# Properties

 Label 80.6.a.a.1.1 Level $80$ Weight $6$ Character 80.1 Self dual yes Analytic conductor $12.831$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$80 = 2^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 80.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$12.8307055850$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 10) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 80.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-24.0000 q^{3} +25.0000 q^{5} +172.000 q^{7} +333.000 q^{9} +O(q^{10})$$ $$q-24.0000 q^{3} +25.0000 q^{5} +172.000 q^{7} +333.000 q^{9} -132.000 q^{11} -946.000 q^{13} -600.000 q^{15} -222.000 q^{17} -500.000 q^{19} -4128.00 q^{21} -3564.00 q^{23} +625.000 q^{25} -2160.00 q^{27} +2190.00 q^{29} -2312.00 q^{31} +3168.00 q^{33} +4300.00 q^{35} -11242.0 q^{37} +22704.0 q^{39} +1242.00 q^{41} -20624.0 q^{43} +8325.00 q^{45} -6588.00 q^{47} +12777.0 q^{49} +5328.00 q^{51} -21066.0 q^{53} -3300.00 q^{55} +12000.0 q^{57} -7980.00 q^{59} +16622.0 q^{61} +57276.0 q^{63} -23650.0 q^{65} -1808.00 q^{67} +85536.0 q^{69} +24528.0 q^{71} +20474.0 q^{73} -15000.0 q^{75} -22704.0 q^{77} +46240.0 q^{79} -29079.0 q^{81} +51576.0 q^{83} -5550.00 q^{85} -52560.0 q^{87} -110310. q^{89} -162712. q^{91} +55488.0 q^{93} -12500.0 q^{95} -78382.0 q^{97} -43956.0 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −24.0000 −1.53960 −0.769800 0.638285i $$-0.779644\pi$$
−0.769800 + 0.638285i $$0.779644\pi$$
$$4$$ 0 0
$$5$$ 25.0000 0.447214
$$6$$ 0 0
$$7$$ 172.000 1.32673 0.663366 0.748295i $$-0.269127\pi$$
0.663366 + 0.748295i $$0.269127\pi$$
$$8$$ 0 0
$$9$$ 333.000 1.37037
$$10$$ 0 0
$$11$$ −132.000 −0.328921 −0.164461 0.986384i $$-0.552588\pi$$
−0.164461 + 0.986384i $$0.552588\pi$$
$$12$$ 0 0
$$13$$ −946.000 −1.55250 −0.776252 0.630423i $$-0.782882\pi$$
−0.776252 + 0.630423i $$0.782882\pi$$
$$14$$ 0 0
$$15$$ −600.000 −0.688530
$$16$$ 0 0
$$17$$ −222.000 −0.186308 −0.0931538 0.995652i $$-0.529695\pi$$
−0.0931538 + 0.995652i $$0.529695\pi$$
$$18$$ 0 0
$$19$$ −500.000 −0.317750 −0.158875 0.987299i $$-0.550787\pi$$
−0.158875 + 0.987299i $$0.550787\pi$$
$$20$$ 0 0
$$21$$ −4128.00 −2.04264
$$22$$ 0 0
$$23$$ −3564.00 −1.40481 −0.702406 0.711777i $$-0.747891\pi$$
−0.702406 + 0.711777i $$0.747891\pi$$
$$24$$ 0 0
$$25$$ 625.000 0.200000
$$26$$ 0 0
$$27$$ −2160.00 −0.570222
$$28$$ 0 0
$$29$$ 2190.00 0.483559 0.241779 0.970331i $$-0.422269\pi$$
0.241779 + 0.970331i $$0.422269\pi$$
$$30$$ 0 0
$$31$$ −2312.00 −0.432099 −0.216050 0.976382i $$-0.569317\pi$$
−0.216050 + 0.976382i $$0.569317\pi$$
$$32$$ 0 0
$$33$$ 3168.00 0.506408
$$34$$ 0 0
$$35$$ 4300.00 0.593333
$$36$$ 0 0
$$37$$ −11242.0 −1.35002 −0.675009 0.737810i $$-0.735860\pi$$
−0.675009 + 0.737810i $$0.735860\pi$$
$$38$$ 0 0
$$39$$ 22704.0 2.39024
$$40$$ 0 0
$$41$$ 1242.00 0.115388 0.0576942 0.998334i $$-0.481625\pi$$
0.0576942 + 0.998334i $$0.481625\pi$$
$$42$$ 0 0
$$43$$ −20624.0 −1.70099 −0.850495 0.525983i $$-0.823697\pi$$
−0.850495 + 0.525983i $$0.823697\pi$$
$$44$$ 0 0
$$45$$ 8325.00 0.612848
$$46$$ 0 0
$$47$$ −6588.00 −0.435020 −0.217510 0.976058i $$-0.569793\pi$$
−0.217510 + 0.976058i $$0.569793\pi$$
$$48$$ 0 0
$$49$$ 12777.0 0.760219
$$50$$ 0 0
$$51$$ 5328.00 0.286839
$$52$$ 0 0
$$53$$ −21066.0 −1.03013 −0.515065 0.857151i $$-0.672232\pi$$
−0.515065 + 0.857151i $$0.672232\pi$$
$$54$$ 0 0
$$55$$ −3300.00 −0.147098
$$56$$ 0 0
$$57$$ 12000.0 0.489209
$$58$$ 0 0
$$59$$ −7980.00 −0.298451 −0.149225 0.988803i $$-0.547678\pi$$
−0.149225 + 0.988803i $$0.547678\pi$$
$$60$$ 0 0
$$61$$ 16622.0 0.571951 0.285975 0.958237i $$-0.407682\pi$$
0.285975 + 0.958237i $$0.407682\pi$$
$$62$$ 0 0
$$63$$ 57276.0 1.81811
$$64$$ 0 0
$$65$$ −23650.0 −0.694301
$$66$$ 0 0
$$67$$ −1808.00 −0.0492052 −0.0246026 0.999697i $$-0.507832\pi$$
−0.0246026 + 0.999697i $$0.507832\pi$$
$$68$$ 0 0
$$69$$ 85536.0 2.16285
$$70$$ 0 0
$$71$$ 24528.0 0.577452 0.288726 0.957412i $$-0.406768\pi$$
0.288726 + 0.957412i $$0.406768\pi$$
$$72$$ 0 0
$$73$$ 20474.0 0.449672 0.224836 0.974397i $$-0.427815\pi$$
0.224836 + 0.974397i $$0.427815\pi$$
$$74$$ 0 0
$$75$$ −15000.0 −0.307920
$$76$$ 0 0
$$77$$ −22704.0 −0.436391
$$78$$ 0 0
$$79$$ 46240.0 0.833585 0.416793 0.909002i $$-0.363154\pi$$
0.416793 + 0.909002i $$0.363154\pi$$
$$80$$ 0 0
$$81$$ −29079.0 −0.492455
