# Properties

 Label 1521.2.a.m.1.1 Level $1521$ Weight $2$ Character 1521.1 Self dual yes Analytic conductor $12.145$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1521 = 3^{2} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1521.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$12.1452461474$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 39) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.56155$$ of defining polynomial Character $$\chi$$ $$=$$ 1521.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.56155 q^{2} +0.438447 q^{4} -3.56155 q^{5} -0.561553 q^{7} +2.43845 q^{8} +O(q^{10})$$ $$q-1.56155 q^{2} +0.438447 q^{4} -3.56155 q^{5} -0.561553 q^{7} +2.43845 q^{8} +5.56155 q^{10} -2.00000 q^{11} +0.876894 q^{14} -4.68466 q^{16} +1.56155 q^{17} -7.12311 q^{19} -1.56155 q^{20} +3.12311 q^{22} -2.00000 q^{23} +7.68466 q^{25} -0.246211 q^{28} -6.68466 q^{29} -2.56155 q^{31} +2.43845 q^{32} -2.43845 q^{34} +2.00000 q^{35} -7.56155 q^{37} +11.1231 q^{38} -8.68466 q^{40} -1.56155 q^{41} +4.56155 q^{43} -0.876894 q^{44} +3.12311 q^{46} +8.24621 q^{47} -6.68466 q^{49} -12.0000 q^{50} +0.684658 q^{53} +7.12311 q^{55} -1.36932 q^{56} +10.4384 q^{58} -2.87689 q^{59} +3.87689 q^{61} +4.00000 q^{62} +5.56155 q^{64} -4.56155 q^{67} +0.684658 q^{68} -3.12311 q^{70} +14.0000 q^{71} +10.1231 q^{73} +11.8078 q^{74} -3.12311 q^{76} +1.12311 q^{77} +5.43845 q^{79} +16.6847 q^{80} +2.43845 q^{82} -0.876894 q^{83} -5.56155 q^{85} -7.12311 q^{86} -4.87689 q^{88} +4.87689 q^{89} -0.876894 q^{92} -12.8769 q^{94} +25.3693 q^{95} +8.56155 q^{97} +10.4384 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} + 5 q^{4} - 3 q^{5} + 3 q^{7} + 9 q^{8}+O(q^{10})$$ 2 * q + q^2 + 5 * q^4 - 3 * q^5 + 3 * q^7 + 9 * q^8 $$2 q + q^{2} + 5 q^{4} - 3 q^{5} + 3 q^{7} + 9 q^{8} + 7 q^{10} - 4 q^{11} + 10 q^{14} + 3 q^{16} - q^{17} - 6 q^{19} + q^{20} - 2 q^{22} - 4 q^{23} + 3 q^{25} + 16 q^{28} - q^{29} - q^{31} + 9 q^{32} - 9 q^{34} + 4 q^{35} - 11 q^{37} + 14 q^{38} - 5 q^{40} + q^{41} + 5 q^{43} - 10 q^{44} - 2 q^{46} - q^{49} - 24 q^{50} - 11 q^{53} + 6 q^{55} + 22 q^{56} + 25 q^{58} - 14 q^{59} + 16 q^{61} + 8 q^{62} + 7 q^{64} - 5 q^{67} - 11 q^{68} + 2 q^{70} + 28 q^{71} + 12 q^{73} + 3 q^{74} + 2 q^{76} - 6 q^{77} + 15 q^{79} + 21 q^{80} + 9 q^{82} - 10 q^{83} - 7 q^{85} - 6 q^{86} - 18 q^{88} + 18 q^{89} - 10 q^{92} - 34 q^{94} + 26 q^{95} + 13 q^{97} + 25 q^{98}+O(q^{100})$$ 2 * q + q^2 + 5 * q^4 - 3 * q^5 + 3 * q^7 + 9 * q^8 + 7 * q^10 - 4 * q^11 + 10 * q^14 + 3 * q^16 - q^17 - 6 * q^19 + q^20 - 2 * q^22 - 4 * q^23 + 3 * q^25 + 16 * q^28 - q^29 - q^31 + 9 * q^32 - 9 * q^34 + 4 * q^35 - 11 * q^37 + 14 * q^38 - 5 * q^40 + q^41 + 5 * q^43 - 10 * q^44 - 2 * q^46 - q^49 - 24 * q^50 - 11 * q^53 + 6 * q^55 + 22 * q^56 + 25 * q^58 - 14 * q^59 + 16 * q^61 + 8 * q^62 + 7 * q^64 - 5 * q^67 - 11 * q^68 + 2 * q^70 + 28 * q^71 + 12 * q^73 + 3 * q^74 + 2 * q^76 - 6 * q^77 + 15 * q^79 + 21 * q^80 + 9 * q^82 - 10 * q^83 - 7 * q^85 - 6 * q^86 - 18 * q^88 + 18 * q^89 - 10 * q^92 - 34 * q^94 + 26 * q^95 + 13 * q^97 + 25 * q^98

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.56155 −1.10418 −0.552092 0.833783i $$-0.686170\pi$$
−0.552092 + 0.833783i $$0.686170\pi$$
$$3$$ 0 0
$$4$$ 0.438447 0.219224
$$5$$ −3.56155 −1.59277 −0.796387 0.604787i $$-0.793258\pi$$
−0.796387 + 0.604787i $$0.793258\pi$$
$$6$$ 0 0
$$7$$ −0.561553 −0.212247 −0.106124 0.994353i $$-0.533844\pi$$
−0.106124 + 0.994353i $$0.533844\pi$$
$$8$$ 2.43845 0.862121
$$9$$ 0 0
$$10$$ 5.56155 1.75872
$$11$$ −2.00000 −0.603023 −0.301511 0.953463i $$-0.597491\pi$$
−0.301511 + 0.953463i $$0.597491\pi$$
$$12$$ 0 0
$$13$$ 0 0
$$14$$ 0.876894 0.234360
$$15$$ 0 0
$$16$$ −4.68466 −1.17116
$$17$$ 1.56155 0.378732 0.189366 0.981907i $$-0.439357\pi$$
0.189366 + 0.981907i $$0.439357\pi$$
$$18$$ 0 0
$$19$$ −7.12311 −1.63415 −0.817076 0.576530i $$-0.804407\pi$$
−0.817076 + 0.576530i $$0.804407\pi$$
$$20$$ −1.56155 −0.349174
$$21$$ 0 0
$$22$$ 3.12311 0.665848
$$23$$ −2.00000 −0.417029 −0.208514 0.978019i $$-0.566863\pi$$
−0.208514 + 0.978019i $$0.566863\pi$$
$$24$$ 0 0
$$25$$ 7.68466 1.53693
$$26$$ 0 0
$$27$$ 0 0
$$28$$ −0.246211 −0.0465296
$$29$$ −6.68466 −1.24131 −0.620655 0.784084i $$-0.713133\pi$$
−0.620655 + 0.784084i $$0.713133\pi$$
$$30$$ 0 0
$$31$$ −2.56155 −0.460068 −0.230034 0.973183i $$-0.573884\pi$$
−0.230034 + 0.973183i $$0.573884\pi$$
$$32$$ 2.43845 0.431061
$$33$$ 0 0
$$34$$ −2.43845 −0.418190
$$35$$ 2.00000 0.338062
$$36$$ 0 0
$$37$$ −7.56155 −1.24311 −0.621556 0.783370i $$-0.713499\pi$$
−0.621556 + 0.783370i $$0.713499\pi$$
$$38$$ 11.1231 1.80441
$$39$$ 0 0
$$40$$ −8.68466 −1.37317
$$41$$ −1.56155 −0.243874 −0.121937 0.992538i $$-0.538911\pi$$
−0.121937 + 0.992538i $$0.538911\pi$$
$$42$$ 0 0
$$43$$ 4.56155 0.695630 0.347815 0.937563i $$-0.386924\pi$$
0.347815 + 0.937563i $$0.386924\pi$$
