# Properties

 Label 8470.2.a.cy.1.1 Level $8470$ Weight $2$ Character 8470.1 Self dual yes Analytic conductor $67.633$ Analytic rank $1$ Dimension $6$ CM no Inner twists $1$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8470.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$67.6332905120$$ Analytic rank: $$1$$ Dimension: $$6$$ Coefficient field: 6.6.13298000.1 Defining polynomial: $$x^{6} - x^{5} - 10 x^{4} + 3 x^{3} + 26 x^{2} + 13 x - 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 770) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-2.27063$$ of defining polynomial Character $$\chi$$ $$=$$ 8470.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -2.80837 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.80837 q^{6} -1.00000 q^{7} -1.00000 q^{8} +4.88695 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} -2.80837 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.80837 q^{6} -1.00000 q^{7} -1.00000 q^{8} +4.88695 q^{9} -1.00000 q^{10} -2.80837 q^{12} +3.05764 q^{13} +1.00000 q^{14} -2.80837 q^{15} +1.00000 q^{16} -6.94459 q^{17} -4.88695 q^{18} +4.69704 q^{19} +1.00000 q^{20} +2.80837 q^{21} -1.73118 q^{23} +2.80837 q^{24} +1.00000 q^{25} -3.05764 q^{26} -5.29926 q^{27} -1.00000 q^{28} +6.74554 q^{29} +2.80837 q^{30} +3.34334 q^{31} -1.00000 q^{32} +6.94459 q^{34} -1.00000 q^{35} +4.88695 q^{36} -7.32736 q^{37} -4.69704 q^{38} -8.58700 q^{39} -1.00000 q^{40} -7.58977 q^{41} -2.80837 q^{42} -7.27592 q^{43} +4.88695 q^{45} +1.73118 q^{46} +4.00555 q^{47} -2.80837 q^{48} +1.00000 q^{49} -1.00000 q^{50} +19.5030 q^{51} +3.05764 q^{52} -1.70116 q^{53} +5.29926 q^{54} +1.00000 q^{56} -13.1910 q^{57} -6.74554 q^{58} +7.64741 q^{59} -2.80837 q^{60} -8.93050 q^{61} -3.34334 q^{62} -4.88695 q^{63} +1.00000 q^{64} +3.05764 q^{65} +10.5718 q^{67} -6.94459 q^{68} +4.86179 q^{69} +1.00000 q^{70} -2.30414 q^{71} -4.88695 q^{72} +13.1113 q^{73} +7.32736 q^{74} -2.80837 q^{75} +4.69704 q^{76} +8.58700 q^{78} -4.25829 q^{79} +1.00000 q^{80} +0.221440 q^{81} +7.58977 q^{82} -3.86073 q^{83} +2.80837 q^{84} -6.94459 q^{85} +7.27592 q^{86} -18.9440 q^{87} -12.3140 q^{89} -4.88695 q^{90} -3.05764 q^{91} -1.73118 q^{92} -9.38933 q^{93} -4.00555 q^{94} +4.69704 q^{95} +2.80837 q^{96} -6.94282 q^{97} -1.00000 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - 6q^{2} + q^{3} + 6q^{4} + 6q^{5} - q^{6} - 6q^{7} - 6q^{8} + 15q^{9} + O(q^{10})$$ $$6q - 6q^{2} + q^{3} + 6q^{4} + 6q^{5} - q^{6} - 6q^{7} - 6q^{8} + 15q^{9} - 6q^{10} + q^{12} - 2q^{13} + 6q^{14} + q^{15} + 6q^{16} - 7q^{17} - 15q^{18} - 11q^{19} + 6q^{20} - q^{21} - 6q^{23} - q^{24} + 6q^{25} + 2q^{26} + 4q^{27} - 6q^{28} - 2q^{29} - q^{30} - 6q^{32} + 7q^{34} - 6q^{35} + 15q^{36} - 14q^{37} + 11q^{38} - 20q^{39} - 6q^{40} - 13q^{41} + q^{42} - 19q^{43} + 15q^{45} + 6q^{46} + 22q^{47} + q^{48} + 6q^{49} - 6q^{50} + 14q^{51} - 2q^{52} - 10q^{53} - 4q^{54} + 6q^{56} - 32q^{57} + 2q^{58} - 7q^{59} + q^{60} - 22q^{61} - 15q^{63} + 6q^{64} - 2q^{65} + 5q^{67} - 7q^{68} - 36q^{69} + 6q^{70} + 8q^{71} - 15q^{72} - 13q^{73} + 14q^{74} + q^{75} - 11q^{76} + 20q^{78} + 16q^{79} + 6q^{80} + 18q^{81} + 13q^{82} + 5q^{83} - q^{84} - 7q^{85} + 19q^{86} - 14q^{87} + q^{89} - 15q^{90} + 2q^{91} - 6q^{92} - 42q^{93} - 22q^{94} - 11q^{95} - q^{96} - 3q^{97} - 6q^{98} + O(q^{100})$$

## 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
$$3$$ −2.80837 −1.62141 −0.810707 0.585452i $$-0.800917\pi$$
−0.810707 + 0.585452i $$0.800917\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 1.00000 0.447214
$$6$$ 2.80837 1.14651
$$7$$ −1.00000 −0.377964
$$8$$ −1.00000 −0.353553
$$9$$ 4.88695 1.62898
$$10$$ −1.00000 −0.316228
$$11$$ 0 0
$$12$$ −2.80837 −0.810707
$$13$$ 3.05764 0.848037 0.424019 0.905653i $$-0.360619\pi$$
0.424019 + 0.905653i $$0.360619\pi$$
$$14$$ 1.00000 0.267261
$$15$$ −2.80837 −0.725118
$$16$$ 1.00000 0.250000
$$17$$ −6.94459 −1.68431 −0.842156 0.539234i $$-0.818714\pi$$
−0.842156 + 0.539234i $$0.818714\pi$$
$$18$$ −4.88695 −1.15187
$$19$$ 4.69704 1.07757 0.538787 0.842442i $$-0.318883\pi$$
0.538787 + 0.842442i $$0.318883\pi$$
$$20$$ 1.00000 0.223607
$$21$$ 2.80837 0.612837
$$22$$ 0 0
$$23$$ −1.73118 −0.360976 −0.180488 0.983577i $$-0.557768\pi$$
−0.180488 + 0.983577i $$0.557768\pi$$
$$24$$ 2.80837 0.573256
$$25$$ 1.00000 0.200000
$$26$$ −3.05764 −0.599653
$$27$$ −5.29926 −1.01984
$$28$$ −1.00000 −0.188982
$$29$$ 6.74554 1.25262 0.626308 0.779576i $$-0.284565\pi$$
0.626308 + 0.779576i $$0.284565\pi$$
$$30$$ 2.80837 0.512736
$$31$$ 3.34334 0.600481 0.300240 0.953864i $$-0.402933\pi$$
0.300240 + 0.953864i $$0.402933\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ 6.94459 1.19099
$$35$$ −1.00000 −0.169031
$$36$$ 4.88695 0.814492
$$37$$ −7.32736 −1.20461 −0.602305 0.798266i $$-0.705751\pi$$
−0.602305 + 0.798266i $$0.705751\pi$$
$$38$$ −4.69704 −0.761960
$$39$$ −8.58700 −1.37502
$$40$$ −1.00000 −0.158114
$$41$$ −7.58977 −1.18532 −0.592662 0.805452i $$-0.701923\pi$$
−0.592662 + 0.805452i $$0.701923\pi$$
$$42$$ −2.80837 −0.433341
$$43$$ −7.27592 −1.10957 −0.554784 0.831995i $$-0.687199\pi$$