$$82$$ 0 0
$$83$$ 51576.0 0.821774 0.410887 0.911686i $$-0.365219\pi$$
0.410887 + 0.911686i $$0.365219\pi$$
$$84$$ 0 0
$$85$$ −5550.00 −0.0833193
$$86$$ 0 0
$$87$$ −52560.0 −0.744487
$$88$$ 0 0
$$89$$ −110310. −1.47618 −0.738091 0.674701i $$-0.764272\pi$$
−0.738091 + 0.674701i $$0.764272\pi$$
$$90$$ 0 0
$$91$$ −162712. −2.05976
$$92$$ 0 0
$$93$$ 55488.0 0.665260
$$94$$ 0 0
$$95$$ −12500.0 −0.142102
$$96$$ 0 0
$$97$$ −78382.0 −0.845838 −0.422919 0.906168i $$-0.638994\pi$$
−0.422919 + 0.906168i $$0.638994\pi$$
$$98$$ 0 0
$$99$$ −43956.0 −0.450744
$$100$$ 0 0
$$101$$ 141942. 1.38455 0.692273 0.721636i $$-0.256609\pi$$
0.692273 + 0.721636i $$0.256609\pi$$
$$102$$ 0 0
$$103$$ 436.000 0.00404943 0.00202471 0.999998i $$-0.499356\pi$$
0.00202471 + 0.999998i $$0.499356\pi$$
$$104$$ 0 0
$$105$$ −103200. −0.913496
$$106$$ 0 0
$$107$$ −232968. −1.96715 −0.983574 0.180508i $$-0.942226\pi$$
−0.983574 + 0.180508i $$0.942226\pi$$
$$108$$ 0 0
$$109$$ −174850. −1.40961 −0.704806 0.709400i $$-0.748966\pi$$
−0.704806 + 0.709400i $$0.748966\pi$$
$$110$$ 0 0
$$111$$ 269808. 2.07849
$$112$$ 0 0
$$113$$ 182994. 1.34816 0.674079 0.738659i $$-0.264541\pi$$
0.674079 + 0.738659i $$0.264541\pi$$
$$114$$ 0 0
$$115$$ −89100.0 −0.628251
$$116$$ 0 0
$$117$$ −315018. −2.12751
$$118$$ 0 0
$$119$$ −38184.0 −0.247180
$$120$$ 0 0
$$121$$ −143627. −0.891811
$$122$$ 0 0
$$123$$ −29808.0 −0.177652
$$124$$ 0 0
$$125$$ 15625.0 0.0894427
$$126$$ 0 0
$$127$$ 122452. 0.673685 0.336842 0.941561i $$-0.390641\pi$$
0.336842 + 0.941561i $$0.390641\pi$$
$$128$$ 0 0
$$129$$ 494976. 2.61885
$$130$$ 0 0
$$131$$ 241908. 1.23161 0.615803 0.787900i $$-0.288832\pi$$
0.615803 + 0.787900i $$0.288832\pi$$
$$132$$ 0 0
$$133$$ −86000.0 −0.421570
$$134$$ 0 0
$$135$$ −54000.0 −0.255011
$$136$$ 0 0
$$137$$ 277098. 1.26134 0.630670 0.776051i $$-0.282780\pi$$
0.630670 + 0.776051i $$0.282780\pi$$
$$138$$ 0 0
$$139$$ 193540. 0.849638 0.424819 0.905278i $$-0.360338\pi$$
0.424819 + 0.905278i $$0.360338\pi$$
$$140$$ 0 0
$$141$$ 158112. 0.669757
$$142$$ 0 0
$$143$$ 124872. 0.510652
$$144$$ 0 0
$$145$$ 54750.0 0.216254
$$146$$ 0 0
$$147$$ −306648. −1.17043
$$148$$ 0 0
$$149$$ 140550. 0.518639 0.259320 0.965792i $$-0.416502\pi$$
0.259320 + 0.965792i $$0.416502\pi$$
$$150$$ 0 0
$$151$$ −433952. −1.54881 −0.774407 0.632688i $$-0.781952\pi$$
−0.774407 + 0.632688i $$0.781952\pi$$
$$152$$ 0 0
$$153$$ −73926.0 −0.255310
$$154$$ 0 0
$$155$$ −57800.0 −0.193241
$$156$$ 0 0
$$157$$ −555922. −1.79997 −0.899984 0.435923i $$-0.856422\pi$$
−0.899984 + 0.435923i $$0.856422\pi$$
$$158$$ 0 0
$$159$$ 505584. 1.58599
$$160$$ 0 0
$$161$$ −613008. −1.86381
$$162$$ 0 0
$$163$$ 66616.0 0.196386 0.0981928 0.995167i $$-0.468694\pi$$
0.0981928 + 0.995167i $$0.468694\pi$$
$$164$$ 0 0
$$165$$ 79200.0 0.226472
$$166$$ 0 0
$$167$$ 205692. 0.570724 0.285362 0.958420i $$-0.407886\pi$$
0.285362 + 0.958420i $$0.407886\pi$$
$$168$$ 0 0
$$169$$ 523623. 1.41027
$$170$$ 0 0
$$171$$ −166500. −0.435436
$$172$$ 0 0
$$173$$ 433854. 1.10212 0.551059 0.834466i $$-0.314224\pi$$
0.551059 + 0.834466i $$0.314224\pi$$
$$174$$ 0 0
$$175$$ 107500. 0.265346
$$176$$ 0 0
$$177$$ 191520. 0.459495
$$178$$ 0 0
$$179$$ 489180. 1.14113 0.570566 0.821252i $$-0.306724\pi$$
0.570566 + 0.821252i $$0.306724\pi$$
$$180$$ 0 0
$$181$$ 719462. 1.63234 0.816172 0.577810i $$-0.196092\pi$$
0.816172 + 0.577810i $$0.196092\pi$$
$$182$$ 0 0
$$183$$ −398928. −0.880576
$$184$$ 0 0
$$185$$ −281050. −0.603746
$$186$$ 0 0
$$187$$ 29304.0 0.0612806
$$188$$ 0 0
$$189$$ −371520. −0.756533
$$190$$ 0 0
$$191$$ 185928. 0.368775 0.184387 0.982854i $$-0.440970\pi$$
0.184387 + 0.982854i $$0.440970\pi$$
$$192$$ 0 0
$$193$$ −591406. −1.14286 −0.571429 0.820651i $$-0.693611\pi$$
−0.571429 + 0.820651i $$0.693611\pi$$
$$194$$ 0 0
$$195$$ 567600. 1.06895
$$196$$ 0 0
$$197$$ 449478. 0.825169 0.412584 0.910919i $$-0.364626\pi$$
0.412584 + 0.910919i $$0.364626\pi$$
$$198$$ 0 0
$$199$$ −157160. −0.281326 −0.140663 0.990058i $$-0.544923\pi$$
−0.140663 + 0.990058i $$0.544923\pi$$
$$200$$ 0 0
$$201$$ 43392.0 0.0757564
$$202$$ 0 0
$$203$$ 376680. 0.641553
$$204$$ 0 0
$$205$$ 31050.0 0.0516032
$$206$$ 0 0
$$207$$ −1.18681e6 −1.92511
$$208$$ 0 0
$$209$$ 66000.0 0.104515
$$210$$ 0 0
$$211$$ −253052. −0.391294 −0.195647 0.980674i $$-0.562681\pi$$
