# Properties

 Label 15.3.d.a.14.1 Level $15$ Weight $3$ Character 15.14 Self dual yes Analytic conductor $0.409$ Analytic rank $0$ Dimension $1$ CM discriminant -15 Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$15 = 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 15.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.408720396540$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## Embedding invariants

 Embedding label 14.1 Character $$\chi$$ $$=$$ 15.14

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +3.00000 q^{3} -3.00000 q^{4} -5.00000 q^{5} -3.00000 q^{6} +7.00000 q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +3.00000 q^{3} -3.00000 q^{4} -5.00000 q^{5} -3.00000 q^{6} +7.00000 q^{8} +9.00000 q^{9} +5.00000 q^{10} -9.00000 q^{12} -15.0000 q^{15} +5.00000 q^{16} +14.0000 q^{17} -9.00000 q^{18} -22.0000 q^{19} +15.0000 q^{20} -34.0000 q^{23} +21.0000 q^{24} +25.0000 q^{25} +27.0000 q^{27} +15.0000 q^{30} +2.00000 q^{31} -33.0000 q^{32} -14.0000 q^{34} -27.0000 q^{36} +22.0000 q^{38} -35.0000 q^{40} -45.0000 q^{45} +34.0000 q^{46} +14.0000 q^{47} +15.0000 q^{48} +49.0000 q^{49} -25.0000 q^{50} +42.0000 q^{51} +86.0000 q^{53} -27.0000 q^{54} -66.0000 q^{57} +45.0000 q^{60} -118.000 q^{61} -2.00000 q^{62} +13.0000 q^{64} -42.0000 q^{68} -102.000 q^{69} +63.0000 q^{72} +75.0000 q^{75} +66.0000 q^{76} +98.0000 q^{79} -25.0000 q^{80} +81.0000 q^{81} -154.000 q^{83} -70.0000 q^{85} +45.0000 q^{90} +102.000 q^{92} +6.00000 q^{93} -14.0000 q^{94} +110.000 q^{95} -99.0000 q^{96} -49.0000 q^{98} +O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/15\mathbb{Z}\right)^\times$$.

 $$n$$ $$7$$ $$11$$ $$\chi(n)$$ $$-1$$ $$-1$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.500000 −0.250000 0.968246i $$-0.580431\pi$$
−0.250000 + 0.968246i $$0.580431\pi$$
$$3$$ 3.00000 1.00000
$$4$$ −3.00000 −0.750000
$$5$$ −5.00000 −1.00000
$$6$$ −3.00000 −0.500000
$$7$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$8$$ 7.00000 0.875000
$$9$$ 9.00000 1.00000
$$10$$ 5.00000 0.500000
$$11$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$12$$ −9.00000 −0.750000
$$13$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$14$$ 0 0
$$15$$ −15.0000 −1.00000
$$16$$ 5.00000 0.312500
$$17$$ 14.0000 0.823529 0.411765 0.911290i $$-0.364913\pi$$
0.411765 + 0.911290i $$0.364913\pi$$
$$18$$ −9.00000 −0.500000
$$19$$ −22.0000 −1.15789 −0.578947 0.815365i $$-0.696536\pi$$
−0.578947 + 0.815365i $$0.696536\pi$$
$$20$$ 15.0000 0.750000
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −34.0000 −1.47826 −0.739130 0.673562i $$-0.764763\pi$$
−0.739130 + 0.673562i $$0.764763\pi$$
$$24$$ 21.0000 0.875000
$$25$$ 25.0000 1.00000
$$26$$ 0 0
$$27$$ 27.0000 1.00000
$$28$$ 0 0
$$29$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$30$$ 15.0000 0.500000
$$31$$ 2.00000 0.0645161 0.0322581 0.999480i $$-0.489730\pi$$
0.0322581 + 0.999480i $$0.489730\pi$$
$$32$$ −33.0000 −1.03125
$$33$$ 0 0
$$34$$ −14.0000 −0.411765
$$35$$ 0 0
$$36$$ −27.0000 −0.750000
$$37$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$38$$ 22.0000 0.578947
$$39$$ 0 0
$$40$$ −35.0000 −0.875000
$$41$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$42$$ 0 0
$$43$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$44$$ 0 0
$$45$$ −45.0000 −1.00000
$$46$$ 34.0000 0.739130
$$47$$ 14.0000 0.297872 0.148936 0.988847i $$-0.452415\pi$$
0.148936 + 0.988847i $$0.452415\pi$$
$$48$$ 15.0000 0.312500
$$49$$ 49.0000 1.00000
$$50$$ −25.0000 −0.500000
$$51$$ 42.0000 0.823529
$$52$$ 0 0
$$53$$ 86.0000 1.62264 0.811321 0.584601i $$-0.198749\pi$$
0.811321 + 0.584601i $$0.198749\pi$$
$$54$$ −27.0000 −0.500000
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −66.0000 −1.15789
$$58$$ 0 0
$$59$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$60$$ 45.0000 0.750000
