# Properties

 Label 25.6.b.b.24.4 Level $25$ Weight $6$ Character 25.24 Analytic conductor $4.010$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: $$\Q(i, \sqrt{241})$$ Defining polynomial: $$x^{4} + 121 x^{2} + 3600$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$5^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 24.4 Root $$8.26209i$$ of defining polynomial Character $$\chi$$ $$=$$ 25.24 Dual form 25.6.b.b.24.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+10.2621i q^{2} +5.52417i q^{3} -73.3104 q^{4} -56.6896 q^{6} +68.9517i q^{7} -423.931i q^{8} +212.483 q^{9} +O(q^{10})$$ $$q+10.2621i q^{2} +5.52417i q^{3} -73.3104 q^{4} -56.6896 q^{6} +68.9517i q^{7} -423.931i q^{8} +212.483 q^{9} -486.104 q^{11} -404.980i q^{12} +428.387i q^{13} -707.588 q^{14} +2004.49 q^{16} +1800.64i q^{17} +2180.52i q^{18} +1046.65 q^{19} -380.901 q^{21} -4988.45i q^{22} -686.855i q^{23} +2341.87 q^{24} -4396.14 q^{26} +2516.17i q^{27} -5054.88i q^{28} +1339.03 q^{29} +7990.30 q^{31} +7004.41i q^{32} -2685.33i q^{33} -18478.4 q^{34} -15577.3 q^{36} -1970.64i q^{37} +10740.9i q^{38} -2366.48 q^{39} +10772.2 q^{41} -3908.84i q^{42} -15017.7i q^{43} +35636.5 q^{44} +7048.57 q^{46} +895.337i q^{47} +11073.1i q^{48} +12052.7 q^{49} -9947.07 q^{51} -31405.2i q^{52} +19327.1i q^{53} -25821.2 q^{54} +29230.8 q^{56} +5781.90i q^{57} +13741.3i q^{58} -21193.7 q^{59} -27722.2 q^{61} +81997.1i q^{62} +14651.1i q^{63} -7736.31 q^{64} +27557.0 q^{66} -7719.33i q^{67} -132006. i q^{68} +3794.31 q^{69} -51410.1 q^{71} -90078.4i q^{72} +43776.4i q^{73} +20222.9 q^{74} -76730.7 q^{76} -33517.7i q^{77} -24285.1i q^{78} +6225.68 q^{79} +37733.7 q^{81} +110545. i q^{82} -52949.9i q^{83} +27924.0 q^{84} +154113. q^{86} +7397.05i q^{87} +206075. i q^{88} -44631.2 q^{89} -29538.0 q^{91} +50353.6i q^{92} +44139.8i q^{93} -9188.03 q^{94} -38693.6 q^{96} -148018. i q^{97} +123686. i q^{98} -103289. q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 138q^{4} - 382q^{6} - 392q^{9} + O(q^{10})$$ $$4q - 138q^{4} - 382q^{6} - 392q^{9} - 392q^{11} - 36q^{14} + 2274q^{16} + 6360q^{19} + 5928q^{21} - 5070q^{24} - 9512q^{26} + 7840q^{29} - 2192q^{31} - 40226q^{34} - 34676q^{36} - 8224q^{39} + 55508q^{41} + 73774q^{44} + 4908q^{46} + 23372q^{49} - 35752q^{51} - 7190q^{54} + 108540q^{56} - 23920q^{59} - 48792q^{61} - 87298q^{64} - 22814q^{66} - 36984q^{69} - 174592q^{71} + 82444q^{74} - 135070q^{76} - 130960q^{79} + 92564q^{81} + 84684q^{84} + 497848q^{86} + 145620q^{89} - 41152q^{91} + 243304q^{94} - 156482q^{96} - 443584q^{99} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 10.2621i 1.81410i 0.421025 + 0.907049i $$0.361670\pi$$
−0.421025 + 0.907049i $$0.638330\pi$$
$$3$$ 5.52417i 0.354376i 0.984177 + 0.177188i $$0.0567001\pi$$
−0.984177 + 0.177188i $$0.943300\pi$$
$$4$$ −73.3104 −2.29095
$$5$$ 0 0
$$6$$ −56.6896 −0.642873
$$7$$ 68.9517i 0.531863i 0.963992 + 0.265931i $$0.0856795\pi$$
−0.963992 + 0.265931i $$0.914321\pi$$
$$8$$ − 423.931i − 2.34191i
$$9$$ 212.483 0.874418
$$10$$ 0 0
$$11$$ −486.104 −1.21129 −0.605645 0.795735i $$-0.707085\pi$$
−0.605645 + 0.795735i $$0.707085\pi$$
$$12$$ − 404.980i − 0.811858i
$$13$$ 428.387i 0.703036i 0.936181 + 0.351518i $$0.114334\pi$$
−0.936181 + 0.351518i $$0.885666\pi$$
$$14$$ −707.588 −0.964851
$$15$$ 0 0
$$16$$ 2004.49 1.95751
$$17$$ 1800.64i 1.51114i 0.655066 + 0.755571i $$0.272641\pi$$
−0.655066 + 0.755571i $$0.727359\pi$$
$$18$$ 2180.52i 1.58628i
$$19$$ 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
$$22$$ − 4988.45i − 2.19740i
$$23$$ − 686.855i − 0.270736i −0.990795 0.135368i $$-0.956778\pi$$
0.990795 0.135368i $$-0.0432216\pi$$
$$24$$ 2341.87 0.829917
$$25$$ 0 0
$$26$$ −4396.14 −1.27538
$$27$$ 2516.17i 0.664249i
$$28$$ − 5054.88i − 1.21847i
$$29$$ 1339.03 0.295663 0.147831 0.989013i $$-0.452771\pi$$
0.147831 + 0.989013i $$0.452771\pi$$
$$30$$ 0 0
$$31$$ 7990.30 1.49334 0.746670 0.665195i $$-0.231651\pi$$
0.746670 + 0.665195i $$0.231651\pi$$
$$32$$ 7004.41i 1.20920i
$$33$$ − 2685.33i − 0.429252i
$$34$$ −18478.4 −2.74136
$$35$$ 0 0
$$36$$ −15577.3 −2.00325
$$37$$ − 1970.64i − 0.236648i −0.992975 0.118324i $$-0.962248\pi$$
0.992975 0.118324i $$-0.0377522\pi$$
$$38$$ 10740.9i 1.20665i
$$39$$ −2366.48 −0.249139
$$40$$ 0 0
$$41$$ 10772.2 1.00079 0.500395 0.865797i $$-0.333188\pi$$
0.500395 + 0.865797i $$0.333188\pi$$
$$42$$ − 3908.84i − 0.341920i
$$43$$ − 15017.7i − 1.23861i −0.785152 0.619303i $$-0.787415\pi$$
0.785152 0.619303i $$-0.212585\pi$$
$$44$$ 35636.5 2.77500
$$45$$ 0 0
$$46$$ 7048.57 0.491141
$$47$$ 895.337i 0.0591210i 0.999563 + 0.0295605i $$0.00941077\pi$$
−0.999563 + 0.0295605i $$0.990589\pi$$
$$48$$ 11073.1i 0.693693i
$$49$$ 12052.7 0.717122
$$50$$ 0 0
$$51$$ −9947.07 −0.535513
$$52$$ − 31405.2i − 1.61062i
$$53$$ 19327.1i 0.945098i 0.881304 + 0.472549i $$0.156666\pi$$
−0.881304 + 0.472549i $$0.843334\pi$$
$$54$$ −25821.2 −1.20501