−0.195647 + 0.980674i $$0.562681\pi$$
$$212$$ 0 0
$$213$$ −588672. −0.889046
$$214$$ 0 0
$$215$$ −515600. −0.760706
$$216$$ 0 0
$$217$$ −397664. −0.573280
$$218$$ 0 0
$$219$$ −491376. −0.692315
$$220$$ 0 0
$$221$$ 210012. 0.289243
$$222$$ 0 0
$$223$$ −1.07344e6 −1.44550 −0.722749 0.691111i $$-0.757122\pi$$
−0.722749 + 0.691111i $$0.757122\pi$$
$$224$$ 0 0
$$225$$ 208125. 0.274074
$$226$$ 0 0
$$227$$ 626832. 0.807396 0.403698 0.914892i $$-0.367725\pi$$
0.403698 + 0.914892i $$0.367725\pi$$
$$228$$ 0 0
$$229$$ −116650. −0.146993 −0.0734964 0.997295i $$-0.523416\pi$$
−0.0734964 + 0.997295i $$0.523416\pi$$
$$230$$ 0 0
$$231$$ 544896. 0.671868
$$232$$ 0 0
$$233$$ −743046. −0.896656 −0.448328 0.893869i $$-0.647980\pi$$
−0.448328 + 0.893869i $$0.647980\pi$$
$$234$$ 0 0
$$235$$ −164700. −0.194547
$$236$$ 0 0
$$237$$ −1.10976e6 −1.28339
$$238$$ 0 0
$$239$$ −978720. −1.10832 −0.554158 0.832411i $$-0.686960\pi$$
−0.554158 + 0.832411i $$0.686960\pi$$
$$240$$ 0 0
$$241$$ −1.13280e6 −1.25635 −0.628174 0.778073i $$-0.716197\pi$$
−0.628174 + 0.778073i $$0.716197\pi$$
$$242$$ 0 0
$$243$$ 1.22278e6 1.32841
$$244$$ 0 0
$$245$$ 319425. 0.339980
$$246$$ 0 0
$$247$$ 473000. 0.493309
$$248$$ 0 0
$$249$$ −1.23782e6 −1.26520
$$250$$ 0 0
$$251$$ −905652. −0.907355 −0.453677 0.891166i $$-0.649888\pi$$
−0.453677 + 0.891166i $$0.649888\pi$$
$$252$$ 0 0
$$253$$ 470448. 0.462073
$$254$$ 0 0
$$255$$ 133200. 0.128278
$$256$$ 0 0
$$257$$ 1.93994e6 1.83212 0.916062 0.401036i $$-0.131350\pi$$
0.916062 + 0.401036i $$0.131350\pi$$
$$258$$ 0 0
$$259$$ −1.93362e6 −1.79111
$$260$$ 0 0
$$261$$ 729270. 0.662654
$$262$$ 0 0
$$263$$ 805476. 0.718064 0.359032 0.933325i $$-0.383107\pi$$
0.359032 + 0.933325i $$0.383107\pi$$
$$264$$ 0 0
$$265$$ −526650. −0.460689
$$266$$ 0 0
$$267$$ 2.64744e6 2.27273
$$268$$ 0 0
$$269$$ −858690. −0.723529 −0.361764 0.932270i $$-0.617825\pi$$
−0.361764 + 0.932270i $$0.617825\pi$$
$$270$$ 0 0
$$271$$ 383608. 0.317296 0.158648 0.987335i $$-0.449287\pi$$
0.158648 + 0.987335i $$0.449287\pi$$
$$272$$ 0 0
$$273$$ 3.90509e6 3.17120
$$274$$ 0 0
$$275$$ −82500.0 −0.0657843
$$276$$ 0 0
$$277$$ 2.01076e6 1.57456 0.787282 0.616593i $$-0.211488\pi$$
0.787282 + 0.616593i $$0.211488\pi$$
$$278$$ 0 0
$$279$$ −769896. −0.592136
$$280$$ 0 0
$$281$$ 202602. 0.153066 0.0765329 0.997067i $$-0.475615\pi$$
0.0765329 + 0.997067i $$0.475615\pi$$
$$282$$ 0 0
$$283$$ 221536. 0.164429 0.0822145 0.996615i $$-0.473801\pi$$
0.0822145 + 0.996615i $$0.473801\pi$$
$$284$$ 0 0
$$285$$ 300000. 0.218781
$$286$$ 0 0
$$287$$ 213624. 0.153089
$$288$$ 0 0
$$289$$ −1.37057e6 −0.965289
$$290$$ 0 0
$$291$$ 1.88117e6 1.30225
$$292$$ 0 0
$$293$$ −322506. −0.219467 −0.109733 0.993961i $$-0.535000\pi$$
−0.109733 + 0.993961i $$0.535000\pi$$
$$294$$ 0 0
$$295$$ −199500. −0.133471
$$296$$ 0 0
$$297$$ 285120. 0.187558
$$298$$ 0 0
$$299$$ 3.37154e6 2.18098
$$300$$ 0 0
$$301$$ −3.54733e6 −2.25676
$$302$$ 0 0
$$303$$ −3.40661e6 −2.13165
$$304$$ 0 0
$$305$$ 415550. 0.255784
$$306$$ 0 0
$$307$$ −1.44301e6 −0.873822 −0.436911 0.899505i $$-0.643927\pi$$
−0.436911 + 0.899505i $$0.643927\pi$$
$$308$$ 0 0
$$309$$ −10464.0 −0.00623450
$$310$$ 0 0
$$311$$ −171312. −0.100435 −0.0502177 0.998738i $$-0.515992\pi$$
−0.0502177 + 0.998738i $$0.515992\pi$$
$$312$$ 0 0
$$313$$ −1.02689e6 −0.592463 −0.296232 0.955116i $$-0.595730\pi$$
−0.296232 + 0.955116i $$0.595730\pi$$
$$314$$ 0 0
$$315$$ 1.43190e6 0.813086
$$316$$ 0 0
$$317$$ 752958. 0.420845 0.210423 0.977610i $$-0.432516\pi$$
0.210423 + 0.977610i $$0.432516\pi$$
$$318$$ 0 0
$$319$$ −289080. −0.159053
$$320$$ 0 0
$$321$$ 5.59123e6 3.02862
$$322$$ 0 0
$$323$$ 111000. 0.0591993
$$324$$ 0 0
$$325$$ −591250. −0.310501
$$326$$ 0 0
$$327$$ 4.19640e6 2.17024
$$328$$ 0 0
$$329$$ −1.13314e6 −0.577155
$$330$$ 0 0
$$331$$ −1.99413e6 −1.00042 −0.500212 0.865903i $$-0.666745\pi$$
−0.500212 + 0.865903i $$0.666745\pi$$
$$332$$ 0 0
$$333$$ −3.74359e6 −1.85002
$$334$$ 0 0
$$335$$ −45200.0 −0.0220053
$$336$$ 0 0
$$337$$ −987022. −0.473426 −0.236713 0.971580i $$-0.576070\pi$$
−0.236713 + 0.971580i $$0.576070\pi$$
$$338$$ 0 0
$$339$$ −4.39186e6 −2.07562
$$340$$ 0 0
$$341$$ 305184. 0.142127
$$342$$ 0 0
$$343$$ −693160. −0.318125
$$344$$ 0 0
$$345$$ 2.13840e6 0.967256
$$346$$ 0 0
$$347$$ −2.20601e6 −0.983520 −0.491760 0.870731i $$-0.663646\pi$$
