# Properties

 Label 930.2.a.p.1.1 Level $930$ Weight $2$ Character 930.1 Self dual yes Analytic conductor $7.426$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$930 = 2 \cdot 3 \cdot 5 \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 930.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$7.42608738798$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.56155$$ of defining polynomial Character $$\chi$$ $$=$$ 930.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -1.56155 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -1.56155 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -1.56155 q^{11} -1.00000 q^{12} +2.00000 q^{13} -1.56155 q^{14} -1.00000 q^{15} +1.00000 q^{16} +5.12311 q^{17} +1.00000 q^{18} +4.68466 q^{19} +1.00000 q^{20} +1.56155 q^{21} -1.56155 q^{22} +5.56155 q^{23} -1.00000 q^{24} +1.00000 q^{25} +2.00000 q^{26} -1.00000 q^{27} -1.56155 q^{28} -1.12311 q^{29} -1.00000 q^{30} +1.00000 q^{31} +1.00000 q^{32} +1.56155 q^{33} +5.12311 q^{34} -1.56155 q^{35} +1.00000 q^{36} +5.12311 q^{37} +4.68466 q^{38} -2.00000 q^{39} +1.00000 q^{40} -1.12311 q^{41} +1.56155 q^{42} -7.80776 q^{43} -1.56155 q^{44} +1.00000 q^{45} +5.56155 q^{46} -3.12311 q^{47} -1.00000 q^{48} -4.56155 q^{49} +1.00000 q^{50} -5.12311 q^{51} +2.00000 q^{52} +11.5616 q^{53} -1.00000 q^{54} -1.56155 q^{55} -1.56155 q^{56} -4.68466 q^{57} -1.12311 q^{58} -4.87689 q^{59} -1.00000 q^{60} +6.00000 q^{61} +1.00000 q^{62} -1.56155 q^{63} +1.00000 q^{64} +2.00000 q^{65} +1.56155 q^{66} +9.36932 q^{67} +5.12311 q^{68} -5.56155 q^{69} -1.56155 q^{70} +4.68466 q^{71} +1.00000 q^{72} +9.80776 q^{73} +5.12311 q^{74} -1.00000 q^{75} +4.68466 q^{76} +2.43845 q^{77} -2.00000 q^{78} -16.6847 q^{79} +1.00000 q^{80} +1.00000 q^{81} -1.12311 q^{82} -2.24621 q^{83} +1.56155 q^{84} +5.12311 q^{85} -7.80776 q^{86} +1.12311 q^{87} -1.56155 q^{88} -1.31534 q^{89} +1.00000 q^{90} -3.12311 q^{91} +5.56155 q^{92} -1.00000 q^{93} -3.12311 q^{94} +4.68466 q^{95} -1.00000 q^{96} -6.00000 q^{97} -4.56155 q^{98} -1.56155 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{6} + q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^5 - 2 * q^6 + q^7 + 2 * q^8 + 2 * q^9 $$2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{6} + q^{7} + 2 q^{8} + 2 q^{9} + 2 q^{10} + q^{11} - 2 q^{12} + 4 q^{13} + q^{14} - 2 q^{15} + 2 q^{16} + 2 q^{17} + 2 q^{18} - 3 q^{19} + 2 q^{20} - q^{21} + q^{22} + 7 q^{23} - 2 q^{24} + 2 q^{25} + 4 q^{26} - 2 q^{27} + q^{28} + 6 q^{29} - 2 q^{30} + 2 q^{31} + 2 q^{32} - q^{33} + 2 q^{34} + q^{35} + 2 q^{36} + 2 q^{37} - 3 q^{38} - 4 q^{39} + 2 q^{40} + 6 q^{41} - q^{42} + 5 q^{43} + q^{44} + 2 q^{45} + 7 q^{46} + 2 q^{47} - 2 q^{48} - 5 q^{49} + 