$$61$$ −118.000 −1.93443 −0.967213 0.253966i $$-0.918265\pi$$
−0.967213 + 0.253966i $$0.918265\pi$$
$$62$$ −2.00000 −0.0322581
$$63$$ 0 0
$$64$$ 13.0000 0.203125
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$68$$ −42.0000 −0.617647
$$69$$ −102.000 −1.47826
$$70$$ 0 0
$$71$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$72$$ 63.0000 0.875000
$$73$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$74$$ 0 0
$$75$$ 75.0000 1.00000
$$76$$ 66.0000 0.868421
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 98.0000 1.24051 0.620253 0.784402i $$-0.287030\pi$$
0.620253 + 0.784402i $$0.287030\pi$$
$$80$$ −25.0000 −0.312500
$$81$$ 81.0000 1.00000
$$82$$ 0 0
$$83$$ −154.000 −1.85542 −0.927711 0.373300i $$-0.878226\pi$$
−0.927711 + 0.373300i $$0.878226\pi$$
$$84$$ 0 0
$$85$$ −70.0000 −0.823529
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$90$$ 45.0000 0.500000
$$91$$ 0 0
$$92$$ 102.000 1.10870
$$93$$ 6.00000 0.0645161
$$94$$ −14.0000 −0.148936
$$95$$ 110.000 1.15789
$$96$$ −99.0000 −1.03125
$$97$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$98$$ −49.0000 −0.500000
$$99$$ 0 0
$$100$$ −75.0000 −0.750000
$$101$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$102$$ −42.0000 −0.411765
$$103$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ −86.0000 −0.811321
$$107$$ −106.000 −0.990654 −0.495327 0.868707i $$-0.664952\pi$$
−0.495327 + 0.868707i $$0.664952\pi$$
$$108$$ −81.0000 −0.750000
$$109$$ −22.0000 −0.201835 −0.100917 0.994895i $$-0.532178\pi$$
−0.100917 + 0.994895i $$0.532178\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 206.000 1.82301 0.911504 0.411290i $$-0.134922\pi$$
0.911504 + 0.411290i $$0.134922\pi$$
$$114$$ 66.0000 0.578947
$$115$$ 170.000 1.47826
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0 0
$$120$$ −105.000 −0.875000
$$121$$ 121.000 1.00000
$$122$$ 118.000 0.967213
$$123$$ 0 0
$$124$$ −6.00000 −0.0483871
$$125$$ −125.000 −1.00000
$$126$$ 0 0
$$127$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$128$$ 119.000 0.929688
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ −135.000 −1.00000
$$136$$ 98.0000 0.720588
$$137$$ −226.000 −1.64964 −0.824818 0.565399i $$-0.808722\pi$$
−0.824818 + 0.565399i $$0.808722\pi$$
$$138$$ 102.000 0.739130
$$139$$ −262.000 −1.88489 −0.942446 0.334358i $$-0.891480\pi$$
−0.942446 + 0.334358i $$0.891480\pi$$
$$140$$ 0 0
$$141$$ 42.0000 0.297872
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 45.0000 0.312500
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 147.000 1.00000
$$148$$ 0 0
$$149$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$150$$ −75.0000 −0.500000
$$151$$ −238.000 −1.57616 −0.788079 0.615574i $$-0.788924\pi$$
−0.788079 + 0.615574i $$0.788924\pi$$
$$152$$ −154.000 −1.01316
$$153$$ 126.000 0.823529
$$154$$ 0 0
$$155$$ −10.0000 −0.0645161
$$156$$ 0 0
$$157$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$158$$ −98.0000 −0.620253
$$159$$ 258.000 1.62264
$$160$$ 165.000 1.03125
$$161$$ 0 0
$$162$$ −81.0000 −0.500000
$$163$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 154.000 0.927711
$$167$$ 254.000 1.52096 0.760479 0.649362i $$-0.224964\pi$$
0.760479 + 0.649362i $$0.224964\pi$$
$$168$$ 0 0
$$169$$ 169.000 1.00000
$$170$$ 70.0000 0.411765
$$171$$ −198.000 −1.15789
$$172$$ 0 0
$$173$$ −154.000 −0.890173 −0.445087 0.895487i $$-0.646827\pi$$
−0.445087 + 0.895487i $$0.646827\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$180$$ 135.000 0.750000
$$181$$ 122.000 0.674033 0.337017 0.941499i $$-0.390582\pi$$
0.337017 + 0.941499i $$0.390582\pi$$
$$182$$ 0 0
$$183$$ −354.000 −1.93443
$$184$$ −238.000 −1.29348
$$185$$ 0 0
$$186$$ −6.00000 −0.0322581
$$187$$ 0 0
$$188$$ −42.0000 −0.223404
$$189$$ 0 0
$$190$$ −110.000 −0.578947
$$191$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$192$$ 39.0000 0.203125