$$44$$ −0.876894 −0.132197
$$45$$ 0 0
$$46$$ 3.12311 0.460477
$$47$$ 8.24621 1.20283 0.601417 0.798935i $$-0.294603\pi$$
0.601417 + 0.798935i $$0.294603\pi$$
$$48$$ 0 0
$$49$$ −6.68466 −0.954951
$$50$$ −12.0000 −1.69706
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 0.684658 0.0940451 0.0470225 0.998894i $$-0.485027\pi$$
0.0470225 + 0.998894i $$0.485027\pi$$
$$54$$ 0 0
$$55$$ 7.12311 0.960479
$$56$$ −1.36932 −0.182983
$$57$$ 0 0
$$58$$ 10.4384 1.37064
$$59$$ −2.87689 −0.374540 −0.187270 0.982309i $$-0.559964\pi$$
−0.187270 + 0.982309i $$0.559964\pi$$
$$60$$ 0 0
$$61$$ 3.87689 0.496385 0.248193 0.968711i $$-0.420163\pi$$
0.248193 + 0.968711i $$0.420163\pi$$
$$62$$ 4.00000 0.508001
$$63$$ 0 0
$$64$$ 5.56155 0.695194
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −4.56155 −0.557282 −0.278641 0.960395i $$-0.589884\pi$$
−0.278641 + 0.960395i $$0.589884\pi$$
$$68$$ 0.684658 0.0830270
$$69$$ 0 0
$$70$$ −3.12311 −0.373283
$$71$$ 14.0000 1.66149 0.830747 0.556650i $$-0.187914\pi$$
0.830747 + 0.556650i $$0.187914\pi$$
$$72$$ 0 0
$$73$$ 10.1231 1.18482 0.592410 0.805637i $$-0.298177\pi$$
0.592410 + 0.805637i $$0.298177\pi$$
$$74$$ 11.8078 1.37262
$$75$$ 0 0
$$76$$ −3.12311 −0.358245
$$77$$ 1.12311 0.127990
$$78$$ 0 0
$$79$$ 5.43845 0.611873 0.305937 0.952052i $$-0.401030\pi$$
0.305937 + 0.952052i $$0.401030\pi$$
$$80$$ 16.6847 1.86540
$$81$$ 0 0
$$82$$ 2.43845 0.269281
$$83$$ −0.876894 −0.0962517 −0.0481258 0.998841i $$-0.515325\pi$$
−0.0481258 + 0.998841i $$0.515325\pi$$
$$84$$ 0 0
$$85$$ −5.56155 −0.603235
$$86$$ −7.12311 −0.768104
$$87$$ 0 0
$$88$$ −4.87689 −0.519879
$$89$$ 4.87689 0.516950 0.258475 0.966018i $$-0.416780\pi$$
0.258475 + 0.966018i $$0.416780\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −0.876894 −0.0914226
$$93$$ 0 0
$$94$$ −12.8769 −1.32815
$$95$$ 25.3693 2.60284
$$96$$ 0 0
$$97$$ 8.56155 0.869294 0.434647 0.900601i $$-0.356873\pi$$
0.434647 + 0.900601i $$0.356873\pi$$
$$98$$ 10.4384 1.05444
$$99$$ 0 0
$$100$$ 3.36932 0.336932
$$101$$ 7.56155 0.752403 0.376201 0.926538i $$-0.377230\pi$$
0.376201 + 0.926538i $$0.377230\pi$$
$$102$$ 0 0
$$103$$ −3.43845 −0.338800 −0.169400 0.985547i $$-0.554183\pi$$
−0.169400 + 0.985547i $$0.554183\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ −1.06913 −0.103843
$$107$$ 8.24621 0.797191 0.398596 0.917127i $$-0.369498\pi$$
0.398596 + 0.917127i $$0.369498\pi$$
$$108$$ 0 0
$$109$$ 2.80776 0.268935 0.134468 0.990918i $$-0.457068\pi$$
0.134468 + 0.990918i $$0.457068\pi$$
$$110$$ −11.1231 −1.06055
$$111$$ 0 0
$$112$$ 2.63068 0.248576
$$113$$ −5.80776 −0.546348 −0.273174 0.961965i $$-0.588074\pi$$
−0.273174 + 0.961965i $$0.588074\pi$$
$$114$$ 0 0
$$115$$ 7.12311 0.664233
$$116$$ −2.93087 −0.272124
$$117$$ 0 0
$$118$$ 4.49242 0.413561
$$119$$ −0.876894 −0.0803848
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ −6.05398 −0.548101
$$123$$ 0 0
$$124$$ −1.12311 −0.100858
$$125$$ −9.56155 −0.855211
$$126$$ 0 0
$$127$$ 5.43845 0.482584 0.241292 0.970453i $$-0.422429\pi$$
0.241292 + 0.970453i $$0.422429\pi$$
$$128$$ −13.5616 −1.19868
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −7.36932 −0.643860 −0.321930 0.946763i $$-0.604332\pi$$
−0.321930 + 0.946763i $$0.604332\pi$$
$$132$$ 0 0
$$133$$ 4.00000 0.346844
$$134$$ 7.12311 0.615343
$$135$$ 0 0
$$136$$ 3.80776 0.326513
$$137$$ −5.56155 −0.475156 −0.237578 0.971369i $$-0.576353\pi$$
−0.237578 + 0.971369i $$0.576353\pi$$
$$138$$ 0 0
$$139$$ −17.9309 −1.52088 −0.760438 0.649410i $$-0.775016\pi$$
−0.760438 + 0.649410i $$0.775016\pi$$
$$140$$ 0.876894 0.0741111
$$141$$ 0 0
$$142$$ −21.8617 −1.83460
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 23.8078 1.97713
$$146$$ −15.8078 −1.30826
$$147$$ 0 0
$$148$$ −3.31534 −0.272519
$$149$$ −2.43845 −0.199765 −0.0998827 0.994999i $$-0.531847\pi$$
−0.0998827 + 0.994999i $$0.531847\pi$$
$$150$$ 0 0
$$151$$ 9.36932 0.762464 0.381232 0.924479i $$-0.375500\pi$$
0.381232 + 0.924479i $$0.375500\pi$$
$$152$$ −17.3693 −1.40884
$$153$$ 0 0
$$154$$ −1.75379 −0.141324
$$155$$ 9.12311 0.732785
$$156$$ 0 0
$$157$$ 20.3693 1.62565 0.812824 0.582509i $$-0.197929\pi$$
0.812824 + 0.582509i $$0.197929\pi$$
$$158$$ −8.49242 −0.675621
$$159$$ 0 0
$$160$$ −8.68466 −0.686583
$$161$$ 1.12311 0.0885131
$$162$$ 0 0
$$163$$ 4.80776 0.376573 0.188287 0.982114i $$-0.439707\pi$$
0.188287 + 0.982114i $$0.439707\pi$$
$$164$$ −0.684658 −0.0534628
$$165$$ 0 0
$$166$$ 1.36932 0.106280
$$167$$ 10.2462 0.792876 0.396438 0.918062i $$-0.370246\pi$$
0.396438 + 0.918062i $$0.370246\pi$$
$$168$$ 0 0
$$169$$ 0 0
$$170$$ 8.68466 0.666083
$$171$$ 0 0
$$172$$ 2.00000 0.152499
$$173$$ −20.2462 −1.53929 −0.769645 0.638471i $$-0.779567\pi$$
−0.769645 + 0.638471i $$0.779567\pi$$
$$174$$ 0 0
$$175$$ −4.31534 −0.326209
$$176$$ 9.36932 0.706239
$$177$$ 0 0
$$178$$ −7.61553 −0.570808
$$179$$ 4.87689 0.364516 0.182258 0.983251i $$-0.441659\pi$$