−0.554784 + 0.831995i $$0.687199\pi$$
$$44$$ 0 0
$$45$$ 4.88695 0.728504
$$46$$ 1.73118 0.255248
$$47$$ 4.00555 0.584270 0.292135 0.956377i $$-0.405634\pi$$
0.292135 + 0.956377i $$0.405634\pi$$
$$48$$ −2.80837 −0.405354
$$49$$ 1.00000 0.142857
$$50$$ −1.00000 −0.141421
$$51$$ 19.5030 2.73097
$$52$$ 3.05764 0.424019
$$53$$ −1.70116 −0.233673 −0.116836 0.993151i $$-0.537275\pi$$
−0.116836 + 0.993151i $$0.537275\pi$$
$$54$$ 5.29926 0.721138
$$55$$ 0 0
$$56$$ 1.00000 0.133631
$$57$$ −13.1910 −1.74719
$$58$$ −6.74554 −0.885733
$$59$$ 7.64741 0.995608 0.497804 0.867289i $$-0.334140\pi$$
0.497804 + 0.867289i $$0.334140\pi$$
$$60$$ −2.80837 −0.362559
$$61$$ −8.93050 −1.14343 −0.571717 0.820451i $$-0.693722\pi$$
−0.571717 + 0.820451i $$0.693722\pi$$
$$62$$ −3.34334 −0.424604
$$63$$ −4.88695 −0.615698
$$64$$ 1.00000 0.125000
$$65$$ 3.05764 0.379254
$$66$$ 0 0
$$67$$ 10.5718 1.29155 0.645773 0.763529i $$-0.276535\pi$$
0.645773 + 0.763529i $$0.276535\pi$$
$$68$$ −6.94459 −0.842156
$$69$$ 4.86179 0.585291
$$70$$ 1.00000 0.119523
$$71$$ −2.30414 −0.273451 −0.136725 0.990609i $$-0.543658\pi$$
−0.136725 + 0.990609i $$0.543658\pi$$
$$72$$ −4.88695 −0.575933
$$73$$ 13.1113 1.53456 0.767278 0.641314i $$-0.221611\pi$$
0.767278 + 0.641314i $$0.221611\pi$$
$$74$$ 7.32736 0.851788
$$75$$ −2.80837 −0.324283
$$76$$ 4.69704 0.538787
$$77$$ 0 0
$$78$$ 8.58700 0.972286
$$79$$ −4.25829 −0.479095 −0.239547 0.970885i $$-0.576999\pi$$
−0.239547 + 0.970885i $$0.576999\pi$$
$$80$$ 1.00000 0.111803
$$81$$ 0.221440 0.0246044
$$82$$ 7.58977 0.838150
$$83$$ −3.86073 −0.423770 −0.211885 0.977295i $$-0.567960\pi$$
−0.211885 + 0.977295i $$0.567960\pi$$
$$84$$ 2.80837 0.306418
$$85$$ −6.94459 −0.753247
$$86$$ 7.27592 0.784583
$$87$$ −18.9440 −2.03101
$$88$$ 0 0
$$89$$ −12.3140 −1.30529 −0.652643 0.757665i $$-0.726340\pi$$
−0.652643 + 0.757665i $$0.726340\pi$$
$$90$$ −4.88695 −0.515130
$$91$$ −3.05764 −0.320528
$$92$$ −1.73118 −0.180488
$$93$$ −9.38933 −0.973628
$$94$$ −4.00555 −0.413141
$$95$$ 4.69704 0.481906
$$96$$ 2.80837 0.286628
$$97$$ −6.94282 −0.704936 −0.352468 0.935824i $$-0.614658\pi$$
−0.352468 + 0.935824i $$0.614658\pi$$
$$98$$ −1.00000 −0.101015
$$99$$ 0 0
$$100$$ 1.00000 0.100000
$$101$$ −13.5915 −1.35240 −0.676202 0.736717i $$-0.736375\pi$$
−0.676202 + 0.736717i $$0.736375\pi$$
$$102$$ −19.5030 −1.93108
$$103$$ 15.8591 1.56264 0.781320 0.624131i $$-0.214547\pi$$
0.781320 + 0.624131i $$0.214547\pi$$
$$104$$ −3.05764 −0.299826
$$105$$ 2.80837 0.274069
$$106$$ 1.70116 0.165232
$$107$$ 4.39685 0.425060 0.212530 0.977155i $$-0.431830\pi$$
0.212530 + 0.977155i $$0.431830\pi$$
$$108$$ −5.29926 −0.509922
$$109$$ 14.4931 1.38819 0.694095 0.719884i $$-0.255805\pi$$
0.694095 + 0.719884i $$0.255805\pi$$
$$110$$ 0 0
$$111$$ 20.5779 1.95317
$$112$$ −1.00000 −0.0944911
$$113$$ −19.3326 −1.81866 −0.909328 0.416081i $$-0.863403\pi$$
−0.909328 + 0.416081i $$0.863403\pi$$
$$114$$ 13.1910 1.23545
$$115$$ −1.73118 −0.161433
$$116$$ 6.74554 0.626308
$$117$$ 14.9425 1.38144
$$118$$ −7.64741 −0.704001
$$119$$ 6.94459 0.636610
$$120$$ 2.80837 0.256368
$$121$$ 0 0
$$122$$ 8.93050 0.808529
$$123$$ 21.3149 1.92190
$$124$$ 3.34334 0.300240
$$125$$ 1.00000 0.0894427
$$126$$ 4.88695 0.435364
$$127$$ −21.5848 −1.91534 −0.957669 0.287870i $$-0.907053\pi$$
−0.957669 + 0.287870i $$0.907053\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 20.4335 1.79907
$$130$$ −3.05764 −0.268173
$$131$$ −5.23468 −0.457356 −0.228678 0.973502i $$-0.573440\pi$$
−0.228678 + 0.973502i $$0.573440\pi$$
$$132$$ 0 0
$$133$$ −4.69704 −0.407285
$$134$$ −10.5718 −0.913261
$$135$$ −5.29926 −0.456088
$$136$$ 6.94459 0.595494
$$137$$ 5.00039 0.427212 0.213606 0.976920i $$-0.431479\pi$$
0.213606 + 0.976920i $$0.431479\pi$$
$$138$$ −4.86179 −0.413863
$$139$$ 3.08024 0.261262 0.130631 0.991431i $$-0.458300\pi$$
0.130631 + 0.991431i $$0.458300\pi$$
$$140$$ −1.00000 −0.0845154
$$141$$ −11.2491 −0.947343
$$142$$ 2.30414 0.193359
$$143$$ 0 0
$$144$$ 4.88695 0.407246
$$145$$ 6.74554 0.560187
$$146$$ −13.1113 −1.08510
$$147$$ −2.80837 −0.231631
$$148$$ −7.32736 −0.602305
$$149$$ 15.8858 1.30141 0.650706 0.759330i $$-0.274473\pi$$
0.650706 + 0.759330i $$0.274473\pi$$
$$150$$ 2.80837 0.229303
$$151$$ 5.45916 0.444260 0.222130 0.975017i $$-0.428699\pi$$
0.222130 + 0.975017i $$0.428699\pi$$
$$152$$ −4.69704 −0.380980
$$153$$ −33.9379 −2.74372
$$154$$ 0 0
$$155$$ 3.34334 0.268543
$$156$$ −8.58700 −0.687510
$$157$$ 8.44070 0.673641 0.336820 0.941569i $$-0.390648\pi$$
0.336820 + 0.941569i $$0.390648\pi$$
$$158$$ 4.25829 0.338771
$$159$$ 4.77750 0.378880
$$160$$ −1.00000 −0.0790569
$$161$$ 1.73118 0.136436
$$162$$ −0.221440 −0.0173980
$$163$$ 9.74355 0.763174 0.381587 0.924333i $$-0.375378\pi$$
0.381587 + 0.924333i $$0.375378\pi$$
$$164$$ −7.58977 −0.592662
$$165$$ 0 0
$$166$$ 3.86073 0.299650
$$167$$ −20.2081 −1.56375 −0.781873 0.623437i $$-0.785736\pi$$
−0.781873 + 0.623437i $$0.785736\pi$$
$$168$$ −2.80837 −0.216671
$$169$$ −3.65082 −0.280833
$$170$$ 6.94459 0.532626
$$171$$ 22.9542 1.75535
$$172$$ −7.27592 −0.554784
$$173$$ −1.06417 −0.0809073 −0.0404537 0.999181i $$-0.512880\pi$$
