# Properties

 Label 525.4.d.l.274.2 Level 525 Weight 4 Character 525.274 Analytic conductor 30.976 Analytic rank 0 Dimension 4 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$525 = 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 525.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$30.9760027530$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 105) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 274.2 Root $$-0.707107 - 0.707107i$$ of defining polynomial Character $$\chi$$ $$=$$ 525.274 Dual form 525.4.d.l.274.3

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.82843i q^{2} -3.00000i q^{3} +4.65685 q^{4} -5.48528 q^{6} +7.00000i q^{7} -23.1421i q^{8} -9.00000 q^{9} +O(q^{10})$$ $$q-1.82843i q^{2} -3.00000i q^{3} +4.65685 q^{4} -5.48528 q^{6} +7.00000i q^{7} -23.1421i q^{8} -9.00000 q^{9} -64.5685 q^{11} -13.9706i q^{12} -32.3431i q^{13} +12.7990 q^{14} -5.05887 q^{16} +56.3431i q^{17} +16.4558i q^{18} +2.74517 q^{19} +21.0000 q^{21} +118.059i q^{22} +88.1665i q^{23} -69.4264 q^{24} -59.1371 q^{26} +27.0000i q^{27} +32.5980i q^{28} -246.735 q^{29} -110.912 q^{31} -175.887i q^{32} +193.706i q^{33} +103.019 q^{34} -41.9117 q^{36} -120.676i q^{37} -5.01934i q^{38} -97.0294 q^{39} -176.274 q^{41} -38.3970i q^{42} -443.362i q^{43} -300.686 q^{44} +161.206 q^{46} +345.206i q^{47} +15.1766i q^{48} -49.0000 q^{49} +169.029 q^{51} -150.617i q^{52} +260.981i q^{53} +49.3675 q^{54} +161.995 q^{56} -8.23550i q^{57} +451.137i q^{58} -628.999 q^{59} -115.206 q^{61} +202.794i q^{62} -63.0000i q^{63} -362.068 q^{64} +354.177 q^{66} +951.480i q^{67} +262.382i q^{68} +264.500 q^{69} +356.264 q^{71} +208.279i q^{72} -656.754i q^{73} -220.648 q^{74} +12.7838 q^{76} -451.980i q^{77} +177.411i q^{78} -440.195 q^{79} +81.0000 q^{81} +322.304i q^{82} -54.4121i q^{83} +97.7939 q^{84} -810.656 q^{86} +740.205i q^{87} +1494.25i q^{88} +1018.78 q^{89} +226.402 q^{91} +410.579i q^{92} +332.735i q^{93} +631.184 q^{94} -527.662 q^{96} +724.108i q^{97} +89.5929i q^{98} +581.117 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{4} + 12q^{6} - 36q^{9} + O(q^{10})$$ $$4q - 4q^{4} + 12q^{6} - 36q^{9} - 32q^{11} - 28q^{14} - 156q^{16} + 192q^{19} + 84q^{21} - 108q^{24} + 216q^{26} - 376q^{29} - 240q^{31} - 312q^{34} + 36q^{36} - 456q^{39} + 200q^{41} - 1248q^{44} + 1120q^{46} - 196q^{49} + 744q^{51} - 108q^{54} + 252q^{56} - 208q^{59} - 936q^{61} - 68q^{64} + 1824q^{66} - 96q^{69} - 272q^{71} - 2376q^{74} - 1216q^{76} + 864q^{79} + 324q^{81} - 84q^{84} - 912q^{86} + 2808q^{89} + 1064q^{91} - 3200q^{94} - 2484q^{96} + 288q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/525\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$176$$ $$451$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 1.82843i − 0.646447i −0.946323 0.323223i $$-0.895234\pi$$
0.946323 0.323223i $$-0.104766\pi$$
$$3$$ − 3.00000i − 0.577350i
$$4$$ 4.65685 0.582107
$$5$$ 0 0
$$6$$ −5.48528 −0.373226
$$7$$ 7.00000i 0.377964i
$$8$$ − 23.1421i − 1.02275i
$$9$$ −9.00000 −0.333333
$$10$$ 0 0
$$11$$ −64.5685 −1.76983 −0.884916 0.465751i $$-0.845784\pi$$
−0.884916 + 0.465751i $$0.845784\pi$$
$$12$$ − 13.9706i − 0.336080i
$$13$$ − 32.3431i − 0.690029i −0.938597 0.345014i $$-0.887874\pi$$
0.938597 0.345014i $$-0.112126\pi$$
$$14$$ 12.7990 0.244334
$$15$$ 0 0
$$16$$ −5.05887 −0.0790449
$$17$$ 56.3431i 0.803836i 0.915676 + 0.401918i $$0.131656\pi$$
−0.915676 + 0.401918i $$0.868344\pi$$
$$18$$ 16.4558i 0.215482i
$$19$$ 2.74517 0.0331465 0.0165733 0.999863i $$-0.494724\pi$$
0.0165733 + 0.999863i $$0.494724\pi$$
$$20$$ 0 0
$$21$$ 21.0000 0.218218
$$22$$ 118.059i 1.14410i
$$23$$ 88.1665i 0.799304i 0.916667 + 0.399652i $$0.130869\pi$$
−0.916667 + 0.399652i $$0.869131\pi$$
$$24$$ −69.4264 −0.590484
$$25$$ 0 0
$$26$$ −59.1371 −0.446067
$$27$$ 27.0000i 0.192450i
$$28$$ 32.5980i 0.220016i
$$29$$ −246.735 −1.57992 −0.789958 0.613161i $$-0.789898\pi$$
−0.789958 + 0.613161i $$0.789898\pi$$
$$30$$ 0 0
$$31$$ −110.912 −0.642591 −0.321296 0.946979i $$-0.604118\pi$$
−0.321296 + 0.946979i $$0.604118\pi$$
$$32$$ − 175.887i − 0.971649i
$$33$$ 193.706i 1.02181i
$$34$$ 103.019 0.519637
$$35$$ 0 0
$$36$$ −41.9117 −0.194036
$$37$$ − 120.676i − 0.536190i −0.963392 0.268095i $$-0.913606\pi$$
0.963392 0.268095i $$-0.0863942\pi$$
$$38$$ − 5.01934i − 0.0214275i
$$39$$ −97.0294 −0.398388
$$40$$ 0 0
$$41$$ −176.274 −0.671449 −0.335724 0.941960i $$-0.608981\pi$$
−0.335724 + 0.941960i $$0.608981\pi$$
$$42$$ − 38.3970i − 0.141066i
$$43$$ − 443.362i − 1.57238i −0.617988 0.786188i $$-0.712052\pi$$
0.617988 0.786188i $$-0.287948\pi$$
$$44$$ −300.686 −1.03023
$$45$$ 0 0
$$46$$ 161.206 0.516707
$$47$$ 345.206i 1.07135i 0.844424 + 0.535675i $$0.179943\pi$$
−0.844424 + 0.535675i $$0.820057\pi$$
$$48$$ 15.1766i 0.0456366i
$$49$$ −49.0000 −0.142857
$$50$$ 0 0
$$51$$ 169.029 0.464095
$$52$$ − 150.617i − 0.401670i
$$53$$ 260.981i 0.676386i 0.941077 + 0.338193i $$0.109816\pi$$
−0.941077 + 0.338193i $$0.890184\pi$$
$$54$$ 49.3675 0.124409
$$55$$ 0 0
$$56$$ 161.995 0.386562
