# Properties

 Label 450.6.c.e.199.2 Level $450$ Weight $6$ Character 450.199 Analytic conductor $72.173$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$450 = 2 \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 450.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$72.1727189158$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 150) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 199.2 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 450.199 Dual form 450.6.c.e.199.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+4.00000i q^{2} -16.0000 q^{4} -47.0000i q^{7} -64.0000i q^{8} +O(q^{10})$$ $$q+4.00000i q^{2} -16.0000 q^{4} -47.0000i q^{7} -64.0000i q^{8} -222.000 q^{11} +101.000i q^{13} +188.000 q^{14} +256.000 q^{16} +162.000i q^{17} -1685.00 q^{19} -888.000i q^{22} -306.000i q^{23} -404.000 q^{26} +752.000i q^{28} +7890.00 q^{29} -8593.00 q^{31} +1024.00i q^{32} -648.000 q^{34} -8642.00i q^{37} -6740.00i q^{38} +18168.0 q^{41} +14351.0i q^{43} +3552.00 q^{44} +1224.00 q^{46} -1098.00i q^{47} +14598.0 q^{49} -1616.00i q^{52} -17916.0i q^{53} -3008.00 q^{56} +31560.0i q^{58} +17610.0 q^{59} -21853.0 q^{61} -34372.0i q^{62} -4096.00 q^{64} -107.000i q^{67} -2592.00i q^{68} +40728.0 q^{71} +34706.0i q^{73} +34568.0 q^{74} +26960.0 q^{76} +10434.0i q^{77} +69160.0 q^{79} +72672.0i q^{82} +108534. i q^{83} -57404.0 q^{86} +14208.0i q^{88} +35040.0 q^{89} +4747.00 q^{91} +4896.00i q^{92} +4392.00 q^{94} +823.000i q^{97} +58392.0i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 32q^{4} + O(q^{10})$$ $$2q - 32q^{4} - 444q^{11} + 376q^{14} + 512q^{16} - 3370q^{19} - 808q^{26} + 15780q^{29} - 17186q^{31} - 1296q^{34} + 36336q^{41} + 7104q^{44} + 2448q^{46} + 29196q^{49} - 6016q^{56} + 35220q^{59} - 43706q^{61} - 8192q^{64} + 81456q^{71} + 69136q^{74} + 53920q^{76} + 138320q^{79} - 114808q^{86} + 70080q^{89} + 9494q^{91} + 8784q^{94} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/450\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 4.00000i 0.707107i
$$3$$ 0 0
$$4$$ −16.0000 −0.500000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ − 47.0000i − 0.362537i −0.983434 0.181269i $$-0.941980\pi$$
0.983434 0.181269i $$-0.0580204\pi$$
$$8$$ − 64.0000i − 0.353553i
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −222.000 −0.553186 −0.276593 0.960987i $$-0.589205\pi$$
−0.276593 + 0.960987i $$0.589205\pi$$
$$12$$ 0 0
$$13$$ 101.000i 0.165754i 0.996560 + 0.0828768i $$0.0264108\pi$$
−0.996560 + 0.0828768i $$0.973589\pi$$
$$14$$ 188.000 0.256353
$$15$$ 0 0
$$16$$ 256.000 0.250000
$$17$$ 162.000i 0.135954i 0.997687 + 0.0679771i $$0.0216545\pi$$
−0.997687 + 0.0679771i $$0.978346\pi$$
$$18$$ 0 0
$$19$$ −1685.00 −1.07082 −0.535409 0.844593i $$-0.679843\pi$$
−0.535409 + 0.844593i $$0.679843\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ − 888.000i − 0.391162i
$$23$$ − 306.000i − 0.120615i −0.998180 0.0603076i $$-0.980792\pi$$
0.998180 0.0603076i $$-0.0192082\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −404.000 −0.117206
$$27$$ 0 0
$$28$$ 752.000i 0.181269i
$$29$$ 7890.00 1.74214 0.871068 0.491163i $$-0.163428\pi$$
0.871068 + 0.491163i $$0.163428\pi$$
$$30$$ 0 0
$$31$$ −8593.00 −1.60598 −0.802991 0.595991i $$-0.796759\pi$$
−0.802991 + 0.595991i $$0.796759\pi$$
$$32$$ 1024.00i 0.176777i
$$33$$ 0 0
$$34$$ −648.000 −0.0961342
$$35$$ 0 0
$$36$$ 0 0
$$37$$ − 8642.00i − 1.03779i −0.854838 0.518896i $$-0.826343\pi$$
0.854838 0.518896i $$-0.173657\pi$$
$$38$$ − 6740.00i − 0.757183i
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 18168.0 1.68790 0.843951 0.536420i $$-0.180223\pi$$
0.843951 + 0.536420i $$0.180223\pi$$
$$42$$ 0 0
$$43$$ 14351.0i 1.18362i 0.806079 + 0.591808i $$0.201586\pi$$
−0.806079 + 0.591808i $$0.798414\pi$$
$$44$$ 3552.00 0.276593
$$45$$ 0 0
$$46$$ 1224.00 0.0852878
$$47$$ − 1098.00i − 0.0725033i −0.999343 0.0362516i $$-0.988458\pi$$
0.999343 0.0362516i $$-0.0115418\pi$$
$$48$$ 0 0
$$49$$ 14598.0 0.868567
$$50$$ 0 0
$$51$$ 0 0
$$52$$ − 1616.00i − 0.0828768i
$$53$$ − 17916.0i − 0.876095i −0.898952 0.438048i $$-0.855670\pi$$
0.898952 0.438048i $$-0.144330\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −3008.00 −0.128176
$$57$$ 0 0
$$58$$ 31560.0i 1.23188i
$$59$$ 17610.0 0.658612 0.329306 0.944223i $$-0.393185\pi$$
0.329306 + 0.944223i $$0.393185\pi$$
$$60$$ 0 0
$$61$$ −21853.0 −0.751946 −0.375973 0.926631i $$-0.622691\pi$$
−0.375973 + 0.926631i $$0.622691\pi$$
$$62$$ − 34372.0i − 1.13560i
$$63$$ 0 0
$$64$$ −4096.00 −0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 107.000i − 0.00291204i −0.999999 0.00145602i $$-0.999537\pi$$
0.999999 0.00145602i $$-0.000463465\pi$$
$$68$$ − 2592.00i − 0.0679771i
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 40728.0 0.958842 0.479421 0.877585i $$-0.340847\pi$$
