Properties

 Label 25.6.b.a.24.2 Level $25$ Weight $6$ Character 25.24 Analytic conductor $4.010$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$25 = 5^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 25.b (of order $$2$$, degree $$1$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$4.00959549532$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 5) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

 Embedding label 24.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 25.24 Dual form 25.6.b.a.24.1

$q$-expansion

 $$f(q)$$ $$=$$ $$q+2.00000i q^{2} +4.00000i q^{3} +28.0000 q^{4} -8.00000 q^{6} +192.000i q^{7} +120.000i q^{8} +227.000 q^{9} +O(q^{10})$$ $$q+2.00000i q^{2} +4.00000i q^{3} +28.0000 q^{4} -8.00000 q^{6} +192.000i q^{7} +120.000i q^{8} +227.000 q^{9} -148.000 q^{11} +112.000i q^{12} -286.000i q^{13} -384.000 q^{14} +656.000 q^{16} -1678.00i q^{17} +454.000i q^{18} -1060.00 q^{19} -768.000 q^{21} -296.000i q^{22} -2976.00i q^{23} -480.000 q^{24} +572.000 q^{26} +1880.00i q^{27} +5376.00i q^{28} +3410.00 q^{29} -2448.00 q^{31} +5152.00i q^{32} -592.000i q^{33} +3356.00 q^{34} +6356.00 q^{36} +182.000i q^{37} -2120.00i q^{38} +1144.00 q^{39} -9398.00 q^{41} -1536.00i q^{42} +1244.00i q^{43} -4144.00 q^{44} +5952.00 q^{46} -12088.0i q^{47} +2624.00i q^{48} -20057.0 q^{49} +6712.00 q^{51} -8008.00i q^{52} -23846.0i q^{53} -3760.00 q^{54} -23040.0 q^{56} -4240.00i q^{57} +6820.00i q^{58} +20020.0 q^{59} +32302.0 q^{61} -4896.00i q^{62} +43584.0i q^{63} +10688.0 q^{64} +1184.00 q^{66} +60972.0i q^{67} -46984.0i q^{68} +11904.0 q^{69} -32648.0 q^{71} +27240.0i q^{72} +38774.0i q^{73} -364.000 q^{74} -29680.0 q^{76} -28416.0i q^{77} +2288.00i q^{78} +33360.0 q^{79} +47641.0 q^{81} -18796.0i q^{82} -16716.0i q^{83} -21504.0 q^{84} -2488.00 q^{86} +13640.0i q^{87} -17760.0i q^{88} -101370. q^{89} +54912.0 q^{91} -83328.0i q^{92} -9792.00i q^{93} +24176.0 q^{94} -20608.0 q^{96} -119038. i q^{97} -40114.0i q^{98} -33596.0 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 56 q^{4} - 16 q^{6} + 454 q^{9} + O(q^{10})$$ $$2 q + 56 q^{4} - 16 q^{6} + 454 q^{9} - 296 q^{11} - 768 q^{14} + 1312 q^{16} - 2120 q^{19} - 1536 q^{21} - 960 q^{24} + 1144 q^{26} + 6820 q^{29} - 4896 q^{31} + 6712 q^{34} + 12712 q^{36} + 2288 q^{39} - 18796 q^{41} - 8288 q^{44} + 11904 q^{46} - 40114 q^{49} + 13424 q^{51} - 7520 q^{54} - 46080 q^{56} + 40040 q^{59} + 64604 q^{61} + 21376 q^{64} + 2368 q^{66} + 23808 q^{69} - 65296 q^{71} - 728 q^{74} - 59360 q^{76} + 66720 q^{79} + 95282 q^{81} - 43008 q^{84} - 4976 q^{86} - 202740 q^{89} + 109824 q^{91} + 48352 q^{94} - 41216 q^{96} - 67192 q^{99} + O(q^{100})$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/25\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-1$$

Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.00000i 0.353553i 0.984251 + 0.176777i $$0.0565670\pi$$
−0.984251 + 0.176777i $$0.943433\pi$$
$$3$$ 4.00000i 0.256600i 0.991735 + 0.128300i $$0.0409521\pi$$
−0.991735 + 0.128300i $$0.959048\pi$$
$$4$$ 28.0000 0.875000
$$5$$ 0 0
$$6$$ −8.00000 −0.0907218
$$7$$ 192.000i 1.48100i 0.672054 + 0.740502i $$0.265412\pi$$
−0.672054 + 0.740502i $$0.734588\pi$$
$$8$$ 120.000i 0.662913i
$$9$$ 227.000 0.934156
$$10$$ 0 0
$$11$$ −148.000 −0.368791 −0.184395 0.982852i $$-0.559033\pi$$
−0.184395 + 0.982852i $$0.559033\pi$$
$$12$$ 112.000i 0.224525i
$$13$$ − 286.000i − 0.469362i −0.972072 0.234681i $$-0.924595\pi$$
0.972072 0.234681i $$-0.0754045\pi$$
$$14$$ −384.000 −0.523614
$$15$$ 0 0
$$16$$ 656.000 0.640625
$$17$$ − 1678.00i − 1.40822i −0.710092 0.704109i $$-0.751347\pi$$
0.710092 0.704109i $$-0.248653\pi$$
$$18$$ 454.000i 0.330274i
$$19$$ −1060.00 −0.673631 −0.336815 0.941571i $$-0.609350\pi$$
−0.336815 + 0.941571i $$0.609350\pi$$
$$20$$ 0 0
$$21$$ −768.000 −0.380026
$$22$$ − 296.000i − 0.130387i
$$23$$ − 2976.00i − 1.17304i −0.809934 0.586521i $$-0.800497\pi$$
0.809934 0.586521i $$-0.199503\pi$$
$$24$$ −480.000 −0.170103
$$25$$ 0 0
$$26$$ 572.000 0.165944
$$27$$ 1880.00i 0.496305i
$$28$$ 5376.00i 1.29588i
$$29$$ 3410.00 0.752938 0.376469 0.926429i $$-0.377138\pi$$
0.376469 + 0.926429i $$0.377138\pi$$
$$30$$ 0 0
$$31$$ −2448.00 −0.457517 −0.228758 0.973483i $$-0.573467\pi$$
−0.228758 + 0.973483i $$0.573467\pi$$
$$32$$ 5152.00i 0.889408i
$$33$$ − 592.000i − 0.0946317i
$$34$$ 3356.00 0.497880
$$35$$ 0 0
$$36$$ 6356.00 0.817387
$$37$$ 182.000i 0.0218558i 0.999940 + 0.0109279i $$0.00347853\pi$$
−0.999940 + 0.0109279i $$0.996521\pi$$
$$38$$ − 2120.00i − 0.238164i
$$39$$ 1144.00 0.120438
$$40$$ 0 0
$$41$$ −9398.00 −0.873124 −0.436562 0.899674i $$-0.643804\pi$$
−0.436562 + 0.899674i $$0.643804\pi$$
$$42$$ − 1536.00i − 0.134359i
$$43$$ 1244.00i 0.102600i 0.998683 + 0.0513002i $$0.0163365\pi$$
−0.998683 + 0.0513002i $$0.983663\pi$$
$$44$$ −4144.00 −0.322692
$$45$$ 0 0
$$46$$ 5952.00 0.414733
$$47$$ − 12088.0i − 0.798196i −0.916908 0.399098i $$-0.869323\pi$$
0.916908 0.399098i $$-0.130677\pi$$
$$48$$ 2624.00i 0.164384i
$$49$$ −20057.0 −1.19337
$$50$$ 0 0
$$51$$ 6712.00 0.361349
$$52$$ − 8008.00i − 0.410691i
$$53$$ − 23846.0i − 1.16607i −0.812446 0.583037i $$-0.801864\pi$$
0.812446 0.583037i $$-0.198136\pi$$
$$54$$ −3760.00 −0.175470
$$55$$ 0 0
