# Properties

 Label 450.6.a.p.1.1 Level $450$ Weight $6$ Character 450.1 Self dual yes Analytic conductor $72.173$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$450 = 2 \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 450.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$72.1727189158$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 150) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 450.1

## $q$-expansion

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

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 4.00000 0.707107
$$3$$ 0 0
$$4$$ 16.0000 0.500000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −47.0000 −0.362537 −0.181269 0.983434i $$-0.558020\pi$$
−0.181269 + 0.983434i $$0.558020\pi$$
$$8$$ 64.0000 0.353553
$$9$$ 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
$$13$$ −101.000 −0.165754 −0.0828768 0.996560i $$-0.526411\pi$$
−0.0828768 + 0.996560i $$0.526411\pi$$
$$14$$ −188.000 −0.256353
$$15$$ 0 0
$$16$$ 256.000 0.250000
$$17$$ 162.000 0.135954 0.0679771 0.997687i $$-0.478346\pi$$
0.0679771 + 0.997687i $$0.478346\pi$$
$$18$$ 0 0
$$19$$ 1685.00 1.07082 0.535409 0.844593i $$-0.320157\pi$$
0.535409 + 0.844593i $$0.320157\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −888.000 −0.391162
$$23$$ 306.000 0.120615 0.0603076 0.998180i $$-0.480792\pi$$
0.0603076 + 0.998180i $$0.480792\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −404.000 −0.117206
$$27$$ 0 0
$$28$$ −752.000 −0.181269
$$29$$ −7890.00 −1.74214 −0.871068 0.491163i $$-0.836572\pi$$
−0.871068 + 0.491163i $$0.836572\pi$$
$$30$$ 0 0
$$31$$ −8593.00 −1.60598 −0.802991 0.595991i $$-0.796759\pi$$
−0.802991 + 0.595991i $$0.796759\pi$$
$$32$$ 1024.00 0.176777
$$33$$ 0 0
$$34$$ 648.000 0.0961342
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −8642.00 −1.03779 −0.518896 0.854838i $$-0.673657\pi$$
−0.518896 + 0.854838i $$0.673657\pi$$
$$38$$ 6740.00 0.757183
$$39$$ 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
$$43$$ −14351.0 −1.18362 −0.591808 0.806079i $$-0.701586\pi$$
−0.591808 + 0.806079i $$0.701586\pi$$
$$44$$ −3552.00 −0.276593
$$45$$ 0 0
$$46$$ 1224.00 0.0852878
$$47$$ −1098.00 −0.0725033 −0.0362516 0.999343i $$-0.511542\pi$$
−0.0362516 + 0.999343i $$0.511542\pi$$
$$48$$ 0 0
$$49$$ −14598.0 −0.868567
$$50$$ 0 0
$$51$$ 0 0
$$52$$ −1616.00 −0.0828768
$$53$$ 17916.0 0.876095 0.438048 0.898952i $$-0.355670\pi$$
0.438048 + 0.898952i $$0.355670\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −3008.00 −0.128176
$$57$$ 0 0
$$58$$ −31560.0 −1.23188
$$59$$ −17610.0 −0.658612 −0.329306 0.944223i $$-0.606815\pi$$
−0.329306 + 0.944223i $$0.606815\pi$$
$$60$$ 0 0
$$61$$ −21853.0 −0.751946 −0.375973 0.926631i $$-0.622691\pi$$
−0.375973 + 0.926631i $$0.622691\pi$$
$$62$$ −34372.0 −1.13560
$$63$$ 0 0
$$64$$ 4096.00 0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −107.000 −0.00291204 −0.00145602 0.999999i $$-0.500463\pi$$
−0.00145602 + 0.999999i $$0.500463\pi$$
$$68$$ 2592.00 0.0679771
$$69$$ 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
$$73$$ −34706.0 −0.762250 −0.381125 0.924524i $$-0.624463\pi$$
−0.381125 + 0.924524i $$0.624463\pi$$
$$74$$ −34568.0 −0.733829
$$75$$ 0 0
$$76$$ 26960.0 0.535409
$$77$$ 10434.0 0.200551
$$78$$ 0 0
$$79$$ −69160.0 −1.24677 −0.623386 0.781914i $$-0.714244\pi$$
−0.623386 + 0.781914i $$0.714244\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 72672.0 1.19353
$$83$$ −108534. −1.72930 −0.864650 0.502374i $$-0.832460\pi$$
−0.864650 + 0.502374i $$0.832460\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −57404.0 −0.836943
$$87$$ 0 0
$$88$$ −14208.0 −0.195581
$$89$$ −35040.0 −0.468910 −0.234455 0.972127i $$-0.575330\pi$$
−0.234455 + 0.972127i $$0.575330\pi$$
$$90$$ 0 0
$$91$$ 4747.00 0.0600919
$$92$$ 4896.00 0.0603076
$$93$$ 0 0
$$94$$ −4392.00 −0.0512676
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 823.000 0.00888118 0.00444059 0.999990i $$-0.498587\pi$$
0.00444059 + 0.999990i $$0.498587\pi$$
$$98$$ −58392.0 −0.614169
$$99$$ 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. 1.23938 0.619692 0.784845i $$-0.287257\pi$$
0.619692 + 0.784845i $$0.287257\pi$$
$$104$$ −6464.00 −0.0586028
$$105$$ 0 0