0.182258 + 0.983251i $$0.441659\pi$$
$$180$$ 0 0
$$181$$ −2.68466 −0.199549 −0.0997745 0.995010i $$-0.531812\pi$$
−0.0997745 + 0.995010i $$0.531812\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ −4.87689 −0.359529
$$185$$ 26.9309 1.98000
$$186$$ 0 0
$$187$$ −3.12311 −0.228384
$$188$$ 3.61553 0.263689
$$189$$ 0 0
$$190$$ −39.6155 −2.87401
$$191$$ 9.12311 0.660125 0.330062 0.943959i $$-0.392930\pi$$
0.330062 + 0.943959i $$0.392930\pi$$
$$192$$ 0 0
$$193$$ 13.4924 0.971206 0.485603 0.874180i $$-0.338600\pi$$
0.485603 + 0.874180i $$0.338600\pi$$
$$194$$ −13.3693 −0.959861
$$195$$ 0 0
$$196$$ −2.93087 −0.209348
$$197$$ 13.3693 0.952524 0.476262 0.879303i $$-0.341991\pi$$
0.476262 + 0.879303i $$0.341991\pi$$
$$198$$ 0 0
$$199$$ 22.1771 1.57209 0.786046 0.618168i $$-0.212125\pi$$
0.786046 + 0.618168i $$0.212125\pi$$
$$200$$ 18.7386 1.32502
$$201$$ 0 0
$$202$$ −11.8078 −0.830791
$$203$$ 3.75379 0.263464
$$204$$ 0 0
$$205$$ 5.56155 0.388436
$$206$$ 5.36932 0.374098
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 14.2462 0.985431
$$210$$ 0 0
$$211$$ 19.6847 1.35515 0.677574 0.735455i $$-0.263031\pi$$
0.677574 + 0.735455i $$0.263031\pi$$
$$212$$ 0.300187 0.0206169
$$213$$ 0 0
$$214$$ −12.8769 −0.880246
$$215$$ −16.2462 −1.10798
$$216$$ 0 0
$$217$$ 1.43845 0.0976482
$$218$$ −4.38447 −0.296954
$$219$$ 0 0
$$220$$ 3.12311 0.210560
$$221$$ 0 0
$$222$$ 0 0
$$223$$ −8.00000 −0.535720 −0.267860 0.963458i $$-0.586316\pi$$
−0.267860 + 0.963458i $$0.586316\pi$$
$$224$$ −1.36932 −0.0914913
$$225$$ 0 0
$$226$$ 9.06913 0.603270
$$227$$ 7.12311 0.472777 0.236389 0.971659i $$-0.424036\pi$$
0.236389 + 0.971659i $$0.424036\pi$$
$$228$$ 0 0
$$229$$ 16.2462 1.07358 0.536790 0.843716i $$-0.319637\pi$$
0.536790 + 0.843716i $$0.319637\pi$$
$$230$$ −11.1231 −0.733436
$$231$$ 0 0
$$232$$ −16.3002 −1.07016
$$233$$ −26.0000 −1.70332 −0.851658 0.524097i $$-0.824403\pi$$
−0.851658 + 0.524097i $$0.824403\pi$$
$$234$$ 0 0
$$235$$ −29.3693 −1.91584
$$236$$ −1.26137 −0.0821079
$$237$$ 0 0
$$238$$ 1.36932 0.0887596
$$239$$ −25.3693 −1.64100 −0.820502 0.571643i $$-0.806306\pi$$
−0.820502 + 0.571643i $$0.806306\pi$$
$$240$$ 0 0
$$241$$ 17.8078 1.14710 0.573549 0.819171i $$-0.305566\pi$$
0.573549 + 0.819171i $$0.305566\pi$$
$$242$$ 10.9309 0.702663
$$243$$ 0 0
$$244$$ 1.69981 0.108819
$$245$$ 23.8078 1.52102
$$246$$ 0 0
$$247$$ 0 0
$$248$$ −6.24621 −0.396635
$$249$$ 0 0
$$250$$ 14.9309 0.944311
$$251$$ −18.7386 −1.18277 −0.591386 0.806389i $$-0.701419\pi$$
−0.591386 + 0.806389i $$0.701419\pi$$
$$252$$ 0 0
$$253$$ 4.00000 0.251478
$$254$$ −8.49242 −0.532862
$$255$$ 0 0
$$256$$ 10.0540 0.628373
$$257$$ −29.1771 −1.82002 −0.910008 0.414590i $$-0.863925\pi$$
−0.910008 + 0.414590i $$0.863925\pi$$
$$258$$ 0 0
$$259$$ 4.24621 0.263847
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 11.5076 0.710941
$$263$$ −9.36932 −0.577737 −0.288868 0.957369i $$-0.593279\pi$$
−0.288868 + 0.957369i $$0.593279\pi$$
$$264$$ 0 0
$$265$$ −2.43845 −0.149793
$$266$$ −6.24621 −0.382980
$$267$$ 0 0
$$268$$ −2.00000 −0.122169
$$269$$ 21.3693 1.30291 0.651455 0.758687i $$-0.274159\pi$$
0.651455 + 0.758687i $$0.274159\pi$$
$$270$$ 0 0
$$271$$ 29.9309 1.81817 0.909085 0.416610i $$-0.136782\pi$$
0.909085 + 0.416610i $$0.136782\pi$$
$$272$$ −7.31534 −0.443558
$$273$$ 0 0
$$274$$ 8.68466 0.524659
$$275$$ −15.3693 −0.926805
$$276$$ 0 0
$$277$$ 5.31534 0.319368 0.159684 0.987168i $$-0.448952\pi$$
0.159684 + 0.987168i $$0.448952\pi$$
$$278$$ 28.0000 1.67933
$$279$$ 0 0
$$280$$ 4.87689 0.291450
$$281$$ 17.8078 1.06232 0.531161 0.847271i $$-0.321756\pi$$
0.531161 + 0.847271i $$0.321756\pi$$
$$282$$ 0 0
$$283$$ −13.6847 −0.813469 −0.406734 0.913547i $$-0.633333\pi$$
−0.406734 + 0.913547i $$0.633333\pi$$
$$284$$ 6.13826 0.364239
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0.876894 0.0517614
$$288$$ 0 0
$$289$$ −14.5616 −0.856562
$$290$$ −37.1771 −2.18311
$$291$$ 0 0
$$292$$ 4.43845 0.259740
$$293$$ 20.4384 1.19403 0.597013 0.802231i $$-0.296354\pi$$
0.597013 + 0.802231i $$0.296354\pi$$
$$294$$ 0 0
$$295$$ 10.2462 0.596557
$$296$$ −18.4384 −1.07171
$$297$$ 0 0
$$298$$ 3.80776 0.220578
$$299$$ 0 0
$$300$$ 0 0
$$301$$ −2.56155 −0.147645
$$302$$ −14.6307 −0.841901
$$303$$ 0 0
$$304$$ 33.3693 1.91386
$$305$$ −13.8078 −0.790630
$$306$$ 0 0
$$307$$ −30.8078 −1.75829 −0.879146 0.476553i $$-0.841886\pi$$
−0.879146 + 0.476553i $$0.841886\pi$$
$$308$$ 0.492423 0.0280584
$$309$$ 0 0
$$310$$ −14.2462 −0.809130
$$311$$ 19.1231 1.08437 0.542186 0.840259i $$-0.317597\pi$$
0.542186 + 0.840259i $$0.317597\pi$$
$$312$$ 0 0
$$313$$ −13.6847 −0.773503 −0.386751 0.922184i $$-0.626403\pi$$
−0.386751 + 0.922184i $$0.626403\pi$$
$$314$$ −31.8078 −1.79502
$$315$$ 0 0
$$316$$ 2.38447 0.134137
$$317$$ 14.0540 0.789350 0.394675 0.918821i $$-0.370857\pi$$
0.394675 + 0.918821i $$0.370857\pi$$
$$318$$ 0 0