0.479421 + 0.877585i $$0.340847\pi$$
$$72$$ 0 0
$$73$$ 34706.0i 0.762250i 0.924524 + 0.381125i $$0.124463\pi$$
−0.924524 + 0.381125i $$0.875537\pi$$
$$74$$ 34568.0 0.733829
$$75$$ 0 0
$$76$$ 26960.0 0.535409
$$77$$ 10434.0i 0.200551i
$$78$$ 0 0
$$79$$ 69160.0 1.24677 0.623386 0.781914i $$-0.285756\pi$$
0.623386 + 0.781914i $$0.285756\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 72672.0i 1.19353i
$$83$$ 108534.i 1.72930i 0.502374 + 0.864650i $$0.332460\pi$$
−0.502374 + 0.864650i $$0.667540\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −57404.0 −0.836943
$$87$$ 0 0
$$88$$ 14208.0i 0.195581i
$$89$$ 35040.0 0.468910 0.234455 0.972127i $$-0.424670\pi$$
0.234455 + 0.972127i $$0.424670\pi$$
$$90$$ 0 0
$$91$$ 4747.00 0.0600919
$$92$$ 4896.00i 0.0603076i
$$93$$ 0 0
$$94$$ 4392.00 0.0512676
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 823.000i 0.00888118i 0.999990 + 0.00444059i $$0.00141349\pi$$
−0.999990 + 0.00444059i $$0.998587\pi$$
$$98$$ 58392.0i 0.614169i
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 33828.0 0.329969 0.164984 0.986296i $$-0.447243\pi$$
0.164984 + 0.986296i $$0.447243\pi$$
$$102$$ 0 0
$$103$$ − 133444.i − 1.23938i −0.784845 0.619692i $$-0.787257\pi$$
0.784845 0.619692i $$-0.212743\pi$$
$$104$$ 6464.00 0.0586028
$$105$$ 0 0
$$106$$ 71664.0 0.619493
$$107$$ 81252.0i 0.686080i 0.939321 + 0.343040i $$0.111457\pi$$
−0.939321 + 0.343040i $$0.888543\pi$$
$$108$$ 0 0
$$109$$ 217015. 1.74954 0.874769 0.484540i $$-0.161013\pi$$
0.874769 + 0.484540i $$0.161013\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ − 12032.0i − 0.0906343i
$$113$$ 138324.i 1.01906i 0.860452 + 0.509532i $$0.170181\pi$$
−0.860452 + 0.509532i $$0.829819\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −126240. −0.871068
$$117$$ 0 0
$$118$$ 70440.0i 0.465709i
$$119$$ 7614.00 0.0492885
$$120$$ 0 0
$$121$$ −111767. −0.693985
$$122$$ − 87412.0i − 0.531706i
$$123$$ 0 0
$$124$$ 137488. 0.802991
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 256048.i 1.40868i 0.709863 + 0.704340i $$0.248757\pi$$
−0.709863 + 0.704340i $$0.751243\pi$$
$$128$$ − 16384.0i − 0.0883883i
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −118452. −0.603065 −0.301533 0.953456i $$-0.597498\pi$$
−0.301533 + 0.953456i $$0.597498\pi$$
$$132$$ 0 0
$$133$$ 79195.0i 0.388212i
$$134$$ 428.000 0.00205912
$$135$$ 0 0
$$136$$ 10368.0 0.0480671
$$137$$ − 13218.0i − 0.0601678i −0.999547 0.0300839i $$-0.990423\pi$$
0.999547 0.0300839i $$-0.00957745\pi$$
$$138$$ 0 0
$$139$$ 350740. 1.53974 0.769872 0.638199i $$-0.220320\pi$$
0.769872 + 0.638199i $$0.220320\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 162912.i 0.678004i
$$143$$ − 22422.0i − 0.0916926i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ −138824. −0.538992
$$147$$ 0 0
$$148$$ 138272.i 0.518896i
$$149$$ 109890. 0.405502 0.202751 0.979230i $$-0.435012\pi$$
0.202751 + 0.979230i $$0.435012\pi$$
$$150$$ 0 0
$$151$$ −172603. −0.616036 −0.308018 0.951381i $$-0.599666\pi$$
−0.308018 + 0.951381i $$0.599666\pi$$
$$152$$ 107840.i 0.378592i
$$153$$ 0 0
$$154$$ −41736.0 −0.141811
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 349993.i 1.13321i 0.823990 + 0.566605i $$0.191743\pi$$
−0.823990 + 0.566605i $$0.808257\pi$$
$$158$$ 276640.i 0.881601i
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −14382.0 −0.0437275
$$162$$ 0 0
$$163$$ 192581.i 0.567733i 0.958864 + 0.283867i $$0.0916173\pi$$
−0.958864 + 0.283867i $$0.908383\pi$$
$$164$$ −290688. −0.843951
$$165$$ 0 0
$$166$$ −434136. −1.22280
$$167$$ 580692.i 1.61122i 0.592447 + 0.805610i $$0.298162\pi$$
−0.592447 + 0.805610i $$0.701838\pi$$
$$168$$ 0 0
$$169$$ 361092. 0.972526
$$170$$ 0 0
$$171$$ 0 0
$$172$$ − 229616.i − 0.591808i
$$173$$ − 738126.i − 1.87506i −0.347904 0.937530i $$-0.613106\pi$$
0.347904 0.937530i $$-0.386894\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −56832.0 −0.138297
$$177$$ 0 0
$$178$$ 140160.i 0.331569i
$$179$$ 497370. 1.16024 0.580119 0.814532i $$-0.303006\pi$$
0.580119 + 0.814532i $$0.303006\pi$$
$$180$$ 0 0
$$181$$ −333163. −0.755893 −0.377947 0.925827i $$-0.623370\pi$$
−0.377947 + 0.925827i $$0.623370\pi$$
$$182$$ 18988.0i 0.0424914i
$$183$$ 0 0
$$184$$ −19584.0 −0.0426439
$$185$$ 0 0
$$186$$ 0 0
$$187$$ − 35964.0i − 0.0752080i
$$188$$ 17568.0i 0.0362516i
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 40638.0 0.0806026 0.0403013 0.999188i $$-0.487168\pi$$
0.0403013 + 0.999188i $$0.487168\pi$$
$$192$$ 0 0
$$193$$ 494651.i 0.955885i 0.878391 + 0.477942i $$0.158617\pi$$
−0.878391 + 0.477942i $$0.841383\pi$$
$$194$$ −3292.00 −0.00627994
$$195$$ 0 0
$$196$$ −233568. −0.434283
$$197$$ 552342.i 1.01401i 0.861943 + 0.507005i $$0.169248\pi$$
−0.861943 + 0.507005i $$0.830752\pi$$
$$198$$ 0 0