$$106$$ 71664.0 0.619493
$$107$$ 81252.0 0.686080 0.343040 0.939321i $$-0.388543\pi$$
0.343040 + 0.939321i $$0.388543\pi$$
$$108$$ 0 0
$$109$$ −217015. −1.74954 −0.874769 0.484540i $$-0.838987\pi$$
−0.874769 + 0.484540i $$0.838987\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −12032.0 −0.0906343
$$113$$ −138324. −1.01906 −0.509532 0.860452i $$-0.670181\pi$$
−0.509532 + 0.860452i $$0.670181\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −126240. −0.871068
$$117$$ 0 0
$$118$$ −70440.0 −0.465709
$$119$$ −7614.00 −0.0492885
$$120$$ 0 0
$$121$$ −111767. −0.693985
$$122$$ −87412.0 −0.531706
$$123$$ 0 0
$$124$$ −137488. −0.802991
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 256048. 1.40868 0.704340 0.709863i $$-0.251243\pi$$
0.704340 + 0.709863i $$0.251243\pi$$
$$128$$ 16384.0 0.0883883
$$129$$ 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
$$133$$ −79195.0 −0.388212
$$134$$ −428.000 −0.00205912
$$135$$ 0 0
$$136$$ 10368.0 0.0480671
$$137$$ −13218.0 −0.0601678 −0.0300839 0.999547i $$-0.509577\pi$$
−0.0300839 + 0.999547i $$0.509577\pi$$
$$138$$ 0 0
$$139$$ −350740. −1.53974 −0.769872 0.638199i $$-0.779680\pi$$
−0.769872 + 0.638199i $$0.779680\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 162912. 0.678004
$$143$$ 22422.0 0.0916926
$$144$$ 0 0
$$145$$ 0 0
$$146$$ −138824. −0.538992
$$147$$ 0 0
$$148$$ −138272. −0.518896
$$149$$ −109890. −0.405502 −0.202751 0.979230i $$-0.564988\pi$$
−0.202751 + 0.979230i $$0.564988\pi$$
$$150$$ 0 0
$$151$$ −172603. −0.616036 −0.308018 0.951381i $$-0.599666\pi$$
−0.308018 + 0.951381i $$0.599666\pi$$
$$152$$ 107840. 0.378592
$$153$$ 0 0
$$154$$ 41736.0 0.141811
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 349993. 1.13321 0.566605 0.823990i $$-0.308257\pi$$
0.566605 + 0.823990i $$0.308257\pi$$
$$158$$ −276640. −0.881601
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −14382.0 −0.0437275
$$162$$ 0 0
$$163$$ −192581. −0.567733 −0.283867 0.958864i $$-0.591617\pi$$
−0.283867 + 0.958864i $$0.591617\pi$$
$$164$$ 290688. 0.843951
$$165$$ 0 0
$$166$$ −434136. −1.22280
$$167$$ 580692. 1.61122 0.805610 0.592447i $$-0.201838\pi$$
0.805610 + 0.592447i $$0.201838\pi$$
$$168$$ 0 0
$$169$$ −361092. −0.972526
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −229616. −0.591808
$$173$$ 738126. 1.87506 0.937530 0.347904i $$-0.113106\pi$$
0.937530 + 0.347904i $$0.113106\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −56832.0 −0.138297
$$177$$ 0 0
$$178$$ −140160. −0.331569
$$179$$ −497370. −1.16024 −0.580119 0.814532i $$-0.696994\pi$$
−0.580119 + 0.814532i $$0.696994\pi$$
$$180$$ 0 0
$$181$$ −333163. −0.755893 −0.377947 0.925827i $$-0.623370\pi$$
−0.377947 + 0.925827i $$0.623370\pi$$
$$182$$ 18988.0 0.0424914
$$183$$ 0 0
$$184$$ 19584.0 0.0426439
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −35964.0 −0.0752080
$$188$$ −17568.0 −0.0362516
$$189$$ 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. −0.955885 −0.477942 0.878391i $$-0.658617\pi$$
−0.477942 + 0.878391i $$0.658617\pi$$
$$194$$ 3292.00 0.00627994
$$195$$ 0 0
$$196$$ −233568. −0.434283
$$197$$ 552342. 1.01401 0.507005 0.861943i $$-0.330752\pi$$
0.507005 + 0.861943i $$0.330752\pi$$
$$198$$ 0 0
$$199$$ 685625. 1.22731 0.613655 0.789575i $$-0.289699\pi$$
0.613655 + 0.789575i $$0.289699\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 135312. 0.233323
$$203$$ 370830. 0.631589
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 533776. 0.876377
$$207$$ 0 0
$$208$$ −25856.0 −0.0414384
$$209$$ −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. 0.438048
$$213$$ 0 0
$$214$$ 325008. 0.485132
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 403871. 0.582228
$$218$$ −868060. −1.23711
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −16362.0 −0.0225349
$$222$$ 0 0
$$223$$ −169271. −0.227940 −0.113970 0.993484i $$-0.536357\pi$$
−0.113970 + 0.993484i $$0.536357\pi$$
$$224$$ −48128.0 −0.0640882
$$225$$ 0 0
$$226$$ −553296. −0.720587
$$227$$ −46488.0 −0.0598792 −0.0299396 0.999552i $$-0.509532\pi$$
−0.0299396 + 0.999552i $$0.509532\pi$$
$$228$$ 0 0
$$229$$ −90115.0 −0.113556 −0.0567778 0.998387i $$-0.518083\pi$$