$$55$$ 0 0
$$56$$ 29230.8 1.24558
$$57$$ 5781.90i 0.235713i
$$58$$ 13741.3i 0.536361i
$$59$$ −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 $$0.658258\pi$$
$$62$$ 81997.1i 2.70906i
$$63$$ 14651.1i 0.465070i
$$64$$ −7736.31 −0.236093
$$65$$ 0 0
$$66$$ 27557.0 0.778705
$$67$$ − 7719.33i − 0.210084i −0.994468 0.105042i $$-0.966502\pi$$
0.994468 0.105042i $$-0.0334977\pi$$
$$68$$ − 132006.i − 3.46195i
$$69$$ 3794.31 0.0959422
$$70$$ 0 0
$$71$$ −51410.1 −1.21033 −0.605163 0.796101i $$-0.706892\pi$$
−0.605163 + 0.796101i $$0.706892\pi$$
$$72$$ − 90078.4i − 2.04781i
$$73$$ 43776.4i 0.961465i 0.876867 + 0.480732i $$0.159629\pi$$
−0.876867 + 0.480732i $$0.840371\pi$$
$$74$$ 20222.9 0.429303
$$75$$ 0 0
$$76$$ −76730.7 −1.52382
$$77$$ − 33517.7i − 0.644240i
$$78$$ − 24285.1i − 0.451963i
$$79$$ 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
$$82$$ 110545.i 1.81553i
$$83$$ − 52949.9i − 0.843664i −0.906674 0.421832i $$-0.861387\pi$$
0.906674 0.421832i $$-0.138613\pi$$
$$84$$ 27924.0 0.431797
$$85$$ 0 0
$$86$$ 154113. 2.24695
$$87$$ 7397.05i 0.104776i
$$88$$ 206075.i 2.83673i
$$89$$ −44631.2 −0.597260 −0.298630 0.954369i $$-0.596530\pi$$
−0.298630 + 0.954369i $$0.596530\pi$$
$$90$$ 0 0
$$91$$ −29538.0 −0.373919
$$92$$ 50353.6i 0.620242i
$$93$$ 44139.8i 0.529204i
$$94$$ −9188.03 −0.107251
$$95$$ 0 0
$$96$$ −38693.6 −0.428510
$$97$$ − 148018.i − 1.59730i −0.601797 0.798649i $$-0.705548\pi$$
0.601797 0.798649i $$-0.294452\pi$$
$$98$$ 123686.i 1.30093i
$$99$$ −103289. −1.05917
$$100$$ 0 0
$$101$$ 148476. 1.44828 0.724141 0.689652i $$-0.242237\pi$$
0.724141 + 0.689652i $$0.242237\pi$$
$$102$$ − 102078.i − 0.971472i
$$103$$ − 188391.i − 1.74972i −0.484378 0.874859i $$-0.660954\pi$$
0.484378 0.874859i $$-0.339046\pi$$
$$104$$ 181607. 1.64645
$$105$$ 0 0
$$106$$ −198336. −1.71450
$$107$$ 67887.7i 0.573234i 0.958045 + 0.286617i $$0.0925307\pi$$
−0.958045 + 0.286617i $$0.907469\pi$$
$$108$$ − 184462.i − 1.52176i
$$109$$ 219292. 1.76790 0.883949 0.467582i $$-0.154875\pi$$
0.883949 + 0.467582i $$0.154875\pi$$
$$110$$ 0 0
$$111$$ 10886.2 0.0838625
$$112$$ 138213.i 1.04112i
$$113$$ 80783.9i 0.595153i 0.954698 + 0.297577i $$0.0961784\pi$$
−0.954698 + 0.297577i $$0.903822\pi$$
$$114$$ −59334.4 −0.427606
$$115$$ 0 0
$$116$$ −98165.1 −0.677348
$$117$$ 91025.1i 0.614747i
$$118$$ − 217492.i − 1.43793i
$$119$$ −124157. −0.803721
$$120$$ 0 0
$$121$$ 75246.5 0.467221
$$122$$ − 284487.i − 1.73047i
$$123$$ 59507.3i 0.354656i
$$124$$ −585772. −3.42117
$$125$$ 0 0
$$126$$ −150351. −0.843683
$$127$$ 161301.i 0.887417i 0.896171 + 0.443708i $$0.146337\pi$$
−0.896171 + 0.443708i $$0.853663\pi$$
$$128$$ 144750.i 0.780899i
$$129$$ 82960.5 0.438932
$$130$$ 0 0
$$131$$ −193006. −0.982636 −0.491318 0.870980i $$-0.663485\pi$$
−0.491318 + 0.870980i $$0.663485\pi$$
$$132$$ 196862.i 0.983395i
$$133$$ 72168.5i 0.353768i
$$134$$ 79216.4 0.381113
$$135$$ 0 0
$$136$$ 763349. 3.53896
$$137$$ 250340.i 1.13954i 0.821806 + 0.569768i $$0.192967\pi$$
−0.821806 + 0.569768i $$0.807033\pi$$
$$138$$ 38937.5i 0.174049i
$$139$$ 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
$$142$$ − 527575.i − 2.19565i
$$143$$ − 208241.i − 0.851580i
$$144$$ 425920. 1.71168
$$145$$ 0 0
$$146$$ −449238. −1.74419
$$147$$ 66581.1i 0.254131i
$$148$$ 144469.i 0.542150i
$$149$$ 38740.0 0.142953 0.0714766 0.997442i $$-0.477229\pi$$
0.0714766 + 0.997442i $$0.477229\pi$$
$$150$$ 0 0
$$151$$ −154945. −0.553013 −0.276507 0.961012i $$-0.589177\pi$$
−0.276507 + 0.961012i $$0.589177\pi$$
$$152$$ − 443709.i − 1.55772i
$$153$$ 382607.i 1.32137i
$$154$$ 343962. 1.16871
$$155$$ 0 0
$$156$$ 173488. 0.570766
$$157$$ − 344442.i − 1.11523i −0.830098 0.557617i $$-0.811716\pi$$
0.830098 0.557617i $$-0.188284\pi$$
$$158$$ 63888.5i 0.203601i
$$159$$ −106766. −0.334920
$$160$$ 0 0
$$161$$ 47359.8 0.143994
$$162$$ 387227.i 1.15925i
$$163$$ − 366203.i − 1.07957i −0.841801 0.539787i $$-0.818505\pi$$
0.841801 0.539787i $$-0.181495\pi$$
$$164$$ −789712. −2.29276
$$165$$ 0 0
$$166$$ 543376. 1.53049
$$167$$ 249272.i 0.691644i 0.938300 + 0.345822i $$0.112400\pi$$
−0.938300 + 0.345822i $$0.887600\pi$$
$$168$$ 161476.i 0.441402i
$$169$$ 187778. 0.505740
$$170$$ 0 0
$$171$$ 222397. 0.581618
$$172$$ 1.10096e6i 2.83758i
$$173$$ 61460.1i 0.156127i 0.996948 + 0.0780635i $$0.0248737\pi$$
−0.996948 + 0.0780635i $$0.975126\pi$$
$$174$$ −75909.2 −0.190073
$$175$$ 0 0
$$176$$ −974389. −2.37111
$$177$$ − 117078.i − 0.280893i
$$178$$ − 458009.i − 1.08349i
$$179$$ −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 $$0.444220\pi$$
$$182$$ − 303121.i − 0.678325i
$$183$$ − 153142.i − 0.338039i
$$184$$ −291179. −0.634039
$$185$$ 0 0
$$186$$ −452966. −0.960027
$$187$$ − 875301.i − 1.83043i
$$188$$ − 65637.6i − 0.135443i
$$189$$ −173494. −0.353289
$$190$$ 0 0
$$191$$ 182315. 0.361608 0.180804 0.983519i $$-0.442130\pi$$
0.180804 + 0.983519i $$0.442130\pi$$
$$192$$ − 42736.7i − 0.0836659i