$$56$$ −23040.0 −0.981776
$$57$$ − 4240.00i − 0.172854i
$$58$$ 6820.00i 0.266204i
$$59$$ 20020.0 0.748745 0.374373 0.927278i $$-0.377858\pi$$
0.374373 + 0.927278i $$0.377858\pi$$
$$60$$ 0 0
$$61$$ 32302.0 1.11149 0.555744 0.831353i $$-0.312433\pi$$
0.555744 + 0.831353i $$0.312433\pi$$
$$62$$ − 4896.00i − 0.161757i
$$63$$ 43584.0i 1.38349i
$$64$$ 10688.0 0.326172
$$65$$ 0 0
$$66$$ 1184.00 0.0334574
$$67$$ 60972.0i 1.65937i 0.558231 + 0.829685i $$0.311480\pi$$
−0.558231 + 0.829685i $$0.688520\pi$$
$$68$$ − 46984.0i − 1.23219i
$$69$$ 11904.0 0.301003
$$70$$ 0 0
$$71$$ −32648.0 −0.768618 −0.384309 0.923204i $$-0.625560\pi$$
−0.384309 + 0.923204i $$0.625560\pi$$
$$72$$ 27240.0i 0.619264i
$$73$$ 38774.0i 0.851596i 0.904818 + 0.425798i $$0.140007\pi$$
−0.904818 + 0.425798i $$0.859993\pi$$
$$74$$ −364.000 −0.00772720
$$75$$ 0 0
$$76$$ −29680.0 −0.589427
$$77$$ − 28416.0i − 0.546180i
$$78$$ 2288.00i 0.0425814i
$$79$$ 33360.0 0.601393 0.300696 0.953720i $$-0.402781\pi$$
0.300696 + 0.953720i $$0.402781\pi$$
$$80$$ 0 0
$$81$$ 47641.0 0.806805
$$82$$ − 18796.0i − 0.308696i
$$83$$ − 16716.0i − 0.266340i −0.991093 0.133170i $$-0.957484\pi$$
0.991093 0.133170i $$-0.0425157\pi$$
$$84$$ −21504.0 −0.332522
$$85$$ 0 0
$$86$$ −2488.00 −0.0362747
$$87$$ 13640.0i 0.193204i
$$88$$ − 17760.0i − 0.244476i
$$89$$ −101370. −1.35655 −0.678273 0.734810i $$-0.737271\pi$$
−0.678273 + 0.734810i $$0.737271\pi$$
$$90$$ 0 0
$$91$$ 54912.0 0.695126
$$92$$ − 83328.0i − 1.02641i
$$93$$ − 9792.00i − 0.117399i
$$94$$ 24176.0 0.282205
$$95$$ 0 0
$$96$$ −20608.0 −0.228222
$$97$$ − 119038.i − 1.28457i −0.766468 0.642283i $$-0.777987\pi$$
0.766468 0.642283i $$-0.222013\pi$$
$$98$$ − 40114.0i − 0.421921i
$$99$$ −33596.0 −0.344508
$$100$$ 0 0
$$101$$ −89898.0 −0.876893 −0.438446 0.898757i $$-0.644471\pi$$
−0.438446 + 0.898757i $$0.644471\pi$$
$$102$$ 13424.0i 0.127756i
$$103$$ 19504.0i 0.181147i 0.995890 + 0.0905734i $$0.0288700\pi$$
−0.995890 + 0.0905734i $$0.971130\pi$$
$$104$$ 34320.0 0.311146
$$105$$ 0 0
$$106$$ 47692.0 0.412269
$$107$$ 158292.i 1.33659i 0.743895 + 0.668297i $$0.232976\pi$$
−0.743895 + 0.668297i $$0.767024\pi$$
$$108$$ 52640.0i 0.434267i
$$109$$ −36830.0 −0.296917 −0.148459 0.988919i $$-0.547431\pi$$
−0.148459 + 0.988919i $$0.547431\pi$$
$$110$$ 0 0
$$111$$ −728.000 −0.00560821
$$112$$ 125952.i 0.948768i
$$113$$ − 11186.0i − 0.0824098i −0.999151 0.0412049i $$-0.986880\pi$$
0.999151 0.0412049i $$-0.0131196\pi$$
$$114$$ 8480.00 0.0611130
$$115$$ 0 0
$$116$$ 95480.0 0.658821
$$117$$ − 64922.0i − 0.438457i
$$118$$ 40040.0i 0.264721i
$$119$$ 322176. 2.08557
$$120$$ 0 0
$$121$$ −139147. −0.863993
$$122$$ 64604.0i 0.392970i
$$123$$ − 37592.0i − 0.224044i
$$124$$ −68544.0 −0.400327
$$125$$ 0 0
$$126$$ −87168.0 −0.489137
$$127$$ 70552.0i 0.388150i 0.980987 + 0.194075i $$0.0621706\pi$$
−0.980987 + 0.194075i $$0.937829\pi$$
$$128$$ 186240.i 1.00473i
$$129$$ −4976.00 −0.0263273
$$130$$ 0 0
$$131$$ 76452.0 0.389234 0.194617 0.980879i $$-0.437654\pi$$
0.194617 + 0.980879i $$0.437654\pi$$
$$132$$ − 16576.0i − 0.0828028i
$$133$$ − 203520.i − 0.997650i
$$134$$ −121944. −0.586676
$$135$$ 0 0
$$136$$ 201360. 0.933525
$$137$$ − 144918.i − 0.659661i −0.944040 0.329831i $$-0.893008\pi$$
0.944040 0.329831i $$-0.106992\pi$$
$$138$$ 23808.0i 0.106420i
$$139$$ −112220. −0.492644 −0.246322 0.969188i $$-0.579222\pi$$
−0.246322 + 0.969188i $$0.579222\pi$$
$$140$$ 0 0
$$141$$ 48352.0 0.204817
$$142$$ − 65296.0i − 0.271748i
$$143$$ 42328.0i 0.173096i
$$144$$ 148912. 0.598444
$$145$$ 0 0
$$146$$ −77548.0 −0.301085
$$147$$ − 80228.0i − 0.306219i
$$148$$ 5096.00i 0.0191238i
$$149$$ −403750. −1.48986 −0.744932 0.667140i $$-0.767518\pi$$
−0.744932 + 0.667140i $$0.767518\pi$$
$$150$$ 0 0
$$151$$ −446648. −1.59413 −0.797064 0.603895i $$-0.793615\pi$$
−0.797064 + 0.603895i $$0.793615\pi$$
$$152$$ − 127200.i − 0.446558i
$$153$$ − 380906.i − 1.31550i
$$154$$ 56832.0 0.193104
$$155$$ 0 0
$$156$$ 32032.0 0.105383
$$157$$ − 262258.i − 0.849141i −0.905395 0.424570i $$-0.860425\pi$$
0.905395 0.424570i $$-0.139575\pi$$
$$158$$ 66720.0i 0.212625i
$$159$$ 95384.0 0.299215
$$160$$ 0 0
$$161$$ 571392. 1.73728
$$162$$ 95282.0i 0.285248i
$$163$$ 154564.i 0.455658i 0.973701 + 0.227829i $$0.0731628\pi$$
−0.973701 + 0.227829i $$0.926837\pi$$
$$164$$ −263144. −0.763983
$$165$$ 0 0
$$166$$ 33432.0 0.0941656
$$167$$ 396672.i 1.10063i 0.834958 + 0.550314i $$0.185492\pi$$
−0.834958 + 0.550314i $$0.814508\pi$$
$$168$$ − 92160.0i − 0.251924i
$$169$$ 289497. 0.779700
$$170$$ 0 0
$$171$$ −240620. −0.629276
$$172$$ 34832.0i 0.0897754i
$$173$$ 573474.i 1.45680i 0.685155 + 0.728398i $$0.259735\pi$$
−0.685155 + 0.728398i $$0.740265\pi$$
$$174$$ −27280.0 −0.0683079
$$175$$ 0 0
$$176$$ −97088.0 −0.236257
$$177$$ 80080.0i 0.192128i
$$178$$ − 202740.i − 0.479611i
$$179$$ 594460. 1.38672 0.693362 0.720589i $$-0.256129\pi$$
0.693362 + 0.720589i $$0.256129\pi$$
$$180$$ 0 0
$$181$$ −107098. −0.242988 −0.121494 0.992592i $$-0.538769\pi$$
−0.121494 + 0.992592i $$0.538769\pi$$
$$182$$ 109824.i 0.245764i
$$183$$ 129208.i 0.285208i
$$184$$ 357120. 0.777624
$$185$$ 0 0
$$186$$ 19584.0 0.0415068
$$187$$ 248344.i 0.519337i
$$188$$ − 338464.i − 0.698422i
$$189$$ −360960. −0.735029
$$190$$ 0 0