$$193$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ −147.000 −0.750000
$$197$$ 374.000 1.89848 0.949239 0.314557i $$-0.101856\pi$$
0.949239 + 0.314557i $$0.101856\pi$$
$$198$$ 0 0
$$199$$ −142.000 −0.713568 −0.356784 0.934187i $$-0.616127\pi$$
−0.356784 + 0.934187i $$0.616127\pi$$
$$200$$ 175.000 0.875000
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 0 0
$$204$$ −126.000 −0.617647
$$205$$ 0 0
$$206$$ 0 0
$$207$$ −306.000 −1.47826
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 362.000 1.71564 0.857820 0.513950i $$-0.171818\pi$$
0.857820 + 0.513950i $$0.171818\pi$$
$$212$$ −258.000 −1.21698
$$213$$ 0 0
$$214$$ 106.000 0.495327
$$215$$ 0 0
$$216$$ 189.000 0.875000
$$217$$ 0 0
$$218$$ 22.0000 0.100917
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$224$$ 0 0
$$225$$ 225.000 1.00000
$$226$$ −206.000 −0.911504
$$227$$ 134.000 0.590308 0.295154 0.955450i $$-0.404629\pi$$
0.295154 + 0.955450i $$0.404629\pi$$
$$228$$ 198.000 0.868421
$$229$$ 218.000 0.951965 0.475983 0.879455i $$-0.342093\pi$$
0.475983 + 0.879455i $$0.342093\pi$$
$$230$$ −170.000 −0.739130
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −34.0000 −0.145923 −0.0729614 0.997335i $$-0.523245\pi$$
−0.0729614 + 0.997335i $$0.523245\pi$$
$$234$$ 0 0
$$235$$ −70.0000 −0.297872
$$236$$ 0 0
$$237$$ 294.000 1.24051
$$238$$ 0 0
$$239$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$240$$ −75.0000 −0.312500
$$241$$ −478.000 −1.98340 −0.991701 0.128564i $$-0.958963\pi$$
−0.991701 + 0.128564i $$0.958963\pi$$
$$242$$ −121.000 −0.500000
$$243$$ 243.000 1.00000
$$244$$ 354.000 1.45082
$$245$$ −245.000 −1.00000
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 14.0000 0.0564516
$$249$$ −462.000 −1.85542
$$250$$ 125.000 0.500000
$$251$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ −210.000 −0.823529
$$256$$ −171.000 −0.667969
$$257$$ −466.000 −1.81323 −0.906615 0.421959i $$-0.861343\pi$$
−0.906615 + 0.421959i $$0.861343\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 446.000 1.69582 0.847909 0.530142i $$-0.177861\pi$$
0.847909 + 0.530142i $$0.177861\pi$$
$$264$$ 0 0
$$265$$ −430.000 −1.62264
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$270$$ 135.000 0.500000
$$271$$ 482.000 1.77860 0.889299 0.457326i $$-0.151193\pi$$
0.889299 + 0.457326i $$0.151193\pi$$
$$272$$ 70.0000 0.257353
$$273$$ 0 0
$$274$$ 226.000 0.824818
$$275$$ 0 0
$$276$$ 306.000 1.10870
$$277$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$278$$ 262.000 0.942446
$$279$$ 18.0000 0.0645161
$$280$$ 0 0
$$281$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$282$$ −42.0000 −0.148936
$$283$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$284$$ 0 0
$$285$$ 330.000 1.15789
$$286$$ 0 0
$$287$$ 0 0
$$288$$ −297.000 −1.03125
$$289$$ −93.0000 −0.321799
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −394.000 −1.34471 −0.672355 0.740229i $$-0.734717\pi$$
−0.672355 + 0.740229i $$0.734717\pi$$
$$294$$ −147.000 −0.500000
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 0 0
$$300$$ −225.000 −0.750000
$$301$$ 0 0
$$302$$ 238.000 0.788079
$$303$$ 0 0
$$304$$ −110.000 −0.361842
$$305$$ 590.000 1.93443
$$306$$ −126.000 −0.411765
$$307$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 10.0000 0.0322581
$$311$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$312$$ 0 0
$$313$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ −294.000 −0.930380
$$317$$ 134.000 0.422713 0.211356 0.977409i $$-0.432212\pi$$
0.211356 + 0.977409i $$0.432212\pi$$
$$318$$ −258.000 −0.811321
$$319$$ 0 0
$$320$$ −65.0000 −0.203125
$$321$$ −318.000 −0.990654
$$322$$ 0 0
$$323$$ −308.000 −0.953560
$$324$$ −243.000 −0.750000
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −66.0000 −0.201835