$$199$$ −685625. −1.22731 −0.613655 0.789575i $$-0.710301\pi$$
−0.613655 + 0.789575i $$0.710301\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 135312.i 0.233323i
$$203$$ − 370830.i − 0.631589i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 533776. 0.876377
$$207$$ 0 0
$$208$$ 25856.0i 0.0414384i
$$209$$ 374070. 0.592362
$$210$$ 0 0
$$211$$ 749477. 1.15892 0.579458 0.815002i $$-0.303264\pi$$
0.579458 + 0.815002i $$0.303264\pi$$
$$212$$ 286656.i 0.438048i
$$213$$ 0 0
$$214$$ −325008. −0.485132
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 403871.i 0.582228i
$$218$$ 868060.i 1.23711i
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −16362.0 −0.0225349
$$222$$ 0 0
$$223$$ 169271.i 0.227940i 0.993484 + 0.113970i $$0.0363568\pi$$
−0.993484 + 0.113970i $$0.963643\pi$$
$$224$$ 48128.0 0.0640882
$$225$$ 0 0
$$226$$ −553296. −0.720587
$$227$$ − 46488.0i − 0.0598792i −0.999552 0.0299396i $$-0.990468\pi$$
0.999552 0.0299396i $$-0.00953150\pi$$
$$228$$ 0 0
$$229$$ 90115.0 0.113556 0.0567778 0.998387i $$-0.481917\pi$$
0.0567778 + 0.998387i $$0.481917\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ − 504960.i − 0.615938i
$$233$$ − 1.06414e6i − 1.28413i −0.766652 0.642063i $$-0.778079\pi$$
0.766652 0.642063i $$-0.221921\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −281760. −0.329306
$$237$$ 0 0
$$238$$ 30456.0i 0.0348522i
$$239$$ 1.15158e6 1.30407 0.652033 0.758191i $$-0.273916\pi$$
0.652033 + 0.758191i $$0.273916\pi$$
$$240$$ 0 0
$$241$$ 856217. 0.949601 0.474801 0.880093i $$-0.342520\pi$$
0.474801 + 0.880093i $$0.342520\pi$$
$$242$$ − 447068.i − 0.490722i
$$243$$ 0 0
$$244$$ 349648. 0.375973
$$245$$ 0 0
$$246$$ 0 0
$$247$$ − 170185.i − 0.177492i
$$248$$ 549952.i 0.567800i
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 207708. 0.208098 0.104049 0.994572i $$-0.466820\pi$$
0.104049 + 0.994572i $$0.466820\pi$$
$$252$$ 0 0
$$253$$ 67932.0i 0.0667226i
$$254$$ −1.02419e6 −0.996087
$$255$$ 0 0
$$256$$ 65536.0 0.0625000
$$257$$ − 1.45319e6i − 1.37243i −0.727401 0.686213i $$-0.759272\pi$$
0.727401 0.686213i $$-0.240728\pi$$
$$258$$ 0 0
$$259$$ −406174. −0.376238
$$260$$ 0 0
$$261$$ 0 0
$$262$$ − 473808.i − 0.426431i
$$263$$ − 169296.i − 0.150924i −0.997149 0.0754618i $$-0.975957\pi$$
0.997149 0.0754618i $$-0.0240431\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −316780. −0.274507
$$267$$ 0 0
$$268$$ 1712.00i 0.00145602i
$$269$$ −1.58109e6 −1.33222 −0.666110 0.745854i $$-0.732042\pi$$
−0.666110 + 0.745854i $$0.732042\pi$$
$$270$$ 0 0
$$271$$ 822512. 0.680329 0.340165 0.940366i $$-0.389517\pi$$
0.340165 + 0.940366i $$0.389517\pi$$
$$272$$ 41472.0i 0.0339886i
$$273$$ 0 0
$$274$$ 52872.0 0.0425451
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 546823.i 0.428201i 0.976812 + 0.214100i $$0.0686820\pi$$
−0.976812 + 0.214100i $$0.931318\pi$$
$$278$$ 1.40296e6i 1.08876i
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 1.09250e6 0.825382 0.412691 0.910871i $$-0.364589\pi$$
0.412691 + 0.910871i $$0.364589\pi$$
$$282$$ 0 0
$$283$$ − 2.48480e6i − 1.84427i −0.386865 0.922136i $$-0.626442\pi$$
0.386865 0.922136i $$-0.373558\pi$$
$$284$$ −651648. −0.479421
$$285$$ 0 0
$$286$$ 89688.0 0.0648365
$$287$$ − 853896.i − 0.611928i
$$288$$ 0 0
$$289$$ 1.39361e6 0.981516
$$290$$ 0 0
$$291$$ 0 0
$$292$$ − 555296.i − 0.381125i
$$293$$ 341394.i 0.232320i 0.993231 + 0.116160i $$0.0370586\pi$$
−0.993231 + 0.116160i $$0.962941\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −553088. −0.366915
$$297$$ 0 0
$$298$$ 439560.i 0.286733i
$$299$$ 30906.0 0.0199924
$$300$$ 0 0
$$301$$ 674497. 0.429105
$$302$$ − 690412.i − 0.435603i
$$303$$ 0 0
$$304$$ −431360. −0.267705
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 2.02898e6i − 1.22866i −0.789050 0.614329i $$-0.789427\pi$$
0.789050 0.614329i $$-0.210573\pi$$
$$308$$ − 166944.i − 0.100275i
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 206598. 0.121123 0.0605613 0.998164i $$-0.480711\pi$$
0.0605613 + 0.998164i $$0.480711\pi$$
$$312$$ 0 0
$$313$$ − 3.34223e6i − 1.92830i −0.265352 0.964152i $$-0.585488\pi$$
0.265352 0.964152i $$-0.414512\pi$$
$$314$$ −1.39997e6 −0.801300
$$315$$ 0 0
$$316$$ −1.10656e6 −0.623386
$$317$$ − 2.53289e6i − 1.41569i −0.706368 0.707844i $$-0.749668\pi$$
0.706368 0.707844i $$-0.250332\pi$$
$$318$$ 0 0
$$319$$ −1.75158e6 −0.963725
$$320$$ 0 0
$$321$$ 0 0
$$322$$ − 57528.0i − 0.0309200i
$$323$$ − 272970.i − 0.145582i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −770324. −0.401448
$$327$$ 0 0
$$328$$ − 1.16275e6i − 0.596764i
$$329$$ −51606.0 −0.0262851
$$330$$ 0 0
$$331$$ 602132. 0.302080 0.151040 0.988528i $$-0.451738\pi$$
0.151040 + 0.988528i $$0.451738\pi$$
$$332$$ − 1.73654e6i − 0.864650i
$$333$$ 0 0
$$334$$ −2.32277e6 −1.13930