$$191$$ 469552. 0.931323 0.465661 0.884963i $$-0.345816\pi$$
0.465661 + 0.884963i $$0.345816\pi$$
$$192$$ 42752.0i 0.0836957i
$$193$$ − 52706.0i − 0.101851i −0.998702 0.0509257i $$-0.983783\pi$$
0.998702 0.0509257i $$-0.0162172\pi$$
$$194$$ 238076. 0.454163
$$195$$ 0 0
$$196$$ −561596. −1.04420
$$197$$ 455862.i 0.836889i 0.908242 + 0.418444i $$0.137425\pi$$
−0.908242 + 0.418444i $$0.862575\pi$$
$$198$$ − 67192.0i − 0.121802i
$$199$$ −865000. −1.54840 −0.774200 0.632940i $$-0.781848\pi$$
−0.774200 + 0.632940i $$0.781848\pi$$
$$200$$ 0 0
$$201$$ −243888. −0.425795
$$202$$ − 179796.i − 0.310028i
$$203$$ 654720.i 1.11510i
$$204$$ 187936. 0.316180
$$205$$ 0 0
$$206$$ −39008.0 −0.0640451
$$207$$ − 675552.i − 1.09580i
$$208$$ − 187616.i − 0.300685i
$$209$$ 156880. 0.248429
$$210$$ 0 0
$$211$$ 1.10565e6 1.70967 0.854835 0.518900i $$-0.173658\pi$$
0.854835 + 0.518900i $$0.173658\pi$$
$$212$$ − 667688.i − 1.02031i
$$213$$ − 130592.i − 0.197228i
$$214$$ −316584. −0.472557
$$215$$ 0 0
$$216$$ −225600. −0.329007
$$217$$ − 470016.i − 0.677584i
$$218$$ − 73660.0i − 0.104976i
$$219$$ −155096. −0.218520
$$220$$ 0 0
$$221$$ −479908. −0.660963
$$222$$ − 1456.00i − 0.00198280i
$$223$$ − 1.12158e6i − 1.51031i −0.655545 0.755156i $$-0.727561\pi$$
0.655545 0.755156i $$-0.272439\pi$$
$$224$$ −989184. −1.31722
$$225$$ 0 0
$$226$$ 22372.0 0.0291363
$$227$$ − 23348.0i − 0.0300736i −0.999887 0.0150368i $$-0.995213\pi$$
0.999887 0.0150368i $$-0.00478654\pi$$
$$228$$ − 118720.i − 0.151247i
$$229$$ 596010. 0.751043 0.375522 0.926814i $$-0.377464\pi$$
0.375522 + 0.926814i $$0.377464\pi$$
$$230$$ 0 0
$$231$$ 113664. 0.140150
$$232$$ 409200.i 0.499132i
$$233$$ 485334.i 0.585667i 0.956163 + 0.292834i $$0.0945982\pi$$
−0.956163 + 0.292834i $$0.905402\pi$$
$$234$$ 129844. 0.155018
$$235$$ 0 0
$$236$$ 560560. 0.655152
$$237$$ 133440.i 0.154317i
$$238$$ 644352.i 0.737362i
$$239$$ 48880.0 0.0553524 0.0276762 0.999617i $$-0.491189\pi$$
0.0276762 + 0.999617i $$0.491189\pi$$
$$240$$ 0 0
$$241$$ −110798. −0.122882 −0.0614411 0.998111i $$-0.519570\pi$$
−0.0614411 + 0.998111i $$0.519570\pi$$
$$242$$ − 278294.i − 0.305468i
$$243$$ 647404.i 0.703331i
$$244$$ 904456. 0.972552
$$245$$ 0 0
$$246$$ 75184.0 0.0792114
$$247$$ 303160.i 0.316176i
$$248$$ − 293760.i − 0.303294i
$$249$$ 66864.0 0.0683430
$$250$$ 0 0
$$251$$ −1.64375e6 −1.64684 −0.823419 0.567434i $$-0.807936\pi$$
−0.823419 + 0.567434i $$0.807936\pi$$
$$252$$ 1.22035e6i 1.21055i
$$253$$ 440448.i 0.432607i
$$254$$ −141104. −0.137232
$$255$$ 0 0
$$256$$ −30464.0 −0.0290527
$$257$$ 1.30624e6i 1.23365i 0.787102 + 0.616823i $$0.211581\pi$$
−0.787102 + 0.616823i $$0.788419\pi$$
$$258$$ − 9952.00i − 0.00930810i
$$259$$ −34944.0 −0.0323685
$$260$$ 0 0
$$261$$ 774070. 0.703362
$$262$$ 152904.i 0.137615i
$$263$$ − 2.12834e6i − 1.89736i −0.316231 0.948682i $$-0.602417\pi$$
0.316231 0.948682i $$-0.397583\pi$$
$$264$$ 71040.0 0.0627326
$$265$$ 0 0
$$266$$ 407040. 0.352722
$$267$$ − 405480.i − 0.348090i
$$268$$ 1.70722e6i 1.45195i
$$269$$ 1.44109e6 1.21426 0.607128 0.794604i $$-0.292321\pi$$
0.607128 + 0.794604i $$0.292321\pi$$
$$270$$ 0 0
$$271$$ −93248.0 −0.0771288 −0.0385644 0.999256i $$-0.512278\pi$$
−0.0385644 + 0.999256i $$0.512278\pi$$
$$272$$ − 1.10077e6i − 0.902139i
$$273$$ 219648.i 0.178370i
$$274$$ 289836. 0.233225
$$275$$ 0 0
$$276$$ 333312. 0.263377
$$277$$ − 110298.i − 0.0863711i −0.999067 0.0431855i $$-0.986249\pi$$
0.999067 0.0431855i $$-0.0137507\pi$$
$$278$$ − 224440.i − 0.174176i
$$279$$ −555696. −0.427392
$$280$$ 0 0
$$281$$ −192198. −0.145205 −0.0726027 0.997361i $$-0.523131\pi$$
−0.0726027 + 0.997361i $$0.523131\pi$$
$$282$$ 96704.0i 0.0724139i
$$283$$ 331884.i 0.246332i 0.992386 + 0.123166i $$0.0393047\pi$$
−0.992386 + 0.123166i $$0.960695\pi$$
$$284$$ −914144. −0.672541
$$285$$ 0 0
$$286$$ −84656.0 −0.0611988
$$287$$ − 1.80442e6i − 1.29310i
$$288$$ 1.16950e6i 0.830846i
$$289$$ −1.39583e6 −0.983076
$$290$$ 0 0
$$291$$ 476152. 0.329620
$$292$$ 1.08567e6i 0.745146i
$$293$$ − 2.19481e6i − 1.49358i −0.665063 0.746788i $$-0.731595\pi$$
0.665063 0.746788i $$-0.268405\pi$$
$$294$$ 160456. 0.108265
$$295$$ 0 0
$$296$$ −21840.0 −0.0144885
$$297$$ − 278240.i − 0.183033i
$$298$$ − 807500.i − 0.526747i
$$299$$ −851136. −0.550581
$$300$$ 0 0
$$301$$ −238848. −0.151952
$$302$$ − 893296.i − 0.563609i
$$303$$ − 359592.i − 0.225011i
$$304$$ −695360. −0.431545
$$305$$ 0 0
$$306$$ 761812. 0.465098
$$307$$ − 2.37751e6i − 1.43971i −0.694123 0.719857i $$-0.744207\pi$$
0.694123 0.719857i $$-0.255793\pi$$
$$308$$ − 795648.i − 0.477908i
$$309$$ −78016.0 −0.0464823
$$310$$ 0 0
$$311$$ −2.37305e6 −1.39125 −0.695626 0.718405i $$-0.744873\pi$$
−0.695626 + 0.718405i $$0.744873\pi$$
$$312$$ 137280.i 0.0798400i
$$313$$ 1.42941e6i 0.824702i 0.911025 + 0.412351i $$0.135292\pi$$
−0.911025 + 0.412351i $$0.864708\pi$$
$$314$$ 524516. 0.300217
$$315$$ 0 0
$$316$$ 934080. 0.526219
$$317$$ 2.12462e6i 1.18750i 0.804650 + 0.593750i $$0.202353\pi$$
−0.804650 + 0.593750i $$0.797647\pi$$
$$318$$ 190768.i 0.105788i
$$319$$ −504680. −0.277677
$$320$$ 0 0
$$321$$ −633168. −0.342970
$$322$$ 1.14278e6i 0.614221i
$$323$$ 1.77868e6i 0.948618i
$$324$$ 1.33395e6 0.705954
$$325$$ 0 0
$$326$$ −309128. −0.161100
$$327$$ − 147320.i − 0.0761890i