$$319$$ 13.3693 0.748538
$$320$$ −19.8078 −1.10729
$$321$$ 0 0
$$322$$ −1.75379 −0.0977348
$$323$$ −11.1231 −0.618906
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −7.50758 −0.415806
$$327$$ 0 0
$$328$$ −3.80776 −0.210249
$$329$$ −4.63068 −0.255298
$$330$$ 0 0
$$331$$ 3.19224 0.175461 0.0877306 0.996144i $$-0.472039\pi$$
0.0877306 + 0.996144i $$0.472039\pi$$
$$332$$ −0.384472 −0.0211006
$$333$$ 0 0
$$334$$ −16.0000 −0.875481
$$335$$ 16.2462 0.887625
$$336$$ 0 0
$$337$$ −6.12311 −0.333547 −0.166773 0.985995i $$-0.553335\pi$$
−0.166773 + 0.985995i $$0.553335\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ −2.43845 −0.132243
$$341$$ 5.12311 0.277432
$$342$$ 0 0
$$343$$ 7.68466 0.414933
$$344$$ 11.1231 0.599718
$$345$$ 0 0
$$346$$ 31.6155 1.69966
$$347$$ −27.6155 −1.48248 −0.741240 0.671241i $$-0.765762\pi$$
−0.741240 + 0.671241i $$0.765762\pi$$
$$348$$ 0 0
$$349$$ −6.80776 −0.364411 −0.182206 0.983260i $$-0.558324\pi$$
−0.182206 + 0.983260i $$0.558324\pi$$
$$350$$ 6.73863 0.360195
$$351$$ 0 0
$$352$$ −4.87689 −0.259939
$$353$$ 5.31534 0.282907 0.141454 0.989945i $$-0.454822\pi$$
0.141454 + 0.989945i $$0.454822\pi$$
$$354$$ 0 0
$$355$$ −49.8617 −2.64639
$$356$$ 2.13826 0.113328
$$357$$ 0 0
$$358$$ −7.61553 −0.402493
$$359$$ −9.36932 −0.494494 −0.247247 0.968953i $$-0.579526\pi$$
−0.247247 + 0.968953i $$0.579526\pi$$
$$360$$ 0 0
$$361$$ 31.7386 1.67045
$$362$$ 4.19224 0.220339
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −36.0540 −1.88715
$$366$$ 0 0
$$367$$ −17.0540 −0.890210 −0.445105 0.895478i $$-0.646834\pi$$
−0.445105 + 0.895478i $$0.646834\pi$$
$$368$$ 9.36932 0.488409
$$369$$ 0 0
$$370$$ −42.0540 −2.18628
$$371$$ −0.384472 −0.0199608
$$372$$ 0 0
$$373$$ 28.3693 1.46891 0.734454 0.678659i $$-0.237438\pi$$
0.734454 + 0.678659i $$0.237438\pi$$
$$374$$ 4.87689 0.252178
$$375$$ 0 0
$$376$$ 20.1080 1.03699
$$377$$ 0 0
$$378$$ 0 0
$$379$$ −23.6847 −1.21660 −0.608300 0.793708i $$-0.708148\pi$$
−0.608300 + 0.793708i $$0.708148\pi$$
$$380$$ 11.1231 0.570603
$$381$$ 0 0
$$382$$ −14.2462 −0.728900
$$383$$ −22.7386 −1.16189 −0.580945 0.813943i $$-0.697317\pi$$
−0.580945 + 0.813943i $$0.697317\pi$$
$$384$$ 0 0
$$385$$ −4.00000 −0.203859
$$386$$ −21.0691 −1.07239
$$387$$ 0 0
$$388$$ 3.75379 0.190570
$$389$$ −34.0540 −1.72661 −0.863303 0.504687i $$-0.831608\pi$$
−0.863303 + 0.504687i $$0.831608\pi$$
$$390$$ 0 0
$$391$$ −3.12311 −0.157942
$$392$$ −16.3002 −0.823284
$$393$$ 0 0
$$394$$ −20.8769 −1.05176
$$395$$ −19.3693 −0.974576
$$396$$ 0 0
$$397$$ −25.0540 −1.25742 −0.628711 0.777639i $$-0.716417\pi$$
−0.628711 + 0.777639i $$0.716417\pi$$
$$398$$ −34.6307 −1.73588
$$399$$ 0 0
$$400$$ −36.0000 −1.80000
$$401$$ 14.4384 0.721022 0.360511 0.932755i $$-0.382602\pi$$
0.360511 + 0.932755i $$0.382602\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 3.31534 0.164944
$$405$$ 0 0
$$406$$ −5.86174 −0.290913
$$407$$ 15.1231 0.749625
$$408$$ 0 0
$$409$$ −6.36932 −0.314942 −0.157471 0.987524i $$-0.550334\pi$$
−0.157471 + 0.987524i $$0.550334\pi$$
$$410$$ −8.68466 −0.428905
$$411$$ 0 0
$$412$$ −1.50758 −0.0742730
$$413$$ 1.61553 0.0794949
$$414$$ 0 0
$$415$$ 3.12311 0.153307
$$416$$ 0 0
$$417$$ 0 0
$$418$$ −22.2462 −1.08810
$$419$$ 34.2462 1.67304 0.836518 0.547939i $$-0.184587\pi$$
0.836518 + 0.547939i $$0.184587\pi$$
$$420$$ 0 0
$$421$$ −31.2462 −1.52285 −0.761424 0.648255i $$-0.775499\pi$$
−0.761424 + 0.648255i $$0.775499\pi$$
$$422$$ −30.7386 −1.49633
$$423$$ 0 0
$$424$$ 1.66950 0.0810783
$$425$$ 12.0000 0.582086
$$426$$ 0 0
$$427$$ −2.17708 −0.105356
$$428$$ 3.61553 0.174763
$$429$$ 0 0
$$430$$ 25.3693 1.22342
$$431$$ −11.1231 −0.535781 −0.267891 0.963449i $$-0.586327\pi$$
−0.267891 + 0.963449i $$0.586327\pi$$
$$432$$ 0 0
$$433$$ 8.75379 0.420680 0.210340 0.977628i $$-0.432543\pi$$
0.210340 + 0.977628i $$0.432543\pi$$
$$434$$ −2.24621 −0.107822
$$435$$ 0 0
$$436$$ 1.23106 0.0589569
$$437$$ 14.2462 0.681489
$$438$$ 0 0
$$439$$ −13.6847 −0.653133 −0.326567 0.945174i $$-0.605892\pi$$
−0.326567 + 0.945174i $$0.605892\pi$$
$$440$$ 17.3693 0.828050
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 34.7386 1.65048 0.825241 0.564781i $$-0.191039\pi$$
0.825241 + 0.564781i $$0.191039\pi$$
$$444$$ 0 0
$$445$$ −17.3693 −0.823385
$$446$$ 12.4924 0.591533
$$447$$ 0 0
$$448$$ −3.12311 −0.147553
$$449$$ −8.24621 −0.389163 −0.194581 0.980886i $$-0.562335\pi$$
−0.194581 + 0.980886i $$0.562335\pi$$
$$450$$ 0 0
$$451$$ 3.12311 0.147061
$$452$$ −2.54640 −0.119772
$$453$$ 0 0
$$454$$ −11.1231 −0.522033
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −12.6155 −0.590130 −0.295065 0.955477i $$-0.595341\pi$$
−0.295065 + 0.955477i $$0.595341\pi$$
$$458$$ −25.3693 −1.18543
$$459$$ 0 0
$$460$$ 3.12311 0.145616
$$461$$ −16.1922 −0.754148 −0.377074 0.926183i $$-0.623070\pi$$
−0.377074 + 0.926183i $$0.623070\pi$$
$$462$$ 0 0