−0.491760 + 0.870731i $$0.663646\pi$$
$$348$$ 0 0
$$349$$ 2.74187e6 1.20499 0.602495 0.798123i $$-0.294173\pi$$
0.602495 + 0.798123i $$0.294173\pi$$
$$350$$ 0 0
$$351$$ 2.04336e6 0.885273
$$352$$ 0 0
$$353$$ −2.38957e6 −1.02066 −0.510331 0.859978i $$-0.670477\pi$$
−0.510331 + 0.859978i $$0.670477\pi$$
$$354$$ 0 0
$$355$$ 613200. 0.258245
$$356$$ 0 0
$$357$$ 916416. 0.380559
$$358$$ 0 0
$$359$$ 279480. 0.114450 0.0572248 0.998361i $$-0.481775\pi$$
0.0572248 + 0.998361i $$0.481775\pi$$
$$360$$ 0 0
$$361$$ −2.22610e6 −0.899035
$$362$$ 0 0
$$363$$ 3.44705e6 1.37303
$$364$$ 0 0
$$365$$ 511850. 0.201099
$$366$$ 0 0
$$367$$ 2.47637e6 0.959734 0.479867 0.877341i $$-0.340685\pi$$
0.479867 + 0.877341i $$0.340685\pi$$
$$368$$ 0 0
$$369$$ 413586. 0.158125
$$370$$ 0 0
$$371$$ −3.62335e6 −1.36671
$$372$$ 0 0
$$373$$ 2.74525e6 1.02167 0.510835 0.859679i $$-0.329336\pi$$
0.510835 + 0.859679i $$0.329336\pi$$
$$374$$ 0 0
$$375$$ −375000. −0.137706
$$376$$ 0 0
$$377$$ −2.07174e6 −0.750727
$$378$$ 0 0
$$379$$ 1.18906e6 0.425212 0.212606 0.977138i $$-0.431805\pi$$
0.212606 + 0.977138i $$0.431805\pi$$
$$380$$ 0 0
$$381$$ −2.93885e6 −1.03721
$$382$$ 0 0
$$383$$ −3.25760e6 −1.13475 −0.567377 0.823458i $$-0.692042\pi$$
−0.567377 + 0.823458i $$0.692042\pi$$
$$384$$ 0 0
$$385$$ −567600. −0.195160
$$386$$ 0 0
$$387$$ −6.86779e6 −2.33099
$$388$$ 0 0
$$389$$ 1.98351e6 0.664600 0.332300 0.943174i $$-0.392175\pi$$
0.332300 + 0.943174i $$0.392175\pi$$
$$390$$ 0 0
$$391$$ 791208. 0.261727
$$392$$ 0 0
$$393$$ −5.80579e6 −1.89618
$$394$$ 0 0
$$395$$ 1.15600e6 0.372791
$$396$$ 0 0
$$397$$ 4.97416e6 1.58396 0.791978 0.610549i $$-0.209051\pi$$
0.791978 + 0.610549i $$0.209051\pi$$
$$398$$ 0 0
$$399$$ 2.06400e6 0.649049
$$400$$ 0 0
$$401$$ −1.34264e6 −0.416963 −0.208482 0.978026i $$-0.566852\pi$$
−0.208482 + 0.978026i $$0.566852\pi$$
$$402$$ 0 0
$$403$$ 2.18715e6 0.670836
$$404$$ 0 0
$$405$$ −726975. −0.220233
$$406$$ 0 0
$$407$$ 1.48394e6 0.444050
$$408$$ 0 0
$$409$$ −1.09423e6 −0.323445 −0.161722 0.986836i $$-0.551705\pi$$
−0.161722 + 0.986836i $$0.551705\pi$$
$$410$$ 0 0
$$411$$ −6.65035e6 −1.94196
$$412$$ 0 0
$$413$$ −1.37256e6 −0.395964
$$414$$ 0 0
$$415$$ 1.28940e6 0.367509
$$416$$ 0 0
$$417$$ −4.64496e6 −1.30810
$$418$$ 0 0
$$419$$ 954060. 0.265485 0.132743 0.991151i $$-0.457622\pi$$
0.132743 + 0.991151i $$0.457622\pi$$
$$420$$ 0 0
$$421$$ −1.59390e6 −0.438284 −0.219142 0.975693i $$-0.570326\pi$$
−0.219142 + 0.975693i $$0.570326\pi$$
$$422$$ 0 0
$$423$$ −2.19380e6 −0.596138
$$424$$ 0 0
$$425$$ −138750. −0.0372615
$$426$$ 0 0
$$427$$ 2.85898e6 0.758826
$$428$$ 0 0
$$429$$ −2.99693e6 −0.786200
$$430$$ 0 0
$$431$$ 2.64665e6 0.686283 0.343141 0.939284i $$-0.388509\pi$$
0.343141 + 0.939284i $$0.388509\pi$$
$$432$$ 0 0
$$433$$ 3.72355e6 0.954416 0.477208 0.878790i $$-0.341649\pi$$
0.477208 + 0.878790i $$0.341649\pi$$
$$434$$ 0 0
$$435$$ −1.31400e6 −0.332945
$$436$$ 0 0
$$437$$ 1.78200e6 0.446379
$$438$$ 0 0
$$439$$ 2.58340e6 0.639780 0.319890 0.947455i $$-0.396354\pi$$
0.319890 + 0.947455i $$0.396354\pi$$
$$440$$ 0 0
$$441$$ 4.25474e6 1.04178
$$442$$ 0 0
$$443$$ −7.56206e6 −1.83076 −0.915379 0.402593i $$-0.868109\pi$$
−0.915379 + 0.402593i $$0.868109\pi$$
$$444$$ 0 0
$$445$$ −2.75775e6 −0.660169
$$446$$ 0 0
$$447$$ −3.37320e6 −0.798497
$$448$$ 0 0
$$449$$ 4.30773e6 1.00840 0.504200 0.863587i $$-0.331788\pi$$
0.504200 + 0.863587i $$0.331788\pi$$
$$450$$ 0 0
$$451$$ −163944. −0.0379537
$$452$$ 0 0
$$453$$ 1.04148e7 2.38456
$$454$$ 0 0
$$455$$ −4.06780e6 −0.921152
$$456$$ 0 0
$$457$$ −2.24354e6 −0.502509 −0.251254 0.967921i $$-0.580843\pi$$
−0.251254 + 0.967921i $$0.580843\pi$$
$$458$$ 0 0
$$459$$ 479520. 0.106237
$$460$$ 0 0
$$461$$ 1.65670e6 0.363071 0.181536 0.983384i $$-0.441893\pi$$
0.181536 + 0.983384i $$0.441893\pi$$
$$462$$ 0 0
$$463$$ 2.89160e6 0.626881 0.313441 0.949608i $$-0.398518\pi$$
0.313441 + 0.949608i $$0.398518\pi$$
$$464$$ 0 0
$$465$$ 1.38720e6 0.297514
$$466$$ 0 0
$$467$$ 6.52699e6 1.38491 0.692454 0.721462i $$-0.256530\pi$$
0.692454 + 0.721462i $$0.256530\pi$$
$$468$$ 0 0
$$469$$ −310976. −0.0652822
$$470$$ 0 0
$$471$$ 1.33421e7 2.77123
$$472$$ 0 0
$$473$$ 2.72237e6 0.559492
$$474$$ 0 0
$$475$$ −312500. −0.0635501
$$476$$ 0 0
$$477$$ −7.01498e6 −1.41166
$$478$$ 0 0