$$328$$ − 1.12776e6i − 0.578805i
$$329$$ 2.32090e6 1.18213
$$330$$ 0 0
$$331$$ 3.09985e6 1.55515 0.777573 0.628793i $$-0.216451\pi$$
0.777573 + 0.628793i $$0.216451\pi$$
$$332$$ − 468048.i − 0.233048i
$$333$$ 41314.0i 0.0204168i
$$334$$ −793344. −0.389131
$$335$$ 0 0
$$336$$ −503808. −0.243454
$$337$$ 2.40008e6i 1.15120i 0.817731 + 0.575601i $$0.195232\pi$$
−0.817731 + 0.575601i $$0.804768\pi$$
$$338$$ 578994.i 0.275665i
$$339$$ 44744.0 0.0211464
$$340$$ 0 0
$$341$$ 362304. 0.168728
$$342$$ − 481240.i − 0.222483i
$$343$$ − 624000.i − 0.286384i
$$344$$ −149280. −0.0680151
$$345$$ 0 0
$$346$$ −1.14695e6 −0.515055
$$347$$ 1.77741e6i 0.792436i 0.918156 + 0.396218i $$0.129678\pi$$
−0.918156 + 0.396218i $$0.870322\pi$$
$$348$$ 381920.i 0.169054i
$$349$$ 2.14805e6 0.944019 0.472010 0.881593i $$-0.343529\pi$$
0.472010 + 0.881593i $$0.343529\pi$$
$$350$$ 0 0
$$351$$ 537680. 0.232946
$$352$$ − 762496.i − 0.328005i
$$353$$ 661854.i 0.282700i 0.989960 + 0.141350i $$0.0451443\pi$$
−0.989960 + 0.141350i $$0.954856\pi$$
$$354$$ −160160. −0.0679275
$$355$$ 0 0
$$356$$ −2.83836e6 −1.18698
$$357$$ 1.28870e6i 0.535159i
$$358$$ 1.18892e6i 0.490281i
$$359$$ 259320. 0.106194 0.0530970 0.998589i $$-0.483091\pi$$
0.0530970 + 0.998589i $$0.483091\pi$$
$$360$$ 0 0
$$361$$ −1.35250e6 −0.546222
$$362$$ − 214196.i − 0.0859093i
$$363$$ − 556588.i − 0.221701i
$$364$$ 1.53754e6 0.608236
$$365$$ 0 0
$$366$$ −258416. −0.100836
$$367$$ − 1.49993e6i − 0.581307i −0.956828 0.290653i $$-0.906127\pi$$
0.956828 0.290653i $$-0.0938726\pi$$
$$368$$ − 1.95226e6i − 0.751480i
$$369$$ −2.13335e6 −0.815634
$$370$$ 0 0
$$371$$ 4.57843e6 1.72696
$$372$$ − 274176.i − 0.102724i
$$373$$ 2.23807e6i 0.832918i 0.909154 + 0.416459i $$0.136729\pi$$
−0.909154 + 0.416459i $$0.863271\pi$$
$$374$$ −496688. −0.183614
$$375$$ 0 0
$$376$$ 1.45056e6 0.529135
$$377$$ − 975260.i − 0.353400i
$$378$$ − 721920.i − 0.259872i
$$379$$ −3.15934e6 −1.12979 −0.564896 0.825162i $$-0.691084\pi$$
−0.564896 + 0.825162i $$0.691084\pi$$
$$380$$ 0 0
$$381$$ −282208. −0.0995994
$$382$$ 939104.i 0.329272i
$$383$$ − 342216.i − 0.119207i −0.998222 0.0596037i $$-0.981016\pi$$
0.998222 0.0596037i $$-0.0189837\pi$$
$$384$$ −744960. −0.257813
$$385$$ 0 0
$$386$$ 105412. 0.0360099
$$387$$ 282388.i 0.0958449i
$$388$$ − 3.33306e6i − 1.12399i
$$389$$ −88470.0 −0.0296430 −0.0148215 0.999890i $$-0.504718\pi$$
−0.0148215 + 0.999890i $$0.504718\pi$$
$$390$$ 0 0
$$391$$ −4.99373e6 −1.65190
$$392$$ − 2.40684e6i − 0.791101i
$$393$$ 305808.i 0.0998775i
$$394$$ −911724. −0.295885
$$395$$ 0 0
$$396$$ −940688. −0.301445
$$397$$ − 5.45674e6i − 1.73763i −0.495138 0.868814i $$-0.664883\pi$$
0.495138 0.868814i $$-0.335117\pi$$
$$398$$ − 1.73000e6i − 0.547442i
$$399$$ 814080. 0.255997
$$400$$ 0 0
$$401$$ 4.04680e6 1.25676 0.628378 0.777908i $$-0.283719\pi$$
0.628378 + 0.777908i $$0.283719\pi$$
$$402$$ − 487776.i − 0.150541i
$$403$$ 700128.i 0.214741i
$$404$$ −2.51714e6 −0.767281
$$405$$ 0 0
$$406$$ −1.30944e6 −0.394249
$$407$$ − 26936.0i − 0.00806022i
$$408$$ 805440.i 0.239543i
$$409$$ 2.71207e6 0.801664 0.400832 0.916151i $$-0.368721\pi$$
0.400832 + 0.916151i $$0.368721\pi$$
$$410$$ 0 0
$$411$$ 579672. 0.169269
$$412$$ 546112.i 0.158503i
$$413$$ 3.84384e6i 1.10889i
$$414$$ 1.35110e6 0.387425
$$415$$ 0 0
$$416$$ 1.47347e6 0.417454
$$417$$ − 448880.i − 0.126413i
$$418$$ 313760.i 0.0878328i
$$419$$ −3.71746e6 −1.03445 −0.517227 0.855848i $$-0.673036\pi$$
−0.517227 + 0.855848i $$0.673036\pi$$
$$420$$ 0 0
$$421$$ 3.55250e6 0.976853 0.488426 0.872605i $$-0.337571\pi$$
0.488426 + 0.872605i $$0.337571\pi$$
$$422$$ 2.21130e6i 0.604460i
$$423$$ − 2.74398e6i − 0.745640i
$$424$$ 2.86152e6 0.773005
$$425$$ 0 0
$$426$$ 261184. 0.0697305
$$427$$ 6.20198e6i 1.64612i
$$428$$ 4.43218e6i 1.16952i
$$429$$ −169312. −0.0444165
$$430$$ 0 0
$$431$$ −4.06205e6 −1.05330 −0.526650 0.850082i $$-0.676552\pi$$
−0.526650 + 0.850082i $$0.676552\pi$$
$$432$$ 1.23328e6i 0.317945i
$$433$$ − 7.26287e6i − 1.86161i −0.365518 0.930804i $$-0.619108\pi$$
0.365518 0.930804i $$-0.380892\pi$$
$$434$$ 940032. 0.239562
$$435$$ 0 0
$$436$$ −1.03124e6 −0.259803
$$437$$ 3.15456e6i 0.790197i
$$438$$ − 310192.i − 0.0772583i
$$439$$ 5.41028e6 1.33986 0.669928 0.742426i $$-0.266325\pi$$
0.669928 + 0.742426i $$0.266325\pi$$
$$440$$ 0 0
$$441$$ −4.55294e6 −1.11480
$$442$$ − 959816.i − 0.233686i
$$443$$ 6.51524e6i 1.57733i 0.614826 + 0.788663i $$0.289226\pi$$
−0.614826 + 0.788663i $$0.710774\pi$$
$$444$$ −20384.0 −0.00490718
$$445$$ 0 0
$$446$$ 2.24315e6 0.533976
$$447$$ − 1.61500e6i − 0.382299i
$$448$$ 2.05210e6i 0.483062i
$$449$$ 509950. 0.119375 0.0596873 0.998217i $$-0.480990\pi$$
0.0596873 + 0.998217i $$0.480990\pi$$
$$450$$ 0 0
$$451$$ 1.39090e6 0.322000
$$452$$ − 313208.i − 0.0721085i
$$453$$ − 1.78659e6i − 0.409053i
$$454$$ 46696.0 0.0106326
$$455$$ 0 0
$$456$$ 508800. 0.114587
$$457$$ 1.22084e6i 0.273444i 0.990609 + 0.136722i $$0.0436568\pi$$
−0.990609 + 0.136722i $$0.956343\pi$$
$$458$$ 1.19202e6i 0.265534i
$$459$$ 3.15464e6 0.698905
$$460$$ 0 0
$$461$$ −4.07210e6 −0.892413 −0.446207 0.894930i $$-0.647225\pi$$
−0.446207 + 0.894930i $$0.647225\pi$$
$$462$$ 227328.i 0.0495505i
$$463$$ − 2.02294e6i − 0.438561i −0.975662 0.219280i $$-0.929629\pi$$
