# Properties

 Label 6030.2.a.bu.1.1 Level 6030 Weight 2 Character 6030.1 Self dual yes Analytic conductor 48.150 Analytic rank 0 Dimension 4 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$6030 = 2 \cdot 3^{2} \cdot 5 \cdot 67$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6030.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$48.1497924188$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.70292.1 Defining polynomial: $$x^{4} - x^{3} - 8 x^{2} + 10 x + 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 2010) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-0.175890$$ of defining polynomial Character $$\chi$$ $$=$$ 6030.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -3.96906 q^{7} +1.00000 q^{8} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -3.96906 q^{7} +1.00000 q^{8} +1.00000 q^{10} -3.04990 q^{11} +0.919164 q^{13} -3.96906 q^{14} +1.00000 q^{16} -0.351780 q^{17} -0.351780 q^{19} +1.00000 q^{20} -3.04990 q^{22} +3.43262 q^{23} +1.00000 q^{25} +0.919164 q^{26} -3.96906 q^{28} -9.37074 q^{29} +2.91916 q^{31} +1.00000 q^{32} -0.351780 q^{34} -3.96906 q^{35} +5.40168 q^{37} -0.351780 q^{38} +1.00000 q^{40} +8.45158 q^{41} +6.09980 q^{43} -3.04990 q^{44} +3.43262 q^{46} +13.2091 q^{47} +8.75346 q^{49} +1.00000 q^{50} +0.919164 q^{52} -5.93813 q^{53} -3.04990 q^{55} -3.96906 q^{56} -9.37074 q^{58} +8.45158 q^{59} -1.96906 q^{61} +2.91916 q^{62} +1.00000 q^{64} +0.919164 q^{65} -1.00000 q^{67} -0.351780 q^{68} -3.96906 q^{70} +5.61728 q^{71} -12.8034 q^{73} +5.40168 q^{74} -0.351780 q^{76} +12.1052 q^{77} +11.8842 q^{79} +1.00000 q^{80} +8.45158 q^{82} -0.105239 q^{83} -0.351780 q^{85} +6.09980 q^{86} -3.04990 q^{88} +16.2050 q^{89} -3.64822 q^{91} +3.43262 q^{92} +13.2091 q^{94} -0.351780 q^{95} +3.04990 q^{97} +8.75346 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{2} + 4q^{4} + 4q^{5} + q^{7} + 4q^{8} + O(q^{10})$$ $$4q + 4q^{2} + 4q^{4} + 4q^{5} + q^{7} + 4q^{8} + 4q^{10} + 3q^{11} + 2q^{13} + q^{14} + 4q^{16} + 2q^{17} + 2q^{19} + 4q^{20} + 3q^{22} + 12q^{23} + 4q^{25} + 2q^{26} + q^{28} - 2q^{29} + 10q^{31} + 4q^{32} + 2q^{34} + q^{35} + 3q^{37} + 2q^{38} + 4q^{40} - 6q^{43} + 3q^{44} + 12q^{46} + 14q^{47} + 13q^{49} + 4q^{50} + 2q^{52} + 10q^{53} + 3q^{55} + q^{56} - 2q^{58} + 9q^{61} + 10q^{62} + 4q^{64} + 2q^{65} - 4q^{67} + 2q^{68} + q^{70} + 9q^{71} - 14q^{73} + 3q^{74} + 2q^{76} + 23q^{77} + 12q^{79} + 4q^{80} + 25q^{83} + 2q^{85} - 6q^{86} + 3q^{88} + 9q^{89} - 18q^{91} + 12q^{92} + 14q^{94} + 2q^{95} - 3q^{97} + 13q^{98} + O(q^{100})$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ 0 0
$$4$$ 1.00000 0.500000
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ −3.96906 −1.50016 −0.750082 0.661344i $$-0.769986\pi$$
−0.750082 + 0.661344i $$0.769986\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 0 0
$$10$$ 1.00000 0.316228
$$11$$ −3.04990 −0.919579 −0.459790 0.888028i $$-0.652075\pi$$
−0.459790 + 0.888028i $$0.652075\pi$$
$$12$$ 0 0
$$13$$ 0.919164 0.254930 0.127465 0.991843i $$-0.459316\pi$$
0.127465 + 0.991843i $$0.459316\pi$$
$$14$$ −3.96906 −1.06078
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −0.351780 −0.0853192 −0.0426596 0.999090i $$-0.513583\pi$$
−0.0426596 + 0.999090i $$0.513583\pi$$
$$18$$ 0 0
$$19$$ −0.351780 −0.0807039 −0.0403519 0.999186i $$-0.512848\pi$$
−0.0403519 + 0.999186i $$0.512848\pi$$
$$20$$ 1.00000 0.223607
$$21$$ 0 0
$$22$$ −3.04990 −0.650241
$$23$$ 3.43262 0.715750 0.357875 0.933770i $$-0.383501\pi$$
0.357875 + 0.933770i $$0.383501\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0.919164 0.180263
$$27$$ 0 0
$$28$$ −3.96906 −0.750082
$$29$$ −9.37074 −1.74010 −0.870051 0.492961i $$-0.835915\pi$$
−0.870051 + 0.492961i $$0.835915\pi$$
$$30$$ 0 0
$$31$$ 2.91916 0.524297 0.262149 0.965027i $$-0.415569\pi$$
0.262149 + 0.965027i $$0.415569\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 0 0
$$34$$ −0.351780 −0.0603298
$$35$$ −3.96906 −0.670894
$$36$$ 0 0
$$37$$ 5.40168 0.888030 0.444015 0.896019i $$-0.353554\pi$$
0.444015 + 0.896019i $$0.353554\pi$$
$$38$$ −0.351780 −0.0570663
$$39$$ 0 0
$$40$$ 1.00000 0.158114
$$41$$ 8.45158 1.31991 0.659957 0.751303i $$-0.270574\pi$$
0.659957 + 0.751303i $$0.270574\pi$$
$$42$$ 0 0
$$43$$ 6.09980 0.930210 0.465105 0.885255i $$-0.346017\pi$$
0.465105 + 0.885255i $$0.346017\pi$$
$$44$$ −3.04990 −0.459790
$$45$$ 0 0
$$46$$ 3.43262 0.506112
$$47$$ 13.2091 1.92674 0.963370 0.268174i $$-0.0864203\pi$$
0.963370 + 0.268174i $$0.0864203\pi$$
$$48$$ 0 0
$$49$$ 8.75346 1.25049
$$50$$ 1.00000 0.141421
$$51$$ 0 0
$$52$$ 0.919164 0.127465