$$193$$ 102080.i 0.197265i 0.995124 + 0.0986323i $$0.0314468\pi$$
−0.995124 + 0.0986323i $$0.968553\pi$$
$$194$$ 1.51898e6 2.89766
$$195$$ 0 0
$$196$$ −883586. −1.64289
$$197$$ − 404656.i − 0.742882i −0.928456 0.371441i $$-0.878864\pi$$
0.928456 0.371441i $$-0.121136\pi$$
$$198$$ − 1.05996e6i − 1.92144i
$$199$$ 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
$$202$$ 1.52367e6i 2.62732i
$$203$$ 92328.5i 0.157252i
$$204$$ 729224. 1.22683
$$205$$ 0 0
$$206$$ 1.93329e6 3.17416
$$207$$ − 145945.i − 0.236736i
$$208$$ 858695.i 1.37620i
$$209$$ −508783. −0.805688
$$210$$ 0 0
$$211$$ −460778. −0.712502 −0.356251 0.934390i $$-0.615945\pi$$
−0.356251 + 0.934390i $$0.615945\pi$$
$$212$$ − 1.41688e6i − 2.16517i
$$213$$ − 283998.i − 0.428911i
$$214$$ −696670. −1.03990
$$215$$ 0 0
$$216$$ 1.06668e6 1.55561
$$217$$ 550944.i 0.794252i
$$218$$ 2.25040e6i 3.20714i
$$219$$ −241829. −0.340720
$$220$$ 0 0
$$221$$ −771372. −1.06239
$$222$$ 111715.i 0.152135i
$$223$$ 1.08298e6i 1.45834i 0.684330 + 0.729172i $$0.260095\pi$$
−0.684330 + 0.729172i $$0.739905\pi$$
$$224$$ −482966. −0.643126
$$225$$ 0 0
$$226$$ −829012. −1.07967
$$227$$ 412201.i 0.530938i 0.964119 + 0.265469i $$0.0855269\pi$$
−0.964119 + 0.265469i $$0.914473\pi$$
$$228$$ − 423874.i − 0.540007i
$$229$$ 433163. 0.545836 0.272918 0.962037i $$-0.412011\pi$$
0.272918 + 0.962037i $$0.412011\pi$$
$$230$$ 0 0
$$231$$ 185158. 0.228303
$$232$$ − 567658.i − 0.692416i
$$233$$ − 760097.i − 0.917232i −0.888635 0.458616i $$-0.848345\pi$$
0.888635 0.458616i $$-0.151655\pi$$
$$234$$ −934108. −1.11521
$$235$$ 0 0
$$236$$ 1.55372e6 1.81590
$$237$$ 34391.7i 0.0397725i
$$238$$ − 1.27411e6i − 1.45803i
$$239$$ 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 $$0.563773\pi$$
$$242$$ 772186.i 0.847585i
$$243$$ 819877.i 0.890703i
$$244$$ 2.03232e6 2.18534
$$245$$ 0 0
$$246$$ −610669. −0.643381
$$247$$ 448373.i 0.467624i
$$248$$ − 3.38734e6i − 3.49727i
$$249$$ 292504. 0.298974
$$250$$ 0 0
$$251$$ −851049. −0.852649 −0.426324 0.904570i $$-0.640192\pi$$
−0.426324 + 0.904570i $$0.640192\pi$$
$$252$$ − 1.07408e6i − 1.06545i
$$253$$ 333883.i 0.327939i
$$254$$ −1.65528e6 −1.60986
$$255$$ 0 0
$$256$$ −1.73300e6 −1.65272
$$257$$ − 76358.4i − 0.0721147i −0.999350 0.0360574i $$-0.988520\pi$$
0.999350 0.0360574i $$-0.0114799\pi$$
$$258$$ 851348.i 0.796266i
$$259$$ 135879. 0.125864
$$260$$ 0 0
$$261$$ 284522. 0.258533
$$262$$ − 1.98064e6i − 1.78260i
$$263$$ − 1.19420e6i − 1.06460i −0.846555 0.532301i $$-0.821327\pi$$
0.846555 0.532301i $$-0.178673\pi$$
$$264$$ −1.13839e6 −1.00527
$$265$$ 0 0
$$266$$ −740600. −0.641770
$$267$$ − 246551.i − 0.211655i
$$268$$ 565907.i 0.481292i
$$269$$ −1.02930e6 −0.867286 −0.433643 0.901085i $$-0.642772\pi$$
−0.433643 + 0.901085i $$0.642772\pi$$
$$270$$ 0 0
$$271$$ 2.12144e6 1.75472 0.877359 0.479834i $$-0.159303\pi$$
0.877359 + 0.479834i $$0.159303\pi$$
$$272$$ 3.60937e6i 2.95807i
$$273$$ − 163173.i − 0.132508i
$$274$$ −2.56901e6 −2.06723
$$275$$ 0 0
$$276$$ −278162. −0.219799
$$277$$ 1.85145e6i 1.44982i 0.688845 + 0.724908i $$0.258118\pi$$
−0.688845 + 0.724908i $$0.741882\pi$$
$$278$$ 2.24381e6i 1.74130i
$$279$$ 1.69781e6 1.30580
$$280$$ 0 0
$$281$$ 90653.2 0.0684884 0.0342442 0.999413i $$-0.489098\pi$$
0.0342442 + 0.999413i $$0.489098\pi$$
$$282$$ − 50756.3i − 0.0380073i
$$283$$ 929308.i 0.689753i 0.938648 + 0.344877i $$0.112079\pi$$
−0.938648 + 0.344877i $$0.887921\pi$$
$$284$$ 3.76890e6 2.77280
$$285$$ 0 0
$$286$$ 2.13698e6 1.54485
$$287$$ 742759.i 0.532283i
$$288$$ 1.48832e6i 1.05734i
$$289$$ −1.82246e6 −1.28355
$$290$$ 0 0
$$291$$ 817679. 0.566044
$$292$$ − 3.20927e6i − 2.20267i
$$293$$ − 2.72733e6i − 1.85596i −0.372632 0.927979i $$-0.621545\pi$$
0.372632 0.927979i $$-0.378455\pi$$
$$294$$ −683261. −0.461018
$$295$$ 0 0
$$296$$ −835417. −0.554209
$$297$$ − 1.22312e6i − 0.804597i
$$298$$ 397553.i 0.259331i
$$299$$ 294240. 0.190337
$$300$$ 0 0
$$301$$ 1.03550e6 0.658768
$$302$$ − 1.59006e6i − 1.00322i
$$303$$ 820208.i 0.513236i
$$304$$ 2.09800e6 1.30203
$$305$$ 0 0
$$306$$ −3.92635e6 −2.39709
$$307$$ − 2.29648e6i − 1.39064i −0.718698 0.695322i $$-0.755262\pi$$
0.718698 0.695322i $$-0.244738\pi$$
$$308$$ 2.45720e6i 1.47592i
$$309$$ 1.04071e6 0.620058
$$310$$ 0 0
$$311$$ 984847. 0.577388 0.288694 0.957421i $$-0.406779\pi$$
0.288694 + 0.957421i $$0.406779\pi$$
$$312$$ 1.00323e6i 0.583462i
$$313$$ − 2.06650e6i − 1.19227i −0.802884 0.596135i $$-0.796702\pi$$
0.802884 0.596135i $$-0.203298\pi$$
$$314$$ 3.53469e6 2.02315
$$315$$ 0 0
$$316$$ −456407. −0.257119
$$317$$ 1.14349e6i 0.639125i 0.947565 + 0.319563i $$0.103536\pi$$
−0.947565 + 0.319563i $$0.896464\pi$$
$$318$$ − 1.09564e6i − 0.607578i
$$319$$ −650910. −0.358133
$$320$$ 0 0
$$321$$ −375024. −0.203140
$$322$$ 486010.i 0.261220i
$$323$$ 1.88465e6i 1.00514i
$$324$$ −2.76628e6 −1.46397
$$325$$ 0 0
$$326$$ 3.75801e6 1.95845
$$327$$ 1.21141e6i 0.626501i
$$328$$ − 4.56666e6i − 2.34376i