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 209777.i − 0.100620i −0.998734 0.0503099i $$-0.983979\pi$$
0.998734 0.0503099i $$-0.0160209\pi$$
$$338$$ 1.44437e6i 0.687680i
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 1.90765e6 0.888407
$$342$$ 0 0
$$343$$ − 1.47603e6i − 0.677425i
$$344$$ 918464. 0.418472
$$345$$ 0 0
$$346$$ 2.95250e6 1.32587
$$347$$ 4.02166e6i 1.79301i 0.443037 + 0.896503i $$0.353901\pi$$
−0.443037 + 0.896503i $$0.646099\pi$$
$$348$$ 0 0
$$349$$ −8330.00 −0.00366085 −0.00183042 0.999998i $$-0.500583\pi$$
−0.00183042 + 0.999998i $$0.500583\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ − 227328.i − 0.0977904i
$$353$$ − 1.95001e6i − 0.832912i −0.909156 0.416456i $$-0.863272\pi$$
0.909156 0.416456i $$-0.136728\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −560640. −0.234455
$$357$$ 0 0
$$358$$ 1.98948e6i 0.820412i
$$359$$ 2.27088e6 0.929947 0.464973 0.885325i $$-0.346064\pi$$
0.464973 + 0.885325i $$0.346064\pi$$
$$360$$ 0 0
$$361$$ 363126. 0.146652
$$362$$ − 1.33265e6i − 0.534497i
$$363$$ 0 0
$$364$$ −75952.0 −0.0300459
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 2.86154e6i − 1.10901i −0.832181 0.554503i $$-0.812908\pi$$
0.832181 0.554503i $$-0.187092\pi$$
$$368$$ − 78336.0i − 0.0301538i
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −842052. −0.317617
$$372$$ 0 0
$$373$$ 615311.i 0.228993i 0.993424 + 0.114497i $$0.0365255\pi$$
−0.993424 + 0.114497i $$0.963474\pi$$
$$374$$ 143856. 0.0531801
$$375$$ 0 0
$$376$$ −70272.0 −0.0256338
$$377$$ 796890.i 0.288765i
$$378$$ 0 0
$$379$$ −5.39878e6 −1.93062 −0.965311 0.261103i $$-0.915914\pi$$
−0.965311 + 0.261103i $$0.915914\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 162552.i 0.0569946i
$$383$$ − 1.08688e6i − 0.378602i −0.981919 0.189301i $$-0.939378\pi$$
0.981919 0.189301i $$-0.0606222\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −1.97860e6 −0.675913
$$387$$ 0 0
$$388$$ − 13168.0i − 0.00444059i
$$389$$ −3.48432e6 −1.16747 −0.583733 0.811946i $$-0.698408\pi$$
−0.583733 + 0.811946i $$0.698408\pi$$
$$390$$ 0 0
$$391$$ 49572.0 0.0163981
$$392$$ − 934272.i − 0.307085i
$$393$$ 0 0
$$394$$ −2.20937e6 −0.717014
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 3.26591e6i − 1.03999i −0.854170 0.519993i $$-0.825935\pi$$
0.854170 0.519993i $$-0.174065\pi$$
$$398$$ − 2.74250e6i − 0.867839i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 4.27319e6 1.32706 0.663531 0.748149i $$-0.269057\pi$$
0.663531 + 0.748149i $$0.269057\pi$$
$$402$$ 0 0
$$403$$ − 867893.i − 0.266197i
$$404$$ −541248. −0.164984
$$405$$ 0 0
$$406$$ 1.48332e6 0.446601
$$407$$ 1.91852e6i 0.574092i
$$408$$ 0 0
$$409$$ 1.45188e6 0.429162 0.214581 0.976706i $$-0.431161\pi$$
0.214581 + 0.976706i $$0.431161\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 2.13510e6i 0.619692i
$$413$$ − 827670.i − 0.238771i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −103424. −0.0293014
$$417$$ 0 0
$$418$$ 1.49628e6i 0.418863i
$$419$$ 559380. 0.155658 0.0778291 0.996967i $$-0.475201\pi$$
0.0778291 + 0.996967i $$0.475201\pi$$
$$420$$ 0 0
$$421$$ −3.91470e6 −1.07645 −0.538224 0.842802i $$-0.680905\pi$$
−0.538224 + 0.842802i $$0.680905\pi$$
$$422$$ 2.99791e6i 0.819478i
$$423$$ 0 0
$$424$$ −1.14662e6 −0.309746
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 1.02709e6i 0.272608i
$$428$$ − 1.30003e6i − 0.343040i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 3.57500e6 0.927006 0.463503 0.886095i $$-0.346592\pi$$
0.463503 + 0.886095i $$0.346592\pi$$
$$432$$ 0 0
$$433$$ 7.15969e6i 1.83516i 0.397548 + 0.917581i $$0.369861\pi$$
−0.397548 + 0.917581i $$0.630139\pi$$
$$434$$ −1.61548e6 −0.411698
$$435$$ 0 0
$$436$$ −3.47224e6 −0.874769
$$437$$ 515610.i 0.129157i
$$438$$ 0 0
$$439$$ −1.71790e6 −0.425437 −0.212719 0.977114i $$-0.568232\pi$$
−0.212719 + 0.977114i $$0.568232\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ − 65448.0i − 0.0159346i
$$443$$ − 3.39670e6i − 0.822332i −0.911560 0.411166i $$-0.865122\pi$$
0.911560 0.411166i $$-0.134878\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −677084. −0.161178
$$447$$ 0 0
$$448$$ 192512.i 0.0453172i
$$449$$ −3.39606e6 −0.794986 −0.397493 0.917605i $$-0.630120\pi$$
−0.397493 + 0.917605i $$0.630120\pi$$
$$450$$ 0 0
$$451$$ −4.03330e6 −0.933724
$$452$$ − 2.21318e6i − 0.509532i
$$453$$ 0 0
$$454$$ 185952. 0.0423410
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 4.52814e6i 1.01421i 0.861883 + 0.507106i $$0.169285\pi$$
−0.861883 + 0.507106i $$0.830715\pi$$
$$458$$ 360460.i 0.0802959i
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 1.27895e6 0.280285 0.140143 0.990131i $$-0.455244\pi$$
0.140143 + 0.990131i $$0.455244\pi$$
$$462$$ 0 0
$$463$$ − 7.19862e6i − 1.56062i −0.625393 0.780310i $$-0.715061\pi$$
0.625393 0.780310i $$-0.284939\pi$$