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 122.000 0.368580 0.184290 0.982872i $$-0.441001\pi$$
0.184290 + 0.982872i $$0.441001\pi$$
$$332$$ 462.000 1.39157
$$333$$ 0 0
$$334$$ −254.000 −0.760479
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$338$$ −169.000 −0.500000
$$339$$ 618.000 1.82301
$$340$$ 210.000 0.617647
$$341$$ 0 0
$$342$$ 198.000 0.578947
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 510.000 1.47826
$$346$$ 154.000 0.445087
$$347$$ −586.000 −1.68876 −0.844380 0.535744i $$-0.820031\pi$$
−0.844380 + 0.535744i $$0.820031\pi$$
$$348$$ 0 0
$$349$$ 458.000 1.31232 0.656160 0.754621i $$-0.272179\pi$$
0.656160 + 0.754621i $$0.272179\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −274.000 −0.776204 −0.388102 0.921616i $$-0.626869\pi$$
−0.388102 + 0.921616i $$0.626869\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$360$$ −315.000 −0.875000
$$361$$ 123.000 0.340720
$$362$$ −122.000 −0.337017
$$363$$ 363.000 1.00000
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 354.000 0.967213
$$367$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$368$$ −170.000 −0.461957
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ −18.0000 −0.0483871
$$373$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$374$$ 0 0
$$375$$ −375.000 −1.00000
$$376$$ 98.0000 0.260638
$$377$$ 0 0
$$378$$ 0 0
$$379$$ −742.000 −1.95778 −0.978892 0.204379i $$-0.934482\pi$$
−0.978892 + 0.204379i $$0.934482\pi$$
$$380$$ −330.000 −0.868421
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 686.000 1.79112 0.895561 0.444938i $$-0.146774\pi$$
0.895561 + 0.444938i $$0.146774\pi$$
$$384$$ 357.000 0.929688
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$390$$ 0 0
$$391$$ −476.000 −1.21739
$$392$$ 343.000 0.875000
$$393$$ 0 0
$$394$$ −374.000 −0.949239
$$395$$ −490.000 −1.24051
$$396$$ 0 0
$$397$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$398$$ 142.000 0.356784
$$399$$ 0 0
$$400$$ 125.000 0.312500
$$401$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ −405.000 −1.00000
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 294.000 0.720588
$$409$$ −142.000 −0.347188 −0.173594 0.984817i $$-0.555538\pi$$
−0.173594 + 0.984817i $$0.555538\pi$$
$$410$$ 0 0
$$411$$ −678.000 −1.64964
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 306.000 0.739130
$$415$$ 770.000 1.85542
$$416$$ 0 0
$$417$$ −786.000 −1.88489
$$418$$ 0 0
$$419$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$420$$ 0 0
$$421$$ 602.000 1.42993 0.714964 0.699161i $$-0.246443\pi$$
0.714964 + 0.699161i $$0.246443\pi$$
$$422$$ −362.000 −0.857820
$$423$$ 126.000 0.297872
$$424$$ 602.000 1.41981
$$425$$ 350.000 0.823529
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 318.000 0.742991
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$432$$ 135.000 0.312500
$$433$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 66.0000 0.151376
$$437$$ 748.000 1.71167
$$438$$ 0 0
$$439$$ −622.000 −1.41686 −0.708428 0.705783i $$-0.750595\pi$$
−0.708428 + 0.705783i $$0.750595\pi$$
$$440$$ 0 0
$$441$$ 441.000 1.00000
$$442$$ 0 0
$$443$$ 566.000 1.27765 0.638826 0.769351i $$-0.279420\pi$$
0.638826 + 0.769351i $$0.279420\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$450$$ −225.000 −0.500000
$$451$$ 0 0
$$452$$ −618.000 −1.36726
$$453$$ −714.000 −1.57616
$$454$$ −134.000 −0.295154
$$455$$ 0 0
$$456$$ −462.000 −1.01316
$$457$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$458$$ −218.000 −0.475983
$$459$$ 378.000 0.823529
$$460$$ −510.000 −1.10870
$$461$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$462$$ 0 0
$$463$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$464$$ 0 0
$$465$$ −30.0000 −0.0645161
$$466$$ 34.0000 0.0729614
$$467$$ −346.000 −0.740899 −0.370450 0.928853i $$-0.620796\pi$$
−0.370450 + 0.928853i $$0.620796\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 70.0000 0.148936