$$463$$ 14.3153 0.665290 0.332645 0.943052i $$-0.392059\pi$$
0.332645 + 0.943052i $$0.392059\pi$$
$$464$$ 31.3153 1.45378
$$465$$ 0 0
$$466$$ 40.6004 1.88078
$$467$$ 26.0000 1.20314 0.601568 0.798821i $$-0.294543\pi$$
0.601568 + 0.798821i $$0.294543\pi$$
$$468$$ 0 0
$$469$$ 2.56155 0.118282
$$470$$ 45.8617 2.11544
$$471$$ 0 0
$$472$$ −7.01515 −0.322899
$$473$$ −9.12311 −0.419481
$$474$$ 0 0
$$475$$ −54.7386 −2.51158
$$476$$ −0.384472 −0.0176222
$$477$$ 0 0
$$478$$ 39.6155 1.81197
$$479$$ 10.2462 0.468161 0.234081 0.972217i $$-0.424792\pi$$
0.234081 + 0.972217i $$0.424792\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ −27.8078 −1.26661
$$483$$ 0 0
$$484$$ −3.06913 −0.139506
$$485$$ −30.4924 −1.38459
$$486$$ 0 0
$$487$$ −7.12311 −0.322779 −0.161389 0.986891i $$-0.551597\pi$$
−0.161389 + 0.986891i $$0.551597\pi$$
$$488$$ 9.45360 0.427944
$$489$$ 0 0
$$490$$ −37.1771 −1.67949
$$491$$ 36.2462 1.63577 0.817884 0.575383i $$-0.195147\pi$$
0.817884 + 0.575383i $$0.195147\pi$$
$$492$$ 0 0
$$493$$ −10.4384 −0.470124
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 12.0000 0.538816
$$497$$ −7.86174 −0.352647
$$498$$ 0 0
$$499$$ −4.49242 −0.201108 −0.100554 0.994932i $$-0.532062\pi$$
−0.100554 + 0.994932i $$0.532062\pi$$
$$500$$ −4.19224 −0.187482
$$501$$ 0 0
$$502$$ 29.2614 1.30600
$$503$$ 28.2462 1.25944 0.629718 0.776824i $$-0.283170\pi$$
0.629718 + 0.776824i $$0.283170\pi$$
$$504$$ 0 0
$$505$$ −26.9309 −1.19841
$$506$$ −6.24621 −0.277678
$$507$$ 0 0
$$508$$ 2.38447 0.105794
$$509$$ −13.8078 −0.612018 −0.306009 0.952029i $$-0.598994\pi$$
−0.306009 + 0.952029i $$0.598994\pi$$
$$510$$ 0 0
$$511$$ −5.68466 −0.251474
$$512$$ 11.4233 0.504843
$$513$$ 0 0
$$514$$ 45.5616 2.00963
$$515$$ 12.2462 0.539633
$$516$$ 0 0
$$517$$ −16.4924 −0.725336
$$518$$ −6.63068 −0.291335
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 9.06913 0.397326 0.198663 0.980068i $$-0.436340\pi$$
0.198663 + 0.980068i $$0.436340\pi$$
$$522$$ 0 0
$$523$$ 33.8617 1.48067 0.740335 0.672238i $$-0.234667\pi$$
0.740335 + 0.672238i $$0.234667\pi$$
$$524$$ −3.23106 −0.141149
$$525$$ 0 0
$$526$$ 14.6307 0.637928
$$527$$ −4.00000 −0.174243
$$528$$ 0 0
$$529$$ −19.0000 −0.826087
$$530$$ 3.80776 0.165399
$$531$$ 0 0
$$532$$ 1.75379 0.0760364
$$533$$ 0 0
$$534$$ 0 0
$$535$$ −29.3693 −1.26975
$$536$$ −11.1231 −0.480445
$$537$$ 0 0
$$538$$ −33.3693 −1.43865
$$539$$ 13.3693 0.575857
$$540$$ 0 0
$$541$$ −19.7386 −0.848630 −0.424315 0.905515i $$-0.639485\pi$$
−0.424315 + 0.905515i $$0.639485\pi$$
$$542$$ −46.7386 −2.00760
$$543$$ 0 0
$$544$$ 3.80776 0.163257
$$545$$ −10.0000 −0.428353
$$546$$ 0 0
$$547$$ 3.93087 0.168072 0.0840359 0.996463i $$-0.473219\pi$$
0.0840359 + 0.996463i $$0.473219\pi$$
$$548$$ −2.43845 −0.104165
$$549$$ 0 0
$$550$$ 24.0000 1.02336
$$551$$ 47.6155 2.02849
$$552$$ 0 0
$$553$$ −3.05398 −0.129868
$$554$$ −8.30019 −0.352641
$$555$$ 0 0
$$556$$ −7.86174 −0.333412
$$557$$ −42.9309 −1.81904 −0.909520 0.415661i $$-0.863550\pi$$
−0.909520 + 0.415661i $$0.863550\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ −9.36932 −0.395926
$$561$$ 0 0
$$562$$ −27.8078 −1.17300
$$563$$ 23.3693 0.984899 0.492450 0.870341i $$-0.336102\pi$$
0.492450 + 0.870341i $$0.336102\pi$$
$$564$$ 0 0
$$565$$ 20.6847 0.870210
$$566$$ 21.3693 0.898219
$$567$$ 0 0
$$568$$ 34.1383 1.43241
$$569$$ 8.73863 0.366343 0.183171 0.983081i $$-0.441364\pi$$
0.183171 + 0.983081i $$0.441364\pi$$
$$570$$ 0 0
$$571$$ −5.36932 −0.224699 −0.112349 0.993669i $$-0.535838\pi$$
−0.112349 + 0.993669i $$0.535838\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ −1.36932 −0.0571542
$$575$$ −15.3693 −0.640945
$$576$$ 0 0
$$577$$ 17.3153 0.720847 0.360424 0.932789i $$-0.382632\pi$$
0.360424 + 0.932789i $$0.382632\pi$$
$$578$$ 22.7386 0.945802
$$579$$ 0 0
$$580$$ 10.4384 0.433433
$$581$$ 0.492423 0.0204291
$$582$$ 0 0
$$583$$ −1.36932 −0.0567113
$$584$$ 24.6847 1.02146
$$585$$ 0 0
$$586$$ −31.9157 −1.31843
$$587$$ 39.3693 1.62495 0.812473 0.582999i $$-0.198121\pi$$
0.812473 + 0.582999i $$0.198121\pi$$
$$588$$ 0 0
$$589$$ 18.2462 0.751822
$$590$$ −16.0000 −0.658710
$$591$$ 0 0
$$592$$ 35.4233 1.45589
$$593$$ −17.4233 −0.715489 −0.357744 0.933820i $$-0.616454\pi$$
−0.357744 + 0.933820i $$0.616454\pi$$
$$594$$ 0 0
$$595$$ 3.12311 0.128035
$$596$$ −1.06913 −0.0437933
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 41.6155 1.70036 0.850182 0.526489i $$-0.176492\pi$$
0.850182 + 0.526489i $$0.176492\pi$$
$$600$$ 0 0
$$601$$ −7.06913 −0.288356 −0.144178 0.989552i $$-0.546054\pi$$
−0.144178 + 0.989552i $$0.546054\pi$$
$$602$$ 4.00000 0.163028
$$603$$ 0 0
$$604$$ 4.10795 0.167150
$$605$$ 24.9309 1.01358
$$606$$ 0 0
$$607$$ −16.0000 −0.649420 −0.324710 0.945814i $$-0.605267\pi$$
−0.324710 + 0.945814i $$0.605267\pi$$
$$608$$ −17.3693 −0.704419
$$609$$ 0 0
$$610$$ 21.5616 0.873002
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 34.8617 1.40805 0.704026 0.710174i $$-0.251384\pi$$