$$464$$ 2.01984e6 0.435534
$$465$$ 0 0
$$466$$ 4.25654e6 0.908014
$$467$$ 4.83034e6i 1.02491i 0.858714 + 0.512455i $$0.171264\pi$$
−0.858714 + 0.512455i $$0.828736\pi$$
$$468$$ 0 0
$$469$$ −5029.00 −0.00105572
$$470$$ 0 0
$$471$$ 0 0
$$472$$ − 1.12704e6i − 0.232854i
$$473$$ − 3.18592e6i − 0.654760i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −121824. −0.0246442
$$477$$ 0 0
$$478$$ 4.60632e6i 0.922113i
$$479$$ 748650. 0.149087 0.0745435 0.997218i $$-0.476250\pi$$
0.0745435 + 0.997218i $$0.476250\pi$$
$$480$$ 0 0
$$481$$ 872842. 0.172018
$$482$$ 3.42487e6i 0.671469i
$$483$$ 0 0
$$484$$ 1.78827e6 0.346993
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 5.16394e6i 0.986641i 0.869848 + 0.493320i $$0.164217\pi$$
−0.869848 + 0.493320i $$0.835783\pi$$
$$488$$ 1.39859e6i 0.265853i
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −8.54287e6 −1.59919 −0.799595 0.600539i $$-0.794953\pi$$
−0.799595 + 0.600539i $$0.794953\pi$$
$$492$$ 0 0
$$493$$ 1.27818e6i 0.236851i
$$494$$ 680740. 0.125506
$$495$$ 0 0
$$496$$ −2.19981e6 −0.401495
$$497$$ − 1.91422e6i − 0.347616i
$$498$$ 0 0
$$499$$ 4.20588e6 0.756145 0.378072 0.925776i $$-0.376587\pi$$
0.378072 + 0.925776i $$0.376587\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 830832.i 0.147148i
$$503$$ 8.18342e6i 1.44217i 0.692849 + 0.721083i $$0.256355\pi$$
−0.692849 + 0.721083i $$0.743645\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −271728. −0.0471800
$$507$$ 0 0
$$508$$ − 4.09677e6i − 0.704340i
$$509$$ 3.85923e6 0.660247 0.330123 0.943938i $$-0.392910\pi$$
0.330123 + 0.943938i $$0.392910\pi$$
$$510$$ 0 0
$$511$$ 1.63118e6 0.276344
$$512$$ 262144.i 0.0441942i
$$513$$ 0 0
$$514$$ 5.81275e6 0.970452
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 243756.i 0.0401078i
$$518$$ − 1.62470e6i − 0.266040i
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −4.55410e6 −0.735036 −0.367518 0.930016i $$-0.619792\pi$$
−0.367518 + 0.930016i $$0.619792\pi$$
$$522$$ 0 0
$$523$$ 4.82224e6i 0.770894i 0.922730 + 0.385447i $$0.125953\pi$$
−0.922730 + 0.385447i $$0.874047\pi$$
$$524$$ 1.89523e6 0.301533
$$525$$ 0 0
$$526$$ 677184. 0.106719
$$527$$ − 1.39207e6i − 0.218340i
$$528$$ 0 0
$$529$$ 6.34271e6 0.985452
$$530$$ 0 0
$$531$$ 0 0
$$532$$ − 1.26712e6i − 0.194106i
$$533$$ 1.83497e6i 0.279776i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −6848.00 −0.00102956
$$537$$ 0 0
$$538$$ − 6.32436e6i − 0.942022i
$$539$$ −3.24076e6 −0.480479
$$540$$ 0 0
$$541$$ 362537. 0.0532549 0.0266274 0.999645i $$-0.491523\pi$$
0.0266274 + 0.999645i $$0.491523\pi$$
$$542$$ 3.29005e6i 0.481065i
$$543$$ 0 0
$$544$$ −165888. −0.0240335
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 3.11439e6i − 0.445046i −0.974927 0.222523i $$-0.928571\pi$$
0.974927 0.222523i $$-0.0714293\pi$$
$$548$$ 211488.i 0.0300839i
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −1.32947e7 −1.86551
$$552$$ 0 0
$$553$$ − 3.25052e6i − 0.452002i
$$554$$ −2.18729e6 −0.302784
$$555$$ 0 0
$$556$$ −5.61184e6 −0.769872
$$557$$ − 7.99304e6i − 1.09163i −0.837907 0.545813i $$-0.816221\pi$$
0.837907 0.545813i $$-0.183779\pi$$
$$558$$ 0 0
$$559$$ −1.44945e6 −0.196189
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 4.36999e6i 0.583633i
$$563$$ 1.23236e7i 1.63857i 0.573385 + 0.819286i $$0.305630\pi$$
−0.573385 + 0.819286i $$0.694370\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 9.93920e6 1.30410
$$567$$ 0 0
$$568$$ − 2.60659e6i − 0.339002i
$$569$$ 1.01364e7 1.31252 0.656258 0.754537i $$-0.272138\pi$$
0.656258 + 0.754537i $$0.272138\pi$$
$$570$$ 0 0
$$571$$ 6.53084e6 0.838260 0.419130 0.907926i $$-0.362335\pi$$
0.419130 + 0.907926i $$0.362335\pi$$
$$572$$ 358752.i 0.0458463i
$$573$$ 0 0
$$574$$ 3.41558e6 0.432698
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 1.24453e6i 0.155621i 0.996968 + 0.0778103i $$0.0247928\pi$$
−0.996968 + 0.0778103i $$0.975207\pi$$
$$578$$ 5.57445e6i 0.694037i
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 5.10110e6 0.626936
$$582$$ 0 0
$$583$$ 3.97735e6i 0.484644i
$$584$$ 2.22118e6 0.269496
$$585$$ 0 0
$$586$$ −1.36558e6 −0.164275
$$587$$ − 1.33403e6i − 0.159797i −0.996803 0.0798987i $$-0.974540\pi$$
0.996803 0.0798987i $$-0.0254597\pi$$
$$588$$ 0 0
$$589$$ 1.44792e7 1.71972
$$590$$ 0 0
$$591$$ 0 0
$$592$$ − 2.21235e6i − 0.259448i
$$593$$ 1.19401e7i 1.39435i 0.716899 + 0.697177i $$0.245561\pi$$
−0.716899 + 0.697177i $$0.754439\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −1.75824e6 −0.202751
$$597$$ 0 0
$$598$$ 123624.i 0.0141368i
$$599$$ −7.16430e6 −0.815843 −0.407922 0.913017i $$-0.633746\pi$$
−0.407922 + 0.913017i $$0.633746\pi$$
$$600$$ 0 0
$$601$$ 1.15163e6 0.130055 0.0650273 0.997883i $$-0.479287\pi$$
0.0650273 + 0.997883i $$0.479287\pi$$
$$602$$ 2.69799e6i 0.303423i