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 0 0
$$474$$ −294.000 −0.620253
$$475$$ −550.000 −1.15789
$$476$$ 0 0
$$477$$ 774.000 1.62264
$$478$$ 0 0
$$479$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$480$$ 495.000 1.03125
$$481$$ 0 0
$$482$$ 478.000 0.991701
$$483$$ 0 0
$$484$$ −363.000 −0.750000
$$485$$ 0 0
$$486$$ −243.000 −0.500000
$$487$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$488$$ −826.000 −1.69262
$$489$$ 0 0
$$490$$ 245.000 0.500000
$$491$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 10.0000 0.0201613
$$497$$ 0 0
$$498$$ 462.000 0.927711
$$499$$ 938.000 1.87976 0.939880 0.341506i $$-0.110937\pi$$
0.939880 + 0.341506i $$0.110937\pi$$
$$500$$ 375.000 0.750000
$$501$$ 762.000 1.52096
$$502$$ 0 0
$$503$$ −994.000 −1.97614 −0.988072 0.153995i $$-0.950786\pi$$
−0.988072 + 0.153995i $$0.950786\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 507.000 1.00000
$$508$$ 0 0
$$509$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$510$$ 210.000 0.411765
$$511$$ 0 0
$$512$$ −305.000 −0.595703
$$513$$ −594.000 −1.15789
$$514$$ 466.000 0.906615
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ −462.000 −0.890173
$$520$$ 0 0
$$521$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$522$$ 0 0
$$523$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ −446.000 −0.847909
$$527$$ 28.0000 0.0531309
$$528$$ 0 0
$$529$$ 627.000 1.18526
$$530$$ 430.000 0.811321
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 530.000 0.990654
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 405.000 0.750000
$$541$$ −1078.00 −1.99261 −0.996303 0.0859072i $$-0.972621\pi$$
−0.996303 + 0.0859072i $$0.972621\pi$$
$$542$$ −482.000 −0.889299
$$543$$ 366.000 0.674033
$$544$$ −462.000 −0.849265
$$545$$ 110.000 0.201835
$$546$$ 0 0
$$547$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$548$$ 678.000 1.23723
$$549$$ −1062.00 −1.93443
$$550$$ 0 0
$$551$$ 0 0
$$552$$ −714.000 −1.29348
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 786.000 1.41367
$$557$$ 614.000 1.10233 0.551167 0.834395i $$-0.314183\pi$$
0.551167 + 0.834395i $$0.314183\pi$$
$$558$$ −18.0000 −0.0322581
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −154.000 −0.273535 −0.136767 0.990603i $$-0.543671\pi$$
−0.136767 + 0.990603i $$0.543671\pi$$
$$564$$ −126.000 −0.223404
$$565$$ −1030.00 −1.82301
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$570$$ −330.000 −0.578947
$$571$$ −358.000 −0.626970 −0.313485 0.949593i $$-0.601497\pi$$
−0.313485 + 0.949593i $$0.601497\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −850.000 −1.47826
$$576$$ 117.000 0.203125
$$577$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$578$$ 93.0000 0.160900
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 394.000 0.672355
$$587$$ 854.000 1.45486 0.727428 0.686184i $$-0.240716\pi$$
0.727428 + 0.686184i $$0.240716\pi$$
$$588$$ −441.000 −0.750000
$$589$$ −44.0000 −0.0747029
$$590$$ 0 0
$$591$$ 1122.00 1.89848
$$592$$ 0 0
$$593$$ 1166.00 1.96627 0.983137 0.182873i $$-0.0585396\pi$$
0.983137 + 0.182873i $$0.0585396\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −426.000 −0.713568
$$598$$ 0 0
$$599$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$600$$ 525.000 0.875000
$$601$$ 242.000 0.402662 0.201331 0.979523i $$-0.435473\pi$$
0.201331 + 0.979523i $$0.435473\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 714.000 1.18212
$$605$$ −605.000 −1.00000
$$606$$ 0 0
$$607$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$608$$ 726.000 1.19408
$$609$$ 0 0
$$610$$ −590.000 −0.967213
$$611$$ 0 0
$$612$$ −378.000 −0.617647
$$613$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −1186.00 −1.92220 −0.961102 0.276193i $$-0.910927\pi$$
−0.961102 + 0.276193i $$0.910927\pi$$
$$618$$ 0 0