$$53$$ −5.93813 −0.815664 −0.407832 0.913057i $$-0.633715\pi$$
−0.407832 + 0.913057i $$0.633715\pi$$
$$54$$ 0 0
$$55$$ −3.04990 −0.411248
$$56$$ −3.96906 −0.530388
$$57$$ 0 0
$$58$$ −9.37074 −1.23044
$$59$$ 8.45158 1.10030 0.550151 0.835065i $$-0.314570\pi$$
0.550151 + 0.835065i $$0.314570\pi$$
$$60$$ 0 0
$$61$$ −1.96906 −0.252113 −0.126056 0.992023i $$-0.540232\pi$$
−0.126056 + 0.992023i $$0.540232\pi$$
$$62$$ 2.91916 0.370734
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 0.919164 0.114008
$$66$$ 0 0
$$67$$ −1.00000 −0.122169
$$68$$ −0.351780 −0.0426596
$$69$$ 0 0
$$70$$ −3.96906 −0.474394
$$71$$ 5.61728 0.666649 0.333324 0.942812i $$-0.391830\pi$$
0.333324 + 0.942812i $$0.391830\pi$$
$$72$$ 0 0
$$73$$ −12.8034 −1.49852 −0.749260 0.662276i $$-0.769591\pi$$
−0.749260 + 0.662276i $$0.769591\pi$$
$$74$$ 5.40168 0.627932
$$75$$ 0 0
$$76$$ −0.351780 −0.0403519
$$77$$ 12.1052 1.37952
$$78$$ 0 0
$$79$$ 11.8842 1.33708 0.668538 0.743678i $$-0.266920\pi$$
0.668538 + 0.743678i $$0.266920\pi$$
$$80$$ 1.00000 0.111803
$$81$$ 0 0
$$82$$ 8.45158 0.933321
$$83$$ −0.105239 −0.0115515 −0.00577573 0.999983i $$-0.501838\pi$$
−0.00577573 + 0.999983i $$0.501838\pi$$
$$84$$ 0 0
$$85$$ −0.351780 −0.0381559
$$86$$ 6.09980 0.657758
$$87$$ 0 0
$$88$$ −3.04990 −0.325120
$$89$$ 16.2050 1.71773 0.858865 0.512202i $$-0.171170\pi$$
0.858865 + 0.512202i $$0.171170\pi$$
$$90$$ 0 0
$$91$$ −3.64822 −0.382437
$$92$$ 3.43262 0.357875
$$93$$ 0 0
$$94$$ 13.2091 1.36241
$$95$$ −0.351780 −0.0360919
$$96$$ 0 0
$$97$$ 3.04990 0.309670 0.154835 0.987940i $$-0.450515\pi$$
0.154835 + 0.987940i $$0.450515\pi$$
$$98$$ 8.75346 0.884233
$$99$$ 0 0
$$100$$ 1.00000 0.100000
$$101$$ 4.69812 0.467480 0.233740 0.972299i $$-0.424904\pi$$
0.233740 + 0.972299i $$0.424904\pi$$
$$102$$ 0 0
$$103$$ 2.09980 0.206899 0.103450 0.994635i $$-0.467012\pi$$
0.103450 + 0.994635i $$0.467012\pi$$
$$104$$ 0.919164 0.0901315
$$105$$ 0 0
$$106$$ −5.93813 −0.576762
$$107$$ 9.83833 0.951107 0.475554 0.879687i $$-0.342248\pi$$
0.475554 + 0.879687i $$0.342248\pi$$
$$108$$ 0 0
$$109$$ 1.26550 0.121213 0.0606066 0.998162i $$-0.480696\pi$$
0.0606066 + 0.998162i $$0.480696\pi$$
$$110$$ −3.04990 −0.290796
$$111$$ 0 0
$$112$$ −3.96906 −0.375041
$$113$$ −9.69158 −0.911708 −0.455854 0.890055i $$-0.650666\pi$$
−0.455854 + 0.890055i $$0.650666\pi$$
$$114$$ 0 0
$$115$$ 3.43262 0.320093
$$116$$ −9.37074 −0.870051
$$117$$ 0 0
$$118$$ 8.45158 0.778031
$$119$$ 1.39624 0.127993
$$120$$ 0 0
$$121$$ −1.69812 −0.154374
$$122$$ −1.96906 −0.178271
$$123$$ 0 0
$$124$$ 2.91916 0.262149
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ −12.9650 −1.15046 −0.575230 0.817992i $$-0.695088\pi$$
−0.575230 + 0.817992i $$0.695088\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 0 0
$$130$$ 0.919164 0.0806160
$$131$$ 12.5135 1.09331 0.546653 0.837359i $$-0.315902\pi$$
0.546653 + 0.837359i $$0.315902\pi$$
$$132$$ 0 0
$$133$$ 1.39624 0.121069
$$134$$ −1.00000 −0.0863868
$$135$$ 0 0
$$136$$ −0.351780 −0.0301649
$$137$$ −1.40168 −0.119753 −0.0598767 0.998206i $$-0.519071\pi$$
−0.0598767 + 0.998206i $$0.519071\pi$$
$$138$$ 0 0
$$139$$ 6.02844 0.511325 0.255663 0.966766i $$-0.417706\pi$$
0.255663 + 0.966766i $$0.417706\pi$$
$$140$$ −3.96906 −0.335447
$$141$$ 0 0
$$142$$ 5.61728 0.471392
$$143$$ −2.80336 −0.234429
$$144$$ 0 0
$$145$$ −9.37074 −0.778198
$$146$$ −12.8034 −1.05961
$$147$$ 0 0
$$148$$ 5.40168 0.444015
$$149$$ −9.37074 −0.767681 −0.383841 0.923399i $$-0.625399\pi$$
−0.383841 + 0.923399i $$0.625399\pi$$
$$150$$ 0 0
$$151$$ −3.14021 −0.255547 −0.127773 0.991803i $$-0.540783\pi$$
−0.127773 + 0.991803i $$0.540783\pi$$
$$152$$ −0.351780 −0.0285331
$$153$$ 0 0
$$154$$ 12.1052 0.975468
$$155$$ 2.91916 0.234473
$$156$$ 0 0
$$157$$ 14.5798 1.16360 0.581798 0.813333i $$-0.302350\pi$$
0.581798 + 0.813333i $$0.302350\pi$$
$$158$$ 11.8842 0.945456
$$159$$ 0 0
$$160$$ 1.00000 0.0790569
$$161$$ −13.6243 −1.07374
$$162$$ 0 0
$$163$$ −4.59832 −0.360168 −0.180084 0.983651i $$-0.557637\pi$$
−0.180084 + 0.983651i $$0.557637\pi$$
$$164$$ 8.45158 0.659957
$$165$$ 0 0
$$166$$ −0.105239 −0.00816811
$$167$$ 6.40571 0.495689 0.247844 0.968800i $$-0.420278\pi$$
0.247844 + 0.968800i $$0.420278\pi$$
$$168$$ 0 0
$$169$$ −12.1551 −0.935011
$$170$$ −0.351780 −0.0269803
$$171$$ 0 0
$$172$$ 6.09980 0.465105
$$173$$ 24.9625 1.89787 0.948933 0.315478i $$-0.102165\pi$$
0.948933 + 0.315478i $$0.102165\pi$$
$$174$$ 0 0
$$175$$ −3.96906 −0.300033
$$176$$ −3.04990 −0.229895
$$177$$ 0 0
$$178$$ 16.2050 1.21462
$$179$$ 14.1282 1.05599 0.527997 0.849246i $$-0.322943\pi$$
