# Properties

 Label 400.6.a.w.1.1 Level $400$ Weight $6$ Character 400.1 Self dual yes Analytic conductor $64.154$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$400 = 2^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 400.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$64.1535279252$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{241})$$ Defining polynomial: $$x^{2} - x - 60$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 25) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$8.26209$$ of defining polynomial Character $$\chi$$ $$=$$ 400.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-5.52417 q^{3} +68.9517 q^{7} -212.483 q^{9} +O(q^{10})$$ $$q-5.52417 q^{3} +68.9517 q^{7} -212.483 q^{9} +486.104 q^{11} +428.387 q^{13} -1800.64 q^{17} +1046.65 q^{19} -380.901 q^{21} +686.855 q^{23} +2516.17 q^{27} -1339.03 q^{29} -7990.30 q^{31} -2685.33 q^{33} +1970.64 q^{37} -2366.48 q^{39} +10772.2 q^{41} +15017.7 q^{43} +895.337 q^{47} -12052.7 q^{49} +9947.07 q^{51} +19327.1 q^{53} -5781.90 q^{57} -21193.7 q^{59} -27722.2 q^{61} -14651.1 q^{63} -7719.33 q^{67} -3794.31 q^{69} +51410.1 q^{71} +43776.4 q^{73} +33517.7 q^{77} +6225.68 q^{79} +37733.7 q^{81} +52949.9 q^{83} +7397.05 q^{87} +44631.2 q^{89} +29538.0 q^{91} +44139.8 q^{93} +148018. q^{97} -103289. q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 20q^{3} + 200q^{7} + 196q^{9} + O(q^{10})$$ $$2q + 20q^{3} + 200q^{7} + 196q^{9} + 196q^{11} + 360q^{13} - 1490q^{17} + 3180q^{19} + 2964q^{21} + 1560q^{23} + 6740q^{27} - 3920q^{29} + 1096q^{31} - 10090q^{33} - 2020q^{37} - 4112q^{39} + 27754q^{41} - 3000q^{43} + 25760q^{47} - 11686q^{49} + 17876q^{51} + 26980q^{53} + 48670q^{57} - 11960q^{59} - 24396q^{61} + 38880q^{63} - 40060q^{67} + 18492q^{69} + 87296q^{71} + 70290q^{73} - 4500q^{77} - 65480q^{79} + 46282q^{81} + 92580q^{83} - 58480q^{87} - 72810q^{89} + 20576q^{91} + 276060q^{93} + 126140q^{97} - 221792q^{99} + O(q^{100})$$

## 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
$$3$$ −5.52417 −0.354376 −0.177188 0.984177i $$-0.556700\pi$$
−0.177188 + 0.984177i $$0.556700\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 68.9517 0.531863 0.265931 0.963992i $$-0.414321\pi$$
0.265931 + 0.963992i $$0.414321\pi$$
$$8$$ 0 0
$$9$$ −212.483 −0.874418
$$10$$ 0 0
$$11$$ 486.104 1.21129 0.605645 0.795735i $$-0.292915\pi$$
0.605645 + 0.795735i $$0.292915\pi$$
$$12$$ 0 0
$$13$$ 428.387 0.703036 0.351518 0.936181i $$-0.385666\pi$$
0.351518 + 0.936181i $$0.385666\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −1800.64 −1.51114 −0.755571 0.655066i $$-0.772641\pi$$
−0.755571 + 0.655066i $$0.772641\pi$$
$$18$$ 0 0
$$19$$ 1046.65 0.665149 0.332575 0.943077i $$-0.392083\pi$$
0.332575 + 0.943077i $$0.392083\pi$$
$$20$$ 0 0
$$21$$ −380.901 −0.188479
$$22$$ 0 0
$$23$$ 686.855 0.270736 0.135368 0.990795i $$-0.456778\pi$$
0.135368 + 0.990795i $$0.456778\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 2516.17 0.664249
$$28$$ 0 0
$$29$$ −1339.03 −0.295663 −0.147831 0.989013i $$-0.547229\pi$$
−0.147831 + 0.989013i $$0.547229\pi$$
$$30$$ 0 0
$$31$$ −7990.30 −1.49334 −0.746670 0.665195i $$-0.768349\pi$$
−0.746670 + 0.665195i $$0.768349\pi$$
$$32$$ 0 0
$$33$$ −2685.33 −0.429252
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 1970.64 0.236648 0.118324 0.992975i $$-0.462248\pi$$
0.118324 + 0.992975i $$0.462248\pi$$
$$38$$ 0 0
$$39$$ −2366.48 −0.249139
$$40$$ 0 0
$$41$$ 10772.2 1.00079 0.500395 0.865797i $$-0.333188\pi$$
0.500395 + 0.865797i $$0.333188\pi$$
$$42$$ 0 0
$$43$$ 15017.7 1.23861 0.619303 0.785152i $$-0.287415\pi$$
0.619303 + 0.785152i $$0.287415\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 895.337 0.0591210 0.0295605 0.999563i $$-0.490589\pi$$
0.0295605 + 0.999563i $$0.490589\pi$$
$$48$$ 0 0
$$49$$ −12052.7 −0.717122
$$50$$ 0 0
$$51$$ 9947.07 0.535513
$$52$$ 0 0
$$53$$ 19327.1 0.945098 0.472549 0.881304i $$-0.343334\pi$$
0.472549 + 0.881304i $$0.343334\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −5781.90 −0.235713
$$58$$ 0 0
$$59$$ −21193.7 −0.792641 −0.396321 0.918112i $$-0.629713\pi$$
−0.396321 + 0.918112i $$0.629713\pi$$
$$60$$ 0 0
$$61$$ −27722.2 −0.953900 −0.476950 0.878931i $$-0.658258\pi$$
−0.476950 + 0.878931i $$0.658258\pi$$