$$619$$ 698.000 1.12763 0.563813 0.825903i $$-0.309334\pi$$
0.563813 + 0.825903i $$0.309334\pi$$
$$620$$ 30.0000 0.0483871
$$621$$ −918.000 −1.47826
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 625.000 1.00000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ −238.000 −0.377179 −0.188590 0.982056i $$-0.560392\pi$$
−0.188590 + 0.982056i $$0.560392\pi$$
$$632$$ 686.000 1.08544
$$633$$ 1086.00 1.71564
$$634$$ −134.000 −0.211356
$$635$$ 0 0
$$636$$ −774.000 −1.21698
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 0 0
$$640$$ −595.000 −0.929688
$$641$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$642$$ 318.000 0.495327
$$643$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 308.000 0.476780
$$647$$ −706.000 −1.09119 −0.545595 0.838049i $$-0.683696\pi$$
−0.545595 + 0.838049i $$0.683696\pi$$
$$648$$ 567.000 0.875000
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −1114.00 −1.70597 −0.852986 0.521933i $$-0.825211\pi$$
−0.852986 + 0.521933i $$0.825211\pi$$
$$654$$ 66.0000 0.100917
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$660$$ 0 0
$$661$$ −838.000 −1.26778 −0.633888 0.773425i $$-0.718542\pi$$
−0.633888 + 0.773425i $$0.718542\pi$$
$$662$$ −122.000 −0.184290
$$663$$ 0 0
$$664$$ −1078.00 −1.62349
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ −762.000 −1.14072
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$674$$ 0 0
$$675$$ 675.000 1.00000
$$676$$ −507.000 −0.750000
$$677$$ 374.000 0.552437 0.276219 0.961095i $$-0.410919\pi$$
0.276219 + 0.961095i $$0.410919\pi$$
$$678$$ −618.000 −0.911504
$$679$$ 0 0
$$680$$ −490.000 −0.720588
$$681$$ 402.000 0.590308
$$682$$ 0 0
$$683$$ 86.0000 0.125915 0.0629575 0.998016i $$-0.479947\pi$$
0.0629575 + 0.998016i $$0.479947\pi$$
$$684$$ 594.000 0.868421
$$685$$ 1130.00 1.64964
$$686$$ 0 0
$$687$$ 654.000 0.951965
$$688$$ 0 0
$$689$$ 0 0
$$690$$ −510.000 −0.739130
$$691$$ 1322.00 1.91317 0.956585 0.291455i $$-0.0941392\pi$$
0.956585 + 0.291455i $$0.0941392\pi$$
$$692$$ 462.000 0.667630
$$693$$ 0 0
$$694$$ 586.000 0.844380
$$695$$ 1310.00 1.88489
$$696$$ 0 0
$$697$$ 0 0
$$698$$ −458.000 −0.656160
$$699$$ −102.000 −0.145923
$$700$$ 0 0
$$701$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 0 0
$$705$$ −210.000 −0.297872
$$706$$ 274.000 0.388102
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −742.000 −1.04654 −0.523272 0.852166i $$-0.675289\pi$$
−0.523272 + 0.852166i $$0.675289\pi$$
$$710$$ 0 0
$$711$$ 882.000 1.24051
$$712$$ 0 0
$$713$$ −68.0000 −0.0953717
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$720$$ −225.000 −0.312500
$$721$$ 0 0
$$722$$ −123.000 −0.170360
$$723$$ −1434.00 −1.98340
$$724$$ −366.000 −0.505525
$$725$$ 0 0
$$726$$ −363.000 −0.500000
$$727$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$728$$ 0 0
$$729$$ 729.000 1.00000
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 1062.00 1.45082
$$733$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$734$$ 0 0
$$735$$ −735.000 −1.00000
$$736$$ 1122.00 1.52446
$$737$$ 0 0
$$738$$ 0 0
$$739$$ −1462.00 −1.97835 −0.989175 0.146744i $$-0.953121\pi$$
−0.989175 + 0.146744i $$0.953121\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −514.000 −0.691790 −0.345895 0.938273i $$-0.612425\pi$$
−0.345895 + 0.938273i $$0.612425\pi$$
$$744$$ 42.0000 0.0564516
$$745$$ 0 0
$$746$$ 0 0
$$747$$ −1386.00 −1.85542
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 375.000 0.500000
$$751$$ −1438.00 −1.91478 −0.957390 0.288798i $$-0.906745\pi$$
−0.957390 + 0.288798i $$0.906745\pi$$
$$752$$ 70.0000 0.0930851
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 1190.00 1.57616
$$756$$ 0 0
$$757$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$758$$ 742.000 0.978892
$$759$$ 0 0
$$760$$ 770.000 1.01316
$$761$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ −630.000 −0.823529