−0.0567778 + 0.998387i $$0.518083\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −504960. −0.615938
$$233$$ 1.06414e6 1.28413 0.642063 0.766652i $$-0.278079\pi$$
0.642063 + 0.766652i $$0.278079\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −281760. −0.329306
$$237$$ 0 0
$$238$$ −30456.0 −0.0348522
$$239$$ −1.15158e6 −1.30407 −0.652033 0.758191i $$-0.726084\pi$$
−0.652033 + 0.758191i $$0.726084\pi$$
$$240$$ 0 0
$$241$$ 856217. 0.949601 0.474801 0.880093i $$-0.342520\pi$$
0.474801 + 0.880093i $$0.342520\pi$$
$$242$$ −447068. −0.490722
$$243$$ 0 0
$$244$$ −349648. −0.375973
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −170185. −0.177492
$$248$$ −549952. −0.567800
$$249$$ 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
$$253$$ −67932.0 −0.0667226
$$254$$ 1.02419e6 0.996087
$$255$$ 0 0
$$256$$ 65536.0 0.0625000
$$257$$ −1.45319e6 −1.37243 −0.686213 0.727401i $$-0.740728\pi$$
−0.686213 + 0.727401i $$0.740728\pi$$
$$258$$ 0 0
$$259$$ 406174. 0.376238
$$260$$ 0 0
$$261$$ 0 0
$$262$$ −473808. −0.426431
$$263$$ 169296. 0.150924 0.0754618 0.997149i $$-0.475957\pi$$
0.0754618 + 0.997149i $$0.475957\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −316780. −0.274507
$$267$$ 0 0
$$268$$ −1712.00 −0.00145602
$$269$$ 1.58109e6 1.33222 0.666110 0.745854i $$-0.267958\pi$$
0.666110 + 0.745854i $$0.267958\pi$$
$$270$$ 0 0
$$271$$ 822512. 0.680329 0.340165 0.940366i $$-0.389517\pi$$
0.340165 + 0.940366i $$0.389517\pi$$
$$272$$ 41472.0 0.0339886
$$273$$ 0 0
$$274$$ −52872.0 −0.0425451
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 546823. 0.428201 0.214100 0.976812i $$-0.431318\pi$$
0.214100 + 0.976812i $$0.431318\pi$$
$$278$$ −1.40296e6 −1.08876
$$279$$ 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
$$283$$ 2.48480e6 1.84427 0.922136 0.386865i $$-0.126442\pi$$
0.922136 + 0.386865i $$0.126442\pi$$
$$284$$ 651648. 0.479421
$$285$$ 0 0
$$286$$ 89688.0 0.0648365
$$287$$ −853896. −0.611928
$$288$$ 0 0
$$289$$ −1.39361e6 −0.981516
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −555296. −0.381125
$$293$$ −341394. −0.232320 −0.116160 0.993231i $$-0.537059\pi$$
−0.116160 + 0.993231i $$0.537059\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −553088. −0.366915
$$297$$ 0 0
$$298$$ −439560. −0.286733
$$299$$ −30906.0 −0.0199924
$$300$$ 0 0
$$301$$ 674497. 0.429105
$$302$$ −690412. −0.435603
$$303$$ 0 0
$$304$$ 431360. 0.267705
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −2.02898e6 −1.22866 −0.614329 0.789050i $$-0.710573\pi$$
−0.614329 + 0.789050i $$0.710573\pi$$
$$308$$ 166944. 0.100275
$$309$$ 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
$$313$$ 3.34223e6 1.92830 0.964152 0.265352i $$-0.0854881\pi$$
0.964152 + 0.265352i $$0.0854881\pi$$
$$314$$ 1.39997e6 0.801300
$$315$$ 0 0
$$316$$ −1.10656e6 −0.623386
$$317$$ −2.53289e6 −1.41569 −0.707844 0.706368i $$-0.750332\pi$$
−0.707844 + 0.706368i $$0.750332\pi$$
$$318$$ 0 0
$$319$$ 1.75158e6 0.963725
$$320$$ 0 0
$$321$$ 0 0
$$322$$ −57528.0 −0.0309200
$$323$$ 272970. 0.145582
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −770324. −0.401448
$$327$$ 0 0
$$328$$ 1.16275e6 0.596764
$$329$$ 51606.0 0.0262851
$$330$$ 0 0
$$331$$ 602132. 0.302080 0.151040 0.988528i $$-0.451738\pi$$
0.151040 + 0.988528i $$0.451738\pi$$
$$332$$ −1.73654e6 −0.864650
$$333$$ 0 0
$$334$$ 2.32277e6 1.13930
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −209777. −0.100620 −0.0503099 0.998734i $$-0.516021\pi$$
−0.0503099 + 0.998734i $$0.516021\pi$$
$$338$$ −1.44437e6 −0.687680
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 1.90765e6 0.888407
$$342$$ 0 0
$$343$$ 1.47603e6 0.677425
$$344$$ −918464. −0.418472
$$345$$ 0 0
$$346$$ 2.95250e6 1.32587
$$347$$ 4.02166e6 1.79301 0.896503 0.443037i $$-0.146099\pi$$
0.896503 + 0.443037i $$0.146099\pi$$
$$348$$ 0 0
$$349$$ 8330.00 0.00366085 0.00183042 0.999998i $$-0.499417\pi$$
0.00183042 + 0.999998i $$0.499417\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −227328. −0.0977904
$$353$$ 1.95001e6 0.832912 0.416456 0.909156i $$-0.363272\pi$$
0.416456 + 0.909156i $$0.363272\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −560640. −0.234455
$$357$$ 0 0
$$358$$ −1.98948e6 −0.820412