0.527997 + 0.849246i $$0.322943\pi$$
$$180$$ 0 0
$$181$$ −11.0729 −0.823042 −0.411521 0.911400i $$-0.635002\pi$$
−0.411521 + 0.911400i $$0.635002\pi$$
$$182$$ −3.64822 −0.270424
$$183$$ 0 0
$$184$$ 3.43262 0.253056
$$185$$ 5.40168 0.397139
$$186$$ 0 0
$$187$$ 1.07289 0.0784577
$$188$$ 13.2091 0.963370
$$189$$ 0 0
$$190$$ −0.351780 −0.0255208
$$191$$ 18.0998 1.30966 0.654828 0.755778i $$-0.272741\pi$$
0.654828 + 0.755778i $$0.272741\pi$$
$$192$$ 0 0
$$193$$ −11.6861 −0.841187 −0.420593 0.907249i $$-0.638178\pi$$
−0.420593 + 0.907249i $$0.638178\pi$$
$$194$$ 3.04990 0.218970
$$195$$ 0 0
$$196$$ 8.75346 0.625247
$$197$$ 0.161672 0.0115186 0.00575932 0.999983i $$-0.498167\pi$$
0.00575932 + 0.999983i $$0.498167\pi$$
$$198$$ 0 0
$$199$$ −8.10524 −0.574565 −0.287283 0.957846i $$-0.592752\pi$$
−0.287283 + 0.957846i $$0.592752\pi$$
$$200$$ 1.00000 0.0707107
$$201$$ 0 0
$$202$$ 4.69812 0.330558
$$203$$ 37.1931 2.61044
$$204$$ 0 0
$$205$$ 8.45158 0.590284
$$206$$ 2.09980 0.146300
$$207$$ 0 0
$$208$$ 0.919164 0.0637326
$$209$$ 1.07289 0.0742136
$$210$$ 0 0
$$211$$ 18.3278 1.26174 0.630870 0.775889i $$-0.282698\pi$$
0.630870 + 0.775889i $$0.282698\pi$$
$$212$$ −5.93813 −0.407832
$$213$$ 0 0
$$214$$ 9.83833 0.672534
$$215$$ 6.09980 0.416003
$$216$$ 0 0
$$217$$ −11.5863 −0.786532
$$218$$ 1.26550 0.0857107
$$219$$ 0 0
$$220$$ −3.04990 −0.205624
$$221$$ −0.323344 −0.0217504
$$222$$ 0 0
$$223$$ −0.442091 −0.0296046 −0.0148023 0.999890i $$-0.504712\pi$$
−0.0148023 + 0.999890i $$0.504712\pi$$
$$224$$ −3.96906 −0.265194
$$225$$ 0 0
$$226$$ −9.69158 −0.644675
$$227$$ 25.3344 1.68150 0.840750 0.541423i $$-0.182114\pi$$
0.840750 + 0.541423i $$0.182114\pi$$
$$228$$ 0 0
$$229$$ 8.06886 0.533205 0.266603 0.963807i $$-0.414099\pi$$
0.266603 + 0.963807i $$0.414099\pi$$
$$230$$ 3.43262 0.226340
$$231$$ 0 0
$$232$$ −9.37074 −0.615219
$$233$$ −3.67121 −0.240509 −0.120255 0.992743i $$-0.538371\pi$$
−0.120255 + 0.992743i $$0.538371\pi$$
$$234$$ 0 0
$$235$$ 13.2091 0.861665
$$236$$ 8.45158 0.550151
$$237$$ 0 0
$$238$$ 1.39624 0.0905046
$$239$$ −12.9650 −0.838638 −0.419319 0.907839i $$-0.637731\pi$$
−0.419319 + 0.907839i $$0.637731\pi$$
$$240$$ 0 0
$$241$$ −21.9531 −1.41412 −0.707060 0.707153i $$-0.749979\pi$$
−0.707060 + 0.707153i $$0.749979\pi$$
$$242$$ −1.69812 −0.109159
$$243$$ 0 0
$$244$$ −1.96906 −0.126056
$$245$$ 8.75346 0.559238
$$246$$ 0 0
$$247$$ −0.323344 −0.0205739
$$248$$ 2.91916 0.185367
$$249$$ 0 0
$$250$$ 1.00000 0.0632456
$$251$$ 4.59832 0.290243 0.145122 0.989414i $$-0.453643\pi$$
0.145122 + 0.989414i $$0.453643\pi$$
$$252$$ 0 0
$$253$$ −10.4691 −0.658189
$$254$$ −12.9650 −0.813498
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −19.7968 −1.23489 −0.617446 0.786613i $$-0.711833\pi$$
−0.617446 + 0.786613i $$0.711833\pi$$
$$258$$ 0 0
$$259$$ −21.4396 −1.33219
$$260$$ 0.919164 0.0570041
$$261$$ 0 0
$$262$$ 12.5135 0.773084
$$263$$ 17.4705 1.07728 0.538640 0.842536i $$-0.318938\pi$$
0.538640 + 0.842536i $$0.318938\pi$$
$$264$$ 0 0
$$265$$ −5.93813 −0.364776
$$266$$ 1.39624 0.0856088
$$267$$ 0 0
$$268$$ −1.00000 −0.0610847
$$269$$ −7.33282 −0.447090 −0.223545 0.974694i $$-0.571763\pi$$
−0.223545 + 0.974694i $$0.571763\pi$$
$$270$$ 0 0
$$271$$ −14.1537 −0.859778 −0.429889 0.902882i $$-0.641447\pi$$
−0.429889 + 0.902882i $$0.641447\pi$$
$$272$$ −0.351780 −0.0213298
$$273$$ 0 0
$$274$$ −1.40168 −0.0846785
$$275$$ −3.04990 −0.183916
$$276$$ 0 0
$$277$$ 13.6631 0.820939 0.410469 0.911874i $$-0.365365\pi$$
0.410469 + 0.911874i $$0.365365\pi$$
$$278$$ 6.02844 0.361562
$$279$$ 0 0
$$280$$ −3.96906 −0.237197
$$281$$ 1.67666 0.100021 0.0500105 0.998749i $$-0.484075\pi$$
0.0500105 + 0.998749i $$0.484075\pi$$
$$282$$ 0 0
$$283$$ 0.978538 0.0581680 0.0290840 0.999577i $$-0.490741\pi$$
0.0290840 + 0.999577i $$0.490741\pi$$
$$284$$ 5.61728 0.333324
$$285$$ 0 0
$$286$$ −2.80336 −0.165766
$$287$$ −33.5448 −1.98009
$$288$$ 0 0
$$289$$ −16.8763 −0.992721
$$290$$ −9.37074 −0.550269
$$291$$ 0 0
$$292$$ −12.8034 −0.749260
$$293$$ −16.5823 −0.968749 −0.484374 0.874861i $$-0.660953\pi$$
−0.484374 + 0.874861i $$0.660953\pi$$
$$294$$ 0 0
$$295$$ 8.45158 0.492070
$$296$$ 5.40168 0.313966
$$297$$ 0 0
$$298$$ −9.37074 −0.542832
$$299$$ 3.15514 0.182466
$$300$$ 0 0
$$301$$ −24.2105 −1.39547
$$302$$ −3.14021 −0.180699
$$303$$ 0 0
$$304$$ −0.351780 −0.0201760
$$305$$ −1.96906 −0.112748
$$306$$ 0 0
$$307$$ −24.3843 −1.39168 −0.695842 0.718195i $$-0.744968\pi$$
−0.695842 + 0.718195i $$0.744968\pi$$
$$308$$ 12.1052 0.689760
$$309$$ 0 0
$$310$$ 2.91916 0.165797
$$311$$ −6.03792 −0.342379 −0.171190 0.985238i $$-0.554761\pi$$