−0.0404537 + 0.999181i $$0.512880\pi$$
$$174$$ 18.9440 1.43614
$$175$$ −1.00000 −0.0755929
$$176$$ 0 0
$$177$$ −21.4768 −1.61429
$$178$$ 12.3140 0.922977
$$179$$ −5.73780 −0.428863 −0.214432 0.976739i $$-0.568790\pi$$
−0.214432 + 0.976739i $$0.568790\pi$$
$$180$$ 4.88695 0.364252
$$181$$ 3.77619 0.280682 0.140341 0.990103i $$-0.455180\pi$$
0.140341 + 0.990103i $$0.455180\pi$$
$$182$$ 3.05764 0.226648
$$183$$ 25.0802 1.85398
$$184$$ 1.73118 0.127624
$$185$$ −7.32736 −0.538718
$$186$$ 9.38933 0.688459
$$187$$ 0 0
$$188$$ 4.00555 0.292135
$$189$$ 5.29926 0.385465
$$190$$ −4.69704 −0.340759
$$191$$ 11.7298 0.848736 0.424368 0.905490i $$-0.360496\pi$$
0.424368 + 0.905490i $$0.360496\pi$$
$$192$$ −2.80837 −0.202677
$$193$$ −18.9879 −1.36678 −0.683391 0.730053i $$-0.739495\pi$$
−0.683391 + 0.730053i $$0.739495\pi$$
$$194$$ 6.94282 0.498465
$$195$$ −8.58700 −0.614928
$$196$$ 1.00000 0.0714286
$$197$$ 3.27508 0.233340 0.116670 0.993171i $$-0.462778\pi$$
0.116670 + 0.993171i $$0.462778\pi$$
$$198$$ 0 0
$$199$$ 14.7438 1.04516 0.522580 0.852590i $$-0.324970\pi$$
0.522580 + 0.852590i $$0.324970\pi$$
$$200$$ −1.00000 −0.0707107
$$201$$ −29.6894 −2.09413
$$202$$ 13.5915 0.956294
$$203$$ −6.74554 −0.473444
$$204$$ 19.5030 1.36548
$$205$$ −7.58977 −0.530093
$$206$$ −15.8591 −1.10495
$$207$$ −8.46019 −0.588024
$$208$$ 3.05764 0.212009
$$209$$ 0 0
$$210$$ −2.80837 −0.193796
$$211$$ 13.6709 0.941143 0.470572 0.882362i $$-0.344048\pi$$
0.470572 + 0.882362i $$0.344048\pi$$
$$212$$ −1.70116 −0.116836
$$213$$ 6.47087 0.443377
$$214$$ −4.39685 −0.300563
$$215$$ −7.27592 −0.496214
$$216$$ 5.29926 0.360569
$$217$$ −3.34334 −0.226960
$$218$$ −14.4931 −0.981598
$$219$$ −36.8213 −2.48815
$$220$$ 0 0
$$221$$ −21.2341 −1.42836
$$222$$ −20.5779 −1.38110
$$223$$ −5.23577 −0.350613 −0.175307 0.984514i $$-0.556092\pi$$
−0.175307 + 0.984514i $$0.556092\pi$$
$$224$$ 1.00000 0.0668153
$$225$$ 4.88695 0.325797
$$226$$ 19.3326 1.28598
$$227$$ 22.5392 1.49598 0.747991 0.663709i $$-0.231019\pi$$
0.747991 + 0.663709i $$0.231019\pi$$
$$228$$ −13.1910 −0.873597
$$229$$ 1.54850 0.102328 0.0511639 0.998690i $$-0.483707\pi$$
0.0511639 + 0.998690i $$0.483707\pi$$
$$230$$ 1.73118 0.114151
$$231$$ 0 0
$$232$$ −6.74554 −0.442867
$$233$$ 6.59831 0.432270 0.216135 0.976364i $$-0.430655\pi$$
0.216135 + 0.976364i $$0.430655\pi$$
$$234$$ −14.9425 −0.976825
$$235$$ 4.00555 0.261293
$$236$$ 7.64741 0.497804
$$237$$ 11.9589 0.776811
$$238$$ −6.94459 −0.450151
$$239$$ −9.32864 −0.603420 −0.301710 0.953400i $$-0.597557\pi$$
−0.301710 + 0.953400i $$0.597557\pi$$
$$240$$ −2.80837 −0.181280
$$241$$ 4.69741 0.302587 0.151293 0.988489i $$-0.451656\pi$$
0.151293 + 0.988489i $$0.451656\pi$$
$$242$$ 0 0
$$243$$ 15.2759 0.979949
$$244$$ −8.93050 −0.571717
$$245$$ 1.00000 0.0638877
$$246$$ −21.3149 −1.35899
$$247$$ 14.3619 0.913823
$$248$$ −3.34334 −0.212302
$$249$$ 10.8424 0.687106
$$250$$ −1.00000 −0.0632456
$$251$$ 25.1086 1.58484 0.792420 0.609976i $$-0.208821\pi$$
0.792420 + 0.609976i $$0.208821\pi$$
$$252$$ −4.88695 −0.307849
$$253$$ 0 0
$$254$$ 21.5848 1.35435
$$255$$ 19.5030 1.22133
$$256$$ 1.00000 0.0625000
$$257$$ 22.2671 1.38898 0.694491 0.719502i $$-0.255630\pi$$
0.694491 + 0.719502i $$0.255630\pi$$
$$258$$ −20.4335 −1.27213
$$259$$ 7.32736 0.455300
$$260$$ 3.05764 0.189627
$$261$$ 32.9651 2.04049
$$262$$ 5.23468 0.323400
$$263$$ 12.5549 0.774170 0.387085 0.922044i $$-0.373482\pi$$
0.387085 + 0.922044i $$0.373482\pi$$
$$264$$ 0 0
$$265$$ −1.70116 −0.104502
$$266$$ 4.69704 0.287994
$$267$$ 34.5824 2.11641
$$268$$ 10.5718 0.645773
$$269$$ −7.54481 −0.460015 −0.230007 0.973189i $$-0.573875\pi$$
−0.230007 + 0.973189i $$0.573875\pi$$
$$270$$ 5.29926 0.322503
$$271$$ −27.7828 −1.68768 −0.843842 0.536592i $$-0.819711\pi$$
−0.843842 + 0.536592i $$0.819711\pi$$
$$272$$ −6.94459 −0.421078
$$273$$ 8.58700 0.519709
$$274$$ −5.00039 −0.302085
$$275$$ 0 0
$$276$$ 4.86179 0.292646
$$277$$ −18.8691 −1.13374 −0.566868 0.823809i $$-0.691845\pi$$
−0.566868 + 0.823809i $$0.691845\pi$$
$$278$$ −3.08024 −0.184740
$$279$$ 16.3387 0.978173
$$280$$ 1.00000 0.0597614
$$281$$ −6.86665 −0.409630 −0.204815 0.978801i $$-0.565659\pi$$
−0.204815 + 0.978801i $$0.565659\pi$$
$$282$$ 11.2491 0.669873
$$283$$ 25.1097 1.49261 0.746307 0.665601i $$-0.231825\pi$$
0.746307 + 0.665601i $$0.231825\pi$$
$$284$$ −2.30414 −0.136725
$$285$$ −13.1910 −0.781369
$$286$$ 0 0
$$287$$ 7.58977 0.448010
$$288$$ −4.88695 −0.287966
$$289$$ 31.2274 1.83690
$$290$$ −6.74554 −0.396112
$$291$$ 19.4980 1.14299
$$292$$ 13.1113 0.767278
$$293$$ −16.5950 −0.969491 −0.484746 0.874655i $$-0.661088\pi$$
−0.484746 + 0.874655i $$0.661088\pi$$
$$294$$ 2.80837 0.163788
$$295$$ 7.64741 0.445250
$$296$$ 7.32736 0.425894
$$297$$ 0 0
$$298$$ −15.8858 −0.920237
$$299$$ −5.29332 −0.306121
$$300$$ −2.80837 −0.162141
$$301$$ 7.27592 0.419377
$$302$$ −5.45916 −0.314139
$$303$$ 38.1699 2.19281
$$304$$ 4.69704 0.269394
$$305$$ −8.93050 −0.511359
$$306$$ 33.9379 1.94010
$$307$$ 31.2886 1.78573 0.892867 0.450321i $$-0.148691\pi$$
0.892867 + 0.450321i $$0.148691\pi$$