$$329$$ −61735.0 −0.0314443
$$330$$ 0 0
$$331$$ 205230. 0.102961 0.0514804 0.998674i $$-0.483606\pi$$
0.0514804 + 0.998674i $$0.483606\pi$$
$$332$$ 3.88178e6i 1.93279i
$$333$$ − 418729.i − 0.206929i
$$334$$ −2.55805e6 −1.25471
$$335$$ 0 0
$$336$$ −763511. −0.368950
$$337$$ 488213.i 0.234172i 0.993122 + 0.117086i $$0.0373553\pi$$
−0.993122 + 0.117086i $$0.962645\pi$$
$$338$$ 1.92699e6i 0.917462i
$$339$$ −446265. −0.210908
$$340$$ 0 0
$$341$$ −3.88412e6 −1.80887
$$342$$ 2.28225e6i 1.05511i
$$343$$ 1.98992e6i 0.913273i
$$344$$ −6.36648e6 −2.90070
$$345$$ 0 0
$$346$$ −630709. −0.283230
$$347$$ − 3.82809e6i − 1.70670i −0.521336 0.853351i $$-0.674566\pi$$
0.521336 0.853351i $$-0.325434\pi$$
$$348$$ − 542281.i − 0.240036i
$$349$$ −1.45476e6 −0.639333 −0.319667 0.947530i $$-0.603571\pi$$
−0.319667 + 0.947530i $$0.603571\pi$$
$$350$$ 0 0
$$351$$ −1.07789e6 −0.466991
$$352$$ − 3.40487e6i − 1.46469i
$$353$$ 778492.i 0.332520i 0.986082 + 0.166260i $$0.0531690\pi$$
−0.986082 + 0.166260i $$0.946831\pi$$
$$354$$ 1.20146e6 0.509567
$$355$$ 0 0
$$356$$ 3.27193e6 1.36829
$$357$$ − 685867.i − 0.284819i
$$358$$ − 6.22707e6i − 2.56789i
$$359$$ 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
$$362$$ 1.57712e6i 0.632549i
$$363$$ 415675.i 0.165572i
$$364$$ 2.16544e6 0.856630
$$365$$ 0 0
$$366$$ 1.57156e6 0.613236
$$367$$ 4.10801e6i 1.59208i 0.605242 + 0.796042i $$0.293077\pi$$
−0.605242 + 0.796042i $$0.706923\pi$$
$$368$$ − 1.37679e6i − 0.529967i
$$369$$ 2.28891e6 0.875109
$$370$$ 0 0
$$371$$ −1.33263e6 −0.502662
$$372$$ − 3.23591e6i − 1.21238i
$$373$$ 4.54570e6i 1.69172i 0.533405 + 0.845860i $$0.320912\pi$$
−0.533405 + 0.845860i $$0.679088\pi$$
$$374$$ 8.98241e6 3.32058
$$375$$ 0 0
$$376$$ 379561. 0.138456
$$377$$ 573624.i 0.207861i
$$378$$ − 1.78041e6i − 0.640901i
$$379$$ −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
$$382$$ 1.87093e6i 0.655993i
$$383$$ − 4.64417e6i − 1.61775i −0.587982 0.808874i $$-0.700078\pi$$
0.587982 0.808874i $$-0.299922\pi$$
$$384$$ −799627. −0.276732
$$385$$ 0 0
$$386$$ −1.04756e6 −0.357857
$$387$$ − 3.19102e6i − 1.08306i
$$388$$ 1.08513e7i 3.65933i
$$389$$ −3.53606e6 −1.18480 −0.592400 0.805644i $$-0.701819\pi$$
−0.592400 + 0.805644i $$0.701819\pi$$
$$390$$ 0 0
$$391$$ 1.23678e6 0.409120
$$392$$ − 5.10950e6i − 1.67944i
$$393$$ − 1.06620e6i − 0.348223i
$$394$$ 4.15261e6 1.34766
$$395$$ 0 0
$$396$$ 7.57217e6 2.42651
$$397$$ − 2.95611e6i − 0.941336i −0.882310 0.470668i $$-0.844013\pi$$
0.882310 0.470668i $$-0.155987\pi$$
$$398$$ 1.71682e6i 0.543272i
$$399$$ −398671. −0.125367
$$400$$ 0 0
$$401$$ −799254. −0.248213 −0.124106 0.992269i $$-0.539606\pi$$
−0.124106 + 0.992269i $$0.539606\pi$$
$$402$$ 437605.i 0.135057i
$$403$$ 3.42294e6i 1.04987i
$$404$$ −1.08848e7 −3.31794
$$405$$ 0 0
$$406$$ −947484. −0.285270
$$407$$ 957937.i 0.286649i
$$408$$ 4.21687e6i 1.25412i
$$409$$ −898422. −0.265566 −0.132783 0.991145i $$-0.542391\pi$$
−0.132783 + 0.991145i $$0.542391\pi$$
$$410$$ 0 0
$$411$$ −1.38292e6 −0.403824
$$412$$ 1.38111e7i 4.00852i
$$413$$ − 1.46134e6i − 0.421576i
$$414$$ 1.49770e6 0.429462
$$415$$ 0 0
$$416$$ −3.00060e6 −0.850108
$$417$$ 1.20786e6i 0.340155i
$$418$$ − 5.22118e6i − 1.46160i
$$419$$ −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 $$0.291379\pi$$
$$422$$ − 4.72855e6i − 1.29255i
$$423$$ 190244.i 0.0516965i
$$424$$ 8.19336e6 2.21334
$$425$$ 0 0
$$426$$ 2.91442e6 0.778086
$$427$$ − 1.91149e6i − 0.507344i
$$428$$ − 4.97688e6i − 1.31325i
$$429$$ 1.15036e6 0.301780
$$430$$ 0 0
$$431$$ 5.97999e6 1.55063 0.775314 0.631576i $$-0.217592\pi$$
0.775314 + 0.631576i $$0.217592\pi$$
$$432$$ 5.04363e6i 1.30027i
$$433$$ − 2.06419e6i − 0.529089i −0.964373 0.264545i $$-0.914778\pi$$
0.964373 0.264545i $$-0.0852217\pi$$
$$434$$ −5.65384e6 −1.44085
$$435$$ 0 0
$$436$$ −1.60764e7 −4.05017
$$437$$ − 718899.i − 0.180080i
$$438$$ − 2.48167e6i − 0.618099i
$$439$$ 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
$$442$$ − 7.91589e6i − 1.92728i
$$443$$ − 2.75822e6i − 0.667759i −0.942616 0.333879i $$-0.891642\pi$$
0.942616 0.333879i $$-0.108358\pi$$
$$444$$ −798070. −0.192125
$$445$$ 0 0
$$446$$ −1.11137e7 −2.64558
$$447$$ 214006.i 0.0506592i
$$448$$ − 533431.i − 0.125569i
$$449$$ 3.76648e6 0.881698 0.440849 0.897581i $$-0.354677\pi$$
0.440849 + 0.897581i $$0.354677\pi$$
$$450$$ 0 0
$$451$$ −5.23640e6 −1.21225
$$452$$ − 5.92231e6i − 1.36347i
$$453$$ − 855944.i − 0.195975i
$$454$$ −4.23004e6 −0.963174
$$455$$ 0 0
$$456$$ 2.45113e6 0.552019
$$457$$ 480604.i 0.107646i 0.998550 + 0.0538229i $$0.0171407\pi$$
−0.998550 + 0.0538229i $$0.982859\pi$$
$$458$$ 4.44515e6i 0.990200i
$$459$$ −4.53073e6 −1.00377
$$460$$ 0 0
$$461$$ 4.52514e6 0.991699 0.495849 0.868409i $$-0.334857\pi$$
0.495849 + 0.868409i $$0.334857\pi$$
$$462$$ 1.90010e6i 0.414164i
$$463$$ − 7.39975e6i − 1.60422i −0.597175 0.802111i $$-0.703710\pi$$
0.597175 0.802111i $$-0.296290\pi$$