−0.171190 + 0.985238i $$0.554761\pi$$
$$312$$ 0 0
$$313$$ 11.4301 0.646068 0.323034 0.946387i $$-0.395297\pi$$
0.323034 + 0.946387i $$0.395297\pi$$
$$314$$ 14.5798 0.822786
$$315$$ 0 0
$$316$$ 11.8842 0.668538
$$317$$ 29.3911 1.65077 0.825385 0.564571i $$-0.190958\pi$$
0.825385 + 0.564571i $$0.190958\pi$$
$$318$$ 0 0
$$319$$ 28.5798 1.60016
$$320$$ 1.00000 0.0559017
$$321$$ 0 0
$$322$$ −13.6243 −0.759251
$$323$$ 0.123749 0.00688559
$$324$$ 0 0
$$325$$ 0.919164 0.0509861
$$326$$ −4.59832 −0.254677
$$327$$ 0 0
$$328$$ 8.45158 0.466660
$$329$$ −52.4276 −2.89043
$$330$$ 0 0
$$331$$ −4.60781 −0.253268 −0.126634 0.991950i $$-0.540417\pi$$
−0.126634 + 0.991950i $$0.540417\pi$$
$$332$$ −0.105239 −0.00577573
$$333$$ 0 0
$$334$$ 6.40571 0.350505
$$335$$ −1.00000 −0.0546358
$$336$$ 0 0
$$337$$ 18.0095 0.981039 0.490520 0.871430i $$-0.336807\pi$$
0.490520 + 0.871430i $$0.336807\pi$$
$$338$$ −12.1551 −0.661152
$$339$$ 0 0
$$340$$ −0.351780 −0.0190780
$$341$$ −8.90315 −0.482133
$$342$$ 0 0
$$343$$ −6.95959 −0.375782
$$344$$ 6.09980 0.328879
$$345$$ 0 0
$$346$$ 24.9625 1.34199
$$347$$ 10.7239 0.575691 0.287845 0.957677i $$-0.407061\pi$$
0.287845 + 0.957677i $$0.407061\pi$$
$$348$$ 0 0
$$349$$ 18.1996 0.974202 0.487101 0.873346i $$-0.338054\pi$$
0.487101 + 0.873346i $$0.338054\pi$$
$$350$$ −3.96906 −0.212155
$$351$$ 0 0
$$352$$ −3.04990 −0.162560
$$353$$ −25.5863 −1.36182 −0.680912 0.732365i $$-0.738416\pi$$
−0.680912 + 0.732365i $$0.738416\pi$$
$$354$$ 0 0
$$355$$ 5.61728 0.298134
$$356$$ 16.2050 0.858865
$$357$$ 0 0
$$358$$ 14.1282 0.746700
$$359$$ −4.59038 −0.242271 −0.121135 0.992636i $$-0.538654\pi$$
−0.121135 + 0.992636i $$0.538654\pi$$
$$360$$ 0 0
$$361$$ −18.8763 −0.993487
$$362$$ −11.0729 −0.581978
$$363$$ 0 0
$$364$$ −3.64822 −0.191219
$$365$$ −12.8034 −0.670158
$$366$$ 0 0
$$367$$ 26.9855 1.40863 0.704316 0.709886i $$-0.251254\pi$$
0.704316 + 0.709886i $$0.251254\pi$$
$$368$$ 3.43262 0.178937
$$369$$ 0 0
$$370$$ 5.40168 0.280820
$$371$$ 23.5688 1.22363
$$372$$ 0 0
$$373$$ −16.7954 −0.869634 −0.434817 0.900519i $$-0.643187\pi$$
−0.434817 + 0.900519i $$0.643187\pi$$
$$374$$ 1.07289 0.0554780
$$375$$ 0 0
$$376$$ 13.2091 0.681206
$$377$$ −8.61325 −0.443605
$$378$$ 0 0
$$379$$ −21.9047 −1.12517 −0.562584 0.826740i $$-0.690193\pi$$
−0.562584 + 0.826740i $$0.690193\pi$$
$$380$$ −0.351780 −0.0180459
$$381$$ 0 0
$$382$$ 18.0998 0.926066
$$383$$ 4.47457 0.228640 0.114320 0.993444i $$-0.463531\pi$$
0.114320 + 0.993444i $$0.463531\pi$$
$$384$$ 0 0
$$385$$ 12.1052 0.616940
$$386$$ −11.6861 −0.594809
$$387$$ 0 0
$$388$$ 3.04990 0.154835
$$389$$ −30.4545 −1.54411 −0.772053 0.635559i $$-0.780770\pi$$
−0.772053 + 0.635559i $$0.780770\pi$$
$$390$$ 0 0
$$391$$ −1.20753 −0.0610672
$$392$$ 8.75346 0.442116
$$393$$ 0 0
$$394$$ 0.161672 0.00814491
$$395$$ 11.8842 0.597959
$$396$$ 0 0
$$397$$ 27.3777 1.37405 0.687024 0.726634i $$-0.258917\pi$$
0.687024 + 0.726634i $$0.258917\pi$$
$$398$$ −8.10524 −0.406279
$$399$$ 0 0
$$400$$ 1.00000 0.0500000
$$401$$ 34.0867 1.70221 0.851105 0.524996i $$-0.175933\pi$$
0.851105 + 0.524996i $$0.175933\pi$$
$$402$$ 0 0
$$403$$ 2.68319 0.133659
$$404$$ 4.69812 0.233740
$$405$$ 0 0
$$406$$ 37.1931 1.84586
$$407$$ −16.4746 −0.816614
$$408$$ 0 0
$$409$$ 8.16167 0.403569 0.201784 0.979430i $$-0.435326\pi$$
0.201784 + 0.979430i $$0.435326\pi$$
$$410$$ 8.45158 0.417394
$$411$$ 0 0
$$412$$ 2.09980 0.103450
$$413$$ −33.5448 −1.65063
$$414$$ 0 0
$$415$$ −0.105239 −0.00516597
$$416$$ 0.919164 0.0450657
$$417$$ 0 0
$$418$$ 1.07289 0.0524769
$$419$$ −27.3628 −1.33676 −0.668380 0.743820i $$-0.733012\pi$$
−0.668380 + 0.743820i $$0.733012\pi$$
$$420$$ 0 0
$$421$$ −15.9760 −0.778625 −0.389312 0.921106i $$-0.627287\pi$$
−0.389312 + 0.921106i $$0.627287\pi$$
$$422$$ 18.3278 0.892185
$$423$$ 0 0
$$424$$ −5.93813 −0.288381
$$425$$ −0.351780 −0.0170638
$$426$$ 0 0
$$427$$ 7.81533 0.378210
$$428$$ 9.83833 0.475554
$$429$$ 0 0
$$430$$ 6.09980 0.294158
$$431$$ −16.7900 −0.808745 −0.404372 0.914594i $$-0.632510\pi$$
−0.404372 + 0.914594i $$0.632510\pi$$
$$432$$ 0 0
$$433$$ 29.6352 1.42417 0.712087 0.702091i $$-0.247750\pi$$
0.712087 + 0.702091i $$0.247750\pi$$
$$434$$ −11.5863 −0.556162
$$435$$ 0 0
$$436$$ 1.26550 0.0606066
$$437$$ −1.20753 −0.0577638
$$438$$ 0 0
$$439$$ 1.99456 0.0951951 0.0475975 0.998867i $$-0.484844\pi$$
0.0475975 + 0.998867i $$0.484844\pi$$
$$440$$ −3.04990 −0.145398
$$441$$ 0 0
$$442$$ −0.323344 −0.0153799
$$443$$ 3.29644 0.156619 0.0783093 0.996929i $$-0.475048\pi$$
0.0783093 + 0.996929i $$0.475048\pi$$
$$444$$ 0 0
$$445$$ 16.2050 0.768192