$$479$$ 5.96232e6 1.18734 0.593672 0.804707i $$-0.297678\pi$$
0.593672 + 0.804707i $$0.297678\pi$$
$$480$$ 0 0
$$481$$ 1.06349e7 2.09591
$$482$$ 0 0
$$483$$ 1.47122e7 2.86952
$$484$$ 0 0
$$485$$ −1.95955e6 −0.378270
$$486$$ 0 0
$$487$$ −2.99191e6 −0.571644 −0.285822 0.958283i $$-0.592267\pi$$
−0.285822 + 0.958283i $$0.592267\pi$$
$$488$$ 0 0
$$489$$ −1.59878e6 −0.302355
$$490$$ 0 0
$$491$$ 1.20419e6 0.225419 0.112710 0.993628i $$-0.464047\pi$$
0.112710 + 0.993628i $$0.464047\pi$$
$$492$$ 0 0
$$493$$ −486180. −0.0900907
$$494$$ 0 0
$$495$$ −1.09890e6 −0.201579
$$496$$ 0 0
$$497$$ 4.21882e6 0.766125
$$498$$ 0 0
$$499$$ −9.20546e6 −1.65499 −0.827493 0.561477i $$-0.810233\pi$$
−0.827493 + 0.561477i $$0.810233\pi$$
$$500$$ 0 0
$$501$$ −4.93661e6 −0.878687
$$502$$ 0 0
$$503$$ 3.35956e6 0.592055 0.296027 0.955179i $$-0.404338\pi$$
0.296027 + 0.955179i $$0.404338\pi$$
$$504$$ 0 0
$$505$$ 3.54855e6 0.619188
$$506$$ 0 0
$$507$$ −1.25670e7 −2.17125
$$508$$ 0 0
$$509$$ −2.53701e6 −0.434038 −0.217019 0.976167i $$-0.569633\pi$$
−0.217019 + 0.976167i $$0.569633\pi$$
$$510$$ 0 0
$$511$$ 3.52153e6 0.596594
$$512$$ 0 0
$$513$$ 1.08000e6 0.181188
$$514$$ 0 0
$$515$$ 10900.0 0.00181096
$$516$$ 0 0
$$517$$ 869616. 0.143087
$$518$$ 0 0
$$519$$ −1.04125e7 −1.69682
$$520$$ 0 0
$$521$$ −9.31580e6 −1.50358 −0.751789 0.659404i $$-0.770809\pi$$
−0.751789 + 0.659404i $$0.770809\pi$$
$$522$$ 0 0
$$523$$ 5.02802e6 0.803790 0.401895 0.915686i $$-0.368352\pi$$
0.401895 + 0.915686i $$0.368352\pi$$
$$524$$ 0 0
$$525$$ −2.58000e6 −0.408528
$$526$$ 0 0
$$527$$ 513264. 0.0805034
$$528$$ 0 0
$$529$$ 6.26575e6 0.973496
$$530$$ 0 0
$$531$$ −2.65734e6 −0.408988
$$532$$ 0 0
$$533$$ −1.17493e6 −0.179141
$$534$$ 0 0
$$535$$ −5.82420e6 −0.879735
$$536$$ 0 0
$$537$$ −1.17403e7 −1.75689
$$538$$ 0 0
$$539$$ −1.68656e6 −0.250052
$$540$$ 0 0
$$541$$ 134222. 0.0197165 0.00985827 0.999951i $$-0.496862\pi$$
0.00985827 + 0.999951i $$0.496862\pi$$
$$542$$ 0 0
$$543$$ −1.72671e7 −2.51316
$$544$$ 0 0
$$545$$ −4.37125e6 −0.630397
$$546$$ 0 0
$$547$$ −605648. −0.0865470 −0.0432735 0.999063i $$-0.513779\pi$$
−0.0432735 + 0.999063i $$0.513779\pi$$
$$548$$ 0 0
$$549$$ 5.53513e6 0.783784
$$550$$ 0 0
$$551$$ −1.09500e6 −0.153651
$$552$$ 0 0
$$553$$ 7.95328e6 1.10594
$$554$$ 0 0
$$555$$ 6.74520e6 0.929528
$$556$$ 0 0
$$557$$ −7.06240e6 −0.964527 −0.482264 0.876026i $$-0.660185\pi$$
−0.482264 + 0.876026i $$0.660185\pi$$
$$558$$ 0 0
$$559$$ 1.95103e7 2.64079
$$560$$ 0 0
$$561$$ −703296. −0.0943476
$$562$$ 0 0
$$563$$ 1.03029e7 1.36990 0.684952 0.728588i $$-0.259823\pi$$
0.684952 + 0.728588i $$0.259823\pi$$
$$564$$ 0 0
$$565$$ 4.57485e6 0.602915
$$566$$ 0 0
$$567$$ −5.00159e6 −0.653357
$$568$$ 0 0
$$569$$ 1.04769e6 0.135660 0.0678300 0.997697i $$-0.478392\pi$$
0.0678300 + 0.997697i $$0.478392\pi$$
$$570$$ 0 0
$$571$$ −1.40765e7 −1.80677 −0.903385 0.428830i $$-0.858926\pi$$
−0.903385 + 0.428830i $$0.858926\pi$$
$$572$$ 0 0
$$573$$ −4.46227e6 −0.567766
$$574$$ 0 0
$$575$$ −2.22750e6 −0.280962
$$576$$ 0 0
$$577$$ 1.62682e6 0.203423 0.101711 0.994814i $$-0.467568\pi$$
0.101711 + 0.994814i $$0.467568\pi$$
$$578$$ 0 0
$$579$$ 1.41937e7 1.75955
$$580$$ 0 0
$$581$$ 8.87107e6 1.09027
$$582$$ 0 0
$$583$$ 2.78071e6 0.338832
$$584$$ 0 0
$$585$$ −7.87545e6 −0.951449
$$586$$ 0 0
$$587$$ −6.96089e6 −0.833814 −0.416907 0.908949i $$-0.636886\pi$$
−0.416907 + 0.908949i $$0.636886\pi$$
$$588$$ 0 0
$$589$$ 1.15600e6 0.137300
$$590$$ 0 0
$$591$$ −1.07875e7 −1.27043
$$592$$ 0 0
$$593$$ −1.13639e7 −1.32706 −0.663529 0.748150i $$-0.730942\pi$$
−0.663529 + 0.748150i $$0.730942\pi$$
$$594$$ 0 0
$$595$$ −954600. −0.110542
$$596$$ 0 0
$$597$$ 3.77184e6 0.433129
$$598$$ 0 0
$$599$$ −1.48688e7 −1.69321 −0.846603 0.532224i $$-0.821356\pi$$
−0.846603 + 0.532224i $$0.821356\pi$$
$$600$$ 0 0
$$601$$ −1.23612e6 −0.139596 −0.0697981 0.997561i $$-0.522236\pi$$
−0.0697981 + 0.997561i $$0.522236\pi$$
$$602$$ 0 0
$$603$$ −602064. −0.0674294
$$604$$ 0 0
$$605$$ −3.59068e6 −0.398830
$$606$$ 0 0
$$607$$ 1.24498e7 1.37149 0.685743 0.727844i $$-0.259478\pi$$
0.685743 + 0.727844i $$0.259478\pi$$
$$608$$ 0 0
$$609$$ −9.04032e6 −0.987735
$$610$$ 0 0
$$611$$ 6.23225e6 0.675370
$$612$$ 0 0
$$613$$ −8.73491e6 −0.938873 −0.469437 0.882966i $$-0.655543\pi$$
−0.469437 + 0.882966i $$0.655543\pi$$
$$614$$ 0 0
$$615$$ −745200. −0.0794484