$$62$$ 0 0
$$63$$ −14651.1 −0.465070
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −7719.33 −0.210084 −0.105042 0.994468i $$-0.533498\pi$$
−0.105042 + 0.994468i $$0.533498\pi$$
$$68$$ 0 0
$$69$$ −3794.31 −0.0959422
$$70$$ 0 0
$$71$$ 51410.1 1.21033 0.605163 0.796101i $$-0.293108\pi$$
0.605163 + 0.796101i $$0.293108\pi$$
$$72$$ 0 0
$$73$$ 43776.4 0.961465 0.480732 0.876867i $$-0.340371\pi$$
0.480732 + 0.876867i $$0.340371\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 33517.7 0.644240
$$78$$ 0 0
$$79$$ 6225.68 0.112233 0.0561163 0.998424i $$-0.482128\pi$$
0.0561163 + 0.998424i $$0.482128\pi$$
$$80$$ 0 0
$$81$$ 37733.7 0.639024
$$82$$ 0 0
$$83$$ 52949.9 0.843664 0.421832 0.906674i $$-0.361387\pi$$
0.421832 + 0.906674i $$0.361387\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 7397.05 0.104776
$$88$$ 0 0
$$89$$ 44631.2 0.597260 0.298630 0.954369i $$-0.403470\pi$$
0.298630 + 0.954369i $$0.403470\pi$$
$$90$$ 0 0
$$91$$ 29538.0 0.373919
$$92$$ 0 0
$$93$$ 44139.8 0.529204
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 148018. 1.59730 0.798649 0.601797i $$-0.205548\pi$$
0.798649 + 0.601797i $$0.205548\pi$$
$$98$$ 0 0
$$99$$ −103289. −1.05917
$$100$$ 0 0
$$101$$ 148476. 1.44828 0.724141 0.689652i $$-0.242237\pi$$
0.724141 + 0.689652i $$0.242237\pi$$
$$102$$ 0 0
$$103$$ 188391. 1.74972 0.874859 0.484378i $$-0.160954\pi$$
0.874859 + 0.484378i $$0.160954\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 67887.7 0.573234 0.286617 0.958045i $$-0.407469\pi$$
0.286617 + 0.958045i $$0.407469\pi$$
$$108$$ 0 0
$$109$$ −219292. −1.76790 −0.883949 0.467582i $$-0.845125\pi$$
−0.883949 + 0.467582i $$0.845125\pi$$
$$110$$ 0 0
$$111$$ −10886.2 −0.0838625
$$112$$ 0 0
$$113$$ 80783.9 0.595153 0.297577 0.954698i $$-0.403822\pi$$
0.297577 + 0.954698i $$0.403822\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −91025.1 −0.614747
$$118$$ 0 0
$$119$$ −124157. −0.803721
$$120$$ 0 0
$$121$$ 75246.5 0.467221
$$122$$ 0 0
$$123$$ −59507.3 −0.354656
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 161301. 0.887417 0.443708 0.896171i $$-0.353663\pi$$
0.443708 + 0.896171i $$0.353663\pi$$
$$128$$ 0 0
$$129$$ −82960.5 −0.438932
$$130$$ 0 0
$$131$$ 193006. 0.982636 0.491318 0.870980i $$-0.336515\pi$$
0.491318 + 0.870980i $$0.336515\pi$$
$$132$$ 0 0
$$133$$ 72168.5 0.353768
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −250340. −1.13954 −0.569768 0.821806i $$-0.692967\pi$$
−0.569768 + 0.821806i $$0.692967\pi$$
$$138$$ 0 0
$$139$$ 218650. 0.959871 0.479935 0.877304i $$-0.340660\pi$$
0.479935 + 0.877304i $$0.340660\pi$$
$$140$$ 0 0
$$141$$ −4946.00 −0.0209511
$$142$$ 0 0
$$143$$ 208241. 0.851580
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 66581.1 0.254131
$$148$$ 0 0
$$149$$ −38740.0 −0.142953 −0.0714766 0.997442i $$-0.522771\pi$$
−0.0714766 + 0.997442i $$0.522771\pi$$
$$150$$ 0 0
$$151$$ 154945. 0.553013 0.276507 0.961012i $$-0.410823\pi$$
0.276507 + 0.961012i $$0.410823\pi$$
$$152$$ 0 0
$$153$$ 382607. 1.32137
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 344442. 1.11523 0.557617 0.830098i $$-0.311716\pi$$
0.557617 + 0.830098i $$0.311716\pi$$
$$158$$ 0 0
$$159$$ −106766. −0.334920
$$160$$ 0 0
$$161$$ 47359.8 0.143994
$$162$$ 0 0
$$163$$ 366203. 1.07957 0.539787 0.841801i $$-0.318505\pi$$
0.539787 + 0.841801i $$0.318505\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 249272. 0.691644 0.345822 0.938300i $$-0.387600\pi$$
0.345822 + 0.938300i $$0.387600\pi$$
$$168$$ 0 0
$$169$$ −187778. −0.505740
$$170$$ 0 0
$$171$$ −222397. −0.581618
$$172$$ 0 0
$$173$$ 61460.1 0.156127 0.0780635 0.996948i $$-0.475126\pi$$
0.0780635 + 0.996948i $$0.475126\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 117078. 0.280893
$$178$$ 0 0
$$179$$ −606803. −1.41552 −0.707759 0.706454i $$-0.750294\pi$$
−0.707759 + 0.706454i $$0.750294\pi$$
$$180$$ 0 0
$$181$$ 153684. 0.348685 0.174343 0.984685i $$-0.444220\pi$$
0.174343 + 0.984685i $$0.444220\pi$$
$$182$$ 0 0
$$183$$ 153142. 0.338039
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −875301. −1.83043
$$188$$ 0 0
$$189$$ 173494. 0.353289
$$190$$ 0 0
$$191$$ −182315. −0.361608 −0.180804 0.983519i $$-0.557870\pi$$
−0.180804 + 0.983519i $$0.557870\pi$$
$$192$$ 0 0