$$446$$ −0.442091 −0.0209336
$$447$$ 0 0
$$448$$ −3.96906 −0.187521
$$449$$ 13.9196 0.656907 0.328454 0.944520i $$-0.393473\pi$$
0.328454 + 0.944520i $$0.393473\pi$$
$$450$$ 0 0
$$451$$ −25.7765 −1.21377
$$452$$ −9.69158 −0.455854
$$453$$ 0 0
$$454$$ 25.3344 1.18900
$$455$$ −3.64822 −0.171031
$$456$$ 0 0
$$457$$ 12.3897 0.579566 0.289783 0.957092i $$-0.406417\pi$$
0.289783 + 0.957092i $$0.406417\pi$$
$$458$$ 8.06886 0.377033
$$459$$ 0 0
$$460$$ 3.43262 0.160047
$$461$$ −38.8537 −1.80960 −0.904799 0.425839i $$-0.859979\pi$$
−0.904799 + 0.425839i $$0.859979\pi$$
$$462$$ 0 0
$$463$$ −5.85076 −0.271908 −0.135954 0.990715i $$-0.543410\pi$$
−0.135954 + 0.990715i $$0.543410\pi$$
$$464$$ −9.37074 −0.435026
$$465$$ 0 0
$$466$$ −3.67121 −0.170066
$$467$$ 6.37477 0.294989 0.147495 0.989063i $$-0.452879\pi$$
0.147495 + 0.989063i $$0.452879\pi$$
$$468$$ 0 0
$$469$$ 3.96906 0.183274
$$470$$ 13.2091 0.609289
$$471$$ 0 0
$$472$$ 8.45158 0.389015
$$473$$ −18.6038 −0.855402
$$474$$ 0 0
$$475$$ −0.351780 −0.0161408
$$476$$ 1.39624 0.0639964
$$477$$ 0 0
$$478$$ −12.9650 −0.593007
$$479$$ −13.5094 −0.617261 −0.308631 0.951182i $$-0.599871\pi$$
−0.308631 + 0.951182i $$0.599871\pi$$
$$480$$ 0 0
$$481$$ 4.96503 0.226386
$$482$$ −21.9531 −0.999934
$$483$$ 0 0
$$484$$ −1.69812 −0.0771872
$$485$$ 3.04990 0.138489
$$486$$ 0 0
$$487$$ 24.4920 1.10984 0.554919 0.831904i $$-0.312749\pi$$
0.554919 + 0.831904i $$0.312749\pi$$
$$488$$ −1.96906 −0.0891353
$$489$$ 0 0
$$490$$ 8.75346 0.395441
$$491$$ −1.59723 −0.0720819 −0.0360410 0.999350i $$-0.511475\pi$$
−0.0360410 + 0.999350i $$0.511475\pi$$
$$492$$ 0 0
$$493$$ 3.29644 0.148464
$$494$$ −0.323344 −0.0145479
$$495$$ 0 0
$$496$$ 2.91916 0.131074
$$497$$ −22.2953 −1.00008
$$498$$ 0 0
$$499$$ 21.5837 0.966220 0.483110 0.875560i $$-0.339507\pi$$
0.483110 + 0.875560i $$0.339507\pi$$
$$500$$ 1.00000 0.0447214
$$501$$ 0 0
$$502$$ 4.59832 0.205233
$$503$$ 34.3428 1.53127 0.765634 0.643277i $$-0.222425\pi$$
0.765634 + 0.643277i $$0.222425\pi$$
$$504$$ 0 0
$$505$$ 4.69812 0.209064
$$506$$ −10.4691 −0.465410
$$507$$ 0 0
$$508$$ −12.9650 −0.575230
$$509$$ −25.3089 −1.12180 −0.560898 0.827885i $$-0.689544\pi$$
−0.560898 + 0.827885i $$0.689544\pi$$
$$510$$ 0 0
$$511$$ 50.8173 2.24803
$$512$$ 1.00000 0.0441942
$$513$$ 0 0
$$514$$ −19.7968 −0.873200
$$515$$ 2.09980 0.0925281
$$516$$ 0 0
$$517$$ −40.2863 −1.77179
$$518$$ −21.4396 −0.942002
$$519$$ 0 0
$$520$$ 0.919164 0.0403080
$$521$$ 23.4545 1.02756 0.513781 0.857922i $$-0.328244\pi$$
0.513781 + 0.857922i $$0.328244\pi$$
$$522$$ 0 0
$$523$$ −19.4586 −0.850863 −0.425432 0.904991i $$-0.639878\pi$$
−0.425432 + 0.904991i $$0.639878\pi$$
$$524$$ 12.5135 0.546653
$$525$$ 0 0
$$526$$ 17.4705 0.761752
$$527$$ −1.02690 −0.0447326
$$528$$ 0 0
$$529$$ −11.2171 −0.487702
$$530$$ −5.93813 −0.257936
$$531$$ 0 0
$$532$$ 1.39624 0.0605346
$$533$$ 7.76839 0.336486
$$534$$ 0 0
$$535$$ 9.83833 0.425348
$$536$$ −1.00000 −0.0431934
$$537$$ 0 0
$$538$$ −7.33282 −0.316140
$$539$$ −26.6972 −1.14993
$$540$$ 0 0
$$541$$ −39.2389 −1.68701 −0.843507 0.537119i $$-0.819513\pi$$
−0.843507 + 0.537119i $$0.819513\pi$$
$$542$$ −14.1537 −0.607955
$$543$$ 0 0
$$544$$ −0.351780 −0.0150824
$$545$$ 1.26550 0.0542082
$$546$$ 0 0
$$547$$ −26.1865 −1.11965 −0.559827 0.828609i $$-0.689133\pi$$
−0.559827 + 0.828609i $$0.689133\pi$$
$$548$$ −1.40168 −0.0598767
$$549$$ 0 0
$$550$$ −3.04990 −0.130048
$$551$$ 3.29644 0.140433
$$552$$ 0 0
$$553$$ −47.1691 −2.00583
$$554$$ 13.6631 0.580492
$$555$$ 0 0
$$556$$ 6.02844 0.255663
$$557$$ 40.1326 1.70047 0.850236 0.526401i $$-0.176459\pi$$
0.850236 + 0.526401i $$0.176459\pi$$
$$558$$ 0 0
$$559$$ 5.60671 0.237139
$$560$$ −3.96906 −0.167724
$$561$$ 0 0
$$562$$ 1.67666 0.0707255
$$563$$ 18.0998 0.762816 0.381408 0.924407i $$-0.375439\pi$$
0.381408 + 0.924407i $$0.375439\pi$$
$$564$$ 0 0
$$565$$ −9.69158 −0.407728
$$566$$ 0.978538 0.0411310
$$567$$ 0 0
$$568$$ 5.61728 0.235696
$$569$$ −14.4905 −0.607472 −0.303736 0.952756i $$-0.598234\pi$$
−0.303736 + 0.952756i $$0.598234\pi$$
$$570$$ 0 0
$$571$$ 0.703560 0.0294431 0.0147215 0.999892i $$-0.495314\pi$$
0.0147215 + 0.999892i $$0.495314\pi$$
$$572$$ −2.80336 −0.117214
$$573$$ 0 0
$$574$$ −33.5448 −1.40013
$$575$$ 3.43262 0.143150
$$576$$ 0 0
$$577$$ −32.8237 −1.36647 −0.683235 0.730199i $$-0.739427\pi$$
−0.683235 + 0.730199i $$0.739427\pi$$
$$578$$ −16.8763 −0.701959
$$579$$ 0 0
$$580$$ −9.37074 −0.389099
$$581$$ 0.417699 0.0173291
$$582$$ 0 0
$$583$$ 18.1107 0.750068
$$584$$ −12.8034 −0.529807
$$585$$ 0 0
$$586$$ −16.5823 −0.685009
$$587$$ 39.2929 1.62179 0.810895 0.585192i $$-0.198981\pi$$