$$616$$ 0 0
$$617$$ 1.25495e7 1.32713 0.663565 0.748119i $$-0.269043\pi$$
0.663565 + 0.748119i $$0.269043\pi$$
$$618$$ 0 0
$$619$$ 1.46658e7 1.53843 0.769216 0.638988i $$-0.220647\pi$$
0.769216 + 0.638988i $$0.220647\pi$$
$$620$$ 0 0
$$621$$ 7.69824e6 0.801055
$$622$$ 0 0
$$623$$ −1.89733e7 −1.95850
$$624$$ 0 0
$$625$$ 390625. 0.0400000
$$626$$ 0 0
$$627$$ −1.58400e6 −0.160911
$$628$$ 0 0
$$629$$ 2.49572e6 0.251519
$$630$$ 0 0
$$631$$ 196288. 0.0196255 0.00981274 0.999952i $$-0.496876\pi$$
0.00981274 + 0.999952i $$0.496876\pi$$
$$632$$ 0 0
$$633$$ 6.07325e6 0.602437
$$634$$ 0 0
$$635$$ 3.06130e6 0.301281
$$636$$ 0 0
$$637$$ −1.20870e7 −1.18024
$$638$$ 0 0
$$639$$ 8.16782e6 0.791324
$$640$$ 0 0
$$641$$ −1.11596e7 −1.07276 −0.536381 0.843976i $$-0.680209\pi$$
−0.536381 + 0.843976i $$0.680209\pi$$
$$642$$ 0 0
$$643$$ 2.25158e6 0.214763 0.107381 0.994218i $$-0.465753\pi$$
0.107381 + 0.994218i $$0.465753\pi$$
$$644$$ 0 0
$$645$$ 1.23744e7 1.17118
$$646$$ 0 0
$$647$$ −8.05319e6 −0.756323 −0.378161 0.925740i $$-0.623444\pi$$
−0.378161 + 0.925740i $$0.623444\pi$$
$$648$$ 0 0
$$649$$ 1.05336e6 0.0981669
$$650$$ 0 0
$$651$$ 9.54394e6 0.882623
$$652$$ 0 0
$$653$$ −416466. −0.0382205 −0.0191103 0.999817i $$-0.506083\pi$$
−0.0191103 + 0.999817i $$0.506083\pi$$
$$654$$ 0 0
$$655$$ 6.04770e6 0.550791
$$656$$ 0 0
$$657$$ 6.81784e6 0.616217
$$658$$ 0 0
$$659$$ −1.31721e7 −1.18152 −0.590761 0.806847i $$-0.701172\pi$$
−0.590761 + 0.806847i $$0.701172\pi$$
$$660$$ 0 0
$$661$$ −1.69494e6 −0.150886 −0.0754432 0.997150i $$-0.524037\pi$$
−0.0754432 + 0.997150i $$0.524037\pi$$
$$662$$ 0 0
$$663$$ −5.04029e6 −0.445319
$$664$$ 0 0
$$665$$ −2.15000e6 −0.188532
$$666$$ 0 0
$$667$$ −7.80516e6 −0.679309
$$668$$ 0 0
$$669$$ 2.57627e7 2.22549
$$670$$ 0 0
$$671$$ −2.19410e6 −0.188127
$$672$$ 0 0
$$673$$ −8.91605e6 −0.758813 −0.379406 0.925230i $$-0.623872\pi$$
−0.379406 + 0.925230i $$0.623872\pi$$
$$674$$ 0 0
$$675$$ −1.35000e6 −0.114044
$$676$$ 0 0
$$677$$ −1.42894e7 −1.19824 −0.599118 0.800661i $$-0.704482\pi$$
−0.599118 + 0.800661i $$0.704482\pi$$
$$678$$ 0 0
$$679$$ −1.34817e7 −1.12220
$$680$$ 0 0
$$681$$ −1.50440e7 −1.24307
$$682$$ 0 0
$$683$$ 5.33314e6 0.437452 0.218726 0.975786i $$-0.429810\pi$$
0.218726 + 0.975786i $$0.429810\pi$$
$$684$$ 0 0
$$685$$ 6.92745e6 0.564088
$$686$$ 0 0
$$687$$ 2.79960e6 0.226310
$$688$$ 0 0
$$689$$ 1.99284e7 1.59928
$$690$$ 0 0
$$691$$ −698252. −0.0556310 −0.0278155 0.999613i $$-0.508855\pi$$
−0.0278155 + 0.999613i $$0.508855\pi$$
$$692$$ 0 0
$$693$$ −7.56043e6 −0.598017
$$694$$ 0 0
$$695$$ 4.83850e6 0.379969
$$696$$ 0 0
$$697$$ −275724. −0.0214977
$$698$$ 0 0
$$699$$ 1.78331e7 1.38049
$$700$$ 0 0
$$701$$ 1.79880e7 1.38257 0.691285 0.722582i $$-0.257045\pi$$
0.691285 + 0.722582i $$0.257045\pi$$
$$702$$ 0 0
$$703$$ 5.62100e6 0.428968
$$704$$ 0 0
$$705$$ 3.95280e6 0.299524
$$706$$ 0 0
$$707$$ 2.44140e7 1.83692
$$708$$ 0 0
$$709$$ −1.39464e7 −1.04195 −0.520975 0.853572i $$-0.674432\pi$$
−0.520975 + 0.853572i $$0.674432\pi$$
$$710$$ 0 0
$$711$$ 1.53979e7 1.14232
$$712$$ 0 0
$$713$$ 8.23997e6 0.607018
$$714$$ 0 0
$$715$$ 3.12180e6 0.228370
$$716$$ 0 0
$$717$$ 2.34893e7 1.70636
$$718$$ 0 0
$$719$$ −6.22272e6 −0.448909 −0.224454 0.974485i $$-0.572060\pi$$
−0.224454 + 0.974485i $$0.572060\pi$$
$$720$$ 0 0
$$721$$ 74992.0 0.00537250
$$722$$ 0 0
$$723$$ 2.71872e7 1.93427
$$724$$ 0 0
$$725$$ 1.36875e6 0.0967117
$$726$$ 0 0
$$727$$ 7.76729e6 0.545047 0.272523 0.962149i $$-0.412142\pi$$
0.272523 + 0.962149i $$0.412142\pi$$
$$728$$ 0 0
$$729$$ −2.22804e7 −1.55276
$$730$$ 0 0
$$731$$ 4.57853e6 0.316907
$$732$$ 0 0
$$733$$ 2.42083e7 1.66420 0.832099 0.554627i $$-0.187139\pi$$
0.832099 + 0.554627i $$0.187139\pi$$
$$734$$ 0 0
$$735$$ −7.66620e6 −0.523434
$$736$$ 0 0
$$737$$ 238656. 0.0161847
$$738$$ 0 0
$$739$$ −1.26850e7 −0.854434 −0.427217 0.904149i $$-0.640506\pi$$
−0.427217 + 0.904149i $$0.640506\pi$$
$$740$$ 0 0
$$741$$ −1.13520e7 −0.759498
$$742$$ 0 0
$$743$$ −1.97632e7 −1.31337 −0.656684 0.754166i $$-0.728041\pi$$
−0.656684 + 0.754166i $$0.728041\pi$$
$$744$$ 0 0
$$745$$ 3.51375e6 0.231942
$$746$$ 0 0
$$747$$ 1.71748e7 1.12613
$$748$$ 0 0
$$749$$ −4.00705e7 −2.60988
$$750$$ 0 0
$$751$$ 9.01761e6 0.583434 0.291717 0.956505i $$-0.405774\pi$$
0.291717 + 0.956505i $$0.405774\pi$$