$$57$$ − 8.23550i − 0.0191372i
$$58$$ 451.137i 1.02133i
$$59$$ −628.999 −1.38794 −0.693972 0.720002i $$-0.744141\pi$$
−0.693972 + 0.720002i $$0.744141\pi$$
$$60$$ 0 0
$$61$$ −115.206 −0.241814 −0.120907 0.992664i $$-0.538580\pi$$
−0.120907 + 0.992664i $$0.538580\pi$$
$$62$$ 202.794i 0.415401i
$$63$$ − 63.0000i − 0.125988i
$$64$$ −362.068 −0.707164
$$65$$ 0 0
$$66$$ 354.177 0.660547
$$67$$ 951.480i 1.73495i 0.497479 + 0.867476i $$0.334259\pi$$
−0.497479 + 0.867476i $$0.665741\pi$$
$$68$$ 262.382i 0.467919i
$$69$$ 264.500 0.461478
$$70$$ 0 0
$$71$$ 356.264 0.595504 0.297752 0.954643i $$-0.403763\pi$$
0.297752 + 0.954643i $$0.403763\pi$$
$$72$$ 208.279i 0.340916i
$$73$$ − 656.754i − 1.05298i −0.850183 0.526488i $$-0.823509\pi$$
0.850183 0.526488i $$-0.176491\pi$$
$$74$$ −220.648 −0.346618
$$75$$ 0 0
$$76$$ 12.7838 0.0192948
$$77$$ − 451.980i − 0.668933i
$$78$$ 177.411i 0.257537i
$$79$$ −440.195 −0.626909 −0.313455 0.949603i $$-0.601486\pi$$
−0.313455 + 0.949603i $$0.601486\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 322.304i 0.434056i
$$83$$ − 54.4121i − 0.0719579i −0.999353 0.0359790i $$-0.988545\pi$$
0.999353 0.0359790i $$-0.0114549\pi$$
$$84$$ 97.7939 0.127026
$$85$$ 0 0
$$86$$ −810.656 −1.01646
$$87$$ 740.205i 0.912165i
$$88$$ 1494.25i 1.81009i
$$89$$ 1018.78 1.21338 0.606690 0.794938i $$-0.292497\pi$$
0.606690 + 0.794938i $$0.292497\pi$$
$$90$$ 0 0
$$91$$ 226.402 0.260806
$$92$$ 410.579i 0.465280i
$$93$$ 332.735i 0.371000i
$$94$$ 631.184 0.692571
$$95$$ 0 0
$$96$$ −527.662 −0.560982
$$97$$ 724.108i 0.757959i 0.925405 + 0.378979i $$0.123725\pi$$
−0.925405 + 0.378979i $$0.876275\pi$$
$$98$$ 89.5929i 0.0923495i
$$99$$ 581.117 0.589944
$$100$$ 0 0
$$101$$ 268.725 0.264744 0.132372 0.991200i $$-0.457741\pi$$
0.132372 + 0.991200i $$0.457741\pi$$
$$102$$ − 309.058i − 0.300013i
$$103$$ − 1840.63i − 1.76080i −0.474233 0.880399i $$-0.657275\pi$$
0.474233 0.880399i $$-0.342725\pi$$
$$104$$ −748.489 −0.705725
$$105$$ 0 0
$$106$$ 477.184 0.437247
$$107$$ 243.087i 0.219627i 0.993952 + 0.109813i $$0.0350253\pi$$
−0.993952 + 0.109813i $$0.964975\pi$$
$$108$$ 125.735i 0.112027i
$$109$$ 405.176 0.356044 0.178022 0.984027i $$-0.443030\pi$$
0.178022 + 0.984027i $$0.443030\pi$$
$$110$$ 0 0
$$111$$ −362.029 −0.309570
$$112$$ − 35.4121i − 0.0298762i
$$113$$ − 28.1766i − 0.0234569i −0.999931 0.0117285i $$-0.996267\pi$$
0.999931 0.0117285i $$-0.00373337\pi$$
$$114$$ −15.0580 −0.0123712
$$115$$ 0 0
$$116$$ −1149.01 −0.919680
$$117$$ 291.088i 0.230010i
$$118$$ 1150.08i 0.897232i
$$119$$ −394.402 −0.303822
$$120$$ 0 0
$$121$$ 2838.10 2.13230
$$122$$ 210.646i 0.156320i
$$123$$ 528.823i 0.387661i
$$124$$ −516.500 −0.374057
$$125$$ 0 0
$$126$$ −115.191 −0.0814446
$$127$$ 2740.90i 1.91508i 0.288298 + 0.957541i $$0.406911\pi$$
−0.288298 + 0.957541i $$0.593089\pi$$
$$128$$ − 745.083i − 0.514505i
$$129$$ −1330.09 −0.907811
$$130$$ 0 0
$$131$$ −1832.04 −1.22188 −0.610938 0.791678i $$-0.709208\pi$$
−0.610938 + 0.791678i $$0.709208\pi$$
$$132$$ 902.059i 0.594804i
$$133$$ 19.2162i 0.0125282i
$$134$$ 1739.71 1.12155
$$135$$ 0 0
$$136$$ 1303.90 0.822122
$$137$$ − 382.747i − 0.238688i −0.992853 0.119344i $$-0.961921\pi$$
0.992853 0.119344i $$-0.0380792\pi$$
$$138$$ − 483.618i − 0.298321i
$$139$$ −3053.60 −1.86333 −0.931667 0.363314i $$-0.881645\pi$$
−0.931667 + 0.363314i $$0.881645\pi$$
$$140$$ 0 0
$$141$$ 1035.62 0.618545
$$142$$ − 651.403i − 0.384961i
$$143$$ 2088.35i 1.22123i
$$144$$ 45.5299 0.0263483
$$145$$ 0 0
$$146$$ −1200.83 −0.680692
$$147$$ 147.000i 0.0824786i
$$148$$ − 561.971i − 0.312120i
$$149$$ −3560.60 −1.95769 −0.978843 0.204611i $$-0.934407\pi$$
−0.978843 + 0.204611i $$0.934407\pi$$
$$150$$ 0 0
$$151$$ 3261.80 1.75789 0.878945 0.476923i $$-0.158248\pi$$
0.878945 + 0.476923i $$0.158248\pi$$
$$152$$ − 63.5290i − 0.0339005i
$$153$$ − 507.088i − 0.267945i
$$154$$ −826.412 −0.432430
$$155$$ 0 0
$$156$$ −451.852 −0.231905
$$157$$ − 2878.46i − 1.46322i −0.681723 0.731611i $$-0.738769\pi$$
0.681723 0.731611i $$-0.261231\pi$$
$$158$$ 804.865i 0.405263i
$$159$$ 782.942 0.390512
$$160$$ 0 0
$$161$$ −617.166 −0.302108
$$162$$ − 148.103i − 0.0718274i
$$163$$ − 927.537i − 0.445708i −0.974852 0.222854i $$-0.928463\pi$$
0.974852 0.222854i $$-0.0715373\pi$$
$$164$$ −820.883 −0.390855
$$165$$ 0 0
$$166$$ −99.4886 −0.0465169
$$167$$ − 1094.52i − 0.507164i −0.967314 0.253582i $$-0.918391\pi$$
0.967314 0.253582i $$-0.0816087\pi$$
$$168$$ − 485.985i − 0.223182i
$$169$$ 1150.92 0.523860
$$170$$ 0 0
$$171$$ −24.7065 −0.0110488
$$172$$ − 2064.67i − 0.915290i
$$173$$ − 1713.25i − 0.752926i −0.926432 0.376463i $$-0.877140\pi$$
0.926432 0.376463i $$-0.122860\pi$$
$$174$$ 1353.41 0.589666
$$175$$ 0 0
$$176$$ 326.644 0.139896
$$177$$ 1887.00i 0.801330i
$$178$$ − 1862.77i − 0.784386i
$$179$$ 4065.58 1.69763 0.848816 0.528689i $$-0.177316\pi$$
0.848816 + 0.528689i $$0.177316\pi$$
$$180$$ 0 0
$$181$$ −2791.40 −1.14631 −0.573157 0.819445i $$-0.694282\pi$$
−0.573157 + 0.819445i $$0.694282\pi$$
$$182$$ − 413.960i − 0.168597i
$$183$$ 345.618i 0.139611i
$$184$$ 2040.36 0.817486