0.810895 + 0.585192i $$0.198981\pi$$
$$588$$ 0 0
$$589$$ −1.02690 −0.0423128
$$590$$ 8.45158 0.347946
$$591$$ 0 0
$$592$$ 5.40168 0.222008
$$593$$ −10.3048 −0.423169 −0.211584 0.977360i $$-0.567862\pi$$
−0.211584 + 0.977360i $$0.567862\pi$$
$$594$$ 0 0
$$595$$ 1.39624 0.0572401
$$596$$ −9.37074 −0.383841
$$597$$ 0 0
$$598$$ 3.15514 0.129023
$$599$$ −28.3183 −1.15706 −0.578528 0.815662i $$-0.696373\pi$$
−0.578528 + 0.815662i $$0.696373\pi$$
$$600$$ 0 0
$$601$$ 24.2335 0.988504 0.494252 0.869319i $$-0.335442\pi$$
0.494252 + 0.869319i $$0.335442\pi$$
$$602$$ −24.2105 −0.986745
$$603$$ 0 0
$$604$$ −3.14021 −0.127773
$$605$$ −1.69812 −0.0690383
$$606$$ 0 0
$$607$$ 27.6148 1.12085 0.560425 0.828205i $$-0.310638\pi$$
0.560425 + 0.828205i $$0.310638\pi$$
$$608$$ −0.351780 −0.0142666
$$609$$ 0 0
$$610$$ −1.96906 −0.0797250
$$611$$ 12.1413 0.491185
$$612$$ 0 0
$$613$$ 37.7525 1.52481 0.762405 0.647101i $$-0.224019\pi$$
0.762405 + 0.647101i $$0.224019\pi$$
$$614$$ −24.3843 −0.984069
$$615$$ 0 0
$$616$$ 12.1052 0.487734
$$617$$ 25.5733 1.02954 0.514771 0.857328i $$-0.327877\pi$$
0.514771 + 0.857328i $$0.327877\pi$$
$$618$$ 0 0
$$619$$ −20.3518 −0.818007 −0.409004 0.912533i $$-0.634124\pi$$
−0.409004 + 0.912533i $$0.634124\pi$$
$$620$$ 2.91916 0.117236
$$621$$ 0 0
$$622$$ −6.03792 −0.242099
$$623$$ −64.3188 −2.57688
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 11.4301 0.456839
$$627$$ 0 0
$$628$$ 14.5798 0.581798
$$629$$ −1.90020 −0.0757660
$$630$$ 0 0
$$631$$ −17.7844 −0.707986 −0.353993 0.935248i $$-0.615176\pi$$
−0.353993 + 0.935248i $$0.615176\pi$$
$$632$$ 11.8842 0.472728
$$633$$ 0 0
$$634$$ 29.3911 1.16727
$$635$$ −12.9650 −0.514501
$$636$$ 0 0
$$637$$ 8.04586 0.318789
$$638$$ 28.5798 1.13149
$$639$$ 0 0
$$640$$ 1.00000 0.0395285
$$641$$ 37.4292 1.47836 0.739181 0.673506i $$-0.235213\pi$$
0.739181 + 0.673506i $$0.235213\pi$$
$$642$$ 0 0
$$643$$ 25.7729 1.01638 0.508191 0.861244i $$-0.330314\pi$$
0.508191 + 0.861244i $$0.330314\pi$$
$$644$$ −13.6243 −0.536871
$$645$$ 0 0
$$646$$ 0.123749 0.00486885
$$647$$ 29.8658 1.17415 0.587073 0.809534i $$-0.300280\pi$$
0.587073 + 0.809534i $$0.300280\pi$$
$$648$$ 0 0
$$649$$ −25.7765 −1.01181
$$650$$ 0.919164 0.0360526
$$651$$ 0 0
$$652$$ −4.59832 −0.180084
$$653$$ 48.8282 1.91080 0.955398 0.295322i $$-0.0954269\pi$$
0.955398 + 0.295322i $$0.0954269\pi$$
$$654$$ 0 0
$$655$$ 12.5135 0.488941
$$656$$ 8.45158 0.329979
$$657$$ 0 0
$$658$$ −52.4276 −2.04384
$$659$$ −8.24110 −0.321028 −0.160514 0.987034i $$-0.551315\pi$$
−0.160514 + 0.987034i $$0.551315\pi$$
$$660$$ 0 0
$$661$$ −24.6053 −0.957036 −0.478518 0.878078i $$-0.658826\pi$$
−0.478518 + 0.878078i $$0.658826\pi$$
$$662$$ −4.60781 −0.179088
$$663$$ 0 0
$$664$$ −0.105239 −0.00408406
$$665$$ 1.39624 0.0541438
$$666$$ 0 0
$$667$$ −32.1662 −1.24548
$$668$$ 6.40571 0.247844
$$669$$ 0 0
$$670$$ −1.00000 −0.0386334
$$671$$ 6.00544 0.231838
$$672$$ 0 0
$$673$$ −14.0822 −0.542831 −0.271415 0.962462i $$-0.587492\pi$$
−0.271415 + 0.962462i $$0.587492\pi$$
$$674$$ 18.0095 0.693699
$$675$$ 0 0
$$676$$ −12.1551 −0.467505
$$677$$ −5.67666 −0.218172 −0.109086 0.994032i $$-0.534792\pi$$
−0.109086 + 0.994032i $$0.534792\pi$$
$$678$$ 0 0
$$679$$ −12.1052 −0.464556
$$680$$ −0.351780 −0.0134901
$$681$$ 0 0
$$682$$ −8.90315 −0.340919
$$683$$ −3.63067 −0.138924 −0.0694618 0.997585i $$-0.522128\pi$$
−0.0694618 + 0.997585i $$0.522128\pi$$
$$684$$ 0 0
$$685$$ −1.40168 −0.0535554
$$686$$ −6.95959 −0.265718
$$687$$ 0 0
$$688$$ 6.09980 0.232553
$$689$$ −5.45811 −0.207937
$$690$$ 0 0
$$691$$ 13.2374 0.503574 0.251787 0.967783i $$-0.418982\pi$$
0.251787 + 0.967783i $$0.418982\pi$$
$$692$$ 24.9625 0.948933
$$693$$ 0 0
$$694$$ 10.7239 0.407075
$$695$$ 6.02844 0.228672
$$696$$ 0 0
$$697$$ −2.97310 −0.112614
$$698$$ 18.1996 0.688865
$$699$$ 0 0
$$700$$ −3.96906 −0.150016
$$701$$ 20.5125 0.774746 0.387373 0.921923i $$-0.373383\pi$$
0.387373 + 0.921923i $$0.373383\pi$$
$$702$$ 0 0
$$703$$ −1.90020 −0.0716675
$$704$$ −3.04990 −0.114947
$$705$$ 0 0
$$706$$ −25.5863 −0.962955
$$707$$ −18.6471 −0.701297
$$708$$ 0 0
$$709$$ 18.1946 0.683312 0.341656 0.939825i $$-0.389012\pi$$
0.341656 + 0.939825i $$0.389012\pi$$
$$710$$ 5.61728 0.210813
$$711$$ 0 0
$$712$$ 16.2050 0.607309
$$713$$ 10.0204 0.375266
$$714$$ 0 0
$$715$$ −2.80336 −0.104840
$$716$$ 14.1282 0.527997
$$717$$ 0 0
$$718$$ −4.59038 −0.171311
$$719$$ −11.4990 −0.428839 −0.214420 0.976742i $$-0.568786\pi$$
−0.214420 + 0.976742i $$0.568786\pi$$
$$720$$ 0 0
$$721$$ −8.33423 −0.310383
$$722$$ −18.8763 −0.702501
$$723$$ 0 0
$$724$$ −11.0729 −0.411521