$$308$$ 0 0
$$309$$ −44.5381 −2.53368
$$310$$ −3.34334 −0.189889
$$311$$ −5.69208 −0.322768 −0.161384 0.986892i $$-0.551596\pi$$
−0.161384 + 0.986892i $$0.551596\pi$$
$$312$$ 8.58700 0.486143
$$313$$ 17.4320 0.985315 0.492658 0.870223i $$-0.336026\pi$$
0.492658 + 0.870223i $$0.336026\pi$$
$$314$$ −8.44070 −0.476336
$$315$$ −4.88695 −0.275349
$$316$$ −4.25829 −0.239547
$$317$$ 21.2720 1.19475 0.597376 0.801961i $$-0.296210\pi$$
0.597376 + 0.801961i $$0.296210\pi$$
$$318$$ −4.77750 −0.267909
$$319$$ 0 0
$$320$$ 1.00000 0.0559017
$$321$$ −12.3480 −0.689198
$$322$$ −1.73118 −0.0964748
$$323$$ −32.6190 −1.81497
$$324$$ 0.221440 0.0123022
$$325$$ 3.05764 0.169607
$$326$$ −9.74355 −0.539645
$$327$$ −40.7021 −2.25083
$$328$$ 7.58977 0.419075
$$329$$ −4.00555 −0.220833
$$330$$ 0 0
$$331$$ 0.712600 0.0391680 0.0195840 0.999808i $$-0.493766\pi$$
0.0195840 + 0.999808i $$0.493766\pi$$
$$332$$ −3.86073 −0.211885
$$333$$ −35.8084 −1.96229
$$334$$ 20.2081 1.10574
$$335$$ 10.5718 0.577597
$$336$$ 2.80837 0.153209
$$337$$ −28.7577 −1.56653 −0.783267 0.621686i $$-0.786448\pi$$
−0.783267 + 0.621686i $$0.786448\pi$$
$$338$$ 3.65082 0.198579
$$339$$ 54.2930 2.94879
$$340$$ −6.94459 −0.376623
$$341$$ 0 0
$$342$$ −22.9542 −1.24122
$$343$$ −1.00000 −0.0539949
$$344$$ 7.27592 0.392291
$$345$$ 4.86179 0.261750
$$346$$ 1.06417 0.0572101
$$347$$ −34.5981 −1.85732 −0.928662 0.370927i $$-0.879040\pi$$
−0.928662 + 0.370927i $$0.879040\pi$$
$$348$$ −18.9440 −1.01550
$$349$$ −15.4593 −0.827518 −0.413759 0.910386i $$-0.635784\pi$$
−0.413759 + 0.910386i $$0.635784\pi$$
$$350$$ 1.00000 0.0534522
$$351$$ −16.2032 −0.864865
$$352$$ 0 0
$$353$$ 2.31479 0.123204 0.0616019 0.998101i $$-0.480379\pi$$
0.0616019 + 0.998101i $$0.480379\pi$$
$$354$$ 21.4768 1.14148
$$355$$ −2.30414 −0.122291
$$356$$ −12.3140 −0.652643
$$357$$ −19.5030 −1.03221
$$358$$ 5.73780 0.303252
$$359$$ −5.56161 −0.293530 −0.146765 0.989171i $$-0.546886\pi$$
−0.146765 + 0.989171i $$0.546886\pi$$
$$360$$ −4.88695 −0.257565
$$361$$ 3.06217 0.161167
$$362$$ −3.77619 −0.198472
$$363$$ 0 0
$$364$$ −3.05764 −0.160264
$$365$$ 13.1113 0.686274
$$366$$ −25.0802 −1.31096
$$367$$ 26.0861 1.36168 0.680842 0.732430i $$-0.261614\pi$$
0.680842 + 0.732430i $$0.261614\pi$$
$$368$$ −1.73118 −0.0902439
$$369$$ −37.0908 −1.93087
$$370$$ 7.32736 0.380931
$$371$$ 1.70116 0.0883200
$$372$$ −9.38933 −0.486814
$$373$$ −28.3183 −1.46627 −0.733134 0.680085i $$-0.761943\pi$$
−0.733134 + 0.680085i $$0.761943\pi$$
$$374$$ 0 0
$$375$$ −2.80837 −0.145024
$$376$$ −4.00555 −0.206571
$$377$$ 20.6255 1.06227
$$378$$ −5.29926 −0.272565
$$379$$ −32.5648 −1.67274 −0.836371 0.548164i $$-0.815327\pi$$
−0.836371 + 0.548164i $$0.815327\pi$$
$$380$$ 4.69704 0.240953
$$381$$ 60.6181 3.10556
$$382$$ −11.7298 −0.600147
$$383$$ −12.3943 −0.633319 −0.316660 0.948539i $$-0.602561\pi$$
−0.316660 + 0.948539i $$0.602561\pi$$
$$384$$ 2.80837 0.143314
$$385$$ 0 0
$$386$$ 18.9879 0.966461
$$387$$ −35.5571 −1.80747
$$388$$ −6.94282 −0.352468
$$389$$ −19.8793 −1.00792 −0.503960 0.863727i $$-0.668124\pi$$
−0.503960 + 0.863727i $$0.668124\pi$$
$$390$$ 8.58700 0.434819
$$391$$ 12.0223 0.607995
$$392$$ −1.00000 −0.0505076
$$393$$ 14.7009 0.741564
$$394$$ −3.27508 −0.164996
$$395$$ −4.25829 −0.214258
$$396$$ 0 0
$$397$$ 27.6288 1.38665 0.693325 0.720625i $$-0.256145\pi$$
0.693325 + 0.720625i $$0.256145\pi$$
$$398$$ −14.7438 −0.739040
$$399$$ 13.1910 0.660377
$$400$$ 1.00000 0.0500000
$$401$$ 30.1562 1.50593 0.752965 0.658061i $$-0.228623\pi$$
0.752965 + 0.658061i $$0.228623\pi$$
$$402$$ 29.6894 1.48077
$$403$$ 10.2227 0.509230
$$404$$ −13.5915 −0.676202
$$405$$ 0.221440 0.0110034
$$406$$ 6.74554 0.334776
$$407$$ 0 0
$$408$$ −19.5030 −0.965542
$$409$$ −18.8749 −0.933302 −0.466651 0.884442i $$-0.654540\pi$$
−0.466651 + 0.884442i $$0.654540\pi$$
$$410$$ 7.58977 0.374832
$$411$$ −14.0430 −0.692688
$$412$$ 15.8591 0.781320
$$413$$ −7.64741 −0.376305
$$414$$ 8.46019 0.415795
$$415$$ −3.86073 −0.189516
$$416$$ −3.05764 −0.149913
$$417$$ −8.65045 −0.423614
$$418$$ 0 0
$$419$$ 8.51801 0.416132 0.208066 0.978115i $$-0.433283\pi$$
0.208066 + 0.978115i $$0.433283\pi$$
$$420$$ 2.80837 0.137035
$$421$$ −13.4733 −0.656647 −0.328323 0.944565i $$-0.606484\pi$$
−0.328323 + 0.944565i $$0.606484\pi$$
$$422$$ −13.6709 −0.665489
$$423$$ 19.5749 0.951766
$$424$$ 1.70116 0.0826158
$$425$$ −6.94459 −0.336862
$$426$$ −6.47087 −0.313515
$$427$$ 8.93050 0.432177
$$428$$ 4.39685 0.212530
$$429$$ 0 0
$$430$$ 7.27592 0.350876
$$431$$ 27.8105 1.33959 0.669793 0.742548i $$-0.266383\pi$$
0.669793 + 0.742548i $$0.266383\pi$$
$$432$$ −5.29926 −0.254961
$$433$$ −19.9578 −0.959111 −0.479555 0.877512i $$-0.659202\pi$$
−0.479555 + 0.877512i $$0.659202\pi$$
$$434$$ 3.34334 0.160485
$$435$$ −18.9440 −0.908295
$$436$$ 14.4931 0.694095
$$437$$ −8.13141 −0.388978
$$438$$ 36.8213 1.75939
$$439$$ 10.7367 0.512436 0.256218 0.966619i $$-0.417524\pi$$
0.256218 + 0.966619i $$0.417524\pi$$
$$440$$ 0 0
$$441$$ 4.88695 0.232712
$$442$$ 21.2341 1.01000
$$443$$ −10.0787 −0.478854 −0.239427 0.970914i $$-0.576960\pi$$
−0.239427 + 0.970914i $$0.576960\pi$$