$$193$$ 102080. 0.197265 0.0986323 0.995124i $$-0.468553\pi$$
0.0986323 + 0.995124i $$0.468553\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 404656. 0.742882 0.371441 0.928456i $$-0.378864\pi$$
0.371441 + 0.928456i $$0.378864\pi$$
$$198$$ 0 0
$$199$$ 167297. 0.299472 0.149736 0.988726i $$-0.452158\pi$$
0.149736 + 0.988726i $$0.452158\pi$$
$$200$$ 0 0
$$201$$ 42642.9 0.0744487
$$202$$ 0 0
$$203$$ −92328.5 −0.157252
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ −145945. −0.236736
$$208$$ 0 0
$$209$$ 508783. 0.805688
$$210$$ 0 0
$$211$$ 460778. 0.712502 0.356251 0.934390i $$-0.384055\pi$$
0.356251 + 0.934390i $$0.384055\pi$$
$$212$$ 0 0
$$213$$ −283998. −0.428911
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −550944. −0.794252
$$218$$ 0 0
$$219$$ −241829. −0.340720
$$220$$ 0 0
$$221$$ −771372. −1.06239
$$222$$ 0 0
$$223$$ −1.08298e6 −1.45834 −0.729172 0.684330i $$-0.760095\pi$$
−0.729172 + 0.684330i $$0.760095\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 412201. 0.530938 0.265469 0.964119i $$-0.414473\pi$$
0.265469 + 0.964119i $$0.414473\pi$$
$$228$$ 0 0
$$229$$ −433163. −0.545836 −0.272918 0.962037i $$-0.587989\pi$$
−0.272918 + 0.962037i $$0.587989\pi$$
$$230$$ 0 0
$$231$$ −185158. −0.228303
$$232$$ 0 0
$$233$$ −760097. −0.917232 −0.458616 0.888635i $$-0.651655\pi$$
−0.458616 + 0.888635i $$0.651655\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −34391.7 −0.0397725
$$238$$ 0 0
$$239$$ 988624. 1.11953 0.559766 0.828651i $$-0.310891\pi$$
0.559766 + 0.828651i $$0.310891\pi$$
$$240$$ 0 0
$$241$$ −358878. −0.398020 −0.199010 0.979997i $$-0.563773\pi$$
−0.199010 + 0.979997i $$0.563773\pi$$
$$242$$ 0 0
$$243$$ −819877. −0.890703
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 448373. 0.467624
$$248$$ 0 0
$$249$$ −292504. −0.298974
$$250$$ 0 0
$$251$$ 851049. 0.852649 0.426324 0.904570i $$-0.359808\pi$$
0.426324 + 0.904570i $$0.359808\pi$$
$$252$$ 0 0
$$253$$ 333883. 0.327939
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 76358.4 0.0721147 0.0360574 0.999350i $$-0.488520\pi$$
0.0360574 + 0.999350i $$0.488520\pi$$
$$258$$ 0 0
$$259$$ 135879. 0.125864
$$260$$ 0 0
$$261$$ 284522. 0.258533
$$262$$ 0 0
$$263$$ 1.19420e6 1.06460 0.532301 0.846555i $$-0.321327\pi$$
0.532301 + 0.846555i $$0.321327\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −246551. −0.211655
$$268$$ 0 0
$$269$$ 1.02930e6 0.867286 0.433643 0.901085i $$-0.357228\pi$$
0.433643 + 0.901085i $$0.357228\pi$$
$$270$$ 0 0
$$271$$ −2.12144e6 −1.75472 −0.877359 0.479834i $$-0.840697\pi$$
−0.877359 + 0.479834i $$0.840697\pi$$
$$272$$ 0 0
$$273$$ −163173. −0.132508
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −1.85145e6 −1.44982 −0.724908 0.688845i $$-0.758118\pi$$
−0.724908 + 0.688845i $$0.758118\pi$$
$$278$$ 0 0
$$279$$ 1.69781e6 1.30580
$$280$$ 0 0
$$281$$ 90653.2 0.0684884 0.0342442 0.999413i $$-0.489098\pi$$
0.0342442 + 0.999413i $$0.489098\pi$$
$$282$$ 0 0
$$283$$ −929308. −0.689753 −0.344877 0.938648i $$-0.612079\pi$$
−0.344877 + 0.938648i $$0.612079\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 742759. 0.532283
$$288$$ 0 0
$$289$$ 1.82246e6 1.28355
$$290$$ 0 0
$$291$$ −817679. −0.566044
$$292$$ 0 0
$$293$$ −2.72733e6 −1.85596 −0.927979 0.372632i $$-0.878455\pi$$
−0.927979 + 0.372632i $$0.878455\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 1.22312e6 0.804597
$$298$$ 0 0
$$299$$ 294240. 0.190337
$$300$$ 0 0
$$301$$ 1.03550e6 0.658768
$$302$$ 0 0
$$303$$ −820208. −0.513236
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −2.29648e6 −1.39064 −0.695322 0.718698i $$-0.744738\pi$$
−0.695322 + 0.718698i $$0.744738\pi$$
$$308$$ 0 0
$$309$$ −1.04071e6 −0.620058
$$310$$ 0 0
$$311$$ −984847. −0.577388 −0.288694 0.957421i $$-0.593221\pi$$
−0.288694 + 0.957421i $$0.593221\pi$$
$$312$$ 0 0
$$313$$ −2.06650e6 −1.19227 −0.596135 0.802884i $$-0.703298\pi$$
−0.596135 + 0.802884i $$0.703298\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −1.14349e6 −0.639125 −0.319563 0.947565i $$-0.603536\pi$$
−0.319563 + 0.947565i $$0.603536\pi$$
$$318$$ 0 0
$$319$$ −650910. −0.358133
$$320$$ 0 0
$$321$$ −375024. −0.203140
$$322$$ 0 0
$$323$$ −1.88465e6 −1.00514
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 1.21141e6 0.626501
$$328$$ 0 0
$$329$$ 61735.0 0.0314443