$$359$$ −2.27088e6 −0.929947 −0.464973 0.885325i $$-0.653936\pi$$
−0.464973 + 0.885325i $$0.653936\pi$$
$$360$$ 0 0
$$361$$ 363126. 0.146652
$$362$$ −1.33265e6 −0.534497
$$363$$ 0 0
$$364$$ 75952.0 0.0300459
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −2.86154e6 −1.10901 −0.554503 0.832181i $$-0.687092\pi$$
−0.554503 + 0.832181i $$0.687092\pi$$
$$368$$ 78336.0 0.0301538
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −842052. −0.317617
$$372$$ 0 0
$$373$$ −615311. −0.228993 −0.114497 0.993424i $$-0.536526\pi$$
−0.114497 + 0.993424i $$0.536526\pi$$
$$374$$ −143856. −0.0531801
$$375$$ 0 0
$$376$$ −70272.0 −0.0256338
$$377$$ 796890. 0.288765
$$378$$ 0 0
$$379$$ 5.39878e6 1.93062 0.965311 0.261103i $$-0.0840863\pi$$
0.965311 + 0.261103i $$0.0840863\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 162552. 0.0569946
$$383$$ 1.08688e6 0.378602 0.189301 0.981919i $$-0.439378\pi$$
0.189301 + 0.981919i $$0.439378\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −1.97860e6 −0.675913
$$387$$ 0 0
$$388$$ 13168.0 0.00444059
$$389$$ 3.48432e6 1.16747 0.583733 0.811946i $$-0.301592\pi$$
0.583733 + 0.811946i $$0.301592\pi$$
$$390$$ 0 0
$$391$$ 49572.0 0.0163981
$$392$$ −934272. −0.307085
$$393$$ 0 0
$$394$$ 2.20937e6 0.717014
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −3.26591e6 −1.03999 −0.519993 0.854170i $$-0.674065\pi$$
−0.519993 + 0.854170i $$0.674065\pi$$
$$398$$ 2.74250e6 0.867839
$$399$$ 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. 0.266197
$$404$$ 541248. 0.164984
$$405$$ 0 0
$$406$$ 1.48332e6 0.446601
$$407$$ 1.91852e6 0.574092
$$408$$ 0 0
$$409$$ −1.45188e6 −0.429162 −0.214581 0.976706i $$-0.568839\pi$$
−0.214581 + 0.976706i $$0.568839\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 2.13510e6 0.619692
$$413$$ 827670. 0.238771
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −103424. −0.0293014
$$417$$ 0 0
$$418$$ −1.49628e6 −0.418863
$$419$$ −559380. −0.155658 −0.0778291 0.996967i $$-0.524799\pi$$
−0.0778291 + 0.996967i $$0.524799\pi$$
$$420$$ 0 0
$$421$$ −3.91470e6 −1.07645 −0.538224 0.842802i $$-0.680905\pi$$
−0.538224 + 0.842802i $$0.680905\pi$$
$$422$$ 2.99791e6 0.819478
$$423$$ 0 0
$$424$$ 1.14662e6 0.309746
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 1.02709e6 0.272608
$$428$$ 1.30003e6 0.343040
$$429$$ 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
$$433$$ −7.15969e6 −1.83516 −0.917581 0.397548i $$-0.869861\pi$$
−0.917581 + 0.397548i $$0.869861\pi$$
$$434$$ 1.61548e6 0.411698
$$435$$ 0 0
$$436$$ −3.47224e6 −0.874769
$$437$$ 515610. 0.129157
$$438$$ 0 0
$$439$$ 1.71790e6 0.425437 0.212719 0.977114i $$-0.431768\pi$$
0.212719 + 0.977114i $$0.431768\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −65448.0 −0.0159346
$$443$$ 3.39670e6 0.822332 0.411166 0.911560i $$-0.365122\pi$$
0.411166 + 0.911560i $$0.365122\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −677084. −0.161178
$$447$$ 0 0
$$448$$ −192512. −0.0453172
$$449$$ 3.39606e6 0.794986 0.397493 0.917605i $$-0.369880\pi$$
0.397493 + 0.917605i $$0.369880\pi$$
$$450$$ 0 0
$$451$$ −4.03330e6 −0.933724
$$452$$ −2.21318e6 −0.509532
$$453$$ 0 0
$$454$$ −185952. −0.0423410
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 4.52814e6 1.01421 0.507106 0.861883i $$-0.330715\pi$$
0.507106 + 0.861883i $$0.330715\pi$$
$$458$$ −360460. −0.0802959
$$459$$ 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
$$463$$ 7.19862e6 1.56062 0.780310 0.625393i $$-0.215061\pi$$
0.780310 + 0.625393i $$0.215061\pi$$
$$464$$ −2.01984e6 −0.435534
$$465$$ 0 0
$$466$$ 4.25654e6 0.908014
$$467$$ 4.83034e6 1.02491 0.512455 0.858714i $$-0.328736\pi$$
0.512455 + 0.858714i $$0.328736\pi$$
$$468$$ 0 0
$$469$$ 5029.00 0.00105572
$$470$$ 0 0
$$471$$ 0 0
$$472$$ −1.12704e6 −0.232854
$$473$$ 3.18592e6 0.654760
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −121824. −0.0246442
$$477$$ 0 0
$$478$$ −4.60632e6 −0.922113
$$479$$ −748650. −0.149087 −0.0745435 0.997218i $$-0.523750\pi$$
−0.0745435 + 0.997218i $$0.523750\pi$$
$$480$$ 0 0
$$481$$ 872842. 0.172018
$$482$$ 3.42487e6 0.671469
$$483$$ 0 0
$$484$$ −1.78827e6 −0.346993
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 5.16394e6 0.986641 0.493320 0.869848i $$-0.335783\pi$$