0.704026 + 0.710174i $$0.251384\pi$$
$$614$$ 48.1080 1.94148
$$615$$ 0 0
$$616$$ 2.73863 0.110343
$$617$$ 9.80776 0.394846 0.197423 0.980318i $$-0.436743\pi$$
0.197423 + 0.980318i $$0.436743\pi$$
$$618$$ 0 0
$$619$$ −29.3002 −1.17767 −0.588837 0.808252i $$-0.700414\pi$$
−0.588837 + 0.808252i $$0.700414\pi$$
$$620$$ 4.00000 0.160644
$$621$$ 0 0
$$622$$ −29.8617 −1.19735
$$623$$ −2.73863 −0.109721
$$624$$ 0 0
$$625$$ −4.36932 −0.174773
$$626$$ 21.3693 0.854090
$$627$$ 0 0
$$628$$ 8.93087 0.356380
$$629$$ −11.8078 −0.470806
$$630$$ 0 0
$$631$$ 18.5616 0.738924 0.369462 0.929246i $$-0.379542\pi$$
0.369462 + 0.929246i $$0.379542\pi$$
$$632$$ 13.2614 0.527509
$$633$$ 0 0
$$634$$ −21.9460 −0.871588
$$635$$ −19.3693 −0.768648
$$636$$ 0 0
$$637$$ 0 0
$$638$$ −20.8769 −0.826524
$$639$$ 0 0
$$640$$ 48.3002 1.90923
$$641$$ 19.1771 0.757449 0.378725 0.925509i $$-0.376363\pi$$
0.378725 + 0.925509i $$0.376363\pi$$
$$642$$ 0 0
$$643$$ 31.5464 1.24407 0.622034 0.782990i $$-0.286306\pi$$
0.622034 + 0.782990i $$0.286306\pi$$
$$644$$ 0.492423 0.0194042
$$645$$ 0 0
$$646$$ 17.3693 0.683387
$$647$$ 6.38447 0.250999 0.125500 0.992094i $$-0.459947\pi$$
0.125500 + 0.992094i $$0.459947\pi$$
$$648$$ 0 0
$$649$$ 5.75379 0.225856
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 2.10795 0.0825537
$$653$$ −23.1231 −0.904877 −0.452439 0.891796i $$-0.649446\pi$$
−0.452439 + 0.891796i $$0.649446\pi$$
$$654$$ 0 0
$$655$$ 26.2462 1.02552
$$656$$ 7.31534 0.285616
$$657$$ 0 0
$$658$$ 7.23106 0.281896
$$659$$ 2.24621 0.0875000 0.0437500 0.999043i $$-0.486070\pi$$
0.0437500 + 0.999043i $$0.486070\pi$$
$$660$$ 0 0
$$661$$ −5.63068 −0.219008 −0.109504 0.993986i $$-0.534926\pi$$
−0.109504 + 0.993986i $$0.534926\pi$$
$$662$$ −4.98485 −0.193742
$$663$$ 0 0
$$664$$ −2.13826 −0.0829806
$$665$$ −14.2462 −0.552444
$$666$$ 0 0
$$667$$ 13.3693 0.517662
$$668$$ 4.49242 0.173817
$$669$$ 0 0
$$670$$ −25.3693 −0.980102
$$671$$ −7.75379 −0.299332
$$672$$ 0 0
$$673$$ −23.2462 −0.896076 −0.448038 0.894015i $$-0.647877\pi$$
−0.448038 + 0.894015i $$0.647877\pi$$
$$674$$ 9.56155 0.368297
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 15.6155 0.600153 0.300077 0.953915i $$-0.402988\pi$$
0.300077 + 0.953915i $$0.402988\pi$$
$$678$$ 0 0
$$679$$ −4.80776 −0.184505
$$680$$ −13.5616 −0.520062
$$681$$ 0 0
$$682$$ −8.00000 −0.306336
$$683$$ −38.1080 −1.45816 −0.729080 0.684428i $$-0.760052\pi$$
−0.729080 + 0.684428i $$0.760052\pi$$
$$684$$ 0 0
$$685$$ 19.8078 0.756816
$$686$$ −12.0000 −0.458162
$$687$$ 0 0
$$688$$ −21.3693 −0.814698
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 51.3002 1.95155 0.975776 0.218774i $$-0.0702058\pi$$
0.975776 + 0.218774i $$0.0702058\pi$$
$$692$$ −8.87689 −0.337449
$$693$$ 0 0
$$694$$ 43.1231 1.63693
$$695$$ 63.8617 2.42241
$$696$$ 0 0
$$697$$ −2.43845 −0.0923628
$$698$$ 10.6307 0.402377
$$699$$ 0 0
$$700$$ −1.89205 −0.0715127
$$701$$ 5.36932 0.202796 0.101398 0.994846i $$-0.467668\pi$$
0.101398 + 0.994846i $$0.467668\pi$$
$$702$$ 0 0
$$703$$ 53.8617 2.03143
$$704$$ −11.1231 −0.419218
$$705$$ 0 0
$$706$$ −8.30019 −0.312382
$$707$$ −4.24621 −0.159695
$$708$$ 0 0
$$709$$ 7.49242 0.281384 0.140692 0.990053i $$-0.455067\pi$$
0.140692 + 0.990053i $$0.455067\pi$$
$$710$$ 77.8617 2.92210
$$711$$ 0 0
$$712$$ 11.8920 0.445673
$$713$$ 5.12311 0.191862
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 2.13826 0.0799106
$$717$$ 0 0
$$718$$ 14.6307 0.546012
$$719$$ 23.3693 0.871528 0.435764 0.900061i $$-0.356478\pi$$
0.435764 + 0.900061i $$0.356478\pi$$
$$720$$ 0 0
$$721$$ 1.93087 0.0719093
$$722$$ −49.5616 −1.84449
$$723$$ 0 0
$$724$$ −1.17708 −0.0437459
$$725$$ −51.3693 −1.90781
$$726$$ 0 0
$$727$$ −38.6695 −1.43417 −0.717086 0.696984i $$-0.754525\pi$$
−0.717086 + 0.696984i $$0.754525\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 56.3002 2.08376
$$731$$ 7.12311 0.263458
$$732$$ 0 0
$$733$$ −20.5076 −0.757465 −0.378732 0.925506i $$-0.623640\pi$$
−0.378732 + 0.925506i $$0.623640\pi$$
$$734$$ 26.6307 0.982956
$$735$$ 0 0
$$736$$ −4.87689 −0.179765
$$737$$ 9.12311 0.336054
$$738$$ 0 0
$$739$$ −10.2462 −0.376913 −0.188456 0.982082i $$-0.560348\pi$$
−0.188456 + 0.982082i $$0.560348\pi$$
$$740$$ 11.8078 0.434062
$$741$$ 0 0
$$742$$ 0.600373 0.0220404
$$743$$ −12.6307 −0.463375 −0.231687 0.972790i $$-0.574425\pi$$
−0.231687 + 0.972790i $$0.574425\pi$$
$$744$$ 0 0
$$745$$ 8.68466 0.318181
$$746$$ −44.3002 −1.62195
$$747$$ 0 0
$$748$$ −1.36932 −0.0500672
$$749$$ −4.63068 −0.169201
$$750$$ 0 0
$$751$$ 44.1080 1.60952 0.804761 0.593599i $$-0.202293\pi$$
0.804761 + 0.593599i $$0.202293\pi$$
$$752$$ −38.6307 −1.40872
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −33.3693 −1.21443
$$756$$ 0 0
$$757$$ 30.0000 1.09037 0.545184 0.838316i $$-0.316460\pi$$
0.545184 + 0.838316i $$0.316460\pi$$
$$758$$ 36.9848 1.34335