$$603$$ 0 0
$$604$$ 2.76165e6 0.308018
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 1.34268e7i 1.47911i 0.673097 + 0.739554i $$0.264963\pi$$
−0.673097 + 0.739554i $$0.735037\pi$$
$$608$$ − 1.72544e6i − 0.189296i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 110898. 0.0120177
$$612$$ 0 0
$$613$$ 1.20184e7i 1.29180i 0.763422 + 0.645900i $$0.223518\pi$$
−0.763422 + 0.645900i $$0.776482\pi$$
$$614$$ 8.11591e6 0.868793
$$615$$ 0 0
$$616$$ 667776. 0.0709054
$$617$$ − 6.98519e6i − 0.738695i −0.929291 0.369348i $$-0.879581\pi$$
0.929291 0.369348i $$-0.120419\pi$$
$$618$$ 0 0
$$619$$ 8.20625e6 0.860832 0.430416 0.902631i $$-0.358367\pi$$
0.430416 + 0.902631i $$0.358367\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 826392.i 0.0856466i
$$623$$ − 1.64688e6i − 0.169997i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 1.33689e7 1.36352
$$627$$ 0 0
$$628$$ − 5.59989e6i − 0.566605i
$$629$$ 1.40000e6 0.141092
$$630$$ 0 0
$$631$$ −1.07686e7 −1.07668 −0.538338 0.842729i $$-0.680947\pi$$
−0.538338 + 0.842729i $$0.680947\pi$$
$$632$$ − 4.42624e6i − 0.440801i
$$633$$ 0 0
$$634$$ 1.01316e7 1.00104
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 1.47440e6i 0.143968i
$$638$$ − 7.00632e6i − 0.681457i
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −1.92571e7 −1.85117 −0.925585 0.378539i $$-0.876427\pi$$
−0.925585 + 0.378539i $$0.876427\pi$$
$$642$$ 0 0
$$643$$ 1.00999e7i 0.963364i 0.876346 + 0.481682i $$0.159974\pi$$
−0.876346 + 0.481682i $$0.840026\pi$$
$$644$$ 230112. 0.0218637
$$645$$ 0 0
$$646$$ 1.09188e6 0.102942
$$647$$ 7.52113e6i 0.706354i 0.935556 + 0.353177i $$0.114899\pi$$
−0.935556 + 0.353177i $$0.885101\pi$$
$$648$$ 0 0
$$649$$ −3.90942e6 −0.364335
$$650$$ 0 0
$$651$$ 0 0
$$652$$ − 3.08130e6i − 0.283867i
$$653$$ 2.67197e6i 0.245216i 0.992455 + 0.122608i $$0.0391258\pi$$
−0.992455 + 0.122608i $$0.960874\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 4.65101e6 0.421976
$$657$$ 0 0
$$658$$ − 206424.i − 0.0185864i
$$659$$ 6.99948e6 0.627845 0.313922 0.949449i $$-0.398357\pi$$
0.313922 + 0.949449i $$0.398357\pi$$
$$660$$ 0 0
$$661$$ 408122. 0.0363318 0.0181659 0.999835i $$-0.494217\pi$$
0.0181659 + 0.999835i $$0.494217\pi$$
$$662$$ 2.40853e6i 0.213603i
$$663$$ 0 0
$$664$$ 6.94618e6 0.611400
$$665$$ 0 0
$$666$$ 0 0
$$667$$ − 2.41434e6i − 0.210128i
$$668$$ − 9.29107e6i − 0.805610i
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 4.85137e6 0.415966
$$672$$ 0 0
$$673$$ 1.74939e7i 1.48885i 0.667709 + 0.744423i $$0.267275\pi$$
−0.667709 + 0.744423i $$0.732725\pi$$
$$674$$ 839108. 0.0711489
$$675$$ 0 0
$$676$$ −5.77747e6 −0.486263
$$677$$ − 8.67440e6i − 0.727391i −0.931518 0.363695i $$-0.881515\pi$$
0.931518 0.363695i $$-0.118485\pi$$
$$678$$ 0 0
$$679$$ 38681.0 0.00321976
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 7.63058e6i 0.628198i
$$683$$ − 1.18478e7i − 0.971822i −0.874008 0.485911i $$-0.838488\pi$$
0.874008 0.485911i $$-0.161512\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 5.90414e6 0.479012
$$687$$ 0 0
$$688$$ 3.67386e6i 0.295904i
$$689$$ 1.80952e6 0.145216
$$690$$ 0 0
$$691$$ 9.47775e6 0.755110 0.377555 0.925987i $$-0.376765\pi$$
0.377555 + 0.925987i $$0.376765\pi$$
$$692$$ 1.18100e7i 0.937530i
$$693$$ 0 0
$$694$$ −1.60866e7 −1.26785
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 2.94322e6i 0.229478i
$$698$$ − 33320.0i − 0.00258861i
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 2.28147e7 1.75355 0.876777 0.480898i $$-0.159689\pi$$
0.876777 + 0.480898i $$0.159689\pi$$
$$702$$ 0 0
$$703$$ 1.45618e7i 1.11129i
$$704$$ 909312. 0.0691483
$$705$$ 0 0
$$706$$ 7.80002e6 0.588958
$$707$$ − 1.58992e6i − 0.119626i
$$708$$ 0 0
$$709$$ −1.27436e7 −0.952090 −0.476045 0.879421i $$-0.657930\pi$$
−0.476045 + 0.879421i $$0.657930\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ − 2.24256e6i − 0.165785i
$$713$$ 2.62946e6i 0.193706i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −7.95792e6 −0.580119
$$717$$ 0 0
$$718$$ 9.08352e6i 0.657572i
$$719$$ 2.44929e6 0.176692 0.0883462 0.996090i $$-0.471842\pi$$
0.0883462 + 0.996090i $$0.471842\pi$$
$$720$$ 0 0
$$721$$ −6.27187e6 −0.449323
$$722$$ 1.45250e6i 0.103699i
$$723$$ 0 0
$$724$$ 5.33061e6 0.377947
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 415033.i 0.0291237i 0.999894 + 0.0145619i $$0.00463535\pi$$
−0.999894 + 0.0145619i $$0.995365\pi$$
$$728$$ − 303808.i − 0.0212457i
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −2.32486e6 −0.160918
$$732$$ 0 0
$$733$$ − 1.72877e7i − 1.18844i −0.804302 0.594221i $$-0.797461\pi$$
0.804302 0.594221i $$-0.202539\pi$$
$$734$$ 1.14461e7 0.784186
$$735$$ 0 0
$$736$$ 313344. 0.0213219
$$737$$ 23754.0i 0.00161090i
$$738$$ 0 0
$$739$$ −5.18834e6 −0.349476 −0.174738 0.984615i $$-0.555908\pi$$