$$330$$ 0 0
$$331$$ −205230. −0.102961 −0.0514804 0.998674i $$-0.516394\pi$$
−0.0514804 + 0.998674i $$0.516394\pi$$
$$332$$ 0 0
$$333$$ −418729. −0.206929
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −488213. −0.234172 −0.117086 0.993122i $$-0.537355\pi$$
−0.117086 + 0.993122i $$0.537355\pi$$
$$338$$ 0 0
$$339$$ −446265. −0.210908
$$340$$ 0 0
$$341$$ −3.88412e6 −1.80887
$$342$$ 0 0
$$343$$ −1.98992e6 −0.913273
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −3.82809e6 −1.70670 −0.853351 0.521336i $$-0.825434\pi$$
−0.853351 + 0.521336i $$0.825434\pi$$
$$348$$ 0 0
$$349$$ 1.45476e6 0.639333 0.319667 0.947530i $$-0.396429\pi$$
0.319667 + 0.947530i $$0.396429\pi$$
$$350$$ 0 0
$$351$$ 1.07789e6 0.466991
$$352$$ 0 0
$$353$$ 778492. 0.332520 0.166260 0.986082i $$-0.446831\pi$$
0.166260 + 0.986082i $$0.446831\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 685867. 0.284819
$$358$$ 0 0
$$359$$ 2.12510e6 0.870247 0.435124 0.900371i $$-0.356705\pi$$
0.435124 + 0.900371i $$0.356705\pi$$
$$360$$ 0 0
$$361$$ −1.38061e6 −0.557577
$$362$$ 0 0
$$363$$ −415675. −0.165572
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 4.10801e6 1.59208 0.796042 0.605242i $$-0.206923\pi$$
0.796042 + 0.605242i $$0.206923\pi$$
$$368$$ 0 0
$$369$$ −2.28891e6 −0.875109
$$370$$ 0 0
$$371$$ 1.33263e6 0.502662
$$372$$ 0 0
$$373$$ 4.54570e6 1.69172 0.845860 0.533405i $$-0.179088\pi$$
0.845860 + 0.533405i $$0.179088\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −573624. −0.207861
$$378$$ 0 0
$$379$$ −1.40554e6 −0.502626 −0.251313 0.967906i $$-0.580862\pi$$
−0.251313 + 0.967906i $$0.580862\pi$$
$$380$$ 0 0
$$381$$ −891054. −0.314479
$$382$$ 0 0
$$383$$ 4.64417e6 1.61775 0.808874 0.587982i $$-0.200078\pi$$
0.808874 + 0.587982i $$0.200078\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −3.19102e6 −1.08306
$$388$$ 0 0
$$389$$ 3.53606e6 1.18480 0.592400 0.805644i $$-0.298181\pi$$
0.592400 + 0.805644i $$0.298181\pi$$
$$390$$ 0 0
$$391$$ −1.23678e6 −0.409120
$$392$$ 0 0
$$393$$ −1.06620e6 −0.348223
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 2.95611e6 0.941336 0.470668 0.882310i $$-0.344013\pi$$
0.470668 + 0.882310i $$0.344013\pi$$
$$398$$ 0 0
$$399$$ −398671. −0.125367
$$400$$ 0 0
$$401$$ −799254. −0.248213 −0.124106 0.992269i $$-0.539606\pi$$
−0.124106 + 0.992269i $$0.539606\pi$$
$$402$$ 0 0
$$403$$ −3.42294e6 −1.04987
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 957937. 0.286649
$$408$$ 0 0
$$409$$ 898422. 0.265566 0.132783 0.991145i $$-0.457609\pi$$
0.132783 + 0.991145i $$0.457609\pi$$
$$410$$ 0 0
$$411$$ 1.38292e6 0.403824
$$412$$ 0 0
$$413$$ −1.46134e6 −0.421576
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −1.20786e6 −0.340155
$$418$$ 0 0
$$419$$ −2.31259e6 −0.643523 −0.321761 0.946821i $$-0.604275\pi$$
−0.321761 + 0.946821i $$0.604275\pi$$
$$420$$ 0 0
$$421$$ 4.43296e6 1.21896 0.609478 0.792803i $$-0.291379\pi$$
0.609478 + 0.792803i $$0.291379\pi$$
$$422$$ 0 0
$$423$$ −190244. −0.0516965
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −1.91149e6 −0.507344
$$428$$ 0 0
$$429$$ −1.15036e6 −0.301780
$$430$$ 0 0
$$431$$ −5.97999e6 −1.55063 −0.775314 0.631576i $$-0.782408\pi$$
−0.775314 + 0.631576i $$0.782408\pi$$
$$432$$ 0 0
$$433$$ −2.06419e6 −0.529089 −0.264545 0.964373i $$-0.585222\pi$$
−0.264545 + 0.964373i $$0.585222\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 718899. 0.180080
$$438$$ 0 0
$$439$$ 4.09148e6 1.01326 0.506628 0.862165i $$-0.330892\pi$$
0.506628 + 0.862165i $$0.330892\pi$$
$$440$$ 0 0
$$441$$ 2.56099e6 0.627064
$$442$$ 0 0
$$443$$ 2.75822e6 0.667759 0.333879 0.942616i $$-0.391642\pi$$
0.333879 + 0.942616i $$0.391642\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 214006. 0.0506592
$$448$$ 0 0
$$449$$ −3.76648e6 −0.881698 −0.440849 0.897581i $$-0.645323\pi$$
−0.440849 + 0.897581i $$0.645323\pi$$
$$450$$ 0 0
$$451$$ 5.23640e6 1.21225
$$452$$ 0 0
$$453$$ −855944. −0.195975
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −480604. −0.107646 −0.0538229 0.998550i $$-0.517141\pi$$
−0.0538229 + 0.998550i $$0.517141\pi$$
$$458$$ 0 0
$$459$$ −4.53073e6 −1.00377
$$460$$ 0 0
$$461$$ 4.52514e6 0.991699 0.495849 0.868409i $$-0.334857\pi$$
0.495849 + 0.868409i $$0.334857\pi$$
$$462$$ 0 0