0.975662 0.219280i $$-0.0703709\pi$$
$$464$$ 2.23696e6 0.482351
$$465$$ 0 0
$$466$$ −970668. −0.207065
$$467$$ 3.25097e6i 0.689797i 0.938640 + 0.344898i $$0.112087\pi$$
−0.938640 + 0.344898i $$0.887913\pi$$
$$468$$ − 1.81782e6i − 0.383650i
$$469$$ −1.17066e7 −2.45753
$$470$$ 0 0
$$471$$ 1.04903e6 0.217890
$$472$$ 2.40240e6i 0.496353i
$$473$$ − 184112.i − 0.0378381i
$$474$$ −266880. −0.0545595
$$475$$ 0 0
$$476$$ 9.02093e6 1.82488
$$477$$ − 5.41304e6i − 1.08929i
$$478$$ 97760.0i 0.0195700i
$$479$$ 3.27936e6 0.653056 0.326528 0.945188i $$-0.394121\pi$$
0.326528 + 0.945188i $$0.394121\pi$$
$$480$$ 0 0
$$481$$ 52052.0 0.0102583
$$482$$ − 221596.i − 0.0434455i
$$483$$ 2.28557e6i 0.445786i
$$484$$ −3.89612e6 −0.755994
$$485$$ 0 0
$$486$$ −1.29481e6 −0.248665
$$487$$ − 8.53197e6i − 1.63015i −0.579357 0.815074i $$-0.696696\pi$$
0.579357 0.815074i $$-0.303304\pi$$
$$488$$ 3.87624e6i 0.736819i
$$489$$ −618256. −0.116922
$$490$$ 0 0
$$491$$ 1.51265e6 0.283162 0.141581 0.989927i $$-0.454781\pi$$
0.141581 + 0.989927i $$0.454781\pi$$
$$492$$ − 1.05258e6i − 0.196038i
$$493$$ − 5.72198e6i − 1.06030i
$$494$$ −606320. −0.111785
$$495$$ 0 0
$$496$$ −1.60589e6 −0.293097
$$497$$ − 6.26842e6i − 1.13833i
$$498$$ 133728.i 0.0241629i
$$499$$ 6.49190e6 1.16713 0.583567 0.812065i $$-0.301657\pi$$
0.583567 + 0.812065i $$0.301657\pi$$
$$500$$ 0 0
$$501$$ −1.58669e6 −0.282421
$$502$$ − 3.28750e6i − 0.582245i
$$503$$ − 8.61770e6i − 1.51870i −0.650684 0.759349i $$-0.725518\pi$$
0.650684 0.759349i $$-0.274482\pi$$
$$504$$ −5.23008e6 −0.917132
$$505$$ 0 0
$$506$$ −880896. −0.152950
$$507$$ 1.15799e6i 0.200071i
$$508$$ 1.97546e6i 0.339632i
$$509$$ −2.67323e6 −0.457343 −0.228671 0.973504i $$-0.573438\pi$$
−0.228671 + 0.973504i $$0.573438\pi$$
$$510$$ 0 0
$$511$$ −7.44461e6 −1.26122
$$512$$ 5.89875e6i 0.994455i
$$513$$ − 1.99280e6i − 0.334326i
$$514$$ −2.61248e6 −0.436160
$$515$$ 0 0
$$516$$ −139328. −0.0230364
$$517$$ 1.78902e6i 0.294367i
$$518$$ − 69888.0i − 0.0114440i
$$519$$ −2.29390e6 −0.373814
$$520$$ 0 0
$$521$$ 6.18500e6 0.998264 0.499132 0.866526i $$-0.333652\pi$$
0.499132 + 0.866526i $$0.333652\pi$$
$$522$$ 1.54814e6i 0.248676i
$$523$$ 6.89452e6i 1.10217i 0.834448 + 0.551087i $$0.185787\pi$$
−0.834448 + 0.551087i $$0.814213\pi$$
$$524$$ 2.14066e6 0.340580
$$525$$ 0 0
$$526$$ 4.25667e6 0.670820
$$527$$ 4.10774e6i 0.644283i
$$528$$ − 388352.i − 0.0606235i
$$529$$ −2.42023e6 −0.376026
$$530$$ 0 0
$$531$$ 4.54454e6 0.699445
$$532$$ − 5.69856e6i − 0.872943i
$$533$$ 2.68783e6i 0.409811i
$$534$$ 810960. 0.123068
$$535$$ 0 0
$$536$$ −7.31664e6 −1.10002
$$537$$ 2.37784e6i 0.355834i
$$538$$ 2.88218e6i 0.429304i
$$539$$ 2.96844e6 0.440104
$$540$$ 0 0
$$541$$ 155502. 0.0228425 0.0114212 0.999935i $$-0.496364\pi$$
0.0114212 + 0.999935i $$0.496364\pi$$
$$542$$ − 186496.i − 0.0272691i
$$543$$ − 428392.i − 0.0623508i
$$544$$ 8.64506e6 1.25248
$$545$$ 0 0
$$546$$ −439296. −0.0630631
$$547$$ 1.26544e7i 1.80831i 0.427201 + 0.904157i $$0.359500\pi$$
−0.427201 + 0.904157i $$0.640500\pi$$
$$548$$ − 4.05770e6i − 0.577204i
$$549$$ 7.33255e6 1.03830
$$550$$ 0 0
$$551$$ −3.61460e6 −0.507202
$$552$$ 1.42848e6i 0.199538i
$$553$$ 6.40512e6i 0.890665i
$$554$$ 220596. 0.0305368
$$555$$ 0 0
$$556$$ −3.14216e6 −0.431064
$$557$$ − 7.07786e6i − 0.966638i −0.875444 0.483319i $$-0.839431\pi$$
0.875444 0.483319i $$-0.160569\pi$$
$$558$$ − 1.11139e6i − 0.151106i
$$559$$ 355784. 0.0481567
$$560$$ 0 0
$$561$$ −993376. −0.133262
$$562$$ − 384396.i − 0.0513379i
$$563$$ − 846636.i − 0.112571i −0.998415 0.0562854i $$-0.982074\pi$$
0.998415 0.0562854i $$-0.0179257\pi$$
$$564$$ 1.35386e6 0.179215
$$565$$ 0 0
$$566$$ −663768. −0.0870914
$$567$$ 9.14707e6i 1.19488i
$$568$$ − 3.91776e6i − 0.509527i
$$569$$ −4.96041e6 −0.642299 −0.321149 0.947029i $$-0.604069\pi$$
−0.321149 + 0.947029i $$0.604069\pi$$
$$570$$ 0 0
$$571$$ 8.96505e6 1.15070 0.575351 0.817907i $$-0.304866\pi$$
0.575351 + 0.817907i $$0.304866\pi$$
$$572$$ 1.18518e6i 0.151459i
$$573$$ 1.87821e6i 0.238978i
$$574$$ 3.60883e6 0.457180
$$575$$ 0 0
$$576$$ 2.42618e6 0.304696
$$577$$ − 2.86080e6i − 0.357724i −0.983874 0.178862i $$-0.942758\pi$$
0.983874 0.178862i $$-0.0572415\pi$$
$$578$$ − 2.79165e6i − 0.347570i
$$579$$ 210824. 0.0261351
$$580$$ 0 0
$$581$$ 3.20947e6 0.394451
$$582$$ 952304.i 0.116538i
$$583$$ 3.52921e6i 0.430037i
$$584$$ −4.65288e6 −0.564534
$$585$$ 0 0
$$586$$ 4.38961e6 0.528059
$$587$$ − 6.74027e6i − 0.807387i −0.914894 0.403694i $$-0.867726\pi$$
0.914894 0.403694i $$-0.132274\pi$$
$$588$$ − 2.24638e6i − 0.267942i
$$589$$ 2.59488e6 0.308197
$$590$$ 0 0
$$591$$ −1.82345e6 −0.214746
$$592$$ 119392.i 0.0140014i
$$593$$ 1.78609e6i 0.208578i 0.994547 + 0.104289i $$0.0332566\pi$$
−0.994547 + 0.104289i $$0.966743\pi$$
$$594$$ 556480. 0.0647118
$$595$$ 0 0
$$596$$ −1.13050e7 −1.30363
$$597$$ − 3.46000e6i − 0.397320i
$$598$$ − 1.70227e6i − 0.194660i
$$599$$ −4.94620e6 −0.563254 −0.281627 0.959524i $$-0.590874\pi$$
−0.281627 + 0.959524i $$0.590874\pi$$
$$600$$ 0 0
$$601$$ −4.58100e6 −0.517337 −0.258669 0.965966i $$-0.583284\pi$$
−0.258669 + 0.965966i $$0.583284\pi$$
$$602$$ − 477696.i − 0.0537230i
$$603$$ 1.38406e7i 1.55011i
$$604$$ −1.25061e7 −1.39486
$$605$$ 0 0
$$606$$ 719184. 0.0795533