$$759$$ 0 0
$$760$$ 61.8617 2.24396
$$761$$ 9.36932 0.339637 0.169819 0.985475i $$-0.445682\pi$$
0.169819 + 0.985475i $$0.445682\pi$$
$$762$$ 0 0
$$763$$ −1.57671 −0.0570807
$$764$$ 4.00000 0.144715
$$765$$ 0 0
$$766$$ 35.5076 1.28294
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 18.0000 0.649097 0.324548 0.945869i $$-0.394788\pi$$
0.324548 + 0.945869i $$0.394788\pi$$
$$770$$ 6.24621 0.225098
$$771$$ 0 0
$$772$$ 5.91571 0.212911
$$773$$ −24.2462 −0.872076 −0.436038 0.899928i $$-0.643619\pi$$
−0.436038 + 0.899928i $$0.643619\pi$$
$$774$$ 0 0
$$775$$ −19.6847 −0.707094
$$776$$ 20.8769 0.749437
$$777$$ 0 0
$$778$$ 53.1771 1.90649
$$779$$ 11.1231 0.398527
$$780$$ 0 0
$$781$$ −28.0000 −1.00192
$$782$$ 4.87689 0.174397
$$783$$ 0 0
$$784$$ 31.3153 1.11841
$$785$$ −72.5464 −2.58929
$$786$$ 0 0
$$787$$ −44.1771 −1.57474 −0.787371 0.616479i $$-0.788559\pi$$
−0.787371 + 0.616479i $$0.788559\pi$$
$$788$$ 5.86174 0.208816
$$789$$ 0 0
$$790$$ 30.2462 1.07611
$$791$$ 3.26137 0.115961
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 39.1231 1.38843
$$795$$ 0 0
$$796$$ 9.72348 0.344640
$$797$$ 0.384472 0.0136187 0.00680935 0.999977i $$-0.497833\pi$$
0.00680935 + 0.999977i $$0.497833\pi$$
$$798$$ 0 0
$$799$$ 12.8769 0.455552
$$800$$ 18.7386 0.662511
$$801$$ 0 0
$$802$$ −22.5464 −0.796141
$$803$$ −20.2462 −0.714473
$$804$$ 0 0
$$805$$ −4.00000 −0.140981
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 18.4384 0.648662
$$809$$ 16.3002 0.573084 0.286542 0.958068i $$-0.407494\pi$$
0.286542 + 0.958068i $$0.407494\pi$$
$$810$$ 0 0
$$811$$ −2.56155 −0.0899483 −0.0449741 0.998988i $$-0.514321\pi$$
−0.0449741 + 0.998988i $$0.514321\pi$$
$$812$$ 1.64584 0.0577576
$$813$$ 0 0
$$814$$ −23.6155 −0.827724
$$815$$ −17.1231 −0.599796
$$816$$ 0 0
$$817$$ −32.4924 −1.13677
$$818$$ 9.94602 0.347755
$$819$$ 0 0
$$820$$ 2.43845 0.0851543
$$821$$ 6.49242 0.226587 0.113294 0.993562i $$-0.463860\pi$$
0.113294 + 0.993562i $$0.463860\pi$$
$$822$$ 0 0
$$823$$ 8.00000 0.278862 0.139431 0.990232i $$-0.455473\pi$$
0.139431 + 0.990232i $$0.455473\pi$$
$$824$$ −8.38447 −0.292087
$$825$$ 0 0
$$826$$ −2.52273 −0.0877771
$$827$$ −14.7386 −0.512513 −0.256256 0.966609i $$-0.582489\pi$$
−0.256256 + 0.966609i $$0.582489\pi$$
$$828$$ 0 0
$$829$$ 13.4924 0.468611 0.234306 0.972163i $$-0.424718\pi$$
0.234306 + 0.972163i $$0.424718\pi$$
$$830$$ −4.87689 −0.169279
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −10.4384 −0.361671
$$834$$ 0 0
$$835$$ −36.4924 −1.26287
$$836$$ 6.24621 0.216030
$$837$$ 0 0
$$838$$ −53.4773 −1.84734
$$839$$ 21.6155 0.746251 0.373125 0.927781i $$-0.378286\pi$$
0.373125 + 0.927781i $$0.378286\pi$$
$$840$$ 0 0
$$841$$ 15.6847 0.540850
$$842$$ 48.7926 1.68150
$$843$$ 0 0
$$844$$ 8.63068 0.297080
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 3.93087 0.135066
$$848$$ −3.20739 −0.110142
$$849$$ 0 0
$$850$$ −18.7386 −0.642730
$$851$$ 15.1231 0.518413
$$852$$ 0 0
$$853$$ −2.12311 −0.0726938 −0.0363469 0.999339i $$-0.511572\pi$$
−0.0363469 + 0.999339i $$0.511572\pi$$
$$854$$ 3.39963 0.116333
$$855$$ 0 0
$$856$$ 20.1080 0.687276
$$857$$ −35.5616 −1.21476 −0.607380 0.794412i $$-0.707779\pi$$
−0.607380 + 0.794412i $$0.707779\pi$$
$$858$$ 0 0
$$859$$ 24.5616 0.838029 0.419015 0.907979i $$-0.362376\pi$$
0.419015 + 0.907979i $$0.362376\pi$$
$$860$$ −7.12311 −0.242896
$$861$$ 0 0
$$862$$ 17.3693 0.591601
$$863$$ 30.4924 1.03797 0.518987 0.854782i $$-0.326309\pi$$
0.518987 + 0.854782i $$0.326309\pi$$
$$864$$ 0 0
$$865$$ 72.1080 2.45174
$$866$$ −13.6695 −0.464509
$$867$$ 0 0
$$868$$ 0.630683 0.0214068
$$869$$ −10.8769 −0.368973
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 6.84658 0.231855
$$873$$ 0 0
$$874$$ −22.2462 −0.752489
$$875$$ 5.36932 0.181516
$$876$$ 0 0
$$877$$ −23.5616 −0.795617 −0.397809 0.917468i $$-0.630229\pi$$
−0.397809 + 0.917468i $$0.630229\pi$$
$$878$$ 21.3693 0.721180
$$879$$ 0 0
$$880$$ −33.3693 −1.12488
$$881$$ 9.06913 0.305547 0.152773 0.988261i $$-0.451180\pi$$
0.152773 + 0.988261i $$0.451180\pi$$
$$882$$ 0 0
$$883$$ 8.80776 0.296405 0.148202 0.988957i $$-0.452651\pi$$
0.148202 + 0.988957i $$0.452651\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ −54.2462 −1.82244
$$887$$ −24.6307 −0.827017 −0.413509 0.910500i $$-0.635697\pi$$
−0.413509 + 0.910500i $$0.635697\pi$$
$$888$$ 0 0
$$889$$ −3.05398 −0.102427
$$890$$ 27.1231 0.909169
$$891$$ 0 0
$$892$$ −3.50758 −0.117442
$$893$$ −58.7386 −1.96561
$$894$$ 0 0
$$895$$ −17.3693 −0.580592
$$896$$ 7.61553 0.254417
$$897$$ 0 0
$$898$$ 12.8769 0.429708
$$899$$ 17.1231 0.571088
$$900$$ 0 0
$$901$$ 1.06913 0.0356179
$$902$$ −4.87689 −0.162383
$$903$$ 0 0
$$904$$ −14.1619 −0.471019
$$905$$ 9.56155 0.317837
$$906$$ 0 0
$$907$$ 28.0000 0.929725 0.464862 0.885383i $$-0.346104\pi$$
0.464862 + 0.885383i $$0.346104\pi$$
$$908$$ 3.12311 0.103644