$$463$$ 7.39975e6 1.60422 0.802111 0.597175i $$-0.203710\pi$$
0.802111 + 0.597175i $$0.203710\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 1.84711e6 0.391923 0.195962 0.980612i $$-0.437217\pi$$
0.195962 + 0.980612i $$0.437217\pi$$
$$468$$ 0 0
$$469$$ −532261. −0.111736
$$470$$ 0 0
$$471$$ −1.90276e6 −0.395212
$$472$$ 0 0
$$473$$ 7.30018e6 1.50031
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −4.10669e6 −0.826410
$$478$$ 0 0
$$479$$ 3.05088e6 0.607555 0.303778 0.952743i $$-0.401752\pi$$
0.303778 + 0.952743i $$0.401752\pi$$
$$480$$ 0 0
$$481$$ 844197. 0.166372
$$482$$ 0 0
$$483$$ −261624. −0.0510281
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 7.28136e6 1.39120 0.695601 0.718429i $$-0.255138\pi$$
0.695601 + 0.718429i $$0.255138\pi$$
$$488$$ 0 0
$$489$$ −2.02297e6 −0.382575
$$490$$ 0 0
$$491$$ 6.60475e6 1.23638 0.618191 0.786028i $$-0.287866\pi$$
0.618191 + 0.786028i $$0.287866\pi$$
$$492$$ 0 0
$$493$$ 2.41112e6 0.446788
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 3.54481e6 0.643727
$$498$$ 0 0
$$499$$ −4.87006e6 −0.875555 −0.437777 0.899083i $$-0.644234\pi$$
−0.437777 + 0.899083i $$0.644234\pi$$
$$500$$ 0 0
$$501$$ −1.37702e6 −0.245102
$$502$$ 0 0
$$503$$ −1.16752e6 −0.205753 −0.102876 0.994694i $$-0.532805\pi$$
−0.102876 + 0.994694i $$0.532805\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 1.03732e6 0.179222
$$508$$ 0 0
$$509$$ −7.41468e6 −1.26852 −0.634261 0.773119i $$-0.718695\pi$$
−0.634261 + 0.773119i $$0.718695\pi$$
$$510$$ 0 0
$$511$$ 3.01846e6 0.511367
$$512$$ 0 0
$$513$$ 2.63356e6 0.441824
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 435227. 0.0716127
$$518$$ 0 0
$$519$$ −339516. −0.0553277
$$520$$ 0 0
$$521$$ −811897. −0.131041 −0.0655204 0.997851i $$-0.520871\pi$$
−0.0655204 + 0.997851i $$0.520871\pi$$
$$522$$ 0 0
$$523$$ 5.06828e6 0.810226 0.405113 0.914267i $$-0.367232\pi$$
0.405113 + 0.914267i $$0.367232\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 1.43877e7 2.25665
$$528$$ 0 0
$$529$$ −5.96457e6 −0.926702
$$530$$ 0 0
$$531$$ 4.50331e6 0.693099
$$532$$ 0 0
$$533$$ 4.61465e6 0.703592
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 3.35209e6 0.501626
$$538$$ 0 0
$$539$$ −5.85886e6 −0.868642
$$540$$ 0 0
$$541$$ 1.52830e6 0.224499 0.112250 0.993680i $$-0.464194\pi$$
0.112250 + 0.993680i $$0.464194\pi$$
$$542$$ 0 0
$$543$$ −848980. −0.123566
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −1.23234e7 −1.76101 −0.880506 0.474036i $$-0.842797\pi$$
−0.880506 + 0.474036i $$0.842797\pi$$
$$548$$ 0 0
$$549$$ 5.89050e6 0.834107
$$550$$ 0 0
$$551$$ −1.40150e6 −0.196660
$$552$$ 0 0
$$553$$ 429271. 0.0596923
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 4.08606e6 0.558042 0.279021 0.960285i $$-0.409990\pi$$
0.279021 + 0.960285i $$0.409990\pi$$
$$558$$ 0 0
$$559$$ 6.43339e6 0.870784
$$560$$ 0 0
$$561$$ 4.83531e6 0.648661
$$562$$ 0 0
$$563$$ 24160.3 0.00321241 0.00160621 0.999999i $$-0.499489\pi$$
0.00160621 + 0.999999i $$0.499489\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 2.60180e6 0.339873
$$568$$ 0 0
$$569$$ 1.42000e7 1.83869 0.919344 0.393454i $$-0.128720\pi$$
0.919344 + 0.393454i $$0.128720\pi$$
$$570$$ 0 0
$$571$$ 767642. 0.0985300 0.0492650 0.998786i $$-0.484312\pi$$
0.0492650 + 0.998786i $$0.484312\pi$$
$$572$$ 0 0
$$573$$ 1.00714e6 0.128145
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 1.51488e6 0.189426 0.0947129 0.995505i $$-0.469807\pi$$
0.0947129 + 0.995505i $$0.469807\pi$$
$$578$$ 0 0
$$579$$ −563910. −0.0699059
$$580$$ 0 0
$$581$$ 3.65098e6 0.448714
$$582$$ 0 0
$$583$$ 9.39498e6 1.14479
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −1.28973e7 −1.54491 −0.772455 0.635070i $$-0.780971\pi$$
−0.772455 + 0.635070i $$0.780971\pi$$
$$588$$ 0 0
$$589$$ −8.36307e6 −0.993294
$$590$$ 0 0
$$591$$ −2.23539e6 −0.263260
$$592$$ 0 0
$$593$$ 5.43125e6 0.634254 0.317127 0.948383i $$-0.397282\pi$$
0.317127 + 0.948383i $$0.397282\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −924180. −0.106126
$$598$$ 0 0
$$599$$ −3.92217e6 −0.446642 −0.223321 0.974745i $$-0.571690\pi$$
−0.223321 + 0.974745i $$0.571690\pi$$
$$600$$ 0 0
$$601$$ −5.64824e6 −0.637863 −0.318931 0.947778i $$-0.603324\pi$$
−0.318931 + 0.947778i $$0.603324\pi$$
$$602$$ 0 0
$$603$$ 1.64023e6 0.183701