$$752$$ 0 0
$$753$$ 2.17356e7 1.39696
$$754$$ 0 0
$$755$$ −1.08488e7 −0.692651
$$756$$ 0 0
$$757$$ −1.12556e6 −0.0713887 −0.0356944 0.999363i $$-0.511364\pi$$
−0.0356944 + 0.999363i $$0.511364\pi$$
$$758$$ 0 0
$$759$$ −1.12908e7 −0.711407
$$760$$ 0 0
$$761$$ 2.25747e7 1.41306 0.706529 0.707684i $$-0.250260\pi$$
0.706529 + 0.707684i $$0.250260\pi$$
$$762$$ 0 0
$$763$$ −3.00742e7 −1.87018
$$764$$ 0 0
$$765$$ −1.84815e6 −0.114178
$$766$$ 0 0
$$767$$ 7.54908e6 0.463346
$$768$$ 0 0
$$769$$ −632350. −0.0385604 −0.0192802 0.999814i $$-0.506137\pi$$
−0.0192802 + 0.999814i $$0.506137\pi$$
$$770$$ 0 0
$$771$$ −4.65585e7 −2.82074
$$772$$ 0 0
$$773$$ −1.25867e7 −0.757643 −0.378822 0.925470i $$-0.623671\pi$$
−0.378822 + 0.925470i $$0.623671\pi$$
$$774$$ 0 0
$$775$$ −1.44500e6 −0.0864199
$$776$$ 0 0
$$777$$ 4.64070e7 2.75760
$$778$$ 0 0
$$779$$ −621000. −0.0366647
$$780$$ 0 0
$$781$$ −3.23770e6 −0.189937
$$782$$ 0 0
$$783$$ −4.73040e6 −0.275736
$$784$$ 0 0
$$785$$ −1.38981e7 −0.804970
$$786$$ 0 0
$$787$$ −2.15792e7 −1.24194 −0.620968 0.783836i $$-0.713260\pi$$
−0.620968 + 0.783836i $$0.713260\pi$$
$$788$$ 0 0
$$789$$ −1.93314e7 −1.10553
$$790$$ 0 0
$$791$$ 3.14750e7 1.78864
$$792$$ 0 0
$$793$$ −1.57244e7 −0.887956
$$794$$ 0 0
$$795$$ 1.26396e7 0.709276
$$796$$ 0 0
$$797$$ −3.09760e7 −1.72735 −0.863673 0.504052i $$-0.831842\pi$$
−0.863673 + 0.504052i $$0.831842\pi$$
$$798$$ 0 0
$$799$$ 1.46254e6 0.0810475
$$800$$ 0 0
$$801$$ −3.67332e7 −2.02292
$$802$$ 0 0
$$803$$ −2.70257e6 −0.147907
$$804$$ 0 0
$$805$$ −1.53252e7 −0.833521
$$806$$ 0 0
$$807$$ 2.06086e7 1.11395
$$808$$ 0 0
$$809$$ 4.24929e6 0.228268 0.114134 0.993465i $$-0.463591\pi$$
0.114134 + 0.993465i $$0.463591\pi$$
$$810$$ 0 0
$$811$$ −3.42333e6 −0.182767 −0.0913833 0.995816i $$-0.529129\pi$$
−0.0913833 + 0.995816i $$0.529129\pi$$
$$812$$ 0 0
$$813$$ −9.20659e6 −0.488509
$$814$$ 0 0
$$815$$ 1.66540e6 0.0878263
$$816$$ 0 0
$$817$$ 1.03120e7 0.540490
$$818$$ 0 0
$$819$$ −5.41831e7 −2.82263
$$820$$ 0 0
$$821$$ 3.10571e7 1.60806 0.804030 0.594588i $$-0.202685\pi$$
0.804030 + 0.594588i $$0.202685\pi$$
$$822$$ 0 0
$$823$$ 3.11904e7 1.60517 0.802584 0.596538i $$-0.203458\pi$$
0.802584 + 0.596538i $$0.203458\pi$$
$$824$$ 0 0
$$825$$ 1.98000e6 0.101282
$$826$$ 0 0
$$827$$ 8.28487e6 0.421233 0.210616 0.977569i $$-0.432453\pi$$
0.210616 + 0.977569i $$0.432453\pi$$
$$828$$ 0 0
$$829$$ −1.81689e7 −0.918208 −0.459104 0.888383i $$-0.651829\pi$$
−0.459104 + 0.888383i $$0.651829\pi$$
$$830$$ 0 0
$$831$$ −4.82582e7 −2.42420
$$832$$ 0 0
$$833$$ −2.83649e6 −0.141635
$$834$$ 0 0
$$835$$ 5.14230e6 0.255236
$$836$$ 0 0
$$837$$ 4.99392e6 0.246393
$$838$$ 0 0
$$839$$ 1.02743e7 0.503902 0.251951 0.967740i $$-0.418928\pi$$
0.251951 + 0.967740i $$0.418928\pi$$
$$840$$ 0 0
$$841$$ −1.57150e7 −0.766171
$$842$$ 0 0
$$843$$ −4.86245e6 −0.235660
$$844$$ 0 0
$$845$$ 1.30906e7 0.630691
$$846$$ 0 0
$$847$$ −2.47038e7 −1.18319
$$848$$ 0 0
$$849$$ −5.31686e6 −0.253155
$$850$$ 0 0
$$851$$ 4.00665e7 1.89652
$$852$$ 0 0
$$853$$ 6.28597e6 0.295801 0.147901 0.989002i $$-0.452748\pi$$
0.147901 + 0.989002i $$0.452748\pi$$
$$854$$ 0 0
$$855$$ −4.16250e6 −0.194733
$$856$$ 0 0
$$857$$ 1.54050e7 0.716490 0.358245 0.933628i $$-0.383375\pi$$
0.358245 + 0.933628i $$0.383375\pi$$
$$858$$ 0 0
$$859$$ −1.43526e7 −0.663664 −0.331832 0.943338i $$-0.607667\pi$$
−0.331832 + 0.943338i $$0.607667\pi$$
$$860$$ 0 0
$$861$$ −5.12698e6 −0.235697
$$862$$ 0 0
$$863$$ −1.33278e7 −0.609158 −0.304579 0.952487i $$-0.598516\pi$$
−0.304579 + 0.952487i $$0.598516\pi$$
$$864$$ 0 0
$$865$$ 1.08464e7 0.492882
$$866$$ 0 0
$$867$$ 3.28938e7 1.48616
$$868$$ 0 0
$$869$$ −6.10368e6 −0.274184
$$870$$ 0 0
$$871$$ 1.71037e6 0.0763913
$$872$$ 0 0
$$873$$ −2.61012e7 −1.15911
$$874$$ 0 0
$$875$$ 2.68750e6 0.118667
$$876$$ 0 0
$$877$$ 3.24846e7 1.42620 0.713098 0.701065i $$-0.247292\pi$$
0.713098 + 0.701065i $$0.247292\pi$$
$$878$$ 0 0
$$879$$ 7.74014e6 0.337891
$$880$$ 0 0
$$881$$ 1.54600e7 0.671073 0.335537 0.942027i $$-0.391082\pi$$
0.335537 + 0.942027i $$0.391082\pi$$
$$882$$ 0 0
$$883$$ 1.69478e6 0.0731494 0.0365747 0.999331i $$-0.488355\pi$$
0.0365747 + 0.999331i $$0.488355\pi$$
$$884$$ 0 0
$$885$$ 4.78800e6 0.205492
$$886$$ 0 0
$$887$$ 2.87257e6 0.122592 0.0612960 0.998120i $$-0.480477\pi$$
0.0612960 + 0.998120i $$0.480477\pi$$