$$766$$ −686.000 −0.895561
$$767$$ 0 0
$$768$$ −513.000 −0.667969
$$769$$ 578.000 0.751625 0.375813 0.926696i $$-0.377364\pi$$
0.375813 + 0.926696i $$0.377364\pi$$
$$770$$ 0 0
$$771$$ −1398.00 −1.81323
$$772$$ 0 0
$$773$$ 1526.00 1.97413 0.987063 0.160330i $$-0.0512560\pi$$
0.987063 + 0.160330i $$0.0512560\pi$$
$$774$$ 0 0
$$775$$ 50.0000 0.0645161
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 476.000 0.608696
$$783$$ 0 0
$$784$$ 245.000 0.312500
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$788$$ −1122.00 −1.42386
$$789$$ 1338.00 1.69582
$$790$$ 490.000 0.620253
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ −1290.00 −1.62264
$$796$$ 426.000 0.535176
$$797$$ −826.000 −1.03639 −0.518193 0.855264i $$-0.673395\pi$$
−0.518193 + 0.855264i $$0.673395\pi$$
$$798$$ 0 0
$$799$$ 196.000 0.245307
$$800$$ −825.000 −1.03125
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$810$$ 405.000 0.500000
$$811$$ 1082.00 1.33416 0.667078 0.744988i $$-0.267545\pi$$
0.667078 + 0.744988i $$0.267545\pi$$
$$812$$ 0 0
$$813$$ 1446.00 1.77860
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 210.000 0.257353
$$817$$ 0 0
$$818$$ 142.000 0.173594
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$822$$ 678.000 0.824818
$$823$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 374.000 0.452237 0.226119 0.974100i $$-0.427396\pi$$
0.226119 + 0.974100i $$0.427396\pi$$
$$828$$ 918.000 1.10870
$$829$$ −502.000 −0.605549 −0.302774 0.953062i $$-0.597913\pi$$
−0.302774 + 0.953062i $$0.597913\pi$$
$$830$$ −770.000 −0.927711
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 686.000 0.823529
$$834$$ 786.000 0.942446
$$835$$ −1270.00 −1.52096
$$836$$ 0 0
$$837$$ 54.0000 0.0645161
$$838$$ 0 0
$$839$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$840$$ 0 0
$$841$$ 841.000 1.00000
$$842$$ −602.000 −0.714964
$$843$$ 0 0
$$844$$ −1086.00 −1.28673
$$845$$ −845.000 −1.00000
$$846$$ −126.000 −0.148936
$$847$$ 0 0
$$848$$ 430.000 0.507075
$$849$$ 0 0
$$850$$ −350.000 −0.411765
$$851$$ 0 0
$$852$$ 0 0
$$853$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$854$$ 0 0
$$855$$ 990.000 1.15789
$$856$$ −742.000 −0.866822
$$857$$ −1666.00 −1.94399 −0.971995 0.235000i $$-0.924491\pi$$
−0.971995 + 0.235000i $$0.924491\pi$$
$$858$$ 0 0
$$859$$ 218.000 0.253783 0.126892 0.991917i $$-0.459500\pi$$
0.126892 + 0.991917i $$0.459500\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −274.000 −0.317497 −0.158749 0.987319i $$-0.550746\pi$$
−0.158749 + 0.987319i $$0.550746\pi$$
$$864$$ −891.000 −1.03125
$$865$$ 770.000 0.890173
$$866$$ 0 0
$$867$$ −279.000 −0.321799
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 0 0
$$872$$ −154.000 −0.176606
$$873$$ 0 0
$$874$$ −748.000 −0.855835
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$878$$ 622.000 0.708428
$$879$$ −1182.00 −1.34471
$$880$$ 0 0
$$881$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$882$$ −441.000 −0.500000
$$883$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ −566.000 −0.638826
$$887$$ 1694.00 1.90981 0.954904 0.296914i $$-0.0959575\pi$$
0.954904 + 0.296914i $$0.0959575\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −308.000 −0.344905
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 0 0
$$900$$ −675.000 −0.750000
$$901$$ 1204.00 1.33629
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 1442.00 1.59513
$$905$$ −610.000 −0.674033
$$906$$ 714.000 0.788079
$$907$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$908$$ −402.000 −0.442731
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$912$$ −330.000 −0.361842
$$913$$ 0 0
$$914$$ 0 0
$$915$$ 1770.00 1.93443
$$916$$ −654.000 −0.713974
$$917$$ 0 0
$$918$$ −378.000 −0.411765
$$919$$ 1298.00 1.41240 0.706202 0.708010i $$-0.250407\pi$$
0.706202 + 0.708010i $$0.250407\pi$$