$$607$$ 7.07999e6i 0.779940i 0.920828 + 0.389970i $$0.127515\pi$$
−0.920828 + 0.389970i $$0.872485\pi$$
$$608$$ − 5.46112e6i − 0.599132i
$$609$$ −2.61888e6 −0.286136
$$610$$ 0 0
$$611$$ −3.45717e6 −0.374643
$$612$$ − 1.06654e7i − 1.15106i
$$613$$ − 5.09609e6i − 0.547754i −0.961765 0.273877i $$-0.911694\pi$$
0.961765 0.273877i $$-0.0883061\pi$$
$$614$$ 4.75502e6 0.509016
$$615$$ 0 0
$$616$$ 3.40992e6 0.362070
$$617$$ − 1.30003e7i − 1.37480i −0.726279 0.687400i $$-0.758752\pi$$
0.726279 0.687400i $$-0.241248\pi$$
$$618$$ − 156032.i − 0.0164340i
$$619$$ −4.84406e6 −0.508139 −0.254070 0.967186i $$-0.581769\pi$$
−0.254070 + 0.967186i $$0.581769\pi$$
$$620$$ 0 0
$$621$$ 5.59488e6 0.582186
$$622$$ − 4.74610e6i − 0.491882i
$$623$$ − 1.94630e7i − 2.00905i
$$624$$ 750464. 0.0771558
$$625$$ 0 0
$$626$$ −2.85883e6 −0.291576
$$627$$ 627520.i 0.0637468i
$$628$$ − 7.34322e6i − 0.742998i
$$629$$ 305396. 0.0307777
$$630$$ 0 0
$$631$$ 6.22775e6 0.622670 0.311335 0.950300i $$-0.399224\pi$$
0.311335 + 0.950300i $$0.399224\pi$$
$$632$$ 4.00320e6i 0.398671i
$$633$$ 4.42261e6i 0.438702i
$$634$$ −4.24924e6 −0.419845
$$635$$ 0 0
$$636$$ 2.67075e6 0.261813
$$637$$ 5.73630e6i 0.560123i
$$638$$ − 1.00936e6i − 0.0981735i
$$639$$ −7.41110e6 −0.718010
$$640$$ 0 0
$$641$$ 1.53280e6 0.147347 0.0736734 0.997282i $$-0.476528\pi$$
0.0736734 + 0.997282i $$0.476528\pi$$
$$642$$ − 1.26634e6i − 0.121258i
$$643$$ 1.74382e7i 1.66332i 0.555287 + 0.831659i $$0.312609\pi$$
−0.555287 + 0.831659i $$0.687391\pi$$
$$644$$ 1.59990e7 1.52012
$$645$$ 0 0
$$646$$ −3.55736e6 −0.335387
$$647$$ − 4.25469e6i − 0.399583i −0.979838 0.199792i $$-0.935974\pi$$
0.979838 0.199792i $$-0.0640265\pi$$
$$648$$ 5.71692e6i 0.534841i
$$649$$ −2.96296e6 −0.276130
$$650$$ 0 0
$$651$$ 1.88006e6 0.173868
$$652$$ 4.32779e6i 0.398701i
$$653$$ − 3.01085e6i − 0.276316i −0.990410 0.138158i $$-0.955882\pi$$
0.990410 0.138158i $$-0.0441181\pi$$
$$654$$ 294640. 0.0269369
$$655$$ 0 0
$$656$$ −6.16509e6 −0.559345
$$657$$ 8.80170e6i 0.795524i
$$658$$ 4.64179e6i 0.417947i
$$659$$ 8.11462e6 0.727871 0.363936 0.931424i $$-0.381433\pi$$
0.363936 + 0.931424i $$0.381433\pi$$
$$660$$ 0 0
$$661$$ 2.47370e6 0.220213 0.110107 0.993920i $$-0.464881\pi$$
0.110107 + 0.993920i $$0.464881\pi$$
$$662$$ 6.19970e6i 0.549827i
$$663$$ − 1.91963e6i − 0.169603i
$$664$$ 2.00592e6 0.176560
$$665$$ 0 0
$$666$$ −82628.0 −0.00721841
$$667$$ − 1.01482e7i − 0.883228i
$$668$$ 1.11068e7i 0.963049i
$$669$$ 4.48630e6 0.387546
$$670$$ 0 0
$$671$$ −4.78070e6 −0.409907
$$672$$ − 3.95674e6i − 0.337998i
$$673$$ − 5.77063e6i − 0.491117i −0.969382 0.245559i $$-0.921029\pi$$
0.969382 0.245559i $$-0.0789714\pi$$
$$674$$ −4.80016e6 −0.407011
$$675$$ 0 0
$$676$$ 8.10592e6 0.682237
$$677$$ 1.67197e7i 1.40203i 0.713147 + 0.701014i $$0.247269\pi$$
−0.713147 + 0.701014i $$0.752731\pi$$
$$678$$ 89488.0i 0.00747637i
$$679$$ 2.28553e7 1.90245
$$680$$ 0 0
$$681$$ 93392.0 0.00771688
$$682$$ 724608.i 0.0596544i
$$683$$ − 7.14532e6i − 0.586097i −0.956098 0.293049i $$-0.905330\pi$$
0.956098 0.293049i $$-0.0946698\pi$$
$$684$$ −6.73736e6 −0.550617
$$685$$ 0 0
$$686$$ 1.24800e6 0.101252
$$687$$ 2.38404e6i 0.192718i
$$688$$ 816064.i 0.0657284i
$$689$$ −6.81996e6 −0.547310
$$690$$ 0 0
$$691$$ −8.78395e6 −0.699833 −0.349917 0.936781i $$-0.613790\pi$$
−0.349917 + 0.936781i $$0.613790\pi$$
$$692$$ 1.60573e7i 1.27470i
$$693$$ − 6.45043e6i − 0.510218i
$$694$$ −3.55482e6 −0.280169
$$695$$ 0 0
$$696$$ −1.63680e6 −0.128077
$$697$$ 1.57698e7i 1.22955i
$$698$$ 4.29610e6i 0.333761i
$$699$$ −1.94134e6 −0.150282
$$700$$ 0 0
$$701$$ −1.60141e7 −1.23086 −0.615428 0.788193i $$-0.711017\pi$$
−0.615428 + 0.788193i $$0.711017\pi$$
$$702$$ 1.07536e6i 0.0823590i
$$703$$ − 192920.i − 0.0147228i
$$704$$ −1.58182e6 −0.120289
$$705$$ 0 0
$$706$$ −1.32371e6 −0.0999495
$$707$$ − 1.72604e7i − 1.29868i
$$708$$ 2.24224e6i 0.168112i
$$709$$ 1.91354e7 1.42962 0.714811 0.699318i $$-0.246513\pi$$
0.714811 + 0.699318i $$0.246513\pi$$
$$710$$ 0 0
$$711$$ 7.57272e6 0.561795
$$712$$ − 1.21644e7i − 0.899271i
$$713$$ 7.28525e6i 0.536686i
$$714$$ −2.57741e6 −0.189207
$$715$$ 0 0
$$716$$ 1.66449e7 1.21338
$$717$$ 195520.i 0.0142034i
$$718$$ 518640.i 0.0375452i
$$719$$ −1.02934e7 −0.742566 −0.371283 0.928520i $$-0.621082\pi$$
−0.371283 + 0.928520i $$0.621082\pi$$
$$720$$ 0 0
$$721$$ −3.74477e6 −0.268279
$$722$$ − 2.70500e6i − 0.193119i
$$723$$ − 443192.i − 0.0315316i
$$724$$ −2.99874e6 −0.212615
$$725$$ 0 0
$$726$$ 1.11318e6 0.0783831
$$727$$ − 1.93264e7i − 1.35618i −0.734981 0.678088i $$-0.762809\pi$$
0.734981 0.678088i $$-0.237191\pi$$
$$728$$ 6.58944e6i 0.460808i
$$729$$ 8.98715e6 0.626330
$$730$$ 0 0
$$731$$ 2.08743e6 0.144484
$$732$$ 3.61782e6i 0.249557i
$$733$$ − 5.26197e6i − 0.361733i −0.983508 0.180866i $$-0.942110\pi$$
0.983508 0.180866i $$-0.0578902\pi$$
$$734$$ 2.99986e6 0.205523
$$735$$ 0 0
$$736$$ 1.53324e7 1.04331
$$737$$ − 9.02386e6i − 0.611961i
$$738$$ − 4.26669e6i − 0.288370i
$$739$$ −2.82944e7 −1.90585 −0.952927 0.303199i $$-0.901945\pi$$
−0.952927 + 0.303199i $$0.901945\pi$$
$$740$$ 0 0
$$741$$ −1.21264e6 −0.0811309
$$742$$ 9.15686e6i 0.610572i
$$743$$ − 2.09863e7i − 1.39464i −0.716759 0.697321i $$-0.754375\pi$$
0.716759 0.697321i $$-0.245625\pi$$
$$744$$ 1.17504e6 0.0778252
$$745$$ 0 0