$$464$$ 2.68407e6 0.578761
$$465$$ 0 0
$$466$$ 7.80018e6 1.66395
$$467$$ 1.84711e6i 0.391923i 0.980612 + 0.195962i $$0.0627828\pi$$
−0.980612 + 0.195962i $$0.937217\pi$$
$$468$$ − 6.67309e6i − 1.40836i
$$469$$ 532261. 0.111736
$$470$$ 0 0
$$471$$ 1.90276e6 0.395212
$$472$$ 8.98467e6i 1.85630i
$$473$$ 7.30018e6i 1.50031i
$$474$$ −352931. −0.0721513
$$475$$ 0 0
$$476$$ 9.10203e6 1.84128
$$477$$ 4.10669e6i 0.826410i
$$478$$ 1.01453e7i 2.03094i
$$479$$ 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
$$482$$ − 3.68284e6i − 0.722047i
$$483$$ 261624.i 0.0510281i
$$484$$ −5.51635e6 −1.07038
$$485$$ 0 0
$$486$$ −8.41365e6 −1.61582
$$487$$ 7.28136e6i 1.39120i 0.718429 + 0.695601i $$0.244862\pi$$
−0.718429 + 0.695601i $$0.755138\pi$$
$$488$$ 1.17523e7i 2.23395i
$$489$$ 2.02297e6 0.382575
$$490$$ 0 0
$$491$$ −6.60475e6 −1.23638 −0.618191 0.786028i $$-0.712134\pi$$
−0.618191 + 0.786028i $$0.712134\pi$$
$$492$$ − 4.36251e6i − 0.812500i
$$493$$ 2.41112e6i 0.446788i
$$494$$ −4.60124e6 −0.848316
$$495$$ 0 0
$$496$$ 1.60164e7 2.92322
$$497$$ − 3.54481e6i − 0.643727i
$$498$$ 3.00171e6i 0.542369i
$$499$$ −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
$$502$$ − 8.73354e6i − 1.54679i
$$503$$ 1.16752e6i 0.205753i 0.994694 + 0.102876i $$0.0328046\pi$$
−0.994694 + 0.102876i $$0.967195\pi$$
$$504$$ 6.21105e6 1.08915
$$505$$ 0 0
$$506$$ −3.42634e6 −0.594914
$$507$$ 1.03732e6i 0.179222i
$$508$$ − 1.18250e7i − 2.03303i
$$509$$ 7.41468e6 1.26852 0.634261 0.773119i $$-0.281305\pi$$
0.634261 + 0.773119i $$0.281305\pi$$
$$510$$ 0 0
$$511$$ −3.01846e6 −0.511367
$$512$$ − 1.31522e7i − 2.21730i
$$513$$ 2.63356e6i 0.441824i
$$514$$ 783596. 0.130823
$$515$$ 0 0
$$516$$ −6.08187e6 −1.00557
$$517$$ − 435227.i − 0.0716127i
$$518$$ 1.39440e6i 0.228330i
$$519$$ −339516. −0.0553277
$$520$$ 0 0
$$521$$ −811897. −0.131041 −0.0655204 0.997851i $$-0.520871\pi$$
−0.0655204 + 0.997851i $$0.520871\pi$$
$$522$$ 2.91979e6i 0.469003i
$$523$$ − 5.06828e6i − 0.810226i −0.914267 0.405113i $$-0.867232\pi$$
0.914267 0.405113i $$-0.132768\pi$$
$$524$$ 1.41494e7 2.25117
$$525$$ 0 0
$$526$$ 1.22550e7 1.93129
$$527$$ 1.43877e7i 2.25665i
$$528$$ − 5.38270e6i − 0.840263i
$$529$$ 5.96457e6 0.926702
$$530$$ 0 0
$$531$$ −4.50331e6 −0.693099
$$532$$ − 5.29071e6i − 0.810465i
$$533$$ 4.61465e6i 0.703592i
$$534$$ 2.53012e6 0.383962
$$535$$ 0 0
$$536$$ −3.27247e6 −0.491998
$$537$$ − 3.35209e6i − 0.501626i
$$538$$ − 1.05628e7i − 1.57334i
$$539$$ −5.85886e6 −0.868642
$$540$$ 0 0
$$541$$ 1.52830e6 0.224499 0.112250 0.993680i $$-0.464194\pi$$
0.112250 + 0.993680i $$0.464194\pi$$
$$542$$ 2.17704e7i 3.18323i
$$543$$ 848980.i 0.123566i
$$544$$ −1.26124e7 −1.82727
$$545$$ 0 0
$$546$$ 1.67449e6 0.240382
$$547$$ − 1.23234e7i − 1.76101i −0.474036 0.880506i $$-0.657203\pi$$
0.474036 0.880506i $$-0.342797\pi$$
$$548$$ − 1.83525e7i − 2.61062i
$$549$$ −5.89050e6 −0.834107
$$550$$ 0 0
$$551$$ 1.40150e6 0.196660
$$552$$ − 1.60853e6i − 0.224688i
$$553$$ 429271.i 0.0596923i
$$554$$ −1.89998e7 −2.63011
$$555$$ 0 0
$$556$$ −1.60293e7 −2.19902
$$557$$ − 4.08606e6i − 0.558042i −0.960285 0.279021i $$-0.909990\pi$$
0.960285 0.279021i $$-0.0900099\pi$$
$$558$$ 1.74230e7i 2.36885i
$$559$$ 6.43339e6 0.870784
$$560$$ 0 0
$$561$$ 4.83531e6 0.648661
$$562$$ 930291.i 0.124245i
$$563$$ − 24160.3i − 0.00321241i −0.999999 0.00160621i $$-0.999489\pi$$
0.999999 0.00160621i $$-0.000511272\pi$$
$$564$$ 362593. 0.0479979
$$565$$ 0 0
$$566$$ −9.53664e6 −1.25128
$$567$$ 2.60180e6i 0.339873i
$$568$$ 2.17943e7i 2.83448i
$$569$$ −1.42000e7 −1.83869 −0.919344 0.393454i $$-0.871280\pi$$
−0.919344 + 0.393454i $$0.871280\pi$$
$$570$$ 0 0
$$571$$ −767642. −0.0985300 −0.0492650 0.998786i $$-0.515688\pi$$
−0.0492650 + 0.998786i $$0.515688\pi$$
$$572$$ 1.52662e7i 1.95093i
$$573$$ 1.00714e6i 0.128145i
$$574$$ −7.62225e6 −0.965614
$$575$$ 0 0
$$576$$ −1.64384e6 −0.206444
$$577$$ − 1.51488e6i − 0.189426i −0.995505 0.0947129i $$-0.969807\pi$$
0.995505 0.0947129i $$-0.0301933\pi$$
$$578$$ − 1.87023e7i − 2.32849i
$$579$$ −563910. −0.0699059
$$580$$ 0 0
$$581$$ 3.65098e6 0.448714
$$582$$ 8.39109e6i 1.02686i
$$583$$ − 9.39498e6i − 1.14479i
$$584$$ 1.85582e7 2.25167
$$585$$ 0 0
$$586$$ 2.79881e7 3.36689
$$587$$ − 1.28973e7i − 1.54491i −0.635070 0.772455i $$-0.719029\pi$$
0.635070 0.772455i $$-0.280971\pi$$
$$588$$ − 4.88109e6i − 0.582201i
$$589$$ 8.36307e6 0.993294
$$590$$ 0 0
$$591$$ 2.23539e6 0.263260
$$592$$ − 3.95012e6i − 0.463240i
$$593$$ 5.43125e6i 0.634254i 0.948383 + 0.317127i $$0.102718\pi$$
−0.948383 + 0.317127i $$0.897282\pi$$
$$594$$ 1.25518e7 1.45962
$$595$$ 0 0
$$596$$ −2.84004e6 −0.327499
$$597$$ 924180.i 0.106126i
$$598$$ 3.01951e6i 0.345290i
$$599$$ −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 $$0.603324\pi$$
$$602$$ 1.06264e7i 1.19507i
$$603$$ − 1.64023e6i − 0.183701i
$$604$$ 1.13591e7 1.26693
$$605$$ 0 0
$$606$$ −8.41704e6 −0.931061
$$607$$ − 1.07148e7i − 1.18035i −0.807274 0.590177i $$-0.799058\pi$$