$$185$$ 0 0
$$186$$ 608.382 0.239832
$$187$$ − 3637.99i − 1.42266i
$$188$$ 1607.57i 0.623640i
$$189$$ −189.000 −0.0727393
$$190$$ 0 0
$$191$$ −634.185 −0.240251 −0.120126 0.992759i $$-0.538330\pi$$
−0.120126 + 0.992759i $$0.538330\pi$$
$$192$$ 1086.20i 0.408281i
$$193$$ − 254.999i − 0.0951049i −0.998869 0.0475524i $$-0.984858\pi$$
0.998869 0.0475524i $$-0.0151421\pi$$
$$194$$ 1323.98 0.489980
$$195$$ 0 0
$$196$$ −228.186 −0.0831581
$$197$$ − 4172.37i − 1.50898i −0.656311 0.754490i $$-0.727884\pi$$
0.656311 0.754490i $$-0.272116\pi$$
$$198$$ − 1062.53i − 0.381367i
$$199$$ 4626.48 1.64805 0.824026 0.566552i $$-0.191723\pi$$
0.824026 + 0.566552i $$0.191723\pi$$
$$200$$ 0 0
$$201$$ 2854.44 1.00168
$$202$$ − 491.344i − 0.171143i
$$203$$ − 1727.15i − 0.597152i
$$204$$ 787.145 0.270153
$$205$$ 0 0
$$206$$ −3365.45 −1.13826
$$207$$ − 793.499i − 0.266435i
$$208$$ 163.620i 0.0545433i
$$209$$ −177.251 −0.0586638
$$210$$ 0 0
$$211$$ −1562.64 −0.509843 −0.254921 0.966962i $$-0.582050\pi$$
−0.254921 + 0.966962i $$0.582050\pi$$
$$212$$ 1215.35i 0.393729i
$$213$$ − 1068.79i − 0.343814i
$$214$$ 444.466 0.141977
$$215$$ 0 0
$$216$$ 624.838 0.196828
$$217$$ − 776.382i − 0.242877i
$$218$$ − 740.834i − 0.230163i
$$219$$ −1970.26 −0.607935
$$220$$ 0 0
$$221$$ 1822.31 0.554670
$$222$$ 661.943i 0.200120i
$$223$$ − 1236.39i − 0.371278i −0.982618 0.185639i $$-0.940564\pi$$
0.982618 0.185639i $$-0.0594355\pi$$
$$224$$ 1231.21 0.367249
$$225$$ 0 0
$$226$$ −51.5189 −0.0151637
$$227$$ − 4181.82i − 1.22272i −0.791353 0.611359i $$-0.790623\pi$$
0.791353 0.611359i $$-0.209377\pi$$
$$228$$ − 38.3515i − 0.0111399i
$$229$$ 484.774 0.139890 0.0699449 0.997551i $$-0.477718\pi$$
0.0699449 + 0.997551i $$0.477718\pi$$
$$230$$ 0 0
$$231$$ −1355.94 −0.386209
$$232$$ 5709.98i 1.61585i
$$233$$ 2080.54i 0.584982i 0.956268 + 0.292491i $$0.0944842\pi$$
−0.956268 + 0.292491i $$0.905516\pi$$
$$234$$ 532.234 0.148689
$$235$$ 0 0
$$236$$ −2929.16 −0.807932
$$237$$ 1320.59i 0.361946i
$$238$$ 721.135i 0.196404i
$$239$$ −6814.10 −1.84422 −0.922108 0.386933i $$-0.873534\pi$$
−0.922108 + 0.386933i $$0.873534\pi$$
$$240$$ 0 0
$$241$$ −3921.84 −1.04825 −0.524125 0.851642i $$-0.675607\pi$$
−0.524125 + 0.851642i $$0.675607\pi$$
$$242$$ − 5189.25i − 1.37842i
$$243$$ − 243.000i − 0.0641500i
$$244$$ −536.498 −0.140761
$$245$$ 0 0
$$246$$ 966.913 0.250602
$$247$$ − 88.7873i − 0.0228721i
$$248$$ 2566.73i 0.657209i
$$249$$ −163.236 −0.0415449
$$250$$ 0 0
$$251$$ −5219.10 −1.31246 −0.656228 0.754562i $$-0.727849\pi$$
−0.656228 + 0.754562i $$0.727849\pi$$
$$252$$ − 293.382i − 0.0733386i
$$253$$ − 5692.78i − 1.41463i
$$254$$ 5011.53 1.23800
$$255$$ 0 0
$$256$$ −4258.88 −1.03976
$$257$$ − 6975.71i − 1.69312i −0.532289 0.846562i $$-0.678668\pi$$
0.532289 0.846562i $$-0.321332\pi$$
$$258$$ 2431.97i 0.586852i
$$259$$ 844.733 0.202661
$$260$$ 0 0
$$261$$ 2220.62 0.526639
$$262$$ 3349.75i 0.789878i
$$263$$ − 3607.36i − 0.845776i −0.906182 0.422888i $$-0.861016\pi$$
0.906182 0.422888i $$-0.138984\pi$$
$$264$$ 4482.76 1.04506
$$265$$ 0 0
$$266$$ 35.1354 0.00809882
$$267$$ − 3056.35i − 0.700546i
$$268$$ 4430.90i 1.00993i
$$269$$ −5.88572 −0.00133405 −0.000667023 1.00000i $$-0.500212\pi$$
−0.000667023 1.00000i $$0.500212\pi$$
$$270$$ 0 0
$$271$$ 6916.32 1.55032 0.775160 0.631765i $$-0.217669\pi$$
0.775160 + 0.631765i $$0.217669\pi$$
$$272$$ − 285.033i − 0.0635392i
$$273$$ − 679.206i − 0.150577i
$$274$$ −699.825 −0.154299
$$275$$ 0 0
$$276$$ 1231.74 0.268630
$$277$$ 2119.46i 0.459733i 0.973222 + 0.229867i $$0.0738290\pi$$
−0.973222 + 0.229867i $$0.926171\pi$$
$$278$$ 5583.29i 1.20455i
$$279$$ 998.205 0.214197
$$280$$ 0 0
$$281$$ −239.917 −0.0509334 −0.0254667 0.999676i $$-0.508107\pi$$
−0.0254667 + 0.999676i $$0.508107\pi$$
$$282$$ − 1893.55i − 0.399856i
$$283$$ − 4542.12i − 0.954067i −0.878885 0.477034i $$-0.841712\pi$$
0.878885 0.477034i $$-0.158288\pi$$
$$284$$ 1659.07 0.346647
$$285$$ 0 0
$$286$$ 3818.40 0.789463
$$287$$ − 1233.92i − 0.253784i
$$288$$ 1582.99i 0.323883i
$$289$$ 1738.45 0.353847
$$290$$ 0 0
$$291$$ 2172.32 0.437608
$$292$$ − 3058.41i − 0.612944i
$$293$$ − 2171.70i − 0.433010i −0.976281 0.216505i $$-0.930534\pi$$
0.976281 0.216505i $$-0.0694658\pi$$
$$294$$ 268.779 0.0533180
$$295$$ 0 0
$$296$$ −2792.70 −0.548387
$$297$$ − 1743.35i − 0.340604i
$$298$$ 6510.29i 1.26554i
$$299$$ 2851.58 0.551543
$$300$$ 0 0
$$301$$ 3103.54 0.594302
$$302$$ − 5963.96i − 1.13638i
$$303$$ − 806.175i − 0.152850i
$$304$$ −13.8875 −0.00262007
$$305$$ 0 0
$$306$$ −927.174 −0.173212
$$307$$ 3508.64i 0.652276i 0.945322 + 0.326138i $$0.105747\pi$$
−0.945322 + 0.326138i $$0.894253\pi$$
$$308$$ − 2104.80i − 0.389391i
$$309$$ −5521.88 −1.01660
$$310$$ 0 0
$$311$$ −3133.25 −0.571287 −0.285643 0.958336i $$-0.592207\pi$$
−0.285643 + 0.958336i $$0.592207\pi$$
$$312$$ 2245.47i 0.407451i
$$313$$ 6389.59i 1.15387i 0.816790 + 0.576935i $$0.195751\pi$$
−0.816790 + 0.576935i $$0.804249\pi$$
$$314$$ −5263.05 −0.945895
$$315$$ 0 0
$$316$$ −2049.92 −0.364928
$$317$$ − 1634.44i − 0.289587i −0.989462 0.144794i $$-0.953748\pi$$