$$746$$ −4.47615e6 −0.294481
$$747$$ − 3.79453e6i − 0.248804i
$$748$$ 6.95363e6i 0.454420i
$$749$$ −3.03921e7 −1.97950
$$750$$ 0 0
$$751$$ −1.89668e7 −1.22714 −0.613572 0.789639i $$-0.710268\pi$$
−0.613572 + 0.789639i $$0.710268\pi$$
$$752$$ − 7.92973e6i − 0.511345i
$$753$$ − 6.57499e6i − 0.422579i
$$754$$ 1.95052e6 0.124946
$$755$$ 0 0
$$756$$ −1.01069e7 −0.643151
$$757$$ − 1.08257e7i − 0.686617i −0.939223 0.343309i $$-0.888452\pi$$
0.939223 0.343309i $$-0.111548\pi$$
$$758$$ − 6.31868e6i − 0.399442i
$$759$$ −1.76179e6 −0.111007
$$760$$ 0 0
$$761$$ 1.90534e7 1.19264 0.596322 0.802745i $$-0.296628\pi$$
0.596322 + 0.802745i $$0.296628\pi$$
$$762$$ − 564416.i − 0.0352137i
$$763$$ − 7.07136e6i − 0.439736i
$$764$$ 1.31475e7 0.814908
$$765$$ 0 0
$$766$$ 684432. 0.0421462
$$767$$ − 5.72572e6i − 0.351432i
$$768$$ − 121856.i − 0.00745494i
$$769$$ 1.57826e7 0.962415 0.481208 0.876607i $$-0.340198\pi$$
0.481208 + 0.876607i $$0.340198\pi$$
$$770$$ 0 0
$$771$$ −5.22497e6 −0.316554
$$772$$ − 1.47577e6i − 0.0891199i
$$773$$ 2.44049e7i 1.46902i 0.678598 + 0.734510i $$0.262588\pi$$
−0.678598 + 0.734510i $$0.737412\pi$$
$$774$$ −564776. −0.0338863
$$775$$ 0 0
$$776$$ 1.42846e7 0.851555
$$777$$ − 139776.i − 0.00830577i
$$778$$ − 176940.i − 0.0104804i
$$779$$ 9.96188e6 0.588163
$$780$$ 0 0
$$781$$ 4.83190e6 0.283459
$$782$$ − 9.98746e6i − 0.584034i
$$783$$ 6.41080e6i 0.373687i
$$784$$ −1.31574e7 −0.764504
$$785$$ 0 0
$$786$$ −611616. −0.0353120
$$787$$ 3.37607e7i 1.94301i 0.237019 + 0.971505i $$0.423830\pi$$
−0.237019 + 0.971505i $$0.576170\pi$$
$$788$$ 1.27641e7i 0.732278i
$$789$$ 8.51334e6 0.486864
$$790$$ 0 0
$$791$$ 2.14771e6 0.122049
$$792$$ − 4.03152e6i − 0.228379i
$$793$$ − 9.23837e6i − 0.521690i
$$794$$ 1.09135e7 0.614344
$$795$$ 0 0
$$796$$ −2.42200e7 −1.35485
$$797$$ 2.19885e7i 1.22617i 0.790019 + 0.613083i $$0.210071\pi$$
−0.790019 + 0.613083i $$0.789929\pi$$
$$798$$ 1.62816e6i 0.0905086i
$$799$$ −2.02837e7 −1.12403
$$800$$ 0 0
$$801$$ −2.30110e7 −1.26723
$$802$$ 8.09360e6i 0.444330i
$$803$$ − 5.73855e6i − 0.314061i
$$804$$ −6.82886e6 −0.372570
$$805$$ 0 0
$$806$$ −1.40026e6 −0.0759224
$$807$$ 5.76436e6i 0.311578i
$$808$$ − 1.07878e7i − 0.581303i
$$809$$ 2.93597e7 1.57717 0.788587 0.614923i $$-0.210813\pi$$
0.788587 + 0.614923i $$0.210813\pi$$
$$810$$ 0 0
$$811$$ 3.17703e7 1.69617 0.848083 0.529863i $$-0.177757\pi$$
0.848083 + 0.529863i $$0.177757\pi$$
$$812$$ 1.83322e7i 0.975716i
$$813$$ − 372992.i − 0.0197912i
$$814$$ 53872.0 0.00284972
$$815$$ 0 0
$$816$$ 4.40307e6 0.231489
$$817$$ − 1.31864e6i − 0.0691148i
$$818$$ 5.42414e6i 0.283431i
$$819$$ 1.24650e7 0.649357
$$820$$ 0 0
$$821$$ −2.71430e6 −0.140540 −0.0702699 0.997528i $$-0.522386\pi$$
−0.0702699 + 0.997528i $$0.522386\pi$$
$$822$$ 1.15934e6i 0.0598457i
$$823$$ 1.25866e7i 0.647753i 0.946099 + 0.323877i $$0.104986\pi$$
−0.946099 + 0.323877i $$0.895014\pi$$
$$824$$ −2.34048e6 −0.120084
$$825$$ 0 0
$$826$$ −7.68768e6 −0.392053
$$827$$ − 8.72355e6i − 0.443537i −0.975099 0.221768i $$-0.928817\pi$$
0.975099 0.221768i $$-0.0711828\pi$$
$$828$$ − 1.89155e7i − 0.958829i
$$829$$ 1.06178e7 0.536597 0.268299 0.963336i $$-0.413539\pi$$
0.268299 + 0.963336i $$0.413539\pi$$
$$830$$ 0 0
$$831$$ 441192. 0.0221628
$$832$$ − 3.05677e6i − 0.153093i
$$833$$ 3.36556e7i 1.68053i
$$834$$ 897760. 0.0446936
$$835$$ 0 0
$$836$$ 4.39264e6 0.217375
$$837$$ − 4.60224e6i − 0.227068i
$$838$$ − 7.43492e6i − 0.365735i
$$839$$ −1.67765e7 −0.822805 −0.411403 0.911454i $$-0.634961\pi$$
−0.411403 + 0.911454i $$0.634961\pi$$
$$840$$ 0 0
$$841$$ −8.88305e6 −0.433084
$$842$$ 7.10500e6i 0.345370i
$$843$$ − 768792.i − 0.0372597i
$$844$$ 3.09583e7 1.49596
$$845$$ 0 0
$$846$$ 5.48795e6 0.263624
$$847$$ − 2.67162e7i − 1.27958i
$$848$$ − 1.56430e7i − 0.747016i
$$849$$ −1.32754e6 −0.0632087
$$850$$ 0 0
$$851$$ 541632. 0.0256378
$$852$$ − 3.65658e6i − 0.172574i
$$853$$ 2.20186e7i 1.03613i 0.855340 + 0.518067i $$0.173348\pi$$
−0.855340 + 0.518067i $$0.826652\pi$$
$$854$$ −1.24040e7 −0.581991
$$855$$ 0 0
$$856$$ −1.89950e7 −0.886045
$$857$$ 3.16676e7i 1.47287i 0.676510 + 0.736434i $$0.263492\pi$$
−0.676510 + 0.736434i $$0.736508\pi$$
$$858$$ − 338624.i − 0.0157036i
$$859$$ −1.58064e7 −0.730886 −0.365443 0.930834i $$-0.619082\pi$$
−0.365443 + 0.930834i $$0.619082\pi$$
$$860$$ 0 0
$$861$$ 7.21766e6 0.331809
$$862$$ − 8.12410e6i − 0.372398i
$$863$$ 1.44287e7i 0.659476i 0.944072 + 0.329738i $$0.106960\pi$$
−0.944072 + 0.329738i $$0.893040\pi$$
$$864$$ −9.68576e6 −0.441417
$$865$$ 0 0
$$866$$ 1.45257e7 0.658178
$$867$$ − 5.58331e6i − 0.252257i
$$868$$ − 1.31604e7i − 0.592886i
$$869$$ −4.93728e6 −0.221788
$$870$$ 0 0
$$871$$ 1.74380e7 0.778845
$$872$$ − 4.41960e6i − 0.196830i
$$873$$ − 2.70216e7i − 1.19999i
$$874$$ −6.30912e6 −0.279377
$$875$$ 0 0
$$876$$ −4.34269e6 −0.191205
$$877$$ 247902.i 0.0108838i 0.999985 + 0.00544191i $$0.00173222\pi$$
−0.999985 + 0.00544191i $$0.998268\pi$$
$$878$$ 1.08206e7i 0.473711i
$$879$$ 8.77922e6 0.383252
$$880$$ 0 0
$$881$$ 4.10268e7 1.78085 0.890426 0.455128i $$-0.150406\pi$$
0.890426 + 0.455128i $$0.150406\pi$$
$$882$$ − 9.10588e6i − 0.394140i
$$883$$ − 4.18015e7i − 1.80422i −0.431503 0.902112i $$-0.642016\pi$$
0.431503 0.902112i $$-0.357984\pi$$
$$884$$ −1.34374e7 −0.578343