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −38.7386 −1.28347 −0.641734 0.766927i $$-0.721785\pi$$
−0.641734 + 0.766927i $$0.721785\pi$$
$$912$$ 0 0
$$913$$ 1.75379 0.0580419
$$914$$ 19.6998 0.651612
$$915$$ 0 0
$$916$$ 7.12311 0.235354
$$917$$ 4.13826 0.136657
$$918$$ 0 0
$$919$$ 11.5076 0.379600 0.189800 0.981823i $$-0.439216\pi$$
0.189800 + 0.981823i $$0.439216\pi$$
$$920$$ 17.3693 0.572649
$$921$$ 0 0
$$922$$ 25.2850 0.832718
$$923$$ 0 0
$$924$$ 0 0
$$925$$ −58.1080 −1.91058
$$926$$ −22.3542 −0.734603
$$927$$ 0 0
$$928$$ −16.3002 −0.535080
$$929$$ −7.80776 −0.256164 −0.128082 0.991764i $$-0.540882\pi$$
−0.128082 + 0.991764i $$0.540882\pi$$
$$930$$ 0 0
$$931$$ 47.6155 1.56054
$$932$$ −11.3996 −0.373407
$$933$$ 0 0
$$934$$ −40.6004 −1.32848
$$935$$ 11.1231 0.363764
$$936$$ 0 0
$$937$$ −7.56155 −0.247025 −0.123513 0.992343i $$-0.539416\pi$$
−0.123513 + 0.992343i $$0.539416\pi$$
$$938$$ −4.00000 −0.130605
$$939$$ 0 0
$$940$$ −12.8769 −0.419998
$$941$$ 30.4924 0.994025 0.497012 0.867744i $$-0.334430\pi$$
0.497012 + 0.867744i $$0.334430\pi$$
$$942$$ 0 0
$$943$$ 3.12311 0.101702
$$944$$ 13.4773 0.438648
$$945$$ 0 0
$$946$$ 14.2462 0.463184
$$947$$ −38.7386 −1.25884 −0.629418 0.777067i $$-0.716707\pi$$
−0.629418 + 0.777067i $$0.716707\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 85.4773 2.77325
$$951$$ 0 0
$$952$$ −2.13826 −0.0693014
$$953$$ −30.9848 −1.00370 −0.501849 0.864955i $$-0.667347\pi$$
−0.501849 + 0.864955i $$0.667347\pi$$
$$954$$ 0 0
$$955$$ −32.4924 −1.05143
$$956$$ −11.1231 −0.359747
$$957$$ 0 0
$$958$$ −16.0000 −0.516937
$$959$$ 3.12311 0.100850
$$960$$ 0 0
$$961$$ −24.4384 −0.788337
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 7.80776 0.251471
$$965$$ −48.0540 −1.54691
$$966$$ 0 0
$$967$$ 0.876894 0.0281990 0.0140995 0.999901i $$-0.495512\pi$$
0.0140995 + 0.999901i $$0.495512\pi$$
$$968$$ −17.0691 −0.548623
$$969$$ 0 0
$$970$$ 47.6155 1.52884
$$971$$ 12.9848 0.416704 0.208352 0.978054i $$-0.433190\pi$$
0.208352 + 0.978054i $$0.433190\pi$$
$$972$$ 0 0
$$973$$ 10.0691 0.322801
$$974$$ 11.1231 0.356407
$$975$$ 0 0
$$976$$ −18.1619 −0.581349
$$977$$ 61.1771 1.95723 0.978614 0.205705i $$-0.0659486\pi$$
0.978614 + 0.205705i $$0.0659486\pi$$
$$978$$ 0 0
$$979$$ −9.75379 −0.311732
$$980$$ 10.4384 0.333444
$$981$$ 0 0
$$982$$ −56.6004 −1.80619
$$983$$ 13.6155 0.434268 0.217134 0.976142i $$-0.430329\pi$$
0.217134 + 0.976142i $$0.430329\pi$$
$$984$$ 0 0
$$985$$ −47.6155 −1.51716
$$986$$ 16.3002 0.519104
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −9.12311 −0.290098
$$990$$ 0 0
$$991$$ −50.3542 −1.59955 −0.799776 0.600298i $$-0.795049\pi$$
−0.799776 + 0.600298i $$0.795049\pi$$
$$992$$ −6.24621 −0.198317
$$993$$ 0 0
$$994$$ 12.2765 0.389388
$$995$$ −78.9848 −2.50399
$$996$$ 0 0
$$997$$ −20.6155 −0.652900 −0.326450 0.945214i $$-0.605853\pi$$
−0.326450 + 0.945214i $$0.605853\pi$$
$$998$$ 7.01515 0.222061
$$999$$ 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 1521.2.a.m.1.1 2
3.2 odd 2 507.2.a.d.1.2 2
12.11 even 2 8112.2.a.bo.1.2 2
13.4 even 6 117.2.g.c.55.1 4
13.5 odd 4 1521.2.b.h.1351.3 4
13.8 odd 4 1521.2.b.h.1351.2 4
13.10 even 6 117.2.g.c.100.1 4
13.12 even 2 1521.2.a.g.1.2 2
39.2 even 12 507.2.j.g.316.3 8
39.5 even 4 507.2.b.d.337.2 4
39.8 even 4 507.2.b.d.337.3 4
39.11 even 12 507.2.j.g.316.2 8
39.17 odd 6 39.2.e.b.16.2 4
39.20 even 12 507.2.j.g.361.3 8
39.23 odd 6 39.2.e.b.22.2 yes 4
39.29 odd 6 507.2.e.g.22.1 4
39.32 even 12 507.2.j.g.361.2 8
39.35 odd 6 507.2.e.g.484.1 4
39.38 odd 2 507.2.a.g.1.1 2
52.23 odd 6 1872.2.t.r.1153.2 4
52.43 odd 6 1872.2.t.r.289.2 4
156.23 even 6 624.2.q.h.529.1 4
156.95 even 6 624.2.q.h.289.1 4
156.155 even 2 8112.2.a.bk.1.1 2
195.17 even 12 975.2.bb.i.874.2 8
195.23 even 12 975.2.bb.i.724.2 8
195.62 even 12 975.2.bb.i.724.3 8
195.134 odd 6 975.2.i.k.601.1 4
195.173 even 12 975.2.bb.i.874.3 8
195.179 odd 6 975.2.i.k.451.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
39.2.e.b.16.2 4 39.17 odd 6
39.2.e.b.22.2 yes 4 39.23 odd 6
117.2.g.c.55.1 4 13.4 even 6
117.2.g.c.100.1 4 13.10 even 6
507.2.a.d.1.2 2 3.2 odd 2
507.2.a.g.1.1 2 39.38 odd 2
507.2.b.d.337.2 4 39.5 even 4
507.2.b.d.337.3 4 39.8 even 4
507.2.e.g.22.1 4 39.29 odd 6
507.2.e.g.484.1 4 39.35 odd 6
507.2.j.g.316.2 8 39.11 even 12
507.2.j.g.316.3 8 39.2 even 12
507.2.j.g.361.2 8 39.32 even 12
507.2.j.g.361.3 8 39.20 even 12
624.2.q.h.289.1 4 156.95 even 6
624.2.q.h.529.1 4 156.23 even 6
975.2.i.k.451.1 4 195.179 odd 6
975.2.i.k.601.1 4 195.134 odd 6
975.2.bb.i.724.2 8 195.23 even 12
975.2.bb.i.724.3 8 195.62 even 12
975.2.bb.i.874.2 8 195.17 even 12
975.2.bb.i.874.3 8 195.173 even 12
1521.2.a.g.1.2 2 13.12 even 2
1521.2.a.m.1.1 2 1.1 even 1 trivial
1521.2.b.h.1351.2 4 13.8 odd 4
1521.2.b.h.1351.3 4 13.5 odd 4
1872.2.t.r.289.2 4 52.43 odd 6
1872.2.t.r.1153.2 4 52.23 odd 6
8112.2.a.bk.1.1 2 156.155 even 2
8112.2.a.bo.1.2 2 12.11 even 2