0.989462 0.144794i $$-0.0462518\pi$$
$$318$$ − 1431.55i − 0.252445i
$$319$$ 15931.3 2.79618
$$320$$ 0 0
$$321$$ 729.260 0.126802
$$322$$ 1128.44i 0.195297i
$$323$$ 154.671i 0.0266444i
$$324$$ 377.205 0.0646785
$$325$$ 0 0
$$326$$ −1695.93 −0.288126
$$327$$ − 1215.53i − 0.205562i
$$328$$ 4079.36i 0.686723i
$$329$$ −2416.44 −0.404932
$$330$$ 0 0
$$331$$ 4386.17 0.728355 0.364177 0.931330i $$-0.381350\pi$$
0.364177 + 0.931330i $$0.381350\pi$$
$$332$$ − 253.389i − 0.0418872i
$$333$$ 1086.09i 0.178730i
$$334$$ −2001.25 −0.327854
$$335$$ 0 0
$$336$$ −106.236 −0.0172490
$$337$$ − 1713.98i − 0.277051i −0.990359 0.138526i $$-0.955764\pi$$
0.990359 0.138526i $$-0.0442363\pi$$
$$338$$ − 2104.38i − 0.338648i
$$339$$ −84.5299 −0.0135429
$$340$$ 0 0
$$341$$ 7161.41 1.13728
$$342$$ 45.1740i 0.00714249i
$$343$$ − 343.000i − 0.0539949i
$$344$$ −10260.4 −1.60814
$$345$$ 0 0
$$346$$ −3132.56 −0.486727
$$347$$ 1744.83i 0.269935i 0.990850 + 0.134967i $$0.0430930\pi$$
−0.990850 + 0.134967i $$0.956907\pi$$
$$348$$ 3447.03i 0.530977i
$$349$$ −7046.78 −1.08082 −0.540409 0.841403i $$-0.681730\pi$$
−0.540409 + 0.841403i $$0.681730\pi$$
$$350$$ 0 0
$$351$$ 873.265 0.132796
$$352$$ 11356.8i 1.71966i
$$353$$ − 12668.5i − 1.91013i −0.296400 0.955064i $$-0.595786\pi$$
0.296400 0.955064i $$-0.404214\pi$$
$$354$$ 3450.24 0.518017
$$355$$ 0 0
$$356$$ 4744.33 0.706317
$$357$$ 1183.21i 0.175411i
$$358$$ − 7433.62i − 1.09743i
$$359$$ −37.7844 −0.00555483 −0.00277742 0.999996i $$-0.500884\pi$$
−0.00277742 + 0.999996i $$0.500884\pi$$
$$360$$ 0 0
$$361$$ −6851.46 −0.998901
$$362$$ 5103.87i 0.741031i
$$363$$ − 8514.29i − 1.23109i
$$364$$ 1054.32 0.151817
$$365$$ 0 0
$$366$$ 631.938 0.0902511
$$367$$ 759.829i 0.108073i 0.998539 + 0.0540364i $$0.0172087\pi$$
−0.998539 + 0.0540364i $$0.982791\pi$$
$$368$$ − 446.023i − 0.0631809i
$$369$$ 1586.47 0.223816
$$370$$ 0 0
$$371$$ −1826.86 −0.255650
$$372$$ 1549.50i 0.215962i
$$373$$ 719.320i 0.0998525i 0.998753 + 0.0499263i $$0.0158986\pi$$
−0.998753 + 0.0499263i $$0.984101\pi$$
$$374$$ −6651.81 −0.919671
$$375$$ 0 0
$$376$$ 7988.81 1.09572
$$377$$ 7980.19i 1.09019i
$$378$$ 345.573i 0.0470221i
$$379$$ −572.559 −0.0775999 −0.0388000 0.999247i $$-0.512354\pi$$
−0.0388000 + 0.999247i $$0.512354\pi$$
$$380$$ 0 0
$$381$$ 8222.69 1.10567
$$382$$ 1159.56i 0.155310i
$$383$$ 4513.18i 0.602122i 0.953605 + 0.301061i $$0.0973408\pi$$
−0.953605 + 0.301061i $$0.902659\pi$$
$$384$$ −2235.25 −0.297050
$$385$$ 0 0
$$386$$ −466.247 −0.0614802
$$387$$ 3990.26i 0.524125i
$$388$$ 3372.06i 0.441213i
$$389$$ 6902.13 0.899619 0.449810 0.893124i $$-0.351492\pi$$
0.449810 + 0.893124i $$0.351492\pi$$
$$390$$ 0 0
$$391$$ −4967.58 −0.642510
$$392$$ 1133.96i 0.146107i
$$393$$ 5496.11i 0.705451i
$$394$$ −7628.88 −0.975475
$$395$$ 0 0
$$396$$ 2706.18 0.343410
$$397$$ − 4124.58i − 0.521427i −0.965416 0.260714i $$-0.916042\pi$$
0.965416 0.260714i $$-0.0839579\pi$$
$$398$$ − 8459.18i − 1.06538i
$$399$$ 57.6485 0.00723317
$$400$$ 0 0
$$401$$ −1002.50 −0.124844 −0.0624219 0.998050i $$-0.519882\pi$$
−0.0624219 + 0.998050i $$0.519882\pi$$
$$402$$ − 5219.14i − 0.647530i
$$403$$ 3587.23i 0.443406i
$$404$$ 1251.41 0.154109
$$405$$ 0 0
$$406$$ −3157.96 −0.386027
$$407$$ 7791.89i 0.948967i
$$408$$ − 3911.70i − 0.474652i
$$409$$ −10335.0 −1.24947 −0.624736 0.780836i $$-0.714794\pi$$
−0.624736 + 0.780836i $$0.714794\pi$$
$$410$$ 0 0
$$411$$ −1148.24 −0.137807
$$412$$ − 8571.53i − 1.02497i
$$413$$ − 4402.99i − 0.524594i
$$414$$ −1450.85 −0.172236
$$415$$ 0 0
$$416$$ −5688.75 −0.670466
$$417$$ 9160.81i 1.07580i
$$418$$ 324.091i 0.0379230i
$$419$$ −3183.21 −0.371145 −0.185573 0.982631i $$-0.559414\pi$$
−0.185573 + 0.982631i $$0.559414\pi$$
$$420$$ 0 0
$$421$$ −6944.34 −0.803911 −0.401956 0.915659i $$-0.631669\pi$$
−0.401956 + 0.915659i $$0.631669\pi$$
$$422$$ 2857.18i 0.329586i
$$423$$ − 3106.85i − 0.357117i
$$424$$ 6039.65 0.691772
$$425$$ 0 0
$$426$$ −1954.21 −0.222258
$$427$$ − 806.442i − 0.0913969i
$$428$$ 1132.02i 0.127846i
$$429$$ 6265.05 0.705080
$$430$$ 0 0
$$431$$ 3868.41 0.432331 0.216166 0.976357i $$-0.430645\pi$$
0.216166 + 0.976357i $$0.430645\pi$$
$$432$$ − 136.590i − 0.0152122i
$$433$$ − 6132.96i − 0.680673i −0.940304 0.340336i $$-0.889459\pi$$
0.940304 0.340336i $$-0.110541\pi$$
$$434$$ −1419.56 −0.157007
$$435$$ 0 0
$$436$$ 1886.84 0.207256
$$437$$ 242.032i 0.0264942i
$$438$$ 3602.48i 0.392998i
$$439$$ 4090.14 0.444673 0.222337 0.974970i $$-0.428632\pi$$
0.222337 + 0.974970i $$0.428632\pi$$
$$440$$ 0 0
$$441$$ 441.000 0.0476190
$$442$$ − 3331.97i − 0.358565i
$$443$$ 12434.5i 1.33359i 0.745241 + 0.666795i $$0.232334\pi$$
−0.745241 + 0.666795i $$0.767666\pi$$
$$444$$ −1685.91 −0.180203
$$445$$ 0 0
$$446$$ −2260.66 −0.240012
$$447$$ 10681.8i 1.13027i
$$448$$ − 2534.48i − 0.267283i
$$449$$ −883.046 −0.0928141 −0.0464071 0.998923i $$-0.514777\pi$$
−0.0464071 + 0.998923i $$0.514777\pi$$
$$450$$ 0 0
$$451$$ 11381.8 1.18835
$$452$$ − 131.214i − 0.0136544i
$$453$$ − 9785.40i − 1.01492i
$$454$$ −7646.15 −0.790422
$$455$$ 0 0