0.493320 + 0.869848i $$0.335783\pi$$
$$488$$ −1.39859e6 −0.265853
$$489$$ 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
$$493$$ −1.27818e6 −0.236851
$$494$$ −680740. −0.125506
$$495$$ 0 0
$$496$$ −2.19981e6 −0.401495
$$497$$ −1.91422e6 −0.347616
$$498$$ 0 0
$$499$$ −4.20588e6 −0.756145 −0.378072 0.925776i $$-0.623413\pi$$
−0.378072 + 0.925776i $$0.623413\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 830832. 0.147148
$$503$$ −8.18342e6 −1.44217 −0.721083 0.692849i $$-0.756355\pi$$
−0.721083 + 0.692849i $$0.756355\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −271728. −0.0471800
$$507$$ 0 0
$$508$$ 4.09677e6 0.704340
$$509$$ −3.85923e6 −0.660247 −0.330123 0.943938i $$-0.607090\pi$$
−0.330123 + 0.943938i $$0.607090\pi$$
$$510$$ 0 0
$$511$$ 1.63118e6 0.276344
$$512$$ 262144. 0.0441942
$$513$$ 0 0
$$514$$ −5.81275e6 −0.970452
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 243756. 0.0401078
$$518$$ 1.62470e6 0.266040
$$519$$ 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
$$523$$ −4.82224e6 −0.770894 −0.385447 0.922730i $$-0.625953\pi$$
−0.385447 + 0.922730i $$0.625953\pi$$
$$524$$ −1.89523e6 −0.301533
$$525$$ 0 0
$$526$$ 677184. 0.106719
$$527$$ −1.39207e6 −0.218340
$$528$$ 0 0
$$529$$ −6.34271e6 −0.985452
$$530$$ 0 0
$$531$$ 0 0
$$532$$ −1.26712e6 −0.194106
$$533$$ −1.83497e6 −0.279776
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −6848.00 −0.00102956
$$537$$ 0 0
$$538$$ 6.32436e6 0.942022
$$539$$ 3.24076e6 0.480479
$$540$$ 0 0
$$541$$ 362537. 0.0532549 0.0266274 0.999645i $$-0.491523\pi$$
0.0266274 + 0.999645i $$0.491523\pi$$
$$542$$ 3.29005e6 0.481065
$$543$$ 0 0
$$544$$ 165888. 0.0240335
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −3.11439e6 −0.445046 −0.222523 0.974927i $$-0.571429\pi$$
−0.222523 + 0.974927i $$0.571429\pi$$
$$548$$ −211488. −0.0300839
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −1.32947e7 −1.86551
$$552$$ 0 0
$$553$$ 3.25052e6 0.452002
$$554$$ 2.18729e6 0.302784
$$555$$ 0 0
$$556$$ −5.61184e6 −0.769872
$$557$$ −7.99304e6 −1.09163 −0.545813 0.837907i $$-0.683779\pi$$
−0.545813 + 0.837907i $$0.683779\pi$$
$$558$$ 0 0
$$559$$ 1.44945e6 0.196189
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 4.36999e6 0.583633
$$563$$ −1.23236e7 −1.63857 −0.819286 0.573385i $$-0.805630\pi$$
−0.819286 + 0.573385i $$0.805630\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 9.93920e6 1.30410
$$567$$ 0 0
$$568$$ 2.60659e6 0.339002
$$569$$ −1.01364e7 −1.31252 −0.656258 0.754537i $$-0.727862\pi$$
−0.656258 + 0.754537i $$0.727862\pi$$
$$570$$ 0 0
$$571$$ 6.53084e6 0.838260 0.419130 0.907926i $$-0.362335\pi$$
0.419130 + 0.907926i $$0.362335\pi$$
$$572$$ 358752. 0.0458463
$$573$$ 0 0
$$574$$ −3.41558e6 −0.432698
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 1.24453e6 0.155621 0.0778103 0.996968i $$-0.475207\pi$$
0.0778103 + 0.996968i $$0.475207\pi$$
$$578$$ −5.57445e6 −0.694037
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 5.10110e6 0.626936
$$582$$ 0 0
$$583$$ −3.97735e6 −0.484644
$$584$$ −2.22118e6 −0.269496
$$585$$ 0 0
$$586$$ −1.36558e6 −0.164275
$$587$$ −1.33403e6 −0.159797 −0.0798987 0.996803i $$-0.525460\pi$$
−0.0798987 + 0.996803i $$0.525460\pi$$
$$588$$ 0 0
$$589$$ −1.44792e7 −1.71972
$$590$$ 0 0
$$591$$ 0 0
$$592$$ −2.21235e6 −0.259448
$$593$$ −1.19401e7 −1.39435 −0.697177 0.716899i $$-0.745561\pi$$
−0.697177 + 0.716899i $$0.745561\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −1.75824e6 −0.202751
$$597$$ 0 0
$$598$$ −123624. −0.0141368
$$599$$ 7.16430e6 0.815843 0.407922 0.913017i $$-0.366254\pi$$
0.407922 + 0.913017i $$0.366254\pi$$
$$600$$ 0 0
$$601$$ 1.15163e6 0.130055 0.0650273 0.997883i $$-0.479287\pi$$
0.0650273 + 0.997883i $$0.479287\pi$$
$$602$$ 2.69799e6 0.303423
$$603$$ 0 0
$$604$$ −2.76165e6 −0.308018
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 1.34268e7 1.47911 0.739554 0.673097i $$-0.235037\pi$$
0.739554 + 0.673097i $$0.235037\pi$$
$$608$$ 1.72544e6 0.189296
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 110898. 0.0120177
$$612$$ 0 0
$$613$$ −1.20184e7 −1.29180 −0.645900 0.763422i $$-0.723518\pi$$
−0.645900 + 0.763422i $$0.723518\pi$$