$$725$$ −9.37074 −0.348021
$$726$$ 0 0
$$727$$ 25.8508 0.958751 0.479376 0.877610i $$-0.340863\pi$$
0.479376 + 0.877610i $$0.340863\pi$$
$$728$$ −3.64822 −0.135212
$$729$$ 0 0
$$730$$ −12.8034 −0.473874
$$731$$ −2.14579 −0.0793648
$$732$$ 0 0
$$733$$ 32.1836 1.18873 0.594364 0.804196i $$-0.297404\pi$$
0.594364 + 0.804196i $$0.297404\pi$$
$$734$$ 26.9855 0.996054
$$735$$ 0 0
$$736$$ 3.43262 0.126528
$$737$$ 3.04990 0.112344
$$738$$ 0 0
$$739$$ −9.32488 −0.343021 −0.171511 0.985182i $$-0.554865\pi$$
−0.171511 + 0.985182i $$0.554865\pi$$
$$740$$ 5.40168 0.198570
$$741$$ 0 0
$$742$$ 23.5688 0.865238
$$743$$ 12.5624 0.460869 0.230435 0.973088i $$-0.425985\pi$$
0.230435 + 0.973088i $$0.425985\pi$$
$$744$$ 0 0
$$745$$ −9.37074 −0.343317
$$746$$ −16.7954 −0.614924
$$747$$ 0 0
$$748$$ 1.07289 0.0392289
$$749$$ −39.0489 −1.42682
$$750$$ 0 0
$$751$$ −16.1996 −0.591132 −0.295566 0.955322i $$-0.595508\pi$$
−0.295566 + 0.955322i $$0.595508\pi$$
$$752$$ 13.2091 0.481685
$$753$$ 0 0
$$754$$ −8.61325 −0.313676
$$755$$ −3.14021 −0.114284
$$756$$ 0 0
$$757$$ 16.8682 0.613084 0.306542 0.951857i $$-0.400828\pi$$
0.306542 + 0.951857i $$0.400828\pi$$
$$758$$ −21.9047 −0.795614
$$759$$ 0 0
$$760$$ −0.351780 −0.0127604
$$761$$ −21.1701 −0.767414 −0.383707 0.923455i $$-0.625353\pi$$
−0.383707 + 0.923455i $$0.625353\pi$$
$$762$$ 0 0
$$763$$ −5.02286 −0.181840
$$764$$ 18.0998 0.654828
$$765$$ 0 0
$$766$$ 4.47457 0.161673
$$767$$ 7.76839 0.280500
$$768$$ 0 0
$$769$$ −11.2884 −0.407069 −0.203535 0.979068i $$-0.565243\pi$$
−0.203535 + 0.979068i $$0.565243\pi$$
$$770$$ 12.1052 0.436243
$$771$$ 0 0
$$772$$ −11.6861 −0.420593
$$773$$ −48.8068 −1.75546 −0.877729 0.479158i $$-0.840942\pi$$
−0.877729 + 0.479158i $$0.840942\pi$$
$$774$$ 0 0
$$775$$ 2.91916 0.104859
$$776$$ 3.04990 0.109485
$$777$$ 0 0
$$778$$ −30.4545 −1.09185
$$779$$ −2.97310 −0.106522
$$780$$ 0 0
$$781$$ −17.1321 −0.613036
$$782$$ −1.20753 −0.0431810
$$783$$ 0 0
$$784$$ 8.75346 0.312624
$$785$$ 14.5798 0.520376
$$786$$ 0 0
$$787$$ 17.4880 0.623379 0.311689 0.950184i $$-0.399105\pi$$
0.311689 + 0.950184i $$0.399105\pi$$
$$788$$ 0.161672 0.00575932
$$789$$ 0 0
$$790$$ 11.8842 0.422821
$$791$$ 38.4665 1.36771
$$792$$ 0 0
$$793$$ −1.80989 −0.0642711
$$794$$ 27.3777 0.971599
$$795$$ 0 0
$$796$$ −8.10524 −0.287283
$$797$$ −40.7200 −1.44238 −0.721189 0.692739i $$-0.756404\pi$$
−0.721189 + 0.692739i $$0.756404\pi$$
$$798$$ 0 0
$$799$$ −4.64669 −0.164388
$$800$$ 1.00000 0.0353553
$$801$$ 0 0
$$802$$ 34.0867 1.20364
$$803$$ 39.0489 1.37801
$$804$$ 0 0
$$805$$ −13.6243 −0.480192
$$806$$ 2.68319 0.0945114
$$807$$ 0 0
$$808$$ 4.69812 0.165279
$$809$$ −26.0000 −0.914111 −0.457056 0.889438i $$-0.651096\pi$$
−0.457056 + 0.889438i $$0.651096\pi$$
$$810$$ 0 0
$$811$$ −1.32725 −0.0466061 −0.0233031 0.999728i $$-0.507418\pi$$
−0.0233031 + 0.999728i $$0.507418\pi$$
$$812$$ 37.1931 1.30522
$$813$$ 0 0
$$814$$ −16.4746 −0.577433
$$815$$ −4.59832 −0.161072
$$816$$ 0 0
$$817$$ −2.14579 −0.0750716
$$818$$ 8.16167 0.285366
$$819$$ 0 0
$$820$$ 8.45158 0.295142
$$821$$ 0.775046 0.0270493 0.0135246 0.999909i $$-0.495695\pi$$
0.0135246 + 0.999909i $$0.495695\pi$$
$$822$$ 0 0
$$823$$ −7.00295 −0.244108 −0.122054 0.992523i $$-0.538948\pi$$
−0.122054 + 0.992523i $$0.538948\pi$$
$$824$$ 2.09980 0.0731499
$$825$$ 0 0
$$826$$ −33.5448 −1.16717
$$827$$ −8.24296 −0.286636 −0.143318 0.989677i $$-0.545777\pi$$
−0.143318 + 0.989677i $$0.545777\pi$$
$$828$$ 0 0
$$829$$ −17.2346 −0.598581 −0.299291 0.954162i $$-0.596750\pi$$
−0.299291 + 0.954162i $$0.596750\pi$$
$$830$$ −0.105239 −0.00365289
$$831$$ 0 0
$$832$$ 0.919164 0.0318663
$$833$$ −3.07929 −0.106691
$$834$$ 0 0
$$835$$ 6.40571 0.221679
$$836$$ 1.07289 0.0371068
$$837$$ 0 0
$$838$$ −27.3628 −0.945232
$$839$$ 11.8951 0.410664 0.205332 0.978692i $$-0.434173\pi$$
0.205332 + 0.978692i $$0.434173\pi$$
$$840$$ 0 0
$$841$$ 58.8108 2.02796
$$842$$ −15.9760 −0.550571
$$843$$ 0 0
$$844$$ 18.3278 0.630870
$$845$$ −12.1551 −0.418149
$$846$$ 0 0
$$847$$ 6.73994 0.231587
$$848$$ −5.93813 −0.203916
$$849$$ 0 0
$$850$$ −0.351780 −0.0120660
$$851$$ 18.5419 0.635608
$$852$$ 0 0
$$853$$ −25.9841 −0.889679 −0.444840 0.895610i $$-0.646739\pi$$
−0.444840 + 0.895610i $$0.646739\pi$$
$$854$$ 7.81533 0.267435
$$855$$ 0 0
$$856$$ 9.83833 0.336267
$$857$$ −43.6785 −1.49203 −0.746015 0.665929i $$-0.768035\pi$$
−0.746015 + 0.665929i $$0.768035\pi$$
$$858$$ 0 0
$$859$$ −13.1682 −0.449293 −0.224647 0.974440i $$-0.572123\pi$$
−0.224647 + 0.974440i $$0.572123\pi$$
$$860$$ 6.09980 0.208001
$$861$$ 0 0
$$862$$ −16.7900 −0.571869