$$456$$ −190.587 −0.0195725
$$457$$ 9068.44i 0.928235i 0.885774 + 0.464118i $$0.153629\pi$$
−0.885774 + 0.464118i $$0.846371\pi$$
$$458$$ − 886.373i − 0.0904312i
$$459$$ −1521.26 −0.154698
$$460$$ 0 0
$$461$$ 12508.9 1.26377 0.631885 0.775063i $$-0.282282\pi$$
0.631885 + 0.775063i $$0.282282\pi$$
$$462$$ 2479.24i 0.249663i
$$463$$ 12688.7i 1.27363i 0.771015 + 0.636817i $$0.219749\pi$$
−0.771015 + 0.636817i $$0.780251\pi$$
$$464$$ 1248.20 0.124884
$$465$$ 0 0
$$466$$ 3804.12 0.378160
$$467$$ 10136.5i 1.00442i 0.864747 + 0.502208i $$0.167479\pi$$
−0.864747 + 0.502208i $$0.832521\pi$$
$$468$$ 1355.56i 0.133890i
$$469$$ −6660.36 −0.655750
$$470$$ 0 0
$$471$$ −8635.37 −0.844791
$$472$$ 14556.4i 1.41952i
$$473$$ 28627.3i 2.78284i
$$474$$ 2414.59 0.233979
$$475$$ 0 0
$$476$$ −1836.67 −0.176857
$$477$$ − 2348.83i − 0.225462i
$$478$$ 12459.1i 1.19219i
$$479$$ 11361.1 1.08372 0.541861 0.840468i $$-0.317720\pi$$
0.541861 + 0.840468i $$0.317720\pi$$
$$480$$ 0 0
$$481$$ −3903.05 −0.369987
$$482$$ 7170.80i 0.677637i
$$483$$ 1851.50i 0.174422i
$$484$$ 13216.6 1.24123
$$485$$ 0 0
$$486$$ −444.308 −0.0414696
$$487$$ − 7929.53i − 0.737826i −0.929464 0.368913i $$-0.879730\pi$$
0.929464 0.368913i $$-0.120270\pi$$
$$488$$ 2666.11i 0.247314i
$$489$$ −2782.61 −0.257329
$$490$$ 0 0
$$491$$ 8111.51 0.745555 0.372777 0.927921i $$-0.378405\pi$$
0.372777 + 0.927921i $$0.378405\pi$$
$$492$$ 2462.65i 0.225660i
$$493$$ − 13901.8i − 1.26999i
$$494$$ −162.341 −0.0147856
$$495$$ 0 0
$$496$$ 561.088 0.0507936
$$497$$ 2493.85i 0.225079i
$$498$$ 298.466i 0.0268566i
$$499$$ −16816.6 −1.50865 −0.754324 0.656502i $$-0.772035\pi$$
−0.754324 + 0.656502i $$0.772035\pi$$
$$500$$ 0 0
$$501$$ −3283.55 −0.292811
$$502$$ 9542.74i 0.848433i
$$503$$ − 17764.6i − 1.57472i −0.616491 0.787362i $$-0.711446\pi$$
0.616491 0.787362i $$-0.288554\pi$$
$$504$$ −1457.95 −0.128854
$$505$$ 0 0
$$506$$ −10408.8 −0.914485
$$507$$ − 3452.76i − 0.302451i
$$508$$ 12764.0i 1.11478i
$$509$$ −13908.8 −1.21120 −0.605598 0.795771i $$-0.707066\pi$$
−0.605598 + 0.795771i $$0.707066\pi$$
$$510$$ 0 0
$$511$$ 4597.27 0.397987
$$512$$ 1826.38i 0.157647i
$$513$$ 74.1195i 0.00637905i
$$514$$ −12754.6 −1.09451
$$515$$ 0 0
$$516$$ −6194.02 −0.528443
$$517$$ − 22289.5i − 1.89611i
$$518$$ − 1544.53i − 0.131009i
$$519$$ −5139.76 −0.434702
$$520$$ 0 0
$$521$$ −8639.68 −0.726510 −0.363255 0.931690i $$-0.618335\pi$$
−0.363255 + 0.931690i $$0.618335\pi$$
$$522$$ − 4060.23i − 0.340444i
$$523$$ 23242.2i 1.94323i 0.236561 + 0.971617i $$0.423980\pi$$
−0.236561 + 0.971617i $$0.576020\pi$$
$$524$$ −8531.53 −0.711263
$$525$$ 0 0
$$526$$ −6595.79 −0.546749
$$527$$ − 6249.11i − 0.516538i
$$528$$ − 979.932i − 0.0807691i
$$529$$ 4393.66 0.361113
$$530$$ 0 0
$$531$$ 5660.99 0.462648
$$532$$ 89.4869i 0.00729276i
$$533$$ 5701.26i 0.463319i
$$534$$ −5588.32 −0.452865
$$535$$ 0 0
$$536$$ 22019.3 1.77442
$$537$$ − 12196.8i − 0.980128i
$$538$$ 10.7616i 0 0.000862390i
$$539$$ 3163.86 0.252833
$$540$$ 0 0
$$541$$ 11395.2 0.905577 0.452789 0.891618i $$-0.350429\pi$$
0.452789 + 0.891618i $$0.350429\pi$$
$$542$$ − 12646.0i − 1.00220i
$$543$$ 8374.19i 0.661825i
$$544$$ 9910.04 0.781047
$$545$$ 0 0
$$546$$ −1241.88 −0.0973398
$$547$$ 7870.21i 0.615184i 0.951518 + 0.307592i $$0.0995232\pi$$
−0.951518 + 0.307592i $$0.900477\pi$$
$$548$$ − 1782.40i − 0.138942i
$$549$$ 1036.85 0.0806045
$$550$$ 0 0
$$551$$ −677.329 −0.0523687
$$552$$ − 6121.08i − 0.471976i
$$553$$ − 3081.37i − 0.236949i
$$554$$ 3875.28 0.297193
$$555$$ 0 0
$$556$$ −14220.2 −1.08466
$$557$$ − 17769.8i − 1.35176i −0.737012 0.675880i $$-0.763764\pi$$
0.737012 0.675880i $$-0.236236\pi$$
$$558$$ − 1825.15i − 0.138467i
$$559$$ −14339.7 −1.08498
$$560$$ 0 0
$$561$$ −10914.0 −0.821370
$$562$$ 438.672i 0.0329257i
$$563$$ 15192.8i 1.13730i 0.822579 + 0.568651i $$0.192534\pi$$
−0.822579 + 0.568651i $$0.807466\pi$$
$$564$$ 4822.72 0.360059
$$565$$ 0 0
$$566$$ −8304.93 −0.616753
$$567$$ 567.000i 0.0419961i
$$568$$ − 8244.71i − 0.609050i
$$569$$ 23300.0 1.71667 0.858335 0.513090i $$-0.171499\pi$$
0.858335 + 0.513090i $$0.171499\pi$$
$$570$$ 0 0
$$571$$ 10638.2 0.779673 0.389837 0.920884i $$-0.372531\pi$$
0.389837 + 0.920884i $$0.372531\pi$$
$$572$$ 9725.14i 0.710889i
$$573$$ 1902.55i 0.138709i
$$574$$ −2256.13 −0.164058
$$575$$ 0 0
$$576$$ 3258.61 0.235721
$$577$$ 897.258i 0.0647372i 0.999476 + 0.0323686i $$0.0103050\pi$$
−0.999476 + 0.0323686i $$0.989695\pi$$
$$578$$ − 3178.63i − 0.228743i
$$579$$ −764.997 −0.0549088
$$580$$ 0 0
$$581$$ 380.885 0.0271975
$$582$$ − 3971.93i − 0.282890i
$$583$$ − 16851.1i − 1.19709i
$$584$$ −15198.7 −1.07693
$$585$$ 0 0
$$586$$ −3970.79 −0.279918
$$587$$ 14712.9i 1.03452i 0.855828 + 0.517261i $$0.173048\pi$$
−0.855828 + 0.517261i $$0.826952\pi$$
$$588$$ 684.558i 0.0480114i
$$589$$ −304.471 −0.0212997
$$590$$ 0 0
$$591$$ −12517.1 −0.871210
$$592$$ 610.486i 0.0423831i
$$593$$ − 7216.29i − 0.499726i −0.968281 0.249863i $$-0.919614\pi$$
0.968281 0.249863i $$-0.0803856\pi$$
$$594$$ −3187.59 −0.220182
$$595$$ 0 0
$$596$$ −16581.2 −1.13958