$$444$$ 20.5779 0.976586
$$445$$ −12.3140 −0.583742
$$446$$ 5.23577 0.247921
$$447$$ −44.6131 −2.11013
$$448$$ −1.00000 −0.0472456
$$449$$ −31.1076 −1.46806 −0.734030 0.679117i $$-0.762363\pi$$
−0.734030 + 0.679117i $$0.762363\pi$$
$$450$$ −4.88695 −0.230373
$$451$$ 0 0
$$452$$ −19.3326 −0.909328
$$453$$ −15.3313 −0.720330
$$454$$ −22.5392 −1.05782
$$455$$ −3.05764 −0.143344
$$456$$ 13.1910 0.617727
$$457$$ −26.2062 −1.22587 −0.612936 0.790132i $$-0.710012\pi$$
−0.612936 + 0.790132i $$0.710012\pi$$
$$458$$ −1.54850 −0.0723566
$$459$$ 36.8012 1.71773
$$460$$ −1.73118 −0.0807166
$$461$$ 5.33572 0.248509 0.124255 0.992250i $$-0.460346\pi$$
0.124255 + 0.992250i $$0.460346\pi$$
$$462$$ 0 0
$$463$$ 28.8817 1.34224 0.671122 0.741347i $$-0.265813\pi$$
0.671122 + 0.741347i $$0.265813\pi$$
$$464$$ 6.74554 0.313154
$$465$$ −9.38933 −0.435420
$$466$$ −6.59831 −0.305661
$$467$$ −37.6575 −1.74258 −0.871291 0.490766i $$-0.836717\pi$$
−0.871291 + 0.490766i $$0.836717\pi$$
$$468$$ 14.9425 0.690720
$$469$$ −10.5718 −0.488159
$$470$$ −4.00555 −0.184762
$$471$$ −23.7046 −1.09225
$$472$$ −7.64741 −0.352001
$$473$$ 0 0
$$474$$ −11.9589 −0.549288
$$475$$ 4.69704 0.215515
$$476$$ 6.94459 0.318305
$$477$$ −8.31350 −0.380649
$$478$$ 9.32864 0.426682
$$479$$ −25.5852 −1.16902 −0.584509 0.811387i $$-0.698713\pi$$
−0.584509 + 0.811387i $$0.698713\pi$$
$$480$$ 2.80837 0.128184
$$481$$ −22.4044 −1.02155
$$482$$ −4.69741 −0.213961
$$483$$ −4.86179 −0.221219
$$484$$ 0 0
$$485$$ −6.94282 −0.315257
$$486$$ −15.2759 −0.692929
$$487$$ 20.1951 0.915125 0.457563 0.889177i $$-0.348723\pi$$
0.457563 + 0.889177i $$0.348723\pi$$
$$488$$ 8.93050 0.404265
$$489$$ −27.3635 −1.23742
$$490$$ −1.00000 −0.0451754
$$491$$ −18.2166 −0.822104 −0.411052 0.911612i $$-0.634839\pi$$
−0.411052 + 0.911612i $$0.634839\pi$$
$$492$$ 21.3149 0.960950
$$493$$ −46.8451 −2.10980
$$494$$ −14.3619 −0.646171
$$495$$ 0 0
$$496$$ 3.34334 0.150120
$$497$$ 2.30414 0.103355
$$498$$ −10.8424 −0.485857
$$499$$ −21.4887 −0.961966 −0.480983 0.876730i $$-0.659720\pi$$
−0.480983 + 0.876730i $$0.659720\pi$$
$$500$$ 1.00000 0.0447214
$$501$$ 56.7518 2.53548
$$502$$ −25.1086 −1.12065
$$503$$ 5.56623 0.248186 0.124093 0.992271i $$-0.460398\pi$$
0.124093 + 0.992271i $$0.460398\pi$$
$$504$$ 4.88695 0.217682
$$505$$ −13.5915 −0.604813
$$506$$ 0 0
$$507$$ 10.2529 0.455346
$$508$$ −21.5848 −0.957669
$$509$$ −7.22200 −0.320110 −0.160055 0.987108i $$-0.551167\pi$$
−0.160055 + 0.987108i $$0.551167\pi$$
$$510$$ −19.5030 −0.863607
$$511$$ −13.1113 −0.580008
$$512$$ −1.00000 −0.0441942
$$513$$ −24.8908 −1.09896
$$514$$ −22.2671 −0.982158
$$515$$ 15.8591 0.698833
$$516$$ 20.4335 0.899534
$$517$$ 0 0
$$518$$ −7.32736 −0.321946
$$519$$ 2.98858 0.131184
$$520$$ −3.05764 −0.134086
$$521$$ −6.64589 −0.291162 −0.145581 0.989346i $$-0.546505\pi$$
−0.145581 + 0.989346i $$0.546505\pi$$
$$522$$ −32.9651 −1.44285
$$523$$ −24.8014 −1.08449 −0.542246 0.840220i $$-0.682426\pi$$
−0.542246 + 0.840220i $$0.682426\pi$$
$$524$$ −5.23468 −0.228678
$$525$$ 2.80837 0.122567
$$526$$ −12.5549 −0.547421
$$527$$ −23.2181 −1.01140
$$528$$ 0 0
$$529$$ −20.0030 −0.869697
$$530$$ 1.70116 0.0738938
$$531$$ 37.3725 1.62183
$$532$$ −4.69704 −0.203642
$$533$$ −23.2068 −1.00520
$$534$$ −34.5824 −1.49653
$$535$$ 4.39685 0.190092
$$536$$ −10.5718 −0.456631
$$537$$ 16.1139 0.695365
$$538$$ 7.54481 0.325280
$$539$$ 0 0
$$540$$ −5.29926 −0.228044
$$541$$ 34.8421 1.49798 0.748990 0.662581i $$-0.230539\pi$$
0.748990 + 0.662581i $$0.230539\pi$$
$$542$$ 27.7828 1.19337
$$543$$ −10.6049 −0.455101
$$544$$ 6.94459 0.297747
$$545$$ 14.4931 0.620817
$$546$$ −8.58700 −0.367489
$$547$$ −22.5413 −0.963798 −0.481899 0.876227i $$-0.660053\pi$$
−0.481899 + 0.876227i $$0.660053\pi$$
$$548$$ 5.00039 0.213606
$$549$$ −43.6429 −1.86263
$$550$$ 0 0
$$551$$ 31.6841 1.34979
$$552$$ −4.86179 −0.206932
$$553$$ 4.25829 0.181081
$$554$$ 18.8691 0.801672
$$555$$ 20.5779 0.873485
$$556$$ 3.08024 0.130631
$$557$$ −0.716288 −0.0303501 −0.0151750 0.999885i $$-0.504831\pi$$
−0.0151750 + 0.999885i $$0.504831\pi$$
$$558$$ −16.3387 −0.691673
$$559$$ −22.2472 −0.940955
$$560$$ −1.00000 −0.0422577
$$561$$ 0 0
$$562$$ 6.86665 0.289652
$$563$$ −44.5640 −1.87815 −0.939075 0.343713i $$-0.888315\pi$$
−0.939075 + 0.343713i $$0.888315\pi$$
$$564$$ −11.2491 −0.473672
$$565$$ −19.3326 −0.813327
$$566$$ −25.1097 −1.05544
$$567$$ −0.221440 −0.00929961
$$568$$ 2.30414 0.0966795
$$569$$ −9.43478 −0.395526 −0.197763 0.980250i $$-0.563368\pi$$
−0.197763 + 0.980250i $$0.563368\pi$$
$$570$$ 13.1910 0.552511
$$571$$ −31.2693 −1.30858 −0.654289 0.756244i $$-0.727032\pi$$
−0.654289 + 0.756244i $$0.727032\pi$$
$$572$$ 0 0
$$573$$ −32.9415 −1.37615
$$574$$ −7.58977 −0.316791
$$575$$ −1.73118 −0.0721951
$$576$$ 4.88695 0.203623
$$577$$ 33.7440 1.40478 0.702390 0.711792i $$-0.252116\pi$$
0.702390 + 0.711792i $$0.252116\pi$$
$$578$$ −31.2274 −1.29889
$$579$$ 53.3252 2.21612
$$580$$ 6.74554 0.280093
$$581$$ 3.86073 0.160170
$$582$$ −19.4980 −0.808219
$$583$$ 0 0
$$584$$ −13.1113 −0.542548
$$585$$ 14.9425 0.617798
$$586$$ 16.5950 0.685534