0.807274 0.590177i $$-0.200942\pi$$
$$608$$ 7.33119e6i 0.804296i
$$609$$ −510039. −0.0557263
$$610$$ 0 0
$$611$$ −383551. −0.0415642
$$612$$ − 2.80491e7i − 3.02719i
$$613$$ − 4.08748e6i − 0.439344i −0.975574 0.219672i $$-0.929501\pi$$
0.975574 0.219672i $$-0.0704987\pi$$
$$614$$ 2.35666e7 2.52276
$$615$$ 0 0
$$616$$ −1.42092e7 −1.50875
$$617$$ − 7.83395e6i − 0.828453i −0.910174 0.414227i $$-0.864052\pi$$
0.910174 0.414227i $$-0.135948\pi$$
$$618$$ 1.06798e7i 1.12485i
$$619$$ −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
$$622$$ 1.01066e7i 1.04744i
$$623$$ − 3.07739e6i − 0.317660i
$$624$$ −4.74358e6 −0.487691
$$625$$ 0 0
$$626$$ 2.12066e7 2.16290
$$627$$ − 2.81061e6i − 0.285516i
$$628$$ 2.52512e7i 2.55495i
$$629$$ 3.54842e6 0.357609
$$630$$ 0 0
$$631$$ −1.31578e6 −0.131556 −0.0657780 0.997834i $$-0.520953\pi$$
−0.0657780 + 0.997834i $$0.520953\pi$$
$$632$$ − 2.63926e6i − 0.262839i
$$633$$ − 2.54542e6i − 0.252494i
$$634$$ −1.17346e7 −1.15944
$$635$$ 0 0
$$636$$ 7.82708e6 0.767285
$$637$$ 5.16320e6i 0.504163i
$$638$$ − 6.67969e6i − 0.649688i
$$639$$ −1.09238e7 −1.05833
$$640$$ 0 0
$$641$$ 6.55744e6 0.630360 0.315180 0.949032i $$-0.397935\pi$$
0.315180 + 0.949032i $$0.397935\pi$$
$$642$$ − 3.84852e6i − 0.368516i
$$643$$ 4.69954e6i 0.448258i 0.974559 + 0.224129i $$0.0719537\pi$$
−0.974559 + 0.224129i $$0.928046\pi$$
$$644$$ −3.47197e6 −0.329884
$$645$$ 0 0
$$646$$ −1.93405e7 −1.82341
$$647$$ 2.05827e7i 1.93305i 0.256580 + 0.966523i $$0.417404\pi$$
−0.256580 + 0.966523i $$0.582596\pi$$
$$648$$ − 1.59965e7i − 1.49654i
$$649$$ 1.03023e7 0.960117
$$650$$ 0 0
$$651$$ −3.04351e6 −0.281464
$$652$$ 2.68465e7i 2.47325i
$$653$$ 1.42466e7i 1.30746i 0.756727 + 0.653731i $$0.226797\pi$$
−0.756727 + 0.653731i $$0.773203\pi$$
$$654$$ −1.24316e7 −1.13653
$$655$$ 0 0
$$656$$ 2.15927e7 1.95905
$$657$$ 9.30177e6i 0.840722i
$$658$$ − 633530.i − 0.0570430i
$$659$$ 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 $$0.697187\pi$$
$$662$$ 2.10609e6i 0.186781i
$$663$$ − 4.26119e6i − 0.376485i
$$664$$ −2.24471e7 −1.97579
$$665$$ 0 0
$$666$$ 4.29703e6 0.375390
$$667$$ − 919721.i − 0.0800464i
$$668$$ − 1.82743e7i − 1.58452i
$$669$$ −5.98260e6 −0.516802
$$670$$ 0 0
$$671$$ 1.34759e7 1.15545
$$672$$ − 2.66799e6i − 0.227908i
$$673$$ − 4.75951e6i − 0.405065i −0.979276 0.202532i $$-0.935083\pi$$
0.979276 0.202532i $$-0.0649171\pi$$
$$674$$ −5.01008e6 −0.424810
$$675$$ 0 0
$$676$$ −1.37661e7 −1.15863
$$677$$ − 1.51397e7i − 1.26954i −0.772701 0.634770i $$-0.781095\pi$$
0.772701 0.634770i $$-0.218905\pi$$
$$678$$ − 4.57961e6i − 0.382608i
$$679$$ 1.02061e7 0.849543
$$680$$ 0 0
$$681$$ −2.27707e6 −0.188152
$$682$$ − 3.98592e7i − 3.28146i
$$683$$ 2.34145e7i 1.92058i 0.278998 + 0.960292i $$0.409998\pi$$
−0.278998 + 0.960292i $$0.590002\pi$$
$$684$$ −1.63040e7 −1.33246
$$685$$ 0 0
$$686$$ −2.04208e7 −1.65677
$$687$$ 2.39287e6i 0.193431i
$$688$$ − 3.01028e7i − 2.42458i
$$689$$ −8.27947e6 −0.664438
$$690$$ 0 0
$$691$$ −1.62194e7 −1.29223 −0.646113 0.763242i $$-0.723606\pi$$
−0.646113 + 0.763242i $$0.723606\pi$$
$$692$$ − 4.50567e6i − 0.357679i
$$693$$ − 7.12196e6i − 0.563334i
$$694$$ 3.92841e7 3.09613
$$695$$ 0 0
$$696$$ 3.13584e6 0.245375
$$697$$ 1.93968e7i 1.51234i
$$698$$ − 1.49289e7i − 1.15981i
$$699$$ 4.19891e6 0.325045
$$700$$ 0 0
$$701$$ −1.89605e7 −1.45732 −0.728659 0.684876i $$-0.759856\pi$$
−0.728659 + 0.684876i $$0.759856\pi$$
$$702$$ − 1.10614e7i − 0.847167i
$$703$$ − 2.06258e6i − 0.157406i
$$704$$ 3.76065e6 0.285977
$$705$$ 0 0
$$706$$ −7.98895e6 −0.603223
$$707$$ 1.02377e7i 0.770287i
$$708$$ 8.58301e6i 0.643512i
$$709$$ −128325. −0.00958732 −0.00479366 0.999989i $$-0.501526\pi$$
−0.00479366 + 0.999989i $$0.501526\pi$$
$$710$$ 0 0
$$711$$ 1.32285e6 0.0981382
$$712$$ 1.89206e7i 1.39873i
$$713$$ − 5.48817e6i − 0.404300i
$$714$$ 7.03843e6 0.516690
$$715$$ 0 0
$$716$$ 4.44850e7 3.24288
$$717$$ 5.46133e6i 0.396735i
$$718$$ 2.18079e7i 1.57871i
$$719$$ 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
$$722$$ − 1.41680e7i − 1.01150i
$$723$$ − 1.98251e6i − 0.141049i
$$724$$ −1.12667e7 −0.798821
$$725$$ 0 0
$$726$$ −4.26569e6 −0.300364
$$727$$ − 513307.i − 0.0360198i −0.999838 0.0180099i $$-0.994267\pi$$
0.999838 0.0180099i $$-0.00573303\pi$$
$$728$$ 1.25221e7i 0.875685i
$$729$$ 4.64015e6 0.323380
$$730$$ 0 0
$$731$$ 2.70416e7 1.87171
$$732$$ 1.12269e7i 0.774431i
$$733$$ − 1.64153e7i − 1.12847i −0.825615 0.564234i $$-0.809172\pi$$
0.825615 0.564234i $$-0.190828\pi$$
$$734$$ −4.21567e7 −2.88820
$$735$$ 0 0
$$736$$ 4.81101e6 0.327372
$$737$$ 3.75240e6i 0.254472i
$$738$$ 2.34890e7i 1.58753i
$$739$$ −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
$$742$$ − 1.36756e7i − 0.911879i
$$743$$ 5.72590e6i 0.380515i 0.981734 + 0.190257i $$0.0609323\pi$$
−0.981734 + 0.190257i $$0.939068\pi$$
$$744$$ 1.87122e7 1.23935
$$745$$ 0 0
$$746$$ −4.66483e7 −3.06894