$$920$$ 1190.00 1.29348
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$930$$ 30.0000 0.0322581
$$931$$ −1078.00 −1.15789
$$932$$ 102.000 0.109442
$$933$$ 0 0
$$934$$ 346.000 0.370450
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 210.000 0.223404
$$941$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 1574.00 1.66209 0.831045 0.556205i $$-0.187743\pi$$
0.831045 + 0.556205i $$0.187743\pi$$
$$948$$ −882.000 −0.930380
$$949$$ 0 0
$$950$$ 550.000 0.578947
$$951$$ 402.000 0.422713
$$952$$ 0 0
$$953$$ −1474.00 −1.54669 −0.773347 0.633983i $$-0.781419\pi$$
−0.773347 + 0.633983i $$0.781419\pi$$
$$954$$ −774.000 −0.811321
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 0 0
$$960$$ −195.000 −0.203125
$$961$$ −957.000 −0.995838
$$962$$ 0 0
$$963$$ −954.000 −0.990654
$$964$$ 1434.00 1.48755
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$968$$ 847.000 0.875000
$$969$$ −924.000 −0.953560
$$970$$ 0 0
$$971$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$972$$ −729.000 −0.750000
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 0 0
$$976$$ −590.000 −0.604508
$$977$$ 1934.00 1.97953 0.989765 0.142710i $$-0.0455815\pi$$
0.989765 + 0.142710i $$0.0455815\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 735.000 0.750000
$$981$$ −198.000 −0.201835
$$982$$ 0 0
$$983$$ −1954.00 −1.98779 −0.993896 0.110319i $$-0.964813\pi$$
−0.993896 + 0.110319i $$0.964813\pi$$
$$984$$ 0 0
$$985$$ −1870.00 −1.89848
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ −958.000 −0.966700 −0.483350 0.875427i $$-0.660580\pi$$
−0.483350 + 0.875427i $$0.660580\pi$$
$$992$$ −66.0000 −0.0665323
$$993$$ 366.000 0.368580
$$994$$ 0 0
$$995$$ 710.000 0.713568
$$996$$ 1386.00 1.39157
$$997$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$998$$ −938.000 −0.939880
$$999$$ 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 15.3.d.a.14.1 1
3.2 odd 2 15.3.d.b.14.1 yes 1
4.3 odd 2 240.3.c.a.209.1 1
5.2 odd 4 75.3.c.d.26.1 2
5.3 odd 4 75.3.c.d.26.2 2
5.4 even 2 15.3.d.b.14.1 yes 1
8.3 odd 2 960.3.c.d.449.1 1
8.5 even 2 960.3.c.b.449.1 1
9.2 odd 6 405.3.h.a.134.1 2
9.4 even 3 405.3.h.b.269.1 2
9.5 odd 6 405.3.h.a.269.1 2
9.7 even 3 405.3.h.b.134.1 2
12.11 even 2 240.3.c.b.209.1 1
15.2 even 4 75.3.c.d.26.2 2
15.8 even 4 75.3.c.d.26.1 2
15.14 odd 2 CM 15.3.d.a.14.1 1
20.3 even 4 1200.3.l.l.401.1 2
20.7 even 4 1200.3.l.l.401.2 2
20.19 odd 2 240.3.c.b.209.1 1
24.5 odd 2 960.3.c.c.449.1 1
24.11 even 2 960.3.c.a.449.1 1
40.19 odd 2 960.3.c.a.449.1 1
40.29 even 2 960.3.c.c.449.1 1
45.4 even 6 405.3.h.a.269.1 2
45.14 odd 6 405.3.h.b.269.1 2
45.29 odd 6 405.3.h.b.134.1 2
45.34 even 6 405.3.h.a.134.1 2
60.23 odd 4 1200.3.l.l.401.2 2
60.47 odd 4 1200.3.l.l.401.1 2
60.59 even 2 240.3.c.a.209.1 1
120.29 odd 2 960.3.c.b.449.1 1
120.59 even 2 960.3.c.d.449.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
15.3.d.a.14.1 1 1.1 even 1 trivial
15.3.d.a.14.1 1 15.14 odd 2 CM
15.3.d.b.14.1 yes 1 3.2 odd 2
15.3.d.b.14.1 yes 1 5.4 even 2
75.3.c.d.26.1 2 5.2 odd 4
75.3.c.d.26.1 2 15.8 even 4
75.3.c.d.26.2 2 5.3 odd 4
75.3.c.d.26.2 2 15.2 even 4
240.3.c.a.209.1 1 4.3 odd 2
240.3.c.a.209.1 1 60.59 even 2
240.3.c.b.209.1 1 12.11 even 2
240.3.c.b.209.1 1 20.19 odd 2
405.3.h.a.134.1 2 9.2 odd 6
405.3.h.a.134.1 2 45.34 even 6
405.3.h.a.269.1 2 9.5 odd 6
405.3.h.a.269.1 2 45.4 even 6
405.3.h.b.134.1 2 9.7 even 3
405.3.h.b.134.1 2 45.29 odd 6
405.3.h.b.269.1 2 9.4 even 3
405.3.h.b.269.1 2 45.14 odd 6
960.3.c.a.449.1 1 24.11 even 2
960.3.c.a.449.1 1 40.19 odd 2
960.3.c.b.449.1 1 8.5 even 2
960.3.c.b.449.1 1 120.29 odd 2
960.3.c.c.449.1 1 24.5 odd 2
960.3.c.c.449.1 1 40.29 even 2
960.3.c.d.449.1 1 8.3 odd 2
960.3.c.d.449.1 1 120.59 even 2
1200.3.l.l.401.1 2 20.3 even 4
1200.3.l.l.401.1 2 60.47 odd 4
1200.3.l.l.401.2 2 20.7 even 4
1200.3.l.l.401.2 2 60.23 odd 4