$$597$$ − 13879.4i − 0.951503i
$$598$$ − 5213.91i − 0.356543i
$$599$$ 20885.5 1.42464 0.712320 0.701855i $$-0.247645\pi$$
0.712320 + 0.701855i $$0.247645\pi$$
$$600$$ 0 0
$$601$$ −11047.7 −0.749823 −0.374911 0.927061i $$-0.622327\pi$$
−0.374911 + 0.927061i $$0.622327\pi$$
$$602$$ − 5674.59i − 0.384185i
$$603$$ − 8563.32i − 0.578317i
$$604$$ 15189.7 1.02328
$$605$$ 0 0
$$606$$ −1474.03 −0.0988093
$$607$$ 9434.94i 0.630894i 0.948943 + 0.315447i $$0.102154\pi$$
−0.948943 + 0.315447i $$0.897846\pi$$
$$608$$ − 482.840i − 0.0322068i
$$609$$ −5181.44 −0.344766
$$610$$ 0 0
$$611$$ 11165.0 0.739263
$$612$$ − 2361.44i − 0.155973i
$$613$$ − 17662.6i − 1.16376i −0.813274 0.581881i $$-0.802317\pi$$
0.813274 0.581881i $$-0.197683\pi$$
$$614$$ 6415.30 0.421662
$$615$$ 0 0
$$616$$ −10459.8 −0.684150
$$617$$ 10817.4i 0.705820i 0.935657 + 0.352910i $$0.114808\pi$$
−0.935657 + 0.352910i $$0.885192\pi$$
$$618$$ 10096.3i 0.657176i
$$619$$ −29073.9 −1.88785 −0.943926 0.330158i $$-0.892898\pi$$
−0.943926 + 0.330158i $$0.892898\pi$$
$$620$$ 0 0
$$621$$ −2380.50 −0.153826
$$622$$ 5728.92i 0.369306i
$$623$$ 7131.49i 0.458615i
$$624$$ 490.860 0.0314906
$$625$$ 0 0
$$626$$ 11682.9 0.745915
$$627$$ 531.754i 0.0338696i
$$628$$ − 13404.5i − 0.851751i
$$629$$ 6799.28 0.431009
$$630$$ 0 0
$$631$$ −2203.17 −0.138996 −0.0694981 0.997582i $$-0.522140\pi$$
−0.0694981 + 0.997582i $$0.522140\pi$$
$$632$$ 10187.1i 0.641170i
$$633$$ 4687.93i 0.294358i
$$634$$ −2988.45 −0.187203
$$635$$ 0 0
$$636$$ 3646.05 0.227319
$$637$$ 1584.81i 0.0985755i
$$638$$ − 29129.3i − 1.80758i
$$639$$ −3206.38 −0.198501
$$640$$ 0 0
$$641$$ 22466.5 1.38436 0.692180 0.721725i $$-0.256651\pi$$
0.692180 + 0.721725i $$0.256651\pi$$
$$642$$ − 1333.40i − 0.0819705i
$$643$$ − 12347.0i − 0.757257i −0.925549 0.378629i $$-0.876396\pi$$
0.925549 0.378629i $$-0.123604\pi$$
$$644$$ −2874.05 −0.175859
$$645$$ 0 0
$$646$$ 282.805 0.0172242
$$647$$ 24114.0i 1.46525i 0.680631 + 0.732626i $$0.261706\pi$$
−0.680631 + 0.732626i $$0.738294\pi$$
$$648$$ − 1874.51i − 0.113639i
$$649$$ 40613.6 2.45643
$$650$$ 0 0
$$651$$ −2329.15 −0.140225
$$652$$ − 4319.41i − 0.259449i
$$653$$ 7843.33i 0.470035i 0.971991 + 0.235018i $$0.0755148\pi$$
−0.971991 + 0.235018i $$0.924485\pi$$
$$654$$ −2222.50 −0.132885
$$655$$ 0 0
$$656$$ 891.749 0.0530746
$$657$$ 5910.78i 0.350992i
$$658$$ 4418.29i 0.261767i
$$659$$ −21242.8 −1.25569 −0.627846 0.778338i $$-0.716063\pi$$
−0.627846 + 0.778338i $$0.716063\pi$$
$$660$$ 0 0
$$661$$ −22221.7 −1.30760 −0.653801 0.756667i $$-0.726827\pi$$
−0.653801 + 0.756667i $$0.726827\pi$$
$$662$$ − 8019.79i − 0.470843i
$$663$$ − 5466.94i − 0.320239i
$$664$$ −1259.21 −0.0735948
$$665$$ 0 0
$$666$$ 1985.83 0.115539
$$667$$ − 21753.8i − 1.26283i
$$668$$ − 5097.01i − 0.295223i
$$669$$ −3709.18 −0.214358
$$670$$ 0 0
$$671$$ 7438.69 0.427969
$$672$$ − 3693.63i − 0.212031i
$$673$$ − 3787.85i − 0.216955i −0.994099 0.108478i $$-0.965402\pi$$
0.994099 0.108478i $$-0.0345976\pi$$
$$674$$ −3133.88 −0.179099
$$675$$ 0 0
$$676$$ 5359.67 0.304943
$$677$$ 11296.8i 0.641314i 0.947195 + 0.320657i $$0.103904\pi$$
−0.947195 + 0.320657i $$0.896096\pi$$
$$678$$ 154.557i 0.00875474i
$$679$$ −5068.75 −0.286481
$$680$$ 0 0
$$681$$ −12545.5 −0.705937
$$682$$ − 13094.1i − 0.735190i
$$683$$ 4807.14i 0.269312i 0.990892 + 0.134656i $$0.0429929\pi$$
−0.990892 + 0.134656i $$0.957007\pi$$
$$684$$ −115.055 −0.00643161
$$685$$ 0 0
$$686$$ −627.151 −0.0349048
$$687$$ − 1454.32i − 0.0807654i
$$688$$ 2242.92i 0.124288i
$$689$$ 8440.94 0.466726
$$690$$ 0 0
$$691$$ 5393.47 0.296928 0.148464 0.988918i $$-0.452567\pi$$
0.148464 + 0.988918i $$0.452567\pi$$
$$692$$ − 7978.37i − 0.438283i
$$693$$ 4067.82i 0.222978i
$$694$$ 3190.29 0.174498
$$695$$ 0 0
$$696$$ 17129.9 0.932914
$$697$$ − 9931.84i − 0.539735i
$$698$$ 12884.5i 0.698691i
$$699$$ 6241.63 0.337740
$$700$$ 0 0
$$701$$ 2404.77 0.129568 0.0647838 0.997899i $$-0.479364\pi$$
0.0647838 + 0.997899i $$0.479364\pi$$
$$702$$ − 1596.70i − 0.0858456i
$$703$$ − 331.276i − 0.0177729i
$$704$$ 23378.2 1.25156
$$705$$ 0 0
$$706$$ −23163.4 −1.23480
$$707$$ 1881.07i 0.100064i
$$708$$ 8787.47i 0.466460i
$$709$$ 21617.3 1.14507 0.572535 0.819881i $$-0.305960\pi$$
0.572535 + 0.819881i $$0.305960\pi$$
$$710$$ 0 0
$$711$$ 3961.76 0.208970
$$712$$ − 23576.8i − 1.24098i
$$713$$ − 9778.70i − 0.513626i
$$714$$ 2163.41 0.113394
$$715$$ 0 0
$$716$$ 18932.8 0.988203
$$717$$ 20442.3i 1.06476i
$$718$$ 69.0860i 0.00359090i
$$719$$ 18228.6 0.945498 0.472749 0.881197i $$-0.343262\pi$$
0.472749 + 0.881197i $$0.343262\pi$$
$$720$$ 0 0
$$721$$ 12884.4 0.665519
$$722$$ 12527.4i 0.645736i
$$723$$ 11765.5i 0.605207i
$$724$$ −12999.1 −0.667278
$$725$$ 0 0
$$726$$ −15567.8 −0.795832
$$727$$ − 20196.5i − 1.03033i −0.857092 0.515164i $$-0.827731\pi$$
0.857092 0.515164i $$-0.172269\pi$$
$$728$$ − 5239.43i − 0.266739i
$$729$$ −729.000 −0.0370370
$$730$$ 0 0
$$731$$ 24980.4 1.26393
$$732$$ 1609.49i 0.0812686i
$$733$$ − 15264.9i − 0.769196i −0.923084 0.384598i $$-0.874340\pi$$
0.923084 0.384598i $$-0.125660\pi$$