$$587$$ 26.2246 1.08240 0.541202 0.840892i $$-0.317969\pi$$
0.541202 + 0.840892i $$0.317969\pi$$
$$588$$ −2.80837 −0.115815
$$589$$ 15.7038 0.647063
$$590$$ −7.64741 −0.314839
$$591$$ −9.19763 −0.378340
$$592$$ −7.32736 −0.301153
$$593$$ −23.6431 −0.970906 −0.485453 0.874263i $$-0.661345\pi$$
−0.485453 + 0.874263i $$0.661345\pi$$
$$594$$ 0 0
$$595$$ 6.94459 0.284701
$$596$$ 15.8858 0.650706
$$597$$ −41.4060 −1.69464
$$598$$ 5.29332 0.216460
$$599$$ −35.9599 −1.46928 −0.734640 0.678457i $$-0.762649\pi$$
−0.734640 + 0.678457i $$0.762649\pi$$
$$600$$ 2.80837 0.114651
$$601$$ 31.4056 1.28106 0.640530 0.767933i $$-0.278715\pi$$
0.640530 + 0.767933i $$0.278715\pi$$
$$602$$ −7.27592 −0.296544
$$603$$ 51.6637 2.10391
$$604$$ 5.45916 0.222130
$$605$$ 0 0
$$606$$ −38.1699 −1.55055
$$607$$ −35.9288 −1.45830 −0.729152 0.684352i $$-0.760085\pi$$
−0.729152 + 0.684352i $$0.760085\pi$$
$$608$$ −4.69704 −0.190490
$$609$$ 18.9440 0.767649
$$610$$ 8.93050 0.361585
$$611$$ 12.2475 0.495482
$$612$$ −33.9379 −1.37186
$$613$$ −47.6086 −1.92289 −0.961446 0.274992i $$-0.911325\pi$$
−0.961446 + 0.274992i $$0.911325\pi$$
$$614$$ −31.2886 −1.26270
$$615$$ 21.3149 0.859500
$$616$$ 0 0
$$617$$ −5.44411 −0.219172 −0.109586 0.993977i $$-0.534952\pi$$
−0.109586 + 0.993977i $$0.534952\pi$$
$$618$$ 44.5381 1.79159
$$619$$ 24.6723 0.991665 0.495833 0.868418i $$-0.334863\pi$$
0.495833 + 0.868418i $$0.334863\pi$$
$$620$$ 3.34334 0.134272
$$621$$ 9.17397 0.368139
$$622$$ 5.69208 0.228231
$$623$$ 12.3140 0.493352
$$624$$ −8.58700 −0.343755
$$625$$ 1.00000 0.0400000
$$626$$ −17.4320 −0.696723
$$627$$ 0 0
$$628$$ 8.44070 0.336820
$$629$$ 50.8855 2.02894
$$630$$ 4.88695 0.194701
$$631$$ −15.8756 −0.631997 −0.315998 0.948760i $$-0.602339\pi$$
−0.315998 + 0.948760i $$0.602339\pi$$
$$632$$ 4.25829 0.169386
$$633$$ −38.3930 −1.52598
$$634$$ −21.2720 −0.844818
$$635$$ −21.5848 −0.856566
$$636$$ 4.77750 0.189440
$$637$$ 3.05764 0.121148
$$638$$ 0 0
$$639$$ −11.2602 −0.445447
$$640$$ −1.00000 −0.0395285
$$641$$ −3.37605 −0.133346 −0.0666730 0.997775i $$-0.521238\pi$$
−0.0666730 + 0.997775i $$0.521238\pi$$
$$642$$ 12.3480 0.487336
$$643$$ 20.7120 0.816803 0.408402 0.912802i $$-0.366086\pi$$
0.408402 + 0.912802i $$0.366086\pi$$
$$644$$ 1.73118 0.0682180
$$645$$ 20.4335 0.804568
$$646$$ 32.6190 1.28338
$$647$$ −6.82833 −0.268449 −0.134225 0.990951i $$-0.542854\pi$$
−0.134225 + 0.990951i $$0.542854\pi$$
$$648$$ −0.221440 −0.00869898
$$649$$ 0 0
$$650$$ −3.05764 −0.119931
$$651$$ 9.38933 0.367997
$$652$$ 9.74355 0.381587
$$653$$ 30.9046 1.20939 0.604696 0.796456i $$-0.293295\pi$$
0.604696 + 0.796456i $$0.293295\pi$$
$$654$$ 40.7021 1.59158
$$655$$ −5.23468 −0.204536
$$656$$ −7.58977 −0.296331
$$657$$ 64.0741 2.49977
$$658$$ 4.00555 0.156153
$$659$$ −17.6185 −0.686318 −0.343159 0.939277i $$-0.611497\pi$$
−0.343159 + 0.939277i $$0.611497\pi$$
$$660$$ 0 0
$$661$$ 11.7396 0.456617 0.228309 0.973589i $$-0.426680\pi$$
0.228309 + 0.973589i $$0.426680\pi$$
$$662$$ −0.712600 −0.0276960
$$663$$ 59.6332 2.31596
$$664$$ 3.86073 0.149825
$$665$$ −4.69704 −0.182143
$$666$$ 35.8084 1.38755
$$667$$ −11.6777 −0.452164
$$668$$ −20.2081 −0.781873
$$669$$ 14.7040 0.568489
$$670$$ −10.5718 −0.408423
$$671$$ 0 0
$$672$$ −2.80837 −0.108335
$$673$$ −17.0012 −0.655347 −0.327673 0.944791i $$-0.606265\pi$$
−0.327673 + 0.944791i $$0.606265\pi$$
$$674$$ 28.7577 1.10771
$$675$$ −5.29926 −0.203969
$$676$$ −3.65082 −0.140416
$$677$$ 3.84431 0.147749 0.0738744 0.997268i $$-0.476464\pi$$
0.0738744 + 0.997268i $$0.476464\pi$$
$$678$$ −54.2930 −2.08511
$$679$$ 6.94282 0.266441
$$680$$ 6.94459 0.266313
$$681$$ −63.2986 −2.42561
$$682$$ 0 0
$$683$$ −2.04787 −0.0783595 −0.0391798 0.999232i $$-0.512474\pi$$
−0.0391798 + 0.999232i $$0.512474\pi$$
$$684$$ 22.9542 0.877676
$$685$$ 5.00039 0.191055
$$686$$ 1.00000 0.0381802
$$687$$ −4.34876 −0.165916
$$688$$ −7.27592 −0.277392
$$689$$ −5.20155 −0.198163
$$690$$ −4.86179 −0.185085
$$691$$ −24.0616 −0.915348 −0.457674 0.889120i $$-0.651317\pi$$
−0.457674 + 0.889120i $$0.651317\pi$$
$$692$$ −1.06417 −0.0404537
$$693$$ 0 0
$$694$$ 34.5981 1.31333
$$695$$ 3.08024 0.116840
$$696$$ 18.9440 0.718070
$$697$$ 52.7079 1.99645
$$698$$ 15.4593 0.585144
$$699$$ −18.5305 −0.700888
$$700$$ −1.00000 −0.0377964
$$701$$ −7.99724 −0.302052 −0.151026 0.988530i $$-0.548258\pi$$
−0.151026 + 0.988530i $$0.548258\pi$$
$$702$$ 16.2032 0.611552
$$703$$ −34.4169 −1.29806
$$704$$ 0 0
$$705$$ −11.2491 −0.423665
$$706$$ −2.31479 −0.0871183
$$707$$ 13.5915 0.511160
$$708$$ −21.4768 −0.807147
$$709$$ 4.28038 0.160753 0.0803765 0.996765i $$-0.474388\pi$$
0.0803765 + 0.996765i $$0.474388\pi$$
$$710$$ 2.30414 0.0864727
$$711$$ −20.8100 −0.780438
$$712$$ 12.3140 0.461488
$$713$$ −5.78791 −0.216759
$$714$$ 19.5030 0.729881
$$715$$ 0 0
$$716$$ −5.73780 −0.214432
$$717$$ 26.1983 0.978393
$$718$$ 5.56161 0.207557
$$719$$ 36.8044 1.37257 0.686286 0.727332i $$-0.259240\pi$$
0.686286 + 0.727332i $$0.259240\pi$$
$$720$$ 4.88695 0.182126
$$721$$ −15.8591 −0.590622
$$722$$ −3.06217 −0.113962
$$723$$ −13.1921 −0.490618
$$724$$ 3.77619 0.140341