$$888$$ 0 0
$$889$$ 2.10617e7 0.893799
$$890$$ 0 0
$$891$$ 3.83843e6 0.161979
$$892$$ 0 0
$$893$$ 3.29400e6 0.138228
$$894$$ 0 0
$$895$$ 1.22295e7 0.510330
$$896$$ 0 0
$$897$$ −8.09171e7 −3.35783
$$898$$ 0 0
$$899$$ −5.06328e6 −0.208945
$$900$$ 0 0
$$901$$ 4.67665e6 0.191921
$$902$$ 0 0
$$903$$ 8.51359e7 3.47451
$$904$$ 0 0
$$905$$ 1.79865e7 0.730006
$$906$$ 0 0
$$907$$ −3.95422e7 −1.59603 −0.798017 0.602635i $$-0.794118\pi$$
−0.798017 + 0.602635i $$0.794118\pi$$
$$908$$ 0 0
$$909$$ 4.72667e7 1.89734
$$910$$ 0 0
$$911$$ −1.13178e7 −0.451819 −0.225909 0.974148i $$-0.572535\pi$$
−0.225909 + 0.974148i $$0.572535\pi$$
$$912$$ 0 0
$$913$$ −6.80803e6 −0.270299
$$914$$ 0 0
$$915$$ −9.97320e6 −0.393806
$$916$$ 0 0
$$917$$ 4.16082e7 1.63401
$$918$$ 0 0
$$919$$ −8.51348e6 −0.332520 −0.166260 0.986082i $$-0.553169\pi$$
−0.166260 + 0.986082i $$0.553169\pi$$
$$920$$ 0 0
$$921$$ 3.46322e7 1.34534
$$922$$ 0 0
$$923$$ −2.32035e7 −0.896497
$$924$$ 0 0
$$925$$ −7.02625e6 −0.270003
$$926$$ 0 0
$$927$$ 145188. 0.00554921
$$928$$ 0 0
$$929$$ −7.54587e6 −0.286860 −0.143430 0.989660i $$-0.545813\pi$$
−0.143430 + 0.989660i $$0.545813\pi$$
$$930$$ 0 0
$$931$$ −6.38850e6 −0.241560
$$932$$ 0 0
$$933$$ 4.11149e6 0.154630
$$934$$ 0 0
$$935$$ 732600. 0.0274055
$$936$$ 0 0
$$937$$ −1.84500e7 −0.686512 −0.343256 0.939242i $$-0.611530\pi$$
−0.343256 + 0.939242i $$0.611530\pi$$
$$938$$ 0 0
$$939$$ 2.46453e7 0.912157
$$940$$ 0 0
$$941$$ 6.75046e6 0.248519 0.124259 0.992250i $$-0.460344\pi$$
0.124259 + 0.992250i $$0.460344\pi$$
$$942$$ 0 0
$$943$$ −4.42649e6 −0.162099
$$944$$ 0 0
$$945$$ −9.28800e6 −0.338332
$$946$$ 0 0
$$947$$ −6.45677e6 −0.233959 −0.116980 0.993134i $$-0.537321\pi$$
−0.116980 + 0.993134i $$0.537321\pi$$
$$948$$ 0 0
$$949$$ −1.93684e7 −0.698117
$$950$$ 0 0
$$951$$ −1.80710e7 −0.647934
$$952$$ 0 0
$$953$$ −3.96648e7 −1.41473 −0.707364 0.706849i $$-0.750116\pi$$
−0.707364 + 0.706849i $$0.750116\pi$$
$$954$$ 0 0
$$955$$ 4.64820e6 0.164921
$$956$$ 0 0
$$957$$ 6.93792e6 0.244878
$$958$$ 0 0
$$959$$ 4.76609e7 1.67346
$$960$$ 0 0
$$961$$ −2.32838e7 −0.813290
$$962$$ 0 0
$$963$$ −7.75783e7 −2.69572
$$964$$ 0 0
$$965$$ −1.47851e7 −0.511102
$$966$$ 0 0
$$967$$ 3.43015e7 1.17963 0.589816 0.807538i $$-0.299200\pi$$
0.589816 + 0.807538i $$0.299200\pi$$
$$968$$ 0 0
$$969$$ −2.66400e6 −0.0911433
$$970$$ 0 0
$$971$$ 5.77115e6 0.196433 0.0982164 0.995165i $$-0.468686\pi$$
0.0982164 + 0.995165i $$0.468686\pi$$
$$972$$ 0 0
$$973$$ 3.32889e7 1.12724
$$974$$ 0 0
$$975$$ 1.41900e7 0.478047
$$976$$ 0 0
$$977$$ 7.08746e6 0.237549 0.118775 0.992921i $$-0.462103\pi$$
0.118775 + 0.992921i $$0.462103\pi$$
$$978$$ 0 0
$$979$$ 1.45609e7 0.485548
$$980$$ 0 0
$$981$$ −5.82251e7 −1.93169
$$982$$ 0 0
$$983$$ −4.59362e7 −1.51625 −0.758126 0.652108i $$-0.773885\pi$$
−0.758126 + 0.652108i $$0.773885\pi$$
$$984$$ 0 0
$$985$$ 1.12369e7 0.369027
$$986$$ 0 0
$$987$$ 2.71953e7 0.888588
$$988$$ 0 0
$$989$$ 7.35039e7 2.38957
$$990$$ 0 0
$$991$$ 4.50298e7 1.45652 0.728260 0.685301i $$-0.240329\pi$$
0.728260 + 0.685301i $$0.240329\pi$$
$$992$$ 0 0
$$993$$ 4.78592e7 1.54025
$$994$$ 0 0
$$995$$ −3.92900e6 −0.125813
$$996$$ 0 0
$$997$$ −2.37364e7 −0.756271 −0.378136 0.925750i $$-0.623435\pi$$
−0.378136 + 0.925750i $$0.623435\pi$$
$$998$$ 0 0
$$999$$ 2.42827e7 0.769810
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 80.6.a.a.1.1 1
3.2 odd 2 720.6.a.j.1.1 1
4.3 odd 2 10.6.a.b.1.1 1
5.2 odd 4 400.6.c.b.49.2 2
5.3 odd 4 400.6.c.b.49.1 2
5.4 even 2 400.6.a.n.1.1 1
8.3 odd 2 320.6.a.b.1.1 1
8.5 even 2 320.6.a.o.1.1 1
12.11 even 2 90.6.a.d.1.1 1
20.3 even 4 50.6.b.a.49.2 2
20.7 even 4 50.6.b.a.49.1 2
20.19 odd 2 50.6.a.d.1.1 1
28.27 even 2 490.6.a.a.1.1 1
60.23 odd 4 450.6.c.h.199.1 2
60.47 odd 4 450.6.c.h.199.2 2
60.59 even 2 450.6.a.l.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
10.6.a.b.1.1 1 4.3 odd 2
50.6.a.d.1.1 1 20.19 odd 2
50.6.b.a.49.1 2 20.7 even 4
50.6.b.a.49.2 2 20.3 even 4
80.6.a.a.1.1 1 1.1 even 1 trivial
90.6.a.d.1.1 1 12.11 even 2
320.6.a.b.1.1 1 8.3 odd 2
320.6.a.o.1.1 1 8.5 even 2
400.6.a.n.1.1 1 5.4 even 2
400.6.c.b.49.1 2 5.3 odd 4
400.6.c.b.49.2 2 5.2 odd 4
450.6.a.l.1.1 1 60.59 even 2
450.6.c.h.199.1 2 60.23 odd 4
450.6.c.h.199.2 2 60.47 odd 4
490.6.a.a.1.1 1 28.27 even 2
720.6.a.j.1.1 1 3.2 odd 2