$$863$$ 12.3059 0.418898 0.209449 0.977820i $$-0.432833\pi$$
0.209449 + 0.977820i $$0.432833\pi$$
$$864$$ 0 0
$$865$$ 24.9625 0.848751
$$866$$ 29.6352 1.00704
$$867$$ 0 0
$$868$$ −11.5863 −0.393266
$$869$$ −36.2456 −1.22955
$$870$$ 0 0
$$871$$ −0.919164 −0.0311447
$$872$$ 1.26550 0.0428553
$$873$$ 0 0
$$874$$ −1.20753 −0.0408452
$$875$$ −3.96906 −0.134179
$$876$$ 0 0
$$877$$ 41.1407 1.38922 0.694611 0.719386i $$-0.255577\pi$$
0.694611 + 0.719386i $$0.255577\pi$$
$$878$$ 1.99456 0.0673131
$$879$$ 0 0
$$880$$ −3.04990 −0.102812
$$881$$ 25.6122 0.862895 0.431448 0.902138i $$-0.358003\pi$$
0.431448 + 0.902138i $$0.358003\pi$$
$$882$$ 0 0
$$883$$ −5.96208 −0.200640 −0.100320 0.994955i $$-0.531987\pi$$
−0.100320 + 0.994955i $$0.531987\pi$$
$$884$$ −0.323344 −0.0108752
$$885$$ 0 0
$$886$$ 3.29644 0.110746
$$887$$ 1.79388 0.0602327 0.0301163 0.999546i $$-0.490412\pi$$
0.0301163 + 0.999546i $$0.490412\pi$$
$$888$$ 0 0
$$889$$ 51.4590 1.72588
$$890$$ 16.2050 0.543194
$$891$$ 0 0
$$892$$ −0.442091 −0.0148023
$$893$$ −4.64669 −0.155495
$$894$$ 0 0
$$895$$ 14.1282 0.472255
$$896$$ −3.96906 −0.132597
$$897$$ 0 0
$$898$$ 13.9196 0.464504
$$899$$ −27.3547 −0.912331
$$900$$ 0 0
$$901$$ 2.08891 0.0695918
$$902$$ −25.7765 −0.858262
$$903$$ 0 0
$$904$$ −9.69158 −0.322337
$$905$$ −11.0729 −0.368075
$$906$$ 0 0
$$907$$ −54.3603 −1.80500 −0.902502 0.430685i $$-0.858272\pi$$
−0.902502 + 0.430685i $$0.858272\pi$$
$$908$$ 25.3344 0.840750
$$909$$ 0 0
$$910$$ −3.64822 −0.120937
$$911$$ −20.1890 −0.668892 −0.334446 0.942415i $$-0.608549\pi$$
−0.334446 + 0.942415i $$0.608549\pi$$
$$912$$ 0 0
$$913$$ 0.320968 0.0106225
$$914$$ 12.3897 0.409815
$$915$$ 0 0
$$916$$ 8.06886 0.266603
$$917$$ −49.6667 −1.64014
$$918$$ 0 0
$$919$$ −37.9221 −1.25094 −0.625468 0.780250i $$-0.715092\pi$$
−0.625468 + 0.780250i $$0.715092\pi$$
$$920$$ 3.43262 0.113170
$$921$$ 0 0
$$922$$ −38.8537 −1.27958
$$923$$ 5.16320 0.169949
$$924$$ 0 0
$$925$$ 5.40168 0.177606
$$926$$ −5.85076 −0.192268
$$927$$ 0 0
$$928$$ −9.37074 −0.307610
$$929$$ 51.0074 1.67350 0.836750 0.547585i $$-0.184453\pi$$
0.836750 + 0.547585i $$0.184453\pi$$
$$930$$ 0 0
$$931$$ −3.07929 −0.100920
$$932$$ −3.67121 −0.120255
$$933$$ 0 0
$$934$$ 6.37477 0.208589
$$935$$ 1.07289 0.0350874
$$936$$ 0 0
$$937$$ −36.9290 −1.20642 −0.603208 0.797584i $$-0.706111\pi$$
−0.603208 + 0.797584i $$0.706111\pi$$
$$938$$ 3.96906 0.129594
$$939$$ 0 0
$$940$$ 13.2091 0.430832
$$941$$ 45.0489 1.46855 0.734277 0.678850i $$-0.237521\pi$$
0.734277 + 0.678850i $$0.237521\pi$$
$$942$$ 0 0
$$943$$ 29.0110 0.944729
$$944$$ 8.45158 0.275075
$$945$$ 0 0
$$946$$ −18.6038 −0.604861
$$947$$ 5.68210 0.184643 0.0923217 0.995729i $$-0.470571\pi$$
0.0923217 + 0.995729i $$0.470571\pi$$
$$948$$ 0 0
$$949$$ −11.7684 −0.382018
$$950$$ −0.351780 −0.0114133
$$951$$ 0 0
$$952$$ 1.39624 0.0452523
$$953$$ −34.3738 −1.11348 −0.556739 0.830688i $$-0.687947\pi$$
−0.556739 + 0.830688i $$0.687947\pi$$
$$954$$ 0 0
$$955$$ 18.0998 0.585696
$$956$$ −12.9650 −0.419319
$$957$$ 0 0
$$958$$ −13.5094 −0.436469
$$959$$ 5.56335 0.179650
$$960$$ 0 0
$$961$$ −22.4785 −0.725112
$$962$$ 4.96503 0.160079
$$963$$ 0 0
$$964$$ −21.9531 −0.707060
$$965$$ −11.6861 −0.376190
$$966$$ 0 0
$$967$$ −55.7314 −1.79220 −0.896100 0.443852i $$-0.853611\pi$$
−0.896100 + 0.443852i $$0.853611\pi$$
$$968$$ −1.69812 −0.0545796
$$969$$ 0 0
$$970$$ 3.04990 0.0979263
$$971$$ 41.3438 1.32679 0.663394 0.748271i $$-0.269116\pi$$
0.663394 + 0.748271i $$0.269116\pi$$
$$972$$ 0 0
$$973$$ −23.9272 −0.767072
$$974$$ 24.4920 0.784774
$$975$$ 0 0
$$976$$ −1.96906 −0.0630282
$$977$$ 61.0643 1.95362 0.976810 0.214107i $$-0.0686842\pi$$
0.976810 + 0.214107i $$0.0686842\pi$$
$$978$$ 0 0
$$979$$ −49.4237 −1.57959
$$980$$ 8.75346 0.279619
$$981$$ 0 0
$$982$$ −1.59723 −0.0509696
$$983$$ −5.76557 −0.183893 −0.0919466 0.995764i $$-0.529309\pi$$
−0.0919466 + 0.995764i $$0.529309\pi$$
$$984$$ 0 0
$$985$$ 0.161672 0.00515129
$$986$$ 3.29644 0.104980
$$987$$ 0 0
$$988$$ −0.323344 −0.0102869
$$989$$ 20.9383 0.665798
$$990$$ 0 0
$$991$$ 22.5339 0.715814 0.357907 0.933757i $$-0.383490\pi$$
0.357907 + 0.933757i $$0.383490\pi$$
$$992$$ 2.91916 0.0926836
$$993$$ 0 0
$$994$$ −22.2953 −0.707165
$$995$$ −8.10524 −0.256953
$$996$$ 0 0
$$997$$ 6.21310 0.196771 0.0983855 0.995148i $$-0.468632\pi$$
0.0983855 + 0.995148i $$0.468632\pi$$
$$998$$ 21.5837 0.683221
$$999$$ 0 0
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 6030.2.a.bu.1.1 4
3.2 odd 2 2010.2.a.r.1.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
2010.2.a.r.1.1 4 3.2 odd 2
6030.2.a.bu.1.1 4 1.1 even 1 trivial