−0.174738 + 0.984615i $$0.555908\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ − 3.36821e6i − 0.224589i
$$743$$ − 4.79572e6i − 0.318700i −0.987222 0.159350i $$-0.949060\pi$$
0.987222 0.159350i $$-0.0509398\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −2.46124e6 −0.161923
$$747$$ 0 0
$$748$$ 575424.i 0.0376040i
$$749$$ 3.81884e6 0.248730
$$750$$ 0 0
$$751$$ −1.85654e7 −1.20117 −0.600585 0.799561i $$-0.705066\pi$$
−0.600585 + 0.799561i $$0.705066\pi$$
$$752$$ − 281088.i − 0.0181258i
$$753$$ 0 0
$$754$$ −3.18756e6 −0.204188
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 2.82068e7i − 1.78902i −0.447053 0.894508i $$-0.647526\pi$$
0.447053 0.894508i $$-0.352474\pi$$
$$758$$ − 2.15951e7i − 1.36516i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −6.56161e6 −0.410723 −0.205361 0.978686i $$-0.565837\pi$$
−0.205361 + 0.978686i $$0.565837\pi$$
$$762$$ 0 0
$$763$$ − 1.01997e7i − 0.634273i
$$764$$ −650208. −0.0403013
$$765$$ 0 0
$$766$$ 4.34750e6 0.267712
$$767$$ 1.77861e6i 0.109167i
$$768$$ 0 0
$$769$$ −2.20930e7 −1.34722 −0.673610 0.739087i $$-0.735257\pi$$
−0.673610 + 0.739087i $$0.735257\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ − 7.91442e6i − 0.477942i
$$773$$ 3.00787e7i 1.81055i 0.424824 + 0.905276i $$0.360336\pi$$
−0.424824 + 0.905276i $$0.639664\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 52672.0 0.00313997
$$777$$ 0 0
$$778$$ − 1.39373e7i − 0.825523i
$$779$$ −3.06131e7 −1.80744
$$780$$ 0 0
$$781$$ −9.04162e6 −0.530418
$$782$$ 198288.i 0.0115952i
$$783$$ 0 0
$$784$$ 3.73709e6 0.217142
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 3.28954e6i 0.189321i 0.995510 + 0.0946605i $$0.0301766\pi$$
−0.995510 + 0.0946605i $$0.969823\pi$$
$$788$$ − 8.83747e6i − 0.507005i
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 6.50123e6 0.369449
$$792$$ 0 0
$$793$$ − 2.20715e6i − 0.124638i
$$794$$ 1.30636e7 0.735381
$$795$$ 0 0
$$796$$ 1.09700e7 0.613655
$$797$$ 6.71053e6i 0.374206i 0.982340 + 0.187103i $$0.0599099\pi$$
−0.982340 + 0.187103i $$0.940090\pi$$
$$798$$ 0 0
$$799$$ 177876. 0.00985713
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 1.70928e7i 0.938374i
$$803$$ − 7.70473e6i − 0.421666i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 3.47157e6 0.188230
$$807$$ 0 0
$$808$$ − 2.16499e6i − 0.116662i
$$809$$ 8.74254e6 0.469641 0.234821 0.972039i $$-0.424550\pi$$
0.234821 + 0.972039i $$0.424550\pi$$
$$810$$ 0 0
$$811$$ −2.48410e7 −1.32622 −0.663112 0.748520i $$-0.730765\pi$$
−0.663112 + 0.748520i $$0.730765\pi$$
$$812$$ 5.93328e6i 0.315795i
$$813$$ 0 0
$$814$$ −7.67410e6 −0.405944
$$815$$ 0 0
$$816$$ 0 0
$$817$$ − 2.41814e7i − 1.26744i
$$818$$ 5.80750e6i 0.303463i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −2.12219e7 −1.09882 −0.549409 0.835554i $$-0.685147\pi$$
−0.549409 + 0.835554i $$0.685147\pi$$
$$822$$ 0 0
$$823$$ − 8.70659e6i − 0.448073i −0.974581 0.224036i $$-0.928077\pi$$
0.974581 0.224036i $$-0.0719234\pi$$
$$824$$ −8.54042e6 −0.438189
$$825$$ 0 0
$$826$$ 3.31068e6 0.168837
$$827$$ 3.71184e7i 1.88723i 0.331040 + 0.943617i $$0.392600\pi$$
−0.331040 + 0.943617i $$0.607400\pi$$
$$828$$ 0 0
$$829$$ −1.01765e6 −0.0514295 −0.0257147 0.999669i $$-0.508186\pi$$
−0.0257147 + 0.999669i $$0.508186\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ − 413696.i − 0.0207192i
$$833$$ 2.36488e6i 0.118085i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −5.98512e6 −0.296181
$$837$$ 0 0
$$838$$ 2.23752e6i 0.110067i
$$839$$ −3.36194e7 −1.64887 −0.824433 0.565960i $$-0.808506\pi$$
−0.824433 + 0.565960i $$0.808506\pi$$
$$840$$ 0 0
$$841$$ 4.17410e7 2.03504
$$842$$ − 1.56588e7i − 0.761164i
$$843$$ 0 0
$$844$$ −1.19916e7 −0.579458
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 5.25305e6i 0.251596i
$$848$$ − 4.58650e6i − 0.219024i
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −2.64445e6 −0.125173
$$852$$ 0 0
$$853$$ − 3.52574e7i − 1.65912i −0.558419 0.829559i $$-0.688592\pi$$
0.558419 0.829559i $$-0.311408\pi$$
$$854$$ −4.10836e6 −0.192763
$$855$$ 0 0
$$856$$ 5.20013e6 0.242566
$$857$$ − 3.14941e7i − 1.46480i −0.680877 0.732398i $$-0.738401\pi$$
0.680877 0.732398i $$-0.261599\pi$$
$$858$$ 0 0
$$859$$ −1.19344e7 −0.551848 −0.275924 0.961180i $$-0.588984\pi$$
−0.275924 + 0.961180i $$0.588984\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 1.43000e7i 0.655492i
$$863$$ − 8.70442e6i − 0.397844i −0.980015 0.198922i $$-0.936256\pi$$
0.980015 0.198922i $$-0.0637440\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −2.86388e7 −1.29766
$$867$$ 0 0
$$868$$ − 6.46194e6i − 0.291114i
$$869$$ −1.53535e7 −0.689697
$$870$$ 0 0
$$871$$ 10807.0 0.000482681 0
$$872$$ − 1.38890e7i − 0.618555i
$$873$$ 0 0
$$874$$ −2.06244e6 −0.0913277
$$875$$ 0 0
$$876$$ 0 0