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −1.07148e7 −1.18035 −0.590177 0.807274i $$-0.700942\pi$$
−0.590177 + 0.807274i $$0.700942\pi$$
$$608$$ 0 0
$$609$$ 510039. 0.0557263
$$610$$ 0 0
$$611$$ 383551. 0.0415642
$$612$$ 0 0
$$613$$ −4.08748e6 −0.439344 −0.219672 0.975574i $$-0.570499\pi$$
−0.219672 + 0.975574i $$0.570499\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 7.83395e6 0.828453 0.414227 0.910174i $$-0.364052\pi$$
0.414227 + 0.910174i $$0.364052\pi$$
$$618$$ 0 0
$$619$$ −1.23423e7 −1.29470 −0.647352 0.762191i $$-0.724124\pi$$
−0.647352 + 0.762191i $$0.724124\pi$$
$$620$$ 0 0
$$621$$ 1.72824e6 0.179836
$$622$$ 0 0
$$623$$ 3.07739e6 0.317660
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ −2.81061e6 −0.285516
$$628$$ 0 0
$$629$$ −3.54842e6 −0.357609
$$630$$ 0 0
$$631$$ 1.31578e6 0.131556 0.0657780 0.997834i $$-0.479047\pi$$
0.0657780 + 0.997834i $$0.479047\pi$$
$$632$$ 0 0
$$633$$ −2.54542e6 −0.252494
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −5.16320e6 −0.504163
$$638$$ 0 0
$$639$$ −1.09238e7 −1.05833
$$640$$ 0 0
$$641$$ 6.55744e6 0.630360 0.315180 0.949032i $$-0.397935\pi$$
0.315180 + 0.949032i $$0.397935\pi$$
$$642$$ 0 0
$$643$$ −4.69954e6 −0.448258 −0.224129 0.974559i $$-0.571954\pi$$
−0.224129 + 0.974559i $$0.571954\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 2.05827e7 1.93305 0.966523 0.256580i $$-0.0825956\pi$$
0.966523 + 0.256580i $$0.0825956\pi$$
$$648$$ 0 0
$$649$$ −1.03023e7 −0.960117
$$650$$ 0 0
$$651$$ 3.04351e6 0.281464
$$652$$ 0 0
$$653$$ 1.42466e7 1.30746 0.653731 0.756727i $$-0.273203\pi$$
0.653731 + 0.756727i $$0.273203\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ −9.30177e6 −0.840722
$$658$$ 0 0
$$659$$ 1.35369e7 1.21425 0.607123 0.794608i $$-0.292324\pi$$
0.607123 + 0.794608i $$0.292324\pi$$
$$660$$ 0 0
$$661$$ −1.30443e7 −1.16122 −0.580612 0.814180i $$-0.697187\pi$$
−0.580612 + 0.814180i $$0.697187\pi$$
$$662$$ 0 0
$$663$$ 4.26119e6 0.376485
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −919721. −0.0800464
$$668$$ 0 0
$$669$$ 5.98260e6 0.516802
$$670$$ 0 0
$$671$$ −1.34759e7 −1.15545
$$672$$ 0 0
$$673$$ −4.75951e6 −0.405065 −0.202532 0.979276i $$-0.564917\pi$$
−0.202532 + 0.979276i $$0.564917\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 1.51397e7 1.26954 0.634770 0.772701i $$-0.281095\pi$$
0.634770 + 0.772701i $$0.281095\pi$$
$$678$$ 0 0
$$679$$ 1.02061e7 0.849543
$$680$$ 0 0
$$681$$ −2.27707e6 −0.188152
$$682$$ 0 0
$$683$$ −2.34145e7 −1.92058 −0.960292 0.278998i $$-0.909998\pi$$
−0.960292 + 0.278998i $$0.909998\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 2.39287e6 0.193431
$$688$$ 0 0
$$689$$ 8.27947e6 0.664438
$$690$$ 0 0
$$691$$ 1.62194e7 1.29223 0.646113 0.763242i $$-0.276394\pi$$
0.646113 + 0.763242i $$0.276394\pi$$
$$692$$ 0 0
$$693$$ −7.12196e6 −0.563334
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −1.93968e7 −1.51234
$$698$$ 0 0
$$699$$ 4.19891e6 0.325045
$$700$$ 0 0
$$701$$ −1.89605e7 −1.45732 −0.728659 0.684876i $$-0.759856\pi$$
−0.728659 + 0.684876i $$0.759856\pi$$
$$702$$ 0 0
$$703$$ 2.06258e6 0.157406
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 1.02377e7 0.770287
$$708$$ 0 0
$$709$$ 128325. 0.00958732 0.00479366 0.999989i $$-0.498474\pi$$
0.00479366 + 0.999989i $$0.498474\pi$$
$$710$$ 0 0
$$711$$ −1.32285e6 −0.0981382
$$712$$ 0 0
$$713$$ −5.48817e6 −0.404300
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −5.46133e6 −0.396735
$$718$$ 0 0
$$719$$ 2.41874e7 1.74489 0.872444 0.488714i $$-0.162534\pi$$
0.872444 + 0.488714i $$0.162534\pi$$
$$720$$ 0 0
$$721$$ 1.29899e7 0.930610
$$722$$ 0 0
$$723$$ 1.98251e6 0.141049
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −513307. −0.0360198 −0.0180099 0.999838i $$-0.505733\pi$$
−0.0180099 + 0.999838i $$0.505733\pi$$
$$728$$ 0 0
$$729$$ −4.64015e6 −0.323380
$$730$$ 0 0
$$731$$ −2.70416e7 −1.87171
$$732$$ 0 0
$$733$$ −1.64153e7 −1.12847 −0.564234 0.825615i $$-0.690828\pi$$
−0.564234 + 0.825615i $$0.690828\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −3.75240e6 −0.254472
$$738$$ 0 0
$$739$$ −1.16112e7 −0.782109 −0.391054 0.920368i $$-0.627890\pi$$
−0.391054 + 0.920368i $$0.627890\pi$$
$$740$$ 0 0
$$741$$ −2.47689e6 −0.165715
$$742$$ 0 0