$$734$$ 1389.29 0.0698633
$$735$$ 0 0
$$736$$ 15507.4 0.776643
$$737$$ − 61435.7i − 3.07057i
$$738$$ − 2900.74i − 0.144685i
$$739$$ −13906.0 −0.692207 −0.346103 0.938196i $$-0.612495\pi$$
−0.346103 + 0.938196i $$0.612495\pi$$
$$740$$ 0 0
$$741$$ −266.362 −0.0132052
$$742$$ 3340.29i 0.165264i
$$743$$ 4592.87i 0.226778i 0.993551 + 0.113389i $$0.0361706\pi$$
−0.993551 + 0.113389i $$0.963829\pi$$
$$744$$ 7700.20 0.379440
$$745$$ 0 0
$$746$$ 1315.22 0.0645493
$$747$$ 489.709i 0.0239860i
$$748$$ − 16941.6i − 0.828137i
$$749$$ −1701.61 −0.0830111
$$750$$ 0 0
$$751$$ −8390.80 −0.407702 −0.203851 0.979002i $$-0.565346\pi$$
−0.203851 + 0.979002i $$0.565346\pi$$
$$752$$ − 1746.35i − 0.0846848i
$$753$$ 15657.3i 0.757747i
$$754$$ 14591.2 0.704748
$$755$$ 0 0
$$756$$ −880.145 −0.0423420
$$757$$ 1368.89i 0.0657240i 0.999460 + 0.0328620i $$0.0104622\pi$$
−0.999460 + 0.0328620i $$0.989538\pi$$
$$758$$ 1046.88i 0.0501642i
$$759$$ −17078.4 −0.816739
$$760$$ 0 0
$$761$$ −1623.77 −0.0773478 −0.0386739 0.999252i $$-0.512313\pi$$
−0.0386739 + 0.999252i $$0.512313\pi$$
$$762$$ − 15034.6i − 0.714759i
$$763$$ 2836.23i 0.134572i
$$764$$ −2953.31 −0.139852
$$765$$ 0 0
$$766$$ 8252.03 0.389240
$$767$$ 20343.8i 0.957722i
$$768$$ 12776.6i 0.600308i
$$769$$ 26842.5 1.25873 0.629366 0.777109i $$-0.283315\pi$$
0.629366 + 0.777109i $$0.283315\pi$$
$$770$$ 0 0
$$771$$ −20927.1 −0.977526
$$772$$ − 1187.49i − 0.0553612i
$$773$$ − 20961.4i − 0.975330i −0.873031 0.487665i $$-0.837849\pi$$
0.873031 0.487665i $$-0.162151\pi$$
$$774$$ 7295.90 0.338819
$$775$$ 0 0
$$776$$ 16757.4 0.775200
$$777$$ − 2534.20i − 0.117006i
$$778$$ − 12620.0i − 0.581556i
$$779$$ −483.902 −0.0222562
$$780$$ 0 0
$$781$$ −23003.5 −1.05394
$$782$$ 9082.86i 0.415348i
$$783$$ − 6661.85i − 0.304055i
$$784$$ 247.885 0.0112921
$$785$$ 0 0
$$786$$ 10049.2 0.456036
$$787$$ − 35333.2i − 1.60037i −0.599751 0.800187i $$-0.704734\pi$$
0.599751 0.800187i $$-0.295266\pi$$
$$788$$ − 19430.1i − 0.878388i
$$789$$ −10822.1 −0.488309
$$790$$ 0 0
$$791$$ 197.236 0.00886589
$$792$$ − 13448.3i − 0.603364i
$$793$$ 3726.13i 0.166858i
$$794$$ −7541.49 −0.337075
$$795$$ 0 0
$$796$$ 21544.8 0.959342
$$797$$ 8137.04i 0.361642i 0.983516 + 0.180821i $$0.0578755\pi$$
−0.983516 + 0.180821i $$0.942125\pi$$
$$798$$ − 105.406i − 0.00467586i
$$799$$ −19450.0 −0.861191
$$800$$ 0 0
$$801$$ −9169.05 −0.404460
$$802$$ 1832.99i 0.0807049i
$$803$$ 42405.6i 1.86359i
$$804$$ 13292.7 0.583082
$$805$$ 0 0
$$806$$ 6558.99 0.286639
$$807$$ 17.6572i 0 0.000770212i
$$808$$ − 6218.87i − 0.270766i
$$809$$ 36281.0 1.57673 0.788364 0.615209i $$-0.210928\pi$$
0.788364 + 0.615209i $$0.210928\pi$$
$$810$$ 0 0
$$811$$ −34237.2 −1.48240 −0.741202 0.671282i $$-0.765744\pi$$
−0.741202 + 0.671282i $$0.765744\pi$$
$$812$$ − 8043.06i − 0.347606i
$$813$$ − 20749.0i − 0.895077i
$$814$$ 14246.9 0.613456
$$815$$ 0 0
$$816$$ −855.099 −0.0366844
$$817$$ − 1217.10i − 0.0521188i
$$818$$ 18896.8i 0.807717i
$$819$$ −2037.62 −0.0869355
$$820$$ 0 0
$$821$$ −23247.8 −0.988250 −0.494125 0.869391i $$-0.664511\pi$$
−0.494125 + 0.869391i $$0.664511\pi$$
$$822$$ 2099.47i 0.0890846i
$$823$$ 42934.0i 1.81845i 0.416306 + 0.909225i $$0.363325\pi$$
−0.416306 + 0.909225i $$0.636675\pi$$
$$824$$ −42596.0 −1.80085
$$825$$ 0 0
$$826$$ −8050.55 −0.339122
$$827$$ 781.391i 0.0328557i 0.999865 + 0.0164278i $$0.00522938\pi$$
−0.999865 + 0.0164278i $$0.994771\pi$$
$$828$$ − 3695.21i − 0.155093i
$$829$$ 33493.3 1.40322 0.701611 0.712561i $$-0.252465\pi$$
0.701611 + 0.712561i $$0.252465\pi$$
$$830$$ 0 0
$$831$$ 6358.39 0.265427
$$832$$ 11710.4i 0.487964i
$$833$$ − 2760.81i − 0.114834i
$$834$$ 16749.9 0.695445
$$835$$ 0 0
$$836$$ −825.434 −0.0341486
$$837$$ − 2994.62i − 0.123667i
$$838$$ 5820.27i 0.239926i
$$839$$ −15155.7 −0.623639 −0.311819 0.950141i $$-0.600938\pi$$
−0.311819 + 0.950141i $$0.600938\pi$$
$$840$$ 0 0
$$841$$ 36489.2 1.49613
$$842$$ 12697.2i 0.519686i
$$843$$ 719.752i 0.0294064i
$$844$$ −7277.01 −0.296783
$$845$$ 0 0
$$846$$ −5680.66 −0.230857
$$847$$ 19866.7i 0.805935i
$$848$$ − 1320.27i − 0.0534649i
$$849$$ −13626.4 −0.550831
$$850$$ 0 0
$$851$$ 10639.6 0.428579
$$852$$ − 4977.21i − 0.200137i
$$853$$ − 2917.48i − 0.117107i −0.998284 0.0585537i $$-0.981351\pi$$
0.998284 0.0585537i $$-0.0186489\pi$$
$$854$$ −1474.52 −0.0590832
$$855$$ 0 0
$$856$$ 5625.54 0.224623
$$857$$ 31560.7i 1.25799i 0.777411 + 0.628993i $$0.216532\pi$$
−0.777411 + 0.628993i $$0.783468\pi$$
$$858$$ − 11455.2i − 0.455797i
$$859$$ 1404.81 0.0557991 0.0278995 0.999611i $$-0.491118\pi$$
0.0278995 + 0.999611i $$0.491118\pi$$
$$860$$ 0 0
$$861$$ −3701.76 −0.146522
$$862$$ − 7073.11i − 0.279479i
$$863$$ − 9808.24i − 0.386879i −0.981112 0.193439i $$-0.938036\pi$$
0.981112 0.193439i $$-0.0619643\pi$$
$$864$$ 4748.96 0.186994
$$865$$ 0 0
$$866$$ −11213.7 −0.440018
$$867$$ − 5215.35i − 0.204294i
$$868$$ − 3615.50i − 0.141380i
$$869$$ 28422.8 1.10952
$$870$$ 0 0
$$871$$ 30773.9 1.19717
$$872$$ − 9376.63i − 0.364143i
$$873$$ − 6516.97i − 0.252653i
$$874$$ 442.537 0.0171271
$$875$$ 0 0
