 −0.908918
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 6.00000i 0.194461i
$$953$$ 41.5692i 1.34656i 0.739388 + 0.673280i $$0.235115\pi$$
−0.739388 + 0.673280i $$0.764885\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −10.3923 −0.336111
$$957$$ 0 0
$$958$$ − 12.0000i − 0.387702i
$$959$$ −6.92820 −0.223723
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 6.92820i 0.223374i
$$963$$ 0 0
$$964$$ 10.0000 0.322078
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 16.0000i 0.514525i 0.966342 + 0.257263i $$0.0828206\pi$$
−0.966342 + 0.257263i $$0.917179\pi$$
$$968$$ 1.73205i 0.0556702i
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −27.7128 −0.889346 −0.444673 0.895693i $$-0.646680\pi$$
−0.444673 + 0.895693i $$0.646680\pi$$
$$972$$ 0 0
$$973$$ − 16.0000i − 0.512936i
$$974$$ −69.2820 −2.21994
$$975$$ 0 0
$$976$$ 50.0000 1.60046
$$977$$ 34.6410i 1.10826i 0.832429 + 0.554132i $$0.186950\pi$$
−0.832429 + 0.554132i $$0.813050\pi$$
$$978$$ 0 0
$$979$$ 12.0000 0.383522
$$980$$ 0 0
$$981$$ 0 0
$$982$$ − 18.0000i − 0.574403i
$$983$$ − 13.8564i − 0.441951i −0.975279 0.220975i $$-0.929076\pi$$
0.975279 0.220975i $$-0.0709240\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ − 8.00000i − 0.254514i
$$989$$ 13.8564 0.440608
$$990$$ 0 0
$$991$$ 56.0000 1.77890 0.889449 0.457034i $$-0.151088\pi$$
0.889449 + 0.457034i $$0.151088\pi$$
$$992$$ 20.7846i 0.659912i
$$993$$ 0 0
$$994$$ 18.0000 0.570925
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 10.0000i 0.316703i 0.987383 + 0.158352i $$0.0506179\pi$$
−0.987383 + 0.158352i $$0.949382\pi$$
$$998$$ 6.92820i 0.219308i
$$999$$ 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 1575.2.d.i.1324.3 4
3.2 odd 2 inner 1575.2.d.i.1324.1 4
5.2 odd 4 63.2.a.b.1.1 2
5.3 odd 4 1575.2.a.q.1.2 2
5.4 even 2 inner 1575.2.d.i.1324.2 4
15.2 even 4 63.2.a.b.1.2 yes 2
15.8 even 4 1575.2.a.q.1.1 2
15.14 odd 2 inner 1575.2.d.i.1324.4 4
20.7 even 4 1008.2.a.n.1.2 2
35.2 odd 12 441.2.e.j.361.2 4
35.12 even 12 441.2.e.i.361.2 4
35.17 even 12 441.2.e.i.226.2 4
35.27 even 4 441.2.a.g.1.1 2
35.32 odd 12 441.2.e.j.226.2 4
40.27 even 4 4032.2.a.bq.1.1 2
40.37 odd 4 4032.2.a.bt.1.1 2
45.2 even 12 567.2.f.j.190.1 4
45.7 odd 12 567.2.f.j.190.2 4
45.22 odd 12 567.2.f.j.379.2 4
45.32 even 12 567.2.f.j.379.1 4
55.32 even 4 7623.2.a.bi.1.2 2
60.47 odd 4 1008.2.a.n.1.1 2
105.2 even 12 441.2.e.j.361.1 4
105.17 odd 12 441.2.e.i.226.1 4
105.32 even 12 441.2.e.j.226.1 4
105.47 odd 12 441.2.e.i.361.1 4
105.62 odd 4 441.2.a.g.1.2 2
120.77 even 4 4032.2.a.bt.1.2 2
120.107 odd 4 4032.2.a.bq.1.2 2
140.27 odd 4 7056.2.a.cm.1.1 2
165.32 odd 4 7623.2.a.bi.1.1 2
420.167 even 4 7056.2.a.cm.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.a.b.1.1 2 5.2 odd 4
63.2.a.b.1.2 yes 2 15.2 even 4
441.2.a.g.1.1 2 35.27 even 4
441.2.a.g.1.2 2 105.62 odd 4
441.2.e.i.226.1 4 105.17 odd 12
441.2.e.i.226.2 4 35.17 even 12
441.2.e.i.361.1 4 105.47 odd 12
441.2.e.i.361.2 4 35.12 even 12
441.2.e.j.226.1 4 105.32 even 12
441.2.e.j.226.2 4 35.32 odd 12
441.2.e.j.361.1 4 105.2 even 12
441.2.e.j.361.2 4 35.2 odd 12
567.2.f.j.190.1 4 45.2 even 12
567.2.f.j.190.2 4 45.7 odd 12
567.2.f.j.379.1 4 45.32 even 12
567.2.f.j.379.2 4 45.22 odd 12
1008.2.a.n.1.1 2 60.47 odd 4
1008.2.a.n.1.2 2 20.7 even 4
1575.2.a.q.1.1 2 15.8 even 4
1575.2.a.q.1.2 2 5.3 odd 4
1575.2.d.i.1324.1 4 3.2 odd 2 inner
1575.2.d.i.1324.2 4 5.4 even 2 inner
1575.2.d.i.1324.3 4 1.1 even 1 trivial
1575.2.d.i.1324.4 4 15.14 odd 2 inner
4032.2.a.bq.1.1 2 40.27 even 4
4032.2.a.bq.1.2 2 120.107 odd 4
4032.2.a.bt.1.1 2 40.37 odd 4
4032.2.a.bt.1.2 2 120.77 even 4
7056.2.a.cm.1.1 2 140.27 odd 4
7056.2.a.cm.1.2 2 420.167 even 4
7623.2.a.bi.1.1 2 165.32 odd 4
7623.2.a.bi.1.2 2 55.32 even 4