$$614$$ −8.11591e6 −0.868793
$$615$$ 0 0
$$616$$ 667776. 0.0709054
$$617$$ −6.98519e6 −0.738695 −0.369348 0.929291i $$-0.620419\pi$$
−0.369348 + 0.929291i $$0.620419\pi$$
$$618$$ 0 0
$$619$$ −8.20625e6 −0.860832 −0.430416 0.902631i $$-0.641633\pi$$
−0.430416 + 0.902631i $$0.641633\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 826392. 0.0856466
$$623$$ 1.64688e6 0.169997
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 1.33689e7 1.36352
$$627$$ 0 0
$$628$$ 5.59989e6 0.566605
$$629$$ −1.40000e6 −0.141092
$$630$$ 0 0
$$631$$ −1.07686e7 −1.07668 −0.538338 0.842729i $$-0.680947\pi$$
−0.538338 + 0.842729i $$0.680947\pi$$
$$632$$ −4.42624e6 −0.440801
$$633$$ 0 0
$$634$$ −1.01316e7 −1.00104
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 1.47440e6 0.143968
$$638$$ 7.00632e6 0.681457
$$639$$ 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
$$643$$ −1.00999e7 −0.963364 −0.481682 0.876346i $$-0.659974\pi$$
−0.481682 + 0.876346i $$0.659974\pi$$
$$644$$ −230112. −0.0218637
$$645$$ 0 0
$$646$$ 1.09188e6 0.102942
$$647$$ 7.52113e6 0.706354 0.353177 0.935556i $$-0.385101\pi$$
0.353177 + 0.935556i $$0.385101\pi$$
$$648$$ 0 0
$$649$$ 3.90942e6 0.364335
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −3.08130e6 −0.283867
$$653$$ −2.67197e6 −0.245216 −0.122608 0.992455i $$-0.539126\pi$$
−0.122608 + 0.992455i $$0.539126\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 4.65101e6 0.421976
$$657$$ 0 0
$$658$$ 206424. 0.0185864
$$659$$ −6.99948e6 −0.627845 −0.313922 0.949449i $$-0.601643\pi$$
−0.313922 + 0.949449i $$0.601643\pi$$
$$660$$ 0 0
$$661$$ 408122. 0.0363318 0.0181659 0.999835i $$-0.494217\pi$$
0.0181659 + 0.999835i $$0.494217\pi$$
$$662$$ 2.40853e6 0.213603
$$663$$ 0 0
$$664$$ −6.94618e6 −0.611400
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −2.41434e6 −0.210128
$$668$$ 9.29107e6 0.805610
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 4.85137e6 0.415966
$$672$$ 0 0
$$673$$ −1.74939e7 −1.48885 −0.744423 0.667709i $$-0.767275\pi$$
−0.744423 + 0.667709i $$0.767275\pi$$
$$674$$ −839108. −0.0711489
$$675$$ 0 0
$$676$$ −5.77747e6 −0.486263
$$677$$ −8.67440e6 −0.727391 −0.363695 0.931518i $$-0.618485\pi$$
−0.363695 + 0.931518i $$0.618485\pi$$
$$678$$ 0 0
$$679$$ −38681.0 −0.00321976
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 7.63058e6 0.628198
$$683$$ 1.18478e7 0.971822 0.485911 0.874008i $$-0.338488\pi$$
0.485911 + 0.874008i $$0.338488\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 5.90414e6 0.479012
$$687$$ 0 0
$$688$$ −3.67386e6 −0.295904
$$689$$ −1.80952e6 −0.145216
$$690$$ 0 0
$$691$$ 9.47775e6 0.755110 0.377555 0.925987i $$-0.376765\pi$$
0.377555 + 0.925987i $$0.376765\pi$$
$$692$$ 1.18100e7 0.937530
$$693$$ 0 0
$$694$$ 1.60866e7 1.26785
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 2.94322e6 0.229478
$$698$$ 33320.0 0.00258861
$$699$$ 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
$$703$$ −1.45618e7 −1.11129
$$704$$ −909312. −0.0691483
$$705$$ 0 0
$$706$$ 7.80002e6 0.588958
$$707$$ −1.58992e6 −0.119626
$$708$$ 0 0
$$709$$ 1.27436e7 0.952090 0.476045 0.879421i $$-0.342070\pi$$
0.476045 + 0.879421i $$0.342070\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ −2.24256e6 −0.165785
$$713$$ −2.62946e6 −0.193706
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −7.95792e6 −0.580119
$$717$$ 0 0
$$718$$ −9.08352e6 −0.657572
$$719$$ −2.44929e6 −0.176692 −0.0883462 0.996090i $$-0.528158\pi$$
−0.0883462 + 0.996090i $$0.528158\pi$$
$$720$$ 0 0
$$721$$ −6.27187e6 −0.449323
$$722$$ 1.45250e6 0.103699
$$723$$ 0 0
$$724$$ −5.33061e6 −0.377947
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 415033. 0.0291237 0.0145619 0.999894i $$-0.495365\pi$$
0.0145619 + 0.999894i $$0.495365\pi$$
$$728$$ 303808. 0.0212457
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −2.32486e6 −0.160918
$$732$$ 0 0
$$733$$ 1.72877e7 1.18844 0.594221 0.804302i $$-0.297461\pi$$
0.594221 + 0.804302i $$0.297461\pi$$
$$734$$ −1.14461e7 −0.784186
$$735$$ 0 0
$$736$$ 313344. 0.0213219
$$737$$ 23754.0 0.00161090
$$738$$ 0 0
$$739$$ 5.18834e6 0.349476 0.174738 0.984615i $$-0.444092\pi$$
0.174738 + 0.984615i $$0.444092\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −3.36821e6 −0.224589