$$725$$ 6.74554 0.250523
$$726$$ 0 0
$$727$$ −46.5408 −1.72610 −0.863051 0.505116i $$-0.831450\pi$$
−0.863051 + 0.505116i $$0.831450\pi$$
$$728$$ 3.05764 0.113324
$$729$$ −43.5647 −1.61351
$$730$$ −13.1113 −0.485269
$$731$$ 50.5283 1.86886
$$732$$ 25.0802 0.926989
$$733$$ −15.6659 −0.578633 −0.289316 0.957234i $$-0.593428\pi$$
−0.289316 + 0.957234i $$0.593428\pi$$
$$734$$ −26.0861 −0.962857
$$735$$ −2.80837 −0.103588
$$736$$ 1.73118 0.0638121
$$737$$ 0 0
$$738$$ 37.0908 1.36533
$$739$$ −6.15005 −0.226233 −0.113117 0.993582i $$-0.536083\pi$$
−0.113117 + 0.993582i $$0.536083\pi$$
$$740$$ −7.32736 −0.269359
$$741$$ −40.3334 −1.48169
$$742$$ −1.70116 −0.0624517
$$743$$ −27.2721 −1.00052 −0.500259 0.865876i $$-0.666762\pi$$
−0.500259 + 0.865876i $$0.666762\pi$$
$$744$$ 9.38933 0.344229
$$745$$ 15.8858 0.582009
$$746$$ 28.3183 1.03681
$$747$$ −18.8672 −0.690314
$$748$$ 0 0
$$749$$ −4.39685 −0.160657
$$750$$ 2.80837 0.102547
$$751$$ 7.95258 0.290194 0.145097 0.989417i $$-0.453651\pi$$
0.145097 + 0.989417i $$0.453651\pi$$
$$752$$ 4.00555 0.146067
$$753$$ −70.5142 −2.56968
$$754$$ −20.6255 −0.751135
$$755$$ 5.45916 0.198679
$$756$$ 5.29926 0.192732
$$757$$ 24.8369 0.902713 0.451356 0.892344i $$-0.350940\pi$$
0.451356 + 0.892344i $$0.350940\pi$$
$$758$$ 32.5648 1.18281
$$759$$ 0 0
$$760$$ −4.69704 −0.170379
$$761$$ −34.0673 −1.23494 −0.617469 0.786595i $$-0.711842\pi$$
−0.617469 + 0.786595i $$0.711842\pi$$
$$762$$ −60.6181 −2.19596
$$763$$ −14.4931 −0.524686
$$764$$ 11.7298 0.424368
$$765$$ −33.9379 −1.22703
$$766$$ 12.3943 0.447824
$$767$$ 23.3831 0.844313
$$768$$ −2.80837 −0.101338
$$769$$ −10.7430 −0.387404 −0.193702 0.981060i $$-0.562050\pi$$
−0.193702 + 0.981060i $$0.562050\pi$$
$$770$$ 0 0
$$771$$ −62.5342 −2.25211
$$772$$ −18.9879 −0.683391
$$773$$ −35.4511 −1.27509 −0.637544 0.770414i $$-0.720050\pi$$
−0.637544 + 0.770414i $$0.720050\pi$$
$$774$$ 35.5571 1.27807
$$775$$ 3.34334 0.120096
$$776$$ 6.94282 0.249233
$$777$$ −20.5779 −0.738230
$$778$$ 19.8793 0.712708
$$779$$ −35.6494 −1.27727
$$780$$ −8.58700 −0.307464
$$781$$ 0 0
$$782$$ −12.0223 −0.429918
$$783$$ −35.7464 −1.27747
$$784$$ 1.00000 0.0357143
$$785$$ 8.44070 0.301261
$$786$$ −14.7009 −0.524365
$$787$$ 22.9148 0.816825 0.408413 0.912797i $$-0.366082\pi$$
0.408413 + 0.912797i $$0.366082\pi$$
$$788$$ 3.27508 0.116670
$$789$$ −35.2589 −1.25525
$$790$$ 4.25829 0.151503
$$791$$ 19.3326 0.687387
$$792$$ 0 0
$$793$$ −27.3063 −0.969674
$$794$$ −27.6288 −0.980510
$$795$$ 4.77750 0.169440
$$796$$ 14.7438 0.522580
$$797$$ −42.9156 −1.52015 −0.760074 0.649836i $$-0.774837\pi$$
−0.760074 + 0.649836i $$0.774837\pi$$
$$798$$ −13.1910 −0.466957
$$799$$ −27.8169 −0.984092
$$800$$ −1.00000 −0.0353553
$$801$$ −60.1782 −2.12629
$$802$$ −30.1562 −1.06485
$$803$$ 0 0
$$804$$ −29.6894 −1.04707
$$805$$ 1.73118 0.0610160
$$806$$ −10.2227 −0.360080
$$807$$ 21.1886 0.745875
$$808$$ 13.5915 0.478147
$$809$$ −16.9974 −0.597598 −0.298799 0.954316i $$-0.596586\pi$$
−0.298799 + 0.954316i $$0.596586\pi$$
$$810$$ −0.221440 −0.00778061
$$811$$ 10.6859 0.375231 0.187616 0.982243i $$-0.439924\pi$$
0.187616 + 0.982243i $$0.439924\pi$$
$$812$$ −6.74554 −0.236722
$$813$$ 78.0243 2.73643
$$814$$ 0 0
$$815$$ 9.74355 0.341302
$$816$$ 19.5030 0.682742
$$817$$ −34.1753 −1.19564
$$818$$ 18.8749 0.659944
$$819$$ −14.9425 −0.522135
$$820$$ −7.58977 −0.265046
$$821$$ 20.3259 0.709379 0.354689 0.934984i $$-0.384587\pi$$
0.354689 + 0.934984i $$0.384587\pi$$
$$822$$ 14.0430 0.489805
$$823$$ −29.8555 −1.04070 −0.520348 0.853954i $$-0.674198\pi$$
−0.520348 + 0.853954i $$0.674198\pi$$
$$824$$ −15.8591 −0.552476
$$825$$ 0 0
$$826$$ 7.64741 0.266088
$$827$$ 4.53541 0.157712 0.0788558 0.996886i $$-0.474873\pi$$
0.0788558 + 0.996886i $$0.474873\pi$$
$$828$$ −8.46019 −0.294012
$$829$$ −29.0225 −1.00799 −0.503997 0.863705i $$-0.668138\pi$$
−0.503997 + 0.863705i $$0.668138\pi$$
$$830$$ 3.86073 0.134008
$$831$$ 52.9915 1.83826
$$832$$ 3.05764 0.106005
$$833$$ −6.94459 −0.240616
$$834$$ 8.65045 0.299540
$$835$$ −20.2081 −0.699329
$$836$$ 0 0
$$837$$ −17.7172 −0.612396
$$838$$ −8.51801 −0.294250
$$839$$ −3.85780 −0.133186 −0.0665930 0.997780i $$-0.521213\pi$$
−0.0665930 + 0.997780i $$0.521213\pi$$
$$840$$ −2.80837 −0.0968980
$$841$$ 16.5024 0.569047
$$842$$ 13.4733 0.464319
$$843$$ 19.2841 0.664180
$$844$$ 13.6709 0.470572
$$845$$ −3.65082 −0.125592
$$846$$ −19.5749 −0.673000
$$847$$ 0 0
$$848$$ −1.70116 −0.0584182
$$849$$ −70.5173 −2.42015
$$850$$ 6.94459 0.238198
$$851$$ 12.6850 0.434835
$$852$$ 6.47087 0.221688
$$853$$ 14.8560 0.508659 0.254329 0.967118i $$-0.418145\pi$$
0.254329 + 0.967118i $$0.418145\pi$$
$$854$$ −8.93050 −0.305595
$$855$$ 22.9542 0.785017
$$856$$ −4.39685 −0.150281
$$857$$ 21.4567 0.732948 0.366474 0.930428i $$-0.380565\pi$$
0.366474 + 0.930428i $$0.380565\pi$$
$$858$$ 0 0
$$859$$ −50.7945 −1.73309 −0.866543 0.499103i $$-0.833663\pi$$
−0.866543 + 0.499103i $$0.833663\pi$$
$$860$$ −7.27592 −0.248107
$$861$$ −21.3149 −0.726410
$$862$$ −27.8105 −0.947231
$$863$$ −13.6152 −0.463466 −0.231733 0.972779i $$-0.574440\pi$$
−0.231733 + 0.972779i $$0.574440\pi$$