$$747$$ − 1.12510e7i − 0.737715i
$$748$$ 6.41687e7i 4.19343i
$$749$$ −4.68097e6 −0.304882
$$750$$ 0 0
$$751$$ 1.15324e7 0.746137 0.373069 0.927804i $$-0.378306\pi$$
0.373069 + 0.927804i $$0.378306\pi$$
$$752$$ 1.79469e6i 0.115730i
$$753$$ − 4.70134e6i − 0.302158i
$$754$$ −5.88658e6 −0.377081
$$755$$ 0 0
$$756$$ 1.27189e7 0.809368
$$757$$ 8.63293e6i 0.547544i 0.961795 + 0.273772i $$0.0882713\pi$$
−0.961795 + 0.273772i $$0.911729\pi$$
$$758$$ − 1.44238e7i − 0.911812i
$$759$$ −1.84443e6 −0.116214
$$760$$ 0 0
$$761$$ −3.52622e6 −0.220723 −0.110361 0.993892i $$-0.535201\pi$$
−0.110361 + 0.993892i $$0.535201\pi$$
$$762$$ − 9.14408e6i − 0.570496i
$$763$$ 1.51206e7i 0.940280i
$$764$$ −1.33656e7 −0.828427
$$765$$ 0 0
$$766$$ 4.76588e7 2.93475
$$767$$ − 9.07910e6i − 0.557255i
$$768$$ − 9.57341e6i − 0.585685i
$$769$$ −1.40471e7 −0.856585 −0.428293 0.903640i $$-0.640885\pi$$
−0.428293 + 0.903640i $$0.640885\pi$$
$$770$$ 0 0
$$771$$ 421817. 0.0255557
$$772$$ − 7.48356e6i − 0.451924i
$$773$$ − 2.44760e7i − 1.47330i −0.676274 0.736651i $$-0.736406\pi$$
0.676274 0.736651i $$-0.263594\pi$$
$$774$$ 3.27465e7 1.96477
$$775$$ 0 0
$$776$$ −6.27496e7 −3.74073
$$777$$ 750619.i 0.0446033i
$$778$$ − 3.62873e7i − 2.14934i
$$779$$ 1.12747e7 0.665675
$$780$$ 0 0
$$781$$ 2.49907e7 1.46606
$$782$$ 1.26920e7i 0.742184i
$$783$$ 3.36924e6i 0.196393i
$$784$$ 2.41594e7 1.40377
$$785$$ 0 0
$$786$$ 1.09414e7 0.631710
$$787$$ − 4.35977e6i − 0.250915i −0.992099 0.125458i $$-0.959960\pi$$
0.992099 0.125458i $$-0.0400399\pi$$
$$788$$ 2.96655e7i 1.70191i
$$789$$ 6.59697e6 0.377270
$$790$$ 0 0
$$791$$ −5.57019e6 −0.316540
$$792$$ 4.37875e7i 2.48049i
$$793$$ − 1.18758e7i − 0.670626i
$$794$$ 3.03359e7 1.70768
$$795$$ 0 0
$$796$$ −1.22647e7 −0.686076
$$797$$ − 1.06887e7i − 0.596044i −0.954559 0.298022i $$-0.903673\pi$$
0.954559 0.298022i $$-0.0963269\pi$$
$$798$$ − 4.09120e6i − 0.227428i
$$799$$ −1.61218e6 −0.0893403
$$800$$ 0 0
$$801$$ −9.48339e6 −0.522255
$$802$$ − 8.20202e6i − 0.450282i
$$803$$ − 2.12799e7i − 1.16461i
$$804$$ −3.12617e6 −0.170558
$$805$$ 0 0
$$806$$ −3.51265e7 −1.90457
$$807$$ − 5.68605e6i − 0.307345i
$$808$$ − 6.29436e7i − 3.39175i
$$809$$ −9.12014e6 −0.489926 −0.244963 0.969532i $$-0.578776\pi$$
−0.244963 + 0.969532i $$0.578776\pi$$
$$810$$ 0 0
$$811$$ 5.22575e6 0.278995 0.139497 0.990222i $$-0.455451\pi$$
0.139497 + 0.990222i $$0.455451\pi$$
$$812$$ − 6.76865e6i − 0.360256i
$$813$$ 1.17192e7i 0.621830i
$$814$$ −9.83044e6 −0.520010
$$815$$ 0 0
$$816$$ −1.99388e7 −1.04827
$$817$$ − 1.57184e7i − 0.823857i
$$818$$ − 9.21968e6i − 0.481762i
$$819$$ −6.27633e6 −0.326961
$$820$$ 0 0
$$821$$ 9.00437e6 0.466225 0.233112 0.972450i $$-0.425109\pi$$
0.233112 + 0.972450i $$0.425109\pi$$
$$822$$ − 1.41916e7i − 0.732576i
$$823$$ 2.78867e7i 1.43515i 0.696482 + 0.717574i $$0.254748\pi$$
−0.696482 + 0.717574i $$0.745252\pi$$
$$824$$ −7.98650e7 −4.09768
$$825$$ 0 0
$$826$$ 1.49964e7 0.764781
$$827$$ − 6.64309e6i − 0.337758i −0.985637 0.168879i $$-0.945985\pi$$
0.985637 0.168879i $$-0.0540148\pi$$
$$828$$ 1.06993e7i 0.542351i
$$829$$ −2.17030e7 −1.09682 −0.548408 0.836211i $$-0.684766\pi$$
−0.548408 + 0.836211i $$0.684766\pi$$
$$830$$ 0 0
$$831$$ −1.02277e7 −0.513780
$$832$$ − 3.31413e6i − 0.165982i
$$833$$ 2.17026e7i 1.08367i
$$834$$ −1.23952e7 −0.617075
$$835$$ 0 0
$$836$$ 3.72991e7 1.84579
$$837$$ 2.01049e7i 0.991949i
$$838$$ − 2.37320e7i − 1.16741i
$$839$$ 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
$$842$$ 4.54914e7i 2.21131i
$$843$$ 500784.i 0.0242707i
$$844$$ 3.37799e7 1.63231
$$845$$ 0 0
$$846$$ −1.95230e6 −0.0937825
$$847$$ 5.18837e6i 0.248498i
$$848$$ 3.87409e7i 1.85003i
$$849$$ −5.13366e6 −0.244432
$$850$$ 0 0
$$851$$ −1.35354e6 −0.0640691
$$852$$ 2.08200e7i 0.982613i
$$853$$ − 1.46326e7i − 0.688573i −0.938865 0.344286i $$-0.888121\pi$$
0.938865 0.344286i $$-0.111879\pi$$
$$854$$ 1.96159e7 0.920371
$$855$$ 0 0
$$856$$ 2.87797e7 1.34246
$$857$$ − 5.52218e6i − 0.256838i −0.991720 0.128419i $$-0.959010\pi$$
0.991720 0.128419i $$-0.0409902\pi$$
$$858$$ 1.18051e7i 0.547458i
$$859$$ 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
$$862$$ 6.13672e7i 2.81299i
$$863$$ 3.06818e7i 1.40234i 0.712992 + 0.701172i $$0.247339\pi$$
−0.712992 + 0.701172i $$0.752661\pi$$
$$864$$ −1.76243e7 −0.803207
$$865$$ 0 0
$$866$$ 2.11829e7 0.959820
$$867$$ − 1.00676e7i − 0.454860i
$$868$$ − 4.03900e7i − 1.81959i
$$869$$ −3.02633e6 −0.135946
$$870$$ 0 0
$$871$$ 3.30686e6 0.147697
$$872$$ − 9.29649e7i − 4.14026i
$$873$$ − 3.14514e7i − 1.39671i
$$874$$ 7.37741e6 0.326682
$$875$$ 0 0
$$876$$ 1.77286e7 0.780573
$$877$$ − 5.17607e6i − 0.227249i −0.993524 0.113624i $$-0.963754\pi$$
0.993524 0.113624i $$-0.0362460\pi$$
$$878$$ 4.19871e7i 1.83815i
$$879$$ 1.50662e7 0.657707
$$880$$ 0 0
$$881$$ −4.25937e7 −1.84887 −0.924433 0.381345i $$-0.875461\pi$$
−0.924433 + 0.381345i $$0.875461\pi$$
$$882$$ 2.62811e7i 1.13756i