$$885$$ 0 0
$$886$$ −1.30305e7 −0.557669
$$887$$ − 2.10476e7i − 0.898241i −0.893471 0.449120i $$-0.851737\pi$$
0.893471 0.449120i $$-0.148263\pi$$
$$888$$ − 87360.0i − 0.00371775i
$$889$$ −1.35460e7 −0.574852
$$890$$ 0 0
$$891$$ −7.05087e6 −0.297542
$$892$$ − 3.14041e7i − 1.32152i
$$893$$ 1.28133e7i 0.537690i
$$894$$ 3.23000e6 0.135163
$$895$$ 0 0
$$896$$ −3.57581e7 −1.48800
$$897$$ − 3.40454e6i − 0.141279i
$$898$$ 1.01990e6i 0.0422053i
$$899$$ −8.34768e6 −0.344482
$$900$$ 0 0
$$901$$ −4.00136e7 −1.64208
$$902$$ 2.78181e6i 0.113844i
$$903$$ − 955392.i − 0.0389908i
$$904$$ 1.34232e6 0.0546305
$$905$$ 0 0
$$906$$ 3.57318e6 0.144622
$$907$$ 7.48309e6i 0.302039i 0.988531 + 0.151019i $$0.0482556\pi$$
−0.988531 + 0.151019i $$0.951744\pi$$
$$908$$ − 653744.i − 0.0263144i
$$909$$ −2.04068e7 −0.819155
$$910$$ 0 0
$$911$$ −6.63165e6 −0.264744 −0.132372 0.991200i $$-0.542259\pi$$
−0.132372 + 0.991200i $$0.542259\pi$$
$$912$$ − 2.78144e6i − 0.110734i
$$913$$ 2.47397e6i 0.0982239i
$$914$$ −2.44168e6 −0.0966772
$$915$$ 0 0
$$916$$ 1.66883e7 0.657163
$$917$$ 1.46788e7i 0.576457i
$$918$$ 6.30928e6i 0.247100i
$$919$$ 1.68976e7 0.659990 0.329995 0.943983i $$-0.392953\pi$$
0.329995 + 0.943983i $$0.392953\pi$$
$$920$$ 0 0
$$921$$ 9.51003e6 0.369431
$$922$$ − 8.14420e6i − 0.315516i
$$923$$ 9.33733e6i 0.360760i
$$924$$ 3.18259e6 0.122631
$$925$$ 0 0
$$926$$ 4.04587e6 0.155055
$$927$$ 4.42741e6i 0.169219i
$$928$$ 1.75683e7i 0.669669i
$$929$$ 1.28653e7 0.489081 0.244541 0.969639i $$-0.421363\pi$$
0.244541 + 0.969639i $$0.421363\pi$$
$$930$$ 0 0
$$931$$ 2.12604e7 0.803892
$$932$$ 1.35894e7i 0.512459i
$$933$$ − 9.49219e6i − 0.356995i
$$934$$ −6.50194e6 −0.243880
$$935$$ 0 0
$$936$$ 7.79064e6 0.290659
$$937$$ 1.06887e7i 0.397718i 0.980028 + 0.198859i $$0.0637236\pi$$
−0.980028 + 0.198859i $$0.936276\pi$$
$$938$$ − 2.34132e7i − 0.868870i
$$939$$ −5.71766e6 −0.211619
$$940$$ 0 0
$$941$$ 2.82455e7 1.03986 0.519930 0.854209i $$-0.325958\pi$$
0.519930 + 0.854209i $$0.325958\pi$$
$$942$$ 2.09806e6i 0.0770356i
$$943$$ 2.79684e7i 1.02421i
$$944$$ 1.31331e7 0.479665
$$945$$ 0 0
$$946$$ 368224. 0.0133778
$$947$$ − 1.70892e7i − 0.619222i −0.950863 0.309611i $$-0.899801\pi$$
0.950863 0.309611i $$-0.100199\pi$$
$$948$$ 3.73632e6i 0.135028i
$$949$$ 1.10894e7 0.399706
$$950$$ 0 0
$$951$$ −8.49849e6 −0.304713
$$952$$ 3.86611e7i 1.38255i
$$953$$ − 2.22259e7i − 0.792735i −0.918092 0.396367i $$-0.870271\pi$$
0.918092 0.396367i $$-0.129729\pi$$
$$954$$ 1.08261e7 0.385124
$$955$$ 0 0
$$956$$ 1.36864e6 0.0484333
$$957$$ − 2.01872e6i − 0.0712519i
$$958$$ 6.55872e6i 0.230890i
$$959$$ 2.78243e7 0.976961
$$960$$ 0 0
$$961$$ −2.26364e7 −0.790678
$$962$$ 104104.i 0.00362685i
$$963$$ 3.59323e7i 1.24859i
$$964$$ −3.10234e6 −0.107522
$$965$$ 0 0
$$966$$ −4.57114e6 −0.157609
$$967$$ 2.41551e7i 0.830696i 0.909663 + 0.415348i $$0.136340\pi$$
−0.909663 + 0.415348i $$0.863660\pi$$
$$968$$ − 1.66976e7i − 0.572752i
$$969$$ −7.11472e6 −0.243416
$$970$$ 0 0
$$971$$ −5.48313e7 −1.86630 −0.933149 0.359491i $$-0.882950\pi$$
−0.933149 + 0.359491i $$0.882950\pi$$
$$972$$ 1.81273e7i 0.615415i
$$973$$ − 2.15462e7i − 0.729608i
$$974$$ 1.70639e7 0.576344
$$975$$ 0 0
$$976$$ 2.11901e7 0.712047
$$977$$ − 1.56612e7i − 0.524915i −0.964944 0.262457i $$-0.915467\pi$$
0.964944 0.262457i $$-0.0845329\pi$$
$$978$$ − 1.23651e6i − 0.0413382i
$$979$$ 1.50028e7 0.500281
$$980$$ 0 0
$$981$$ −8.36041e6 −0.277367
$$982$$ 3.02530e6i 0.100113i
$$983$$ 1.63420e7i 0.539412i 0.962943 + 0.269706i $$0.0869266\pi$$
−0.962943 + 0.269706i $$0.913073\pi$$
$$984$$ 4.51104e6 0.148521
$$985$$ 0 0
$$986$$ 1.14440e7 0.374873
$$987$$ 9.28358e6i 0.303335i
$$988$$ 8.48848e6i 0.276654i
$$989$$ 3.70214e6 0.120355
$$990$$ 0 0
$$991$$ 1.37576e7 0.444997 0.222498 0.974933i $$-0.428579\pi$$
0.222498 + 0.974933i $$0.428579\pi$$
$$992$$ − 1.26121e7i − 0.406919i
$$993$$ 1.23994e7i 0.399050i
$$994$$ 1.25368e7 0.402459
$$995$$ 0 0
$$996$$ 1.87219e6 0.0598001
$$997$$ − 1.29097e7i − 0.411320i −0.978624 0.205660i $$-0.934066\pi$$
0.978624 0.205660i $$-0.0659341\pi$$
$$998$$ 1.29838e7i 0.412644i
$$999$$ −342160. −0.0108471
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 25.6.b.a.24.2 2
3.2 odd 2 225.6.b.e.199.1 2
4.3 odd 2 400.6.c.j.49.1 2
5.2 odd 4 25.6.a.a.1.1 1
5.3 odd 4 5.6.a.a.1.1 1
5.4 even 2 inner 25.6.b.a.24.1 2
15.2 even 4 225.6.a.f.1.1 1
15.8 even 4 45.6.a.b.1.1 1
15.14 odd 2 225.6.b.e.199.2 2
20.3 even 4 80.6.a.e.1.1 1
20.7 even 4 400.6.a.g.1.1 1
20.19 odd 2 400.6.c.j.49.2 2
35.13 even 4 245.6.a.b.1.1 1
40.3 even 4 320.6.a.g.1.1 1
40.13 odd 4 320.6.a.j.1.1 1
55.43 even 4 605.6.a.a.1.1 1
60.23 odd 4 720.6.a.a.1.1 1
65.38 odd 4 845.6.a.b.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
5.6.a.a.1.1 1 5.3 odd 4
25.6.a.a.1.1 1 5.2 odd 4
25.6.b.a.24.1 2 5.4 even 2 inner
25.6.b.a.24.2 2 1.1 even 1 trivial
45.6.a.b.1.1 1 15.8 even 4
80.6.a.e.1.1 1 20.3 even 4
225.6.a.f.1.1 1 15.2 even 4
225.6.b.e.199.1 2 3.2 odd 2
225.6.b.e.199.2 2 15.14 odd 2
245.6.a.b.1.1 1 35.13 even 4
320.6.a.g.1.1 1 40.3 even 4
320.6.a.j.1.1 1 40.13 odd 4
400.6.a.g.1.1 1 20.7 even 4
400.6.c.j.49.1 2 4.3 odd 2
400.6.c.j.49.2 2 20.19 odd 2
605.6.a.a.1.1 1 55.43 even 4
720.6.a.a.1.1 1 60.23 odd 4
845.6.a.b.1.1 1 65.38 odd 4