$$864$$ 5.29926 0.180285
$$865$$ −1.06417 −0.0361829
$$866$$ 19.9578 0.678194
$$867$$ −87.6981 −2.97838
$$868$$ −3.34334 −0.113480
$$869$$ 0 0
$$870$$ 18.9440 0.642262
$$871$$ 32.3247 1.09528
$$872$$ −14.4931 −0.490799
$$873$$ −33.9292 −1.14833
$$874$$ 8.13141 0.275049
$$875$$ −1.00000 −0.0338062
$$876$$ −36.8213 −1.24408
$$877$$ −8.90340 −0.300647 −0.150323 0.988637i $$-0.548031\pi$$
−0.150323 + 0.988637i $$0.548031\pi$$
$$878$$ −10.7367 −0.362347
$$879$$ 46.6050 1.57195
$$880$$ 0 0
$$881$$ −2.11899 −0.0713904 −0.0356952 0.999363i $$-0.511365\pi$$
−0.0356952 + 0.999363i $$0.511365\pi$$
$$882$$ −4.88695 −0.164552
$$883$$ −10.2827 −0.346041 −0.173021 0.984918i $$-0.555353\pi$$
−0.173021 + 0.984918i $$0.555353\pi$$
$$884$$ −21.2341 −0.714179
$$885$$ −21.4768 −0.721934
$$886$$ 10.0787 0.338601
$$887$$ 5.63590 0.189235 0.0946175 0.995514i $$-0.469837\pi$$
0.0946175 + 0.995514i $$0.469837\pi$$
$$888$$ −20.5779 −0.690551
$$889$$ 21.5848 0.723930
$$890$$ 12.3140 0.412768
$$891$$ 0 0
$$892$$ −5.23577 −0.175307
$$893$$ 18.8142 0.629594
$$894$$ 44.6131 1.49209
$$895$$ −5.73780 −0.191793
$$896$$ 1.00000 0.0334077
$$897$$ 14.8656 0.496349
$$898$$ 31.1076 1.03808
$$899$$ 22.5526 0.752172
$$900$$ 4.88695 0.162898
$$901$$ 11.8139 0.393578
$$902$$ 0 0
$$903$$ −20.4335 −0.679984
$$904$$ 19.3326 0.642992
$$905$$ 3.77619 0.125525
$$906$$ 15.3313 0.509350
$$907$$ −39.0697 −1.29729 −0.648643 0.761092i $$-0.724663\pi$$
−0.648643 + 0.761092i $$0.724663\pi$$
$$908$$ 22.5392 0.747991
$$909$$ −66.4209 −2.20304
$$910$$ 3.05764 0.101360
$$911$$ 22.8604 0.757398 0.378699 0.925520i $$-0.376372\pi$$
0.378699 + 0.925520i $$0.376372\pi$$
$$912$$ −13.1910 −0.436799
$$913$$ 0 0
$$914$$ 26.2062 0.866823
$$915$$ 25.0802 0.829124
$$916$$ 1.54850 0.0511639
$$917$$ 5.23468 0.172864
$$918$$ −36.8012 −1.21462
$$919$$ 13.0335 0.429935 0.214968 0.976621i $$-0.431035\pi$$
0.214968 + 0.976621i $$0.431035\pi$$
$$920$$ 1.73118 0.0570753
$$921$$ −87.8699 −2.89541
$$922$$ −5.33572 −0.175723
$$923$$ −7.04523 −0.231896
$$924$$ 0 0
$$925$$ −7.32736 −0.240922
$$926$$ −28.8817 −0.949110
$$927$$ 77.5024 2.54551
$$928$$ −6.74554 −0.221433
$$929$$ 27.3966 0.898853 0.449427 0.893317i $$-0.351628\pi$$
0.449427 + 0.893317i $$0.351628\pi$$
$$930$$ 9.38933 0.307888
$$931$$ 4.69704 0.153939
$$932$$ 6.59831 0.216135
$$933$$ 15.9855 0.523341
$$934$$ 37.6575 1.23219
$$935$$ 0 0
$$936$$ −14.9425 −0.488412
$$937$$ −22.1523 −0.723684 −0.361842 0.932240i $$-0.617852\pi$$
−0.361842 + 0.932240i $$0.617852\pi$$
$$938$$ 10.5718 0.345180
$$939$$ −48.9556 −1.59760
$$940$$ 4.00555 0.130647
$$941$$ 12.0202 0.391847 0.195924 0.980619i $$-0.437229\pi$$
0.195924 + 0.980619i $$0.437229\pi$$
$$942$$ 23.7046 0.772338
$$943$$ 13.1392 0.427873
$$944$$ 7.64741 0.248902
$$945$$ 5.29926 0.172385
$$946$$ 0 0
$$947$$ −29.5380 −0.959857 −0.479929 0.877307i $$-0.659337\pi$$
−0.479929 + 0.877307i $$0.659337\pi$$
$$948$$ 11.9589 0.388406
$$949$$ 40.0895 1.30136
$$950$$ −4.69704 −0.152392
$$951$$ −59.7396 −1.93719
$$952$$ −6.94459 −0.225076
$$953$$ −38.2985 −1.24061 −0.620306 0.784360i $$-0.712991\pi$$
−0.620306 + 0.784360i $$0.712991\pi$$
$$954$$ 8.31350 0.269160
$$955$$ 11.7298 0.379566
$$956$$ −9.32864 −0.301710
$$957$$ 0 0
$$958$$ 25.5852 0.826621
$$959$$ −5.00039 −0.161471
$$960$$ −2.80837 −0.0906398
$$961$$ −19.8221 −0.639423
$$962$$ 22.4044 0.722348
$$963$$ 21.4872 0.692415
$$964$$ 4.69741 0.151293
$$965$$ −18.9879 −0.611243
$$966$$ 4.86179 0.156426
$$967$$ −2.04155 −0.0656517 −0.0328259 0.999461i $$-0.510451\pi$$
−0.0328259 + 0.999461i $$0.510451\pi$$
$$968$$ 0 0
$$969$$ 91.6063 2.94282
$$970$$ 6.94282 0.222920
$$971$$ −5.03293 −0.161514 −0.0807572 0.996734i $$-0.525734\pi$$
−0.0807572 + 0.996734i $$0.525734\pi$$
$$972$$ 15.2759 0.489975
$$973$$ −3.08024 −0.0987478
$$974$$ −20.1951 −0.647091
$$975$$ −8.58700 −0.275004
$$976$$ −8.93050 −0.285858
$$977$$ −10.1256 −0.323945 −0.161973 0.986795i $$-0.551786\pi$$
−0.161973 + 0.986795i $$0.551786\pi$$
$$978$$ 27.3635 0.874989
$$979$$ 0 0
$$980$$ 1.00000 0.0319438
$$981$$ 70.8272 2.26134
$$982$$ 18.2166 0.581315
$$983$$ 33.8126 1.07846 0.539228 0.842160i $$-0.318716\pi$$
0.539228 + 0.842160i $$0.318716\pi$$
$$984$$ −21.3149 −0.679494
$$985$$ 3.27508 0.104353
$$986$$ 46.8451 1.49185
$$987$$ 11.2491 0.358062
$$988$$ 14.3619 0.456912
$$989$$ 12.5959 0.400527
$$990$$ 0 0
$$991$$ 13.8867 0.441127 0.220563 0.975373i $$-0.429210\pi$$
0.220563 + 0.975373i $$0.429210\pi$$
$$992$$ −3.34334 −0.106151
$$993$$ −2.00124 −0.0635076
$$994$$ −2.30414 −0.0730828
$$995$$ 14.7438 0.467410
$$996$$ 10.8424 0.343553
$$997$$ −5.72382 −0.181275 −0.0906375 0.995884i $$-0.528890\pi$$
−0.0906375 + 0.995884i $$0.528890\pi$$
$$998$$ 21.4887 0.680213
$$999$$ 38.8296 1.22851
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 8470.2.a.cy.1.1 6
11.2 odd 10 770.2.n.i.631.1 yes 12
11.6 odd 10 770.2.n.i.421.1 12
11.10 odd 2 8470.2.a.de.1.1 6

By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.n.i.421.1 12 11.6 odd 10
770.2.n.i.631.1 yes 12 11.2 odd 10
8470.2.a.cy.1.1 6 1.1 even 1 trivial
8470.2.a.de.1.1 6 11.10 odd 2