$$876$$ −9175.22 −0.353883
$$877$$ 7196.05i 0.277073i 0.990357 + 0.138537i $$0.0442399\pi$$
−0.990357 + 0.138537i $$0.955760\pi$$
$$878$$ − 7478.52i − 0.287458i
$$879$$ −6515.10 −0.249999
$$880$$ 0 0
$$881$$ −3183.27 −0.121733 −0.0608667 0.998146i $$-0.519386\pi$$
−0.0608667 + 0.998146i $$0.519386\pi$$
$$882$$ − 806.336i − 0.0307832i
$$883$$ 25392.5i 0.967751i 0.875137 + 0.483876i $$0.160771\pi$$
−0.875137 + 0.483876i $$0.839229\pi$$
$$884$$ 8486.25 0.322877
$$885$$ 0 0
$$886$$ 22735.6 0.862095
$$887$$ 30634.2i 1.15964i 0.814746 + 0.579818i $$0.196876\pi$$
−0.814746 + 0.579818i $$0.803124\pi$$
$$888$$ 8378.11i 0.316612i
$$889$$ −19186.3 −0.723833
$$890$$ 0 0
$$891$$ −5230.05 −0.196648
$$892$$ − 5757.71i − 0.216124i
$$893$$ 947.648i 0.0355116i
$$894$$ 19530.9 0.730660
$$895$$ 0 0
$$896$$ 5215.58 0.194465
$$897$$ − 8554.75i − 0.318433i
$$898$$ 1614.59i 0.0599994i
$$899$$ 27365.8 1.01524
$$900$$ 0 0
$$901$$ −14704.5 −0.543704
$$902$$ − 20810.7i − 0.768206i
$$903$$ − 9310.61i − 0.343120i
$$904$$ −652.067 −0.0239905
$$905$$ 0 0
$$906$$ −17891.9 −0.656091
$$907$$ 28089.9i 1.02834i 0.857687 + 0.514172i $$0.171901\pi$$
−0.857687 + 0.514172i $$0.828099\pi$$
$$908$$ − 19474.1i − 0.711753i
$$909$$ −2418.52 −0.0882480
$$910$$ 0 0
$$911$$ −36102.7 −1.31299 −0.656495 0.754330i $$-0.727962\pi$$
−0.656495 + 0.754330i $$0.727962\pi$$
$$912$$ 41.6624i 0.00151270i
$$913$$ 3513.31i 0.127353i
$$914$$ 16581.0 0.600055
$$915$$ 0 0
$$916$$ 2257.52 0.0814308
$$917$$ − 12824.3i − 0.461826i
$$918$$ 2781.52i 0.100004i
$$919$$ 14533.8 0.521682 0.260841 0.965382i $$-0.416000\pi$$
0.260841 + 0.965382i $$0.416000\pi$$
$$920$$ 0 0
$$921$$ 10525.9 0.376592
$$922$$ − 22871.6i − 0.816959i
$$923$$ − 11522.7i − 0.410915i
$$924$$ −6314.41 −0.224815
$$925$$ 0 0
$$926$$ 23200.3 0.823336
$$927$$ 16565.6i 0.586933i
$$928$$ 43397.6i 1.53512i
$$929$$ −16539.6 −0.584118 −0.292059 0.956400i $$-0.594340\pi$$
−0.292059 + 0.956400i $$0.594340\pi$$
$$930$$ 0 0
$$931$$ −134.513 −0.00473522
$$932$$ 9688.79i 0.340522i
$$933$$ 9399.74i 0.329833i
$$934$$ 18533.9 0.649301
$$935$$ 0 0
$$936$$ 6736.41 0.235242
$$937$$ − 30212.3i − 1.05335i −0.850065 0.526677i $$-0.823438\pi$$
0.850065 0.526677i $$-0.176562\pi$$
$$938$$ 12178.0i 0.423908i
$$939$$ 19168.8 0.666187
$$940$$ 0 0
$$941$$ −26414.4 −0.915074 −0.457537 0.889191i $$-0.651268\pi$$
−0.457537 + 0.889191i $$0.651268\pi$$
$$942$$ 15789.1i 0.546112i
$$943$$ − 15541.5i − 0.536692i
$$944$$ 3182.03 0.109710
$$945$$ 0 0
$$946$$ 52342.9 1.79896
$$947$$ − 10187.3i − 0.349570i −0.984607 0.174785i $$-0.944077\pi$$
0.984607 0.174785i $$-0.0559231\pi$$
$$948$$ 6149.77i 0.210691i
$$949$$ −21241.5 −0.726583
$$950$$ 0 0
$$951$$ −4903.31 −0.167193
$$952$$ 9127.31i 0.310733i
$$953$$ 2211.39i 0.0751669i 0.999293 + 0.0375834i $$0.0119660\pi$$
−0.999293 + 0.0375834i $$0.988034\pi$$
$$954$$ −4294.66 −0.145749
$$955$$ 0 0
$$956$$ −31732.3 −1.07353
$$957$$ − 47794.0i − 1.61438i
$$958$$ − 20773.0i − 0.700569i
$$959$$ 2679.23 0.0902156
$$960$$ 0 0
$$961$$ −17489.6 −0.587077
$$962$$ 7136.44i 0.239177i
$$963$$ − 2187.78i − 0.0732089i
$$964$$ −18263.4 −0.610193
$$965$$ 0 0
$$966$$ 3385.33 0.112755
$$967$$ − 7955.89i − 0.264575i −0.991211 0.132287i $$-0.957768\pi$$
0.991211 0.132287i $$-0.0422322\pi$$
$$968$$ − 65679.6i − 2.18081i
$$969$$ 464.014 0.0153832
$$970$$ 0 0
$$971$$ 53071.2 1.75400 0.877001 0.480488i $$-0.159541\pi$$
0.877001 + 0.480488i $$0.159541\pi$$
$$972$$ − 1131.62i − 0.0373422i
$$973$$ − 21375.2i − 0.704274i
$$974$$ −14498.6 −0.476965
$$975$$ 0 0
$$976$$ 582.813 0.0191141
$$977$$ 22448.2i 0.735089i 0.930006 + 0.367545i $$0.119801\pi$$
−0.930006 + 0.367545i $$0.880199\pi$$
$$978$$ 5087.80i 0.166350i
$$979$$ −65781.4 −2.14748
$$980$$ 0 0
$$981$$ −3646.58 −0.118681
$$982$$ − 14831.3i − 0.481961i
$$983$$ 21712.8i 0.704509i 0.935904 + 0.352254i $$0.114585\pi$$
−0.935904 + 0.352254i $$0.885415\pi$$
$$984$$ 12238.1 0.396479
$$985$$ 0 0
$$986$$ −25418.5 −0.820983
$$987$$ 7249.33i 0.233788i
$$988$$ − 413.470i − 0.0133140i
$$989$$ 39089.7 1.25681
$$990$$ 0 0
$$991$$ −37849.2 −1.21324 −0.606620 0.794992i $$-0.707475\pi$$
−0.606620 + 0.794992i $$0.707475\pi$$
$$992$$ 19508.0i 0.624373i
$$993$$ − 13158.5i − 0.420516i
$$994$$ 4559.82 0.145502
$$995$$ 0 0
$$996$$ −760.168 −0.0241836
$$997$$ 39573.1i 1.25707i 0.777783 + 0.628533i $$0.216344\pi$$
−0.777783 + 0.628533i $$0.783656\pi$$
$$998$$ 30748.0i 0.975261i
$$999$$ 3258.26 0.103190
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 525.4.d.l.274.2 4
5.2 odd 4 105.4.a.e.1.2 2
5.3 odd 4 525.4.a.l.1.1 2
5.4 even 2 inner 525.4.d.l.274.3 4
15.2 even 4 315.4.a.k.1.1 2
15.8 even 4 1575.4.a.q.1.2 2
20.7 even 4 1680.4.a.bo.1.2 2
35.27 even 4 735.4.a.o.1.2 2
105.62 odd 4 2205.4.a.bb.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
105.4.a.e.1.2 2 5.2 odd 4
315.4.a.k.1.1 2 15.2 even 4
525.4.a.l.1.1 2 5.3 odd 4
525.4.d.l.274.2 4 1.1 even 1 trivial
525.4.d.l.274.3 4 5.4 even 2 inner
735.4.a.o.1.2 2 35.27 even 4
1575.4.a.q.1.2 2 15.8 even 4
1680.4.a.bo.1.2 2 20.7 even 4
2205.4.a.bb.1.1 2 105.62 odd 4