$$743$$ −5.72590e6 −0.380515 −0.190257 0.981734i $$-0.560932\pi$$
−0.190257 + 0.981734i $$0.560932\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ −1.12510e7 −0.737715
$$748$$ 0 0
$$749$$ 4.68097e6 0.304882
$$750$$ 0 0
$$751$$ −1.15324e7 −0.746137 −0.373069 0.927804i $$-0.621694\pi$$
−0.373069 + 0.927804i $$0.621694\pi$$
$$752$$ 0 0
$$753$$ −4.70134e6 −0.302158
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −8.63293e6 −0.547544 −0.273772 0.961795i $$-0.588271\pi$$
−0.273772 + 0.961795i $$0.588271\pi$$
$$758$$ 0 0
$$759$$ −1.84443e6 −0.116214
$$760$$ 0 0
$$761$$ −3.52622e6 −0.220723 −0.110361 0.993892i $$-0.535201\pi$$
−0.110361 + 0.993892i $$0.535201\pi$$
$$762$$ 0 0
$$763$$ −1.51206e7 −0.940280
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −9.07910e6 −0.557255
$$768$$ 0 0
$$769$$ 1.40471e7 0.856585 0.428293 0.903640i $$-0.359115\pi$$
0.428293 + 0.903640i $$0.359115\pi$$
$$770$$ 0 0
$$771$$ −421817. −0.0255557
$$772$$ 0 0
$$773$$ −2.44760e7 −1.47330 −0.736651 0.676274i $$-0.763594\pi$$
−0.736651 + 0.676274i $$0.763594\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ −750619. −0.0446033
$$778$$ 0 0
$$779$$ 1.12747e7 0.665675
$$780$$ 0 0
$$781$$ 2.49907e7 1.46606
$$782$$ 0 0
$$783$$ −3.36924e6 −0.196393
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −4.35977e6 −0.250915 −0.125458 0.992099i $$-0.540040\pi$$
−0.125458 + 0.992099i $$0.540040\pi$$
$$788$$ 0 0
$$789$$ −6.59697e6 −0.377270
$$790$$ 0 0
$$791$$ 5.57019e6 0.316540
$$792$$ 0 0
$$793$$ −1.18758e7 −0.670626
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 1.06887e7 0.596044 0.298022 0.954559i $$-0.403673\pi$$
0.298022 + 0.954559i $$0.403673\pi$$
$$798$$ 0 0
$$799$$ −1.61218e6 −0.0893403
$$800$$ 0 0
$$801$$ −9.48339e6 −0.522255
$$802$$ 0 0
$$803$$ 2.12799e7 1.16461
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −5.68605e6 −0.307345
$$808$$ 0 0
$$809$$ 9.12014e6 0.489926 0.244963 0.969532i $$-0.421224\pi$$
0.244963 + 0.969532i $$0.421224\pi$$
$$810$$ 0 0
$$811$$ −5.22575e6 −0.278995 −0.139497 0.990222i $$-0.544549\pi$$
−0.139497 + 0.990222i $$0.544549\pi$$
$$812$$ 0 0
$$813$$ 1.17192e7 0.621830
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 1.57184e7 0.823857
$$818$$ 0 0
$$819$$ −6.27633e6 −0.326961
$$820$$ 0 0
$$821$$ 9.00437e6 0.466225 0.233112 0.972450i $$-0.425109\pi$$
0.233112 + 0.972450i $$0.425109\pi$$
$$822$$ 0 0
$$823$$ −2.78867e7 −1.43515 −0.717574 0.696482i $$-0.754748\pi$$
−0.717574 + 0.696482i $$0.754748\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −6.64309e6 −0.337758 −0.168879 0.985637i $$-0.554015\pi$$
−0.168879 + 0.985637i $$0.554015\pi$$
$$828$$ 0 0
$$829$$ 2.17030e7 1.09682 0.548408 0.836211i $$-0.315234\pi$$
0.548408 + 0.836211i $$0.315234\pi$$
$$830$$ 0 0
$$831$$ 1.02277e7 0.513780
$$832$$ 0 0
$$833$$ 2.17026e7 1.08367
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −2.01049e7 −0.991949
$$838$$ 0 0
$$839$$ 1.01238e7 0.496520 0.248260 0.968693i $$-0.420141\pi$$
0.248260 + 0.968693i $$0.420141\pi$$
$$840$$ 0 0
$$841$$ −1.87181e7 −0.912584
$$842$$ 0 0
$$843$$ −500784. −0.0242707
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 5.18837e6 0.248498
$$848$$ 0 0
$$849$$ 5.13366e6 0.244432
$$850$$ 0 0
$$851$$ 1.35354e6 0.0640691
$$852$$ 0 0
$$853$$ −1.46326e7 −0.688573 −0.344286 0.938865i $$-0.611879\pi$$
−0.344286 + 0.938865i $$0.611879\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 5.52218e6 0.256838 0.128419 0.991720i $$-0.459010\pi$$
0.128419 + 0.991720i $$0.459010\pi$$
$$858$$ 0 0
$$859$$ 3.02260e6 0.139765 0.0698824 0.997555i $$-0.477738\pi$$
0.0698824 + 0.997555i $$0.477738\pi$$
$$860$$ 0 0
$$861$$ −4.10313e6 −0.188628
$$862$$ 0 0
$$863$$ −3.06818e7 −1.40234 −0.701172 0.712992i $$-0.747339\pi$$
−0.701172 + 0.712992i $$0.747339\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ −1.00676e7 −0.454860
$$868$$ 0 0
$$869$$ 3.02633e6 0.135946
$$870$$ 0 0
$$871$$ −3.30686e6 −0.147697
$$872$$ 0 0
$$873$$ −3.14514e7 −1.39671
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 5.17607e6 0.227249 0.113624 0.993524i $$-0.463754\pi$$
0.113624 + 0.993524i $$0.463754\pi$$
$$878$$ 0 0
$$879$$ 1.50662e7 0.657707
$$880$$ 0 0
$$881$$ −4.25937e7 −1.84887 −0.924433 0.381345i $$-0.875461\pi$$
−0.924433 + 0.381345i $$0.875461\pi$$