$$877$$ − 1.17999e7i − 0.518059i −0.965869 0.259029i $$-0.916597\pi$$
0.965869 0.259029i $$-0.0834026\pi$$
$$878$$ − 6.87158e6i − 0.300829i
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 2.73840e7 1.18866 0.594330 0.804221i $$-0.297417\pi$$
0.594330 + 0.804221i $$0.297417\pi$$
$$882$$ 0 0
$$883$$ − 8.80577e6i − 0.380072i −0.981777 0.190036i $$-0.939140\pi$$
0.981777 0.190036i $$-0.0608604\pi$$
$$884$$ 261792. 0.0112675
$$885$$ 0 0
$$886$$ 1.35868e7 0.581477
$$887$$ 250122.i 0.0106744i 0.999986 + 0.00533719i $$0.00169889\pi$$
−0.999986 + 0.00533719i $$0.998301\pi$$
$$888$$ 0 0
$$889$$ 1.20343e7 0.510699
$$890$$ 0 0
$$891$$ 0 0
$$892$$ − 2.70834e6i − 0.113970i
$$893$$ 1.85013e6i 0.0776379i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −770048. −0.0320441
$$897$$ 0 0
$$898$$ − 1.35842e7i − 0.562140i
$$899$$ −6.77988e7 −2.79784
$$900$$ 0 0
$$901$$ 2.90239e6 0.119109
$$902$$ − 1.61332e7i − 0.660243i
$$903$$ 0 0
$$904$$ 8.85274e6 0.360294
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 3.24955e7i 1.31161i 0.754929 + 0.655806i $$0.227671\pi$$
−0.754929 + 0.655806i $$0.772329\pi$$
$$908$$ 743808.i 0.0299396i
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −4.24595e7 −1.69504 −0.847518 0.530766i $$-0.821904\pi$$
−0.847518 + 0.530766i $$0.821904\pi$$
$$912$$ 0 0
$$913$$ − 2.40945e7i − 0.956625i
$$914$$ −1.81126e7 −0.717157
$$915$$ 0 0
$$916$$ −1.44184e6 −0.0567778
$$917$$ 5.56724e6i 0.218634i
$$918$$ 0 0
$$919$$ −1.41629e7 −0.553176 −0.276588 0.960989i $$-0.589204\pi$$
−0.276588 + 0.960989i $$0.589204\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 5.11579e6i 0.198192i
$$923$$ 4.11353e6i 0.158932i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 2.87945e7 1.10352
$$927$$ 0 0
$$928$$ 8.07936e6i 0.307969i
$$929$$ 4.37292e7 1.66239 0.831194 0.555982i $$-0.187658\pi$$
0.831194 + 0.555982i $$0.187658\pi$$
$$930$$ 0 0
$$931$$ −2.45976e7 −0.930077
$$932$$ 1.70262e7i 0.642063i
$$933$$ 0 0
$$934$$ −1.93214e7 −0.724721
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 5.73509e6i − 0.213398i −0.994291 0.106699i $$-0.965972\pi$$
0.994291 0.106699i $$-0.0340282\pi$$
$$938$$ − 20116.0i 0 0.000746508i
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 3.37395e7 1.24212 0.621061 0.783762i $$-0.286702\pi$$
0.621061 + 0.783762i $$0.286702\pi$$
$$942$$ 0 0
$$943$$ − 5.55941e6i − 0.203587i
$$944$$ 4.50816e6 0.164653
$$945$$ 0 0
$$946$$ 1.27437e7 0.462985
$$947$$ − 3.07342e7i − 1.11365i −0.830631 0.556823i $$-0.812020\pi$$
0.830631 0.556823i $$-0.187980\pi$$
$$948$$ 0 0
$$949$$ −3.50531e6 −0.126346
$$950$$ 0 0
$$951$$ 0 0
$$952$$ − 487296.i − 0.0174261i
$$953$$ 2.51847e7i 0.898264i 0.893465 + 0.449132i $$0.148267\pi$$
−0.893465 + 0.449132i $$0.851733\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −1.84253e7 −0.652033
$$957$$ 0 0
$$958$$ 2.99460e6i 0.105420i
$$959$$ −621246. −0.0218131
$$960$$ 0 0
$$961$$ 4.52105e7 1.57918
$$962$$ 3.49137e6i 0.121635i
$$963$$ 0 0
$$964$$ −1.36995e7 −0.474801
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 1.44556e7i − 0.497130i −0.968615 0.248565i $$-0.920041\pi$$
0.968615 0.248565i $$-0.0799589\pi$$
$$968$$ 7.15309e6i 0.245361i
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 1.06974e7 0.364109 0.182054 0.983288i $$-0.441725\pi$$
0.182054 + 0.983288i $$0.441725\pi$$
$$972$$ 0 0
$$973$$ − 1.64848e7i − 0.558214i
$$974$$ −2.06558e7 −0.697660
$$975$$ 0 0
$$976$$ −5.59437e6 −0.187986
$$977$$ − 8.41568e6i − 0.282067i −0.990005 0.141034i $$-0.954957\pi$$
0.990005 0.141034i $$-0.0450426\pi$$
$$978$$ 0 0
$$979$$ −7.77888e6 −0.259394
$$980$$ 0 0
$$981$$ 0 0
$$982$$ − 3.41715e7i − 1.13080i
$$983$$ − 3.89409e7i − 1.28535i −0.766138 0.642676i $$-0.777824\pi$$
0.766138 0.642676i $$-0.222176\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −5.11272e6 −0.167479
$$987$$ 0 0
$$988$$ 2.72296e6i 0.0887460i
$$989$$ 4.39141e6 0.142762
$$990$$ 0 0
$$991$$ 4.84592e7 1.56745 0.783723 0.621111i $$-0.213318\pi$$
0.783723 + 0.621111i $$0.213318\pi$$
$$992$$ − 8.79923e6i − 0.283900i
$$993$$ 0 0
$$994$$ 7.65686e6 0.245802
$$995$$ 0 0
$$996$$ 0 0
$$997$$ − 3.84733e7i − 1.22581i −0.790158 0.612903i $$-0.790002\pi$$
0.790158 0.612903i $$-0.209998\pi$$
$$998$$ 1.68235e7i 0.534675i
$$999$$ 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 450.6.c.e.199.2 2
3.2 odd 2 150.6.c.g.49.1 2
5.2 odd 4 450.6.a.i.1.1 1
5.3 odd 4 450.6.a.p.1.1 1
5.4 even 2 inner 450.6.c.e.199.1 2
15.2 even 4 150.6.a.m.1.1 yes 1
15.8 even 4 150.6.a.a.1.1 1
15.14 odd 2 150.6.c.g.49.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
150.6.a.a.1.1 1 15.8 even 4
150.6.a.m.1.1 yes 1 15.2 even 4
150.6.c.g.49.1 2 3.2 odd 2
150.6.c.g.49.2 2 15.14 odd 2
450.6.a.i.1.1 1 5.2 odd 4
450.6.a.p.1.1 1 5.3 odd 4
450.6.c.e.199.1 2 5.4 even 2 inner
450.6.c.e.199.2 2 1.1 even 1 trivial