$$743$$ 4.79572e6 0.318700 0.159350 0.987222i $$-0.449060\pi$$
0.159350 + 0.987222i $$0.449060\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −2.46124e6 −0.161923
$$747$$ 0 0
$$748$$ −575424. −0.0376040
$$749$$ −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. −0.0181258
$$753$$ 0 0
$$754$$ 3.18756e6 0.204188
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −2.82068e7 −1.78902 −0.894508 0.447053i $$-0.852474\pi$$
−0.894508 + 0.447053i $$0.852474\pi$$
$$758$$ 2.15951e7 1.36516
$$759$$ 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
$$763$$ 1.01997e7 0.634273
$$764$$ 650208. 0.0403013
$$765$$ 0 0
$$766$$ 4.34750e6 0.267712
$$767$$ 1.77861e6 0.109167
$$768$$ 0 0
$$769$$ 2.20930e7 1.34722 0.673610 0.739087i $$-0.264743\pi$$
0.673610 + 0.739087i $$0.264743\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −7.91442e6 −0.477942
$$773$$ −3.00787e7 −1.81055 −0.905276 0.424824i $$-0.860336\pi$$
−0.905276 + 0.424824i $$0.860336\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 52672.0 0.00313997
$$777$$ 0 0
$$778$$ 1.39373e7 0.825523
$$779$$ 3.06131e7 1.80744
$$780$$ 0 0
$$781$$ −9.04162e6 −0.530418
$$782$$ 198288. 0.0115952
$$783$$ 0 0
$$784$$ −3.73709e6 −0.217142
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 3.28954e6 0.189321 0.0946605 0.995510i $$-0.469823\pi$$
0.0946605 + 0.995510i $$0.469823\pi$$
$$788$$ 8.83747e6 0.507005
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 6.50123e6 0.369449
$$792$$ 0 0
$$793$$ 2.20715e6 0.124638
$$794$$ −1.30636e7 −0.735381
$$795$$ 0 0
$$796$$ 1.09700e7 0.613655
$$797$$ 6.71053e6 0.374206 0.187103 0.982340i $$-0.440090\pi$$
0.187103 + 0.982340i $$0.440090\pi$$
$$798$$ 0 0
$$799$$ −177876. −0.00985713
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 1.70928e7 0.938374
$$803$$ 7.70473e6 0.421666
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 3.47157e6 0.188230
$$807$$ 0 0
$$808$$ 2.16499e6 0.116662
$$809$$ −8.74254e6 −0.469641 −0.234821 0.972039i $$-0.575450\pi$$
−0.234821 + 0.972039i $$0.575450\pi$$
$$810$$ 0 0
$$811$$ −2.48410e7 −1.32622 −0.663112 0.748520i $$-0.730765\pi$$
−0.663112 + 0.748520i $$0.730765\pi$$
$$812$$ 5.93328e6 0.315795
$$813$$ 0 0
$$814$$ 7.67410e6 0.405944
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −2.41814e7 −1.26744
$$818$$ −5.80750e6 −0.303463
$$819$$ 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
$$823$$ 8.70659e6 0.448073 0.224036 0.974581i $$-0.428077\pi$$
0.224036 + 0.974581i $$0.428077\pi$$
$$824$$ 8.54042e6 0.438189
$$825$$ 0 0
$$826$$ 3.31068e6 0.168837
$$827$$ 3.71184e7 1.88723 0.943617 0.331040i $$-0.107400\pi$$
0.943617 + 0.331040i $$0.107400\pi$$
$$828$$ 0 0
$$829$$ 1.01765e6 0.0514295 0.0257147 0.999669i $$-0.491814\pi$$
0.0257147 + 0.999669i $$0.491814\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ −413696. −0.0207192
$$833$$ −2.36488e6 −0.118085
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −5.98512e6 −0.296181
$$837$$ 0 0
$$838$$ −2.23752e6 −0.110067
$$839$$ 3.36194e7 1.64887 0.824433 0.565960i $$-0.191494\pi$$
0.824433 + 0.565960i $$0.191494\pi$$
$$840$$ 0 0
$$841$$ 4.17410e7 2.03504
$$842$$ −1.56588e7 −0.761164
$$843$$ 0 0
$$844$$ 1.19916e7 0.579458
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 5.25305e6 0.251596
$$848$$ 4.58650e6 0.219024
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −2.64445e6 −0.125173
$$852$$ 0 0
$$853$$ 3.52574e7 1.65912 0.829559 0.558419i $$-0.188592\pi$$
0.829559 + 0.558419i $$0.188592\pi$$
$$854$$ 4.10836e6 0.192763
$$855$$ 0 0
$$856$$ 5.20013e6 0.242566
$$857$$ −3.14941e7 −1.46480 −0.732398 0.680877i $$-0.761599\pi$$
−0.732398 + 0.680877i $$0.761599\pi$$
$$858$$ 0 0
$$859$$ 1.19344e7 0.551848 0.275924 0.961180i $$-0.411016\pi$$
0.275924 + 0.961180i $$0.411016\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 1.43000e7 0.655492
$$863$$ 8.70442e6 0.397844 0.198922 0.980015i $$-0.436256\pi$$
0.198922 + 0.980015i $$0.436256\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −2.86388e7 −1.29766
$$867$$ 0 0
$$868$$ 6.46194e6 0.291114
$$869$$ 1.53535e7 0.689697
$$870$$ 0 0
$$871$$ 10807.0 0.000482681 0
$$872$$ −1.38890e7 −0.618555
$$873$$ 0 0