$$883$$ 1.72076e7i 0.742709i 0.928491 + 0.371354i $$0.121107\pi$$
−0.928491 + 0.371354i $$0.878893\pi$$
$$884$$ 5.65496e7 2.43388
$$885$$ 0 0
$$886$$ 2.83051e7 1.21138
$$887$$ 2.53773e6i 0.108302i 0.998533 + 0.0541510i $$0.0172452\pi$$
−0.998533 + 0.0541510i $$0.982755\pi$$
$$888$$ − 4.61499e6i − 0.196398i
$$889$$ −1.11220e7 −0.471984
$$890$$ 0 0
$$891$$ −1.83425e7 −0.774043
$$892$$ − 7.93941e7i − 3.34100i
$$893$$ 937108.i 0.0393243i
$$894$$ −2.19615e6 −0.0919007
$$895$$ 0 0
$$896$$ −9.98078e6 −0.415331
$$897$$ 1.62543e6i 0.0674508i
$$898$$ 3.86519e7i 1.59949i
$$899$$ 1.06993e7 0.441525
$$900$$ 0 0
$$901$$ −3.48012e7 −1.42818
$$902$$ − 5.37364e7i − 2.19913i
$$903$$ 5.72026e6i 0.233452i
$$904$$ 3.42468e7 1.39380
$$905$$ 0 0
$$906$$ 8.78377e6 0.355517
$$907$$ 2.60899e7i 1.05306i 0.850156 + 0.526531i $$0.176508\pi$$
−0.850156 + 0.526531i $$0.823492\pi$$
$$908$$ − 3.02186e7i − 1.21635i
$$909$$ 3.15487e7 1.26640
$$910$$ 0 0
$$911$$ 1.44818e7 0.578130 0.289065 0.957309i $$-0.406656\pi$$
0.289065 + 0.957309i $$0.406656\pi$$
$$912$$ 1.15897e7i 0.461409i
$$913$$ 2.57392e7i 1.02192i
$$914$$ −4.93200e6 −0.195280
$$915$$ 0 0
$$916$$ −3.17553e7 −1.25048
$$917$$ − 1.33081e7i − 0.522627i
$$918$$ − 4.64947e7i − 1.82095i
$$919$$ −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
$$922$$ 4.64374e7i 1.79904i
$$923$$ − 2.20234e7i − 0.850903i
$$924$$ −1.35740e7 −0.523031
$$925$$ 0 0
$$926$$ 7.59369e7 2.91022
$$927$$ − 4.00301e7i − 1.52998i
$$928$$ 9.37914e6i 0.357514i
$$929$$ 1.81557e7 0.690197 0.345098 0.938567i $$-0.387846\pi$$
0.345098 + 0.938567i $$0.387846\pi$$
$$930$$ 0 0
$$931$$ 1.26150e7 0.476993
$$932$$ 5.57230e7i 2.10133i
$$933$$ 5.44047e6i 0.204612i
$$934$$ −1.89552e7 −0.710987
$$935$$ 0 0
$$936$$ 3.85884e7 1.43968
$$937$$ 1.78946e7i 0.665844i 0.942954 + 0.332922i $$0.108035\pi$$
−0.942954 + 0.332922i $$0.891965\pi$$
$$938$$ 5.46210e6i 0.202700i
$$939$$ 1.14157e7 0.422512
$$940$$ 0 0
$$941$$ −3.04463e6 −0.112088 −0.0560441 0.998428i $$-0.517849\pi$$
−0.0560441 + 0.998428i $$0.517849\pi$$
$$942$$ 1.95262e7i 0.716954i
$$943$$ − 7.39891e6i − 0.270950i
$$944$$ −4.24825e7 −1.55160
$$945$$ 0 0
$$946$$ −7.49151e7 −2.72171
$$947$$ 3.17110e7i 1.14904i 0.818491 + 0.574519i $$0.194811\pi$$
−0.818491 + 0.574519i $$0.805189\pi$$
$$948$$ − 2.52127e6i − 0.0911169i
$$949$$ −1.87532e7 −0.675944
$$950$$ 0 0
$$951$$ −6.31686e6 −0.226491
$$952$$ 5.26342e7i 1.88224i
$$953$$ − 1.01913e7i − 0.363494i −0.983345 0.181747i $$-0.941825\pi$$
0.983345 0.181747i $$-0.0581752\pi$$
$$954$$ −4.21432e7 −1.49919
$$955$$ 0 0
$$956$$ −7.24764e7 −2.56479
$$957$$ − 3.59574e6i − 0.126914i
$$958$$ 3.13084e7i 1.10216i
$$959$$ −1.72613e7 −0.606077
$$960$$ 0 0
$$961$$ 3.52157e7 1.23006
$$962$$ 8.66322e6i 0.301816i
$$963$$ 1.44250e7i 0.501246i
$$964$$ 2.63095e7 0.911844
$$965$$ 0 0
$$966$$ −2.68481e6 −0.0925699
$$967$$ − 3.21125e7i − 1.10435i −0.833727 0.552177i $$-0.813797\pi$$
0.833727 0.552177i $$-0.186203\pi$$
$$968$$ − 3.18993e7i − 1.09419i
$$969$$ −1.04111e7 −0.356196
$$970$$ 0 0
$$971$$ 2.29867e7 0.782399 0.391200 0.920306i $$-0.372060\pi$$
0.391200 + 0.920306i $$0.372060\pi$$
$$972$$ − 6.01055e7i − 2.04056i
$$973$$ 1.50763e7i 0.510519i
$$974$$ −7.47219e7 −2.52378
$$975$$ 0 0
$$976$$ −5.55687e7 −1.86726
$$977$$ − 2.47331e7i − 0.828978i −0.910054 0.414489i $$-0.863960\pi$$
0.910054 0.414489i $$-0.136040\pi$$
$$978$$ 2.07599e7i 0.694029i
$$979$$ 2.16954e7 0.723455
$$980$$ 0 0
$$981$$ 4.65960e7 1.54588
$$982$$ − 6.77785e7i − 2.24292i
$$983$$ − 5.57031e7i − 1.83863i −0.393518 0.919317i $$-0.628742\pi$$
0.393518 0.919317i $$-0.371258\pi$$
$$984$$ 2.52270e7 0.830574
$$985$$ 0 0
$$986$$ −2.47431e7 −0.810518
$$987$$ − 341035.i − 0.0111431i
$$988$$ − 3.28704e7i − 1.07130i
$$989$$ −1.03150e7 −0.335335
$$990$$ 0 0
$$991$$ −6.86029e6 −0.221901 −0.110950 0.993826i $$-0.535389\pi$$
−0.110950 + 0.993826i $$0.535389\pi$$
$$992$$ 5.59673e7i 1.80574i
$$993$$ 1.13373e6i 0.0364868i
$$994$$ 3.63772e7 1.16778
$$995$$ 0 0
$$996$$ −2.14436e7 −0.684936
$$997$$ 6.03725e7i 1.92354i 0.273856 + 0.961771i $$0.411701\pi$$
−0.273856 + 0.961771i $$0.588299\pi$$
$$998$$ − 4.99770e7i − 1.58834i
$$999$$ 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 25.6.b.b.24.4 4
3.2 odd 2 225.6.b.i.199.1 4
4.3 odd 2 400.6.c.n.49.2 4
5.2 odd 4 25.6.a.b.1.1 2
5.3 odd 4 25.6.a.d.1.2 yes 2
5.4 even 2 inner 25.6.b.b.24.1 4
15.2 even 4 225.6.a.s.1.2 2
15.8 even 4 225.6.a.l.1.1 2
15.14 odd 2 225.6.b.i.199.4 4
20.3 even 4 400.6.a.o.1.2 2
20.7 even 4 400.6.a.w.1.1 2
20.19 odd 2 400.6.c.n.49.3 4

By twisted newform
Twist Min Dim Char Parity Ord Type
25.6.a.b.1.1 2 5.2 odd 4
25.6.a.d.1.2 yes 2 5.3 odd 4
25.6.b.b.24.1 4 5.4 even 2 inner
25.6.b.b.24.4 4 1.1 even 1 trivial
225.6.a.l.1.1 2 15.8 even 4
225.6.a.s.1.2 2 15.2 even 4
225.6.b.i.199.1 4 3.2 odd 2
225.6.b.i.199.4 4 15.14 odd 2
400.6.a.o.1.2 2 20.3 even 4
400.6.a.w.1.1 2 20.7 even 4
400.6.c.n.49.2 4 4.3 odd 2
400.6.c.n.49.3 4 20.19 odd 2