$$882$$ 0 0
$$883$$ −1.72076e7 −0.742709 −0.371354 0.928491i $$-0.621107\pi$$
−0.371354 + 0.928491i $$0.621107\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 2.53773e6 0.108302 0.0541510 0.998533i $$-0.482755\pi$$
0.0541510 + 0.998533i $$0.482755\pi$$
$$888$$ 0 0
$$889$$ 1.11220e7 0.471984
$$890$$ 0 0
$$891$$ 1.83425e7 0.774043
$$892$$ 0 0
$$893$$ 937108. 0.0393243
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ −1.62543e6 −0.0674508
$$898$$ 0 0
$$899$$ 1.06993e7 0.441525
$$900$$ 0 0
$$901$$ −3.48012e7 −1.42818
$$902$$ 0 0
$$903$$ −5.72026e6 −0.233452
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 2.60899e7 1.05306 0.526531 0.850156i $$-0.323492\pi$$
0.526531 + 0.850156i $$0.323492\pi$$
$$908$$ 0 0
$$909$$ −3.15487e7 −1.26640
$$910$$ 0 0
$$911$$ −1.44818e7 −0.578130 −0.289065 0.957309i $$-0.593344\pi$$
−0.289065 + 0.957309i $$0.593344\pi$$
$$912$$ 0 0
$$913$$ 2.57392e7 1.02192
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 1.33081e7 0.522627
$$918$$ 0 0
$$919$$ −4.61041e7 −1.80074 −0.900369 0.435127i $$-0.856704\pi$$
−0.900369 + 0.435127i $$0.856704\pi$$
$$920$$ 0 0
$$921$$ 1.26861e7 0.492811
$$922$$ 0 0
$$923$$ 2.20234e7 0.850903
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ −4.00301e7 −1.52998
$$928$$ 0 0
$$929$$ −1.81557e7 −0.690197 −0.345098 0.938567i $$-0.612154\pi$$
−0.345098 + 0.938567i $$0.612154\pi$$
$$930$$ 0 0
$$931$$ −1.26150e7 −0.476993
$$932$$ 0 0
$$933$$ 5.44047e6 0.204612
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −1.78946e7 −0.665844 −0.332922 0.942954i $$-0.608035\pi$$
−0.332922 + 0.942954i $$0.608035\pi$$
$$938$$ 0 0
$$939$$ 1.14157e7 0.422512
$$940$$ 0 0
$$941$$ −3.04463e6 −0.112088 −0.0560441 0.998428i $$-0.517849\pi$$
−0.0560441 + 0.998428i $$0.517849\pi$$
$$942$$ 0 0
$$943$$ 7.39891e6 0.270950
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 3.17110e7 1.14904 0.574519 0.818491i $$-0.305189\pi$$
0.574519 + 0.818491i $$0.305189\pi$$
$$948$$ 0 0
$$949$$ 1.87532e7 0.675944
$$950$$ 0 0
$$951$$ 6.31686e6 0.226491
$$952$$ 0 0
$$953$$ −1.01913e7 −0.363494 −0.181747 0.983345i $$-0.558175\pi$$
−0.181747 + 0.983345i $$0.558175\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 3.59574e6 0.126914
$$958$$ 0 0
$$959$$ −1.72613e7 −0.606077
$$960$$ 0 0
$$961$$ 3.52157e7 1.23006
$$962$$ 0 0
$$963$$ −1.44250e7 −0.501246
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −3.21125e7 −1.10435 −0.552177 0.833727i $$-0.686203\pi$$
−0.552177 + 0.833727i $$0.686203\pi$$
$$968$$ 0 0
$$969$$ 1.04111e7 0.356196
$$970$$ 0 0
$$971$$ −2.29867e7 −0.782399 −0.391200 0.920306i $$-0.627940\pi$$
−0.391200 + 0.920306i $$0.627940\pi$$
$$972$$ 0 0
$$973$$ 1.50763e7 0.510519
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 2.47331e7 0.828978 0.414489 0.910054i $$-0.363960\pi$$
0.414489 + 0.910054i $$0.363960\pi$$
$$978$$ 0 0
$$979$$ 2.16954e7 0.723455
$$980$$ 0 0
$$981$$ 4.65960e7 1.54588
$$982$$ 0 0
$$983$$ 5.57031e7 1.83863 0.919317 0.393518i $$-0.128742\pi$$
0.919317 + 0.393518i $$0.128742\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ −341035. −0.0111431
$$988$$ 0 0
$$989$$ 1.03150e7 0.335335
$$990$$ 0 0
$$991$$ 6.86029e6 0.221901 0.110950 0.993826i $$-0.464611\pi$$
0.110950 + 0.993826i $$0.464611\pi$$
$$992$$ 0 0
$$993$$ 1.13373e6 0.0364868
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −6.03725e7 −1.92354 −0.961771 0.273856i $$-0.911701\pi$$
−0.961771 + 0.273856i $$0.911701\pi$$
$$998$$ 0 0
$$999$$ 4.95847e6 0.157193
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 400.6.a.w.1.1 2
4.3 odd 2 25.6.a.b.1.1 2
5.2 odd 4 400.6.c.n.49.3 4
5.3 odd 4 400.6.c.n.49.2 4
5.4 even 2 400.6.a.o.1.2 2
12.11 even 2 225.6.a.s.1.2 2
20.3 even 4 25.6.b.b.24.4 4
20.7 even 4 25.6.b.b.24.1 4
20.19 odd 2 25.6.a.d.1.2 yes 2
60.23 odd 4 225.6.b.i.199.1 4
60.47 odd 4 225.6.b.i.199.4 4
60.59 even 2 225.6.a.l.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
25.6.a.b.1.1 2 4.3 odd 2
25.6.a.d.1.2 yes 2 20.19 odd 2
25.6.b.b.24.1 4 20.7 even 4
25.6.b.b.24.4 4 20.3 even 4
225.6.a.l.1.1 2 60.59 even 2
225.6.a.s.1.2 2 12.11 even 2
225.6.b.i.199.1 4 60.23 odd 4
225.6.b.i.199.4 4 60.47 odd 4
400.6.a.o.1.2 2 5.4 even 2
400.6.a.w.1.1 2 1.1 even 1 trivial
400.6.c.n.49.2 4 5.3 odd 4
400.6.c.n.49.3 4 5.2 odd 4