$$874$$ 2.06244e6 0.0913277
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −1.17999e7 −0.518059 −0.259029 0.965869i $$-0.583403\pi$$
−0.259029 + 0.965869i $$0.583403\pi$$
$$878$$ 6.87158e6 0.300829
$$879$$ 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
$$883$$ 8.80577e6 0.380072 0.190036 0.981777i $$-0.439140\pi$$
0.190036 + 0.981777i $$0.439140\pi$$
$$884$$ −261792. −0.0112675
$$885$$ 0 0
$$886$$ 1.35868e7 0.581477
$$887$$ 250122. 0.0106744 0.00533719 0.999986i $$-0.498301\pi$$
0.00533719 + 0.999986i $$0.498301\pi$$
$$888$$ 0 0
$$889$$ −1.20343e7 −0.510699
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −2.70834e6 −0.113970
$$893$$ −1.85013e6 −0.0776379
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −770048. −0.0320441
$$897$$ 0 0
$$898$$ 1.35842e7 0.562140
$$899$$ 6.77988e7 2.79784
$$900$$ 0 0
$$901$$ 2.90239e6 0.119109
$$902$$ −1.61332e7 −0.660243
$$903$$ 0 0
$$904$$ −8.85274e6 −0.360294
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 3.24955e7 1.31161 0.655806 0.754929i $$-0.272329\pi$$
0.655806 + 0.754929i $$0.272329\pi$$
$$908$$ −743808. −0.0299396
$$909$$ 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
$$913$$ 2.40945e7 0.956625
$$914$$ 1.81126e7 0.717157
$$915$$ 0 0
$$916$$ −1.44184e6 −0.0567778
$$917$$ 5.56724e6 0.218634
$$918$$ 0 0
$$919$$ 1.41629e7 0.553176 0.276588 0.960989i $$-0.410796\pi$$
0.276588 + 0.960989i $$0.410796\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 5.11579e6 0.198192
$$923$$ −4.11353e6 −0.158932
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 2.87945e7 1.10352
$$927$$ 0 0
$$928$$ −8.07936e6 −0.307969
$$929$$ −4.37292e7 −1.66239 −0.831194 0.555982i $$-0.812342\pi$$
−0.831194 + 0.555982i $$0.812342\pi$$
$$930$$ 0 0
$$931$$ −2.45976e7 −0.930077
$$932$$ 1.70262e7 0.642063
$$933$$ 0 0
$$934$$ 1.93214e7 0.724721
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −5.73509e6 −0.213398 −0.106699 0.994291i $$-0.534028\pi$$
−0.106699 + 0.994291i $$0.534028\pi$$
$$938$$ 20116.0 0.000746508 0
$$939$$ 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
$$943$$ 5.55941e6 0.203587
$$944$$ −4.50816e6 −0.164653
$$945$$ 0 0
$$946$$ 1.27437e7 0.462985
$$947$$ −3.07342e7 −1.11365 −0.556823 0.830631i $$-0.687980\pi$$
−0.556823 + 0.830631i $$0.687980\pi$$
$$948$$ 0 0
$$949$$ 3.50531e6 0.126346
$$950$$ 0 0
$$951$$ 0 0
$$952$$ −487296. −0.0174261
$$953$$ −2.51847e7 −0.898264 −0.449132 0.893465i $$-0.648267\pi$$
−0.449132 + 0.893465i $$0.648267\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −1.84253e7 −0.652033
$$957$$ 0 0
$$958$$ −2.99460e6 −0.105420
$$959$$ 621246. 0.0218131
$$960$$ 0 0
$$961$$ 4.52105e7 1.57918
$$962$$ 3.49137e6 0.121635
$$963$$ 0 0
$$964$$ 1.36995e7 0.474801
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −1.44556e7 −0.497130 −0.248565 0.968615i $$-0.579959\pi$$
−0.248565 + 0.968615i $$0.579959\pi$$
$$968$$ −7.15309e6 −0.245361
$$969$$ 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
$$973$$ 1.64848e7 0.558214
$$974$$ 2.06558e7 0.697660
$$975$$ 0 0
$$976$$ −5.59437e6 −0.187986
$$977$$ −8.41568e6 −0.282067 −0.141034 0.990005i $$-0.545043\pi$$
−0.141034 + 0.990005i $$0.545043\pi$$
$$978$$ 0 0
$$979$$ 7.77888e6 0.259394
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −3.41715e7 −1.13080
$$983$$ 3.89409e7 1.28535 0.642676 0.766138i $$-0.277824\pi$$
0.642676 + 0.766138i $$0.277824\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −5.11272e6 −0.167479
$$987$$ 0 0
$$988$$ −2.72296e6 −0.0887460
$$989$$ −4.39141e6 −0.142762
$$990$$ 0 0
$$991$$ 4.84592e7 1.56745 0.783723 0.621111i $$-0.213318\pi$$
0.783723 + 0.621111i $$0.213318\pi$$
$$992$$ −8.79923e6 −0.283900
$$993$$ 0 0
$$994$$ −7.65686e6 −0.245802
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −3.84733e7 −1.22581 −0.612903 0.790158i $$-0.709998\pi$$
−0.612903 + 0.790158i $$0.709998\pi$$
$$998$$ −1.68235e7 −0.534675
$$999$$ 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.a.p.1.1 1
3.2 odd 2 150.6.a.a.1.1 1
5.2 odd 4 450.6.c.e.199.2 2
5.3 odd 4 450.6.c.e.199.1 2
5.4 even 2 450.6.a.i.1.1 1
15.2 even 4 150.6.c.g.49.1 2
15.8 even 4 150.6.c.g.49.2 2
15.14 odd 2 150.6.a.m.1.1 yes 1

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