2 q^{50} - 2 q^{51} + 4 q^{52} + 19 q^{53} - 2 q^{54} + q^{55} + q^{56} + 3 q^{57} + 6 q^{58} - 18 q^{59} - 2 q^{60} + 12 q^{61} + 2 q^{62} + q^{63} + 2 q^{64} + 4 q^{65} - q^{66} - 6 q^{67} + 2 q^{68} - 7 q^{69} + q^{70} - 3 q^{71} + 2 q^{72} - q^{73} + 2 q^{74} - 2 q^{75} - 3 q^{76} + 9 q^{77} - 4 q^{78} - 21 q^{79} + 2 q^{80} + 2 q^{81} + 6 q^{82} + 12 q^{83} - q^{84} + 2 q^{85} + 5 q^{86} - 6 q^{87} + q^{88} - 15 q^{89} + 2 q^{90} + 2 q^{91} + 7 q^{92} - 2 q^{93} + 2 q^{94} - 3 q^{95} - 2 q^{96} - 12 q^{97} - 5 q^{98} + q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^5 - 2 * q^6 + q^7 + 2 * q^8 + 2 * q^9 + 2 * q^10 + q^11 - 2 * q^12 + 4 * q^13 + q^14 - 2 * q^15 + 2 * q^16 + 2 * q^17 + 2 * q^18 - 3 * q^19 + 2 * q^20 - q^21 + q^22 + 7 * q^23 - 2 * q^24 + 2 * q^25 + 4 * q^26 - 2 * q^27 + q^28 + 6 * q^29 - 2 * q^30 + 2 * q^31 + 2 * q^32 - q^33 + 2 * q^34 + q^35 + 2 * q^36 + 2 * q^37 - 3 * q^38 - 4 * q^39 + 2 * q^40 + 6 * q^41 - q^42 + 5 * q^43 + q^44 + 2 * q^45 + 7 * q^46 + 2 * q^47 - 2 * q^48 - 5 * q^49 + 2 * q^50 - 2 * q^51 + 4 * q^52 + 19 * q^53 - 2 * q^54 + q^55 + q^56 + 3 * q^57 + 6 * q^58 - 18 * q^59 - 2 * q^60 + 12 * q^61 + 2 * q^62 + q^63 + 2 * q^64 + 4 * q^65 - q^66 - 6 * q^67 + 2 * q^68 - 7 * q^69 + q^70 - 3 * q^71 + 2 * q^72 - q^73 + 2 * q^74 - 2 * q^75 - 3 * q^76 + 9 * q^77 - 4 * q^78 - 21 * q^79 + 2 * q^80 + 2 * q^81 + 6 * q^82 + 12 * q^83 - q^84 + 2 * q^85 + 5 * q^86 - 6 * q^87 + q^88 - 15 * q^89 + 2 * q^90 + 2 * q^91 + 7 * q^92 - 2 * q^93 + 2 * q^94 - 3 * q^95 - 2 * q^96 - 12 * q^97 - 5 * q^98 + q^99

## 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
$$3$$ −1.00000 −0.577350
$$4$$ 1.00000 0.500000
$$5$$ 1.00000 0.447214
$$6$$ −1.00000 −0.408248
$$7$$ −1.56155 −0.590211 −0.295106 0.955465i $$-0.595355\pi$$
−0.295106 + 0.955465i $$0.595355\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 1.00000 0.333333
$$10$$ 1.00000 0.316228
$$11$$ −1.56155 −0.470826 −0.235413 0.971895i $$-0.575644\pi$$
−0.235413 + 0.971895i $$0.575644\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ 2.00000 0.554700 0.277350 0.960769i $$-0.410544\pi$$
0.277350 + 0.960769i $$0.410544\pi$$
$$14$$ −1.56155 −0.417343
$$15$$ −1.00000 −0.258199
$$16$$ 1.00000 0.250000
$$17$$ 5.12311 1.24254 0.621268 0.783598i $$-0.286618\pi$$
0.621268 + 0.783598i $$0.286618\pi$$
$$18$$ 1.00000 0.235702
$$19$$ 4.68466 1.07473 0.537367 0.843348i $$-0.319419\pi$$
0.537367 + 0.843348i $$0.319419\pi$$
$$20$$ 1.00000 0.223607
$$21$$ 1.56155 0.340759
$$22$$ −1.56155 −0.332924
$$23$$ 5.56155 1.15966 0.579832 0.814736i $$-0.303118\pi$$
0.579832 + 0.814736i $$0.303118\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ 1.00000 0.200000
$$26$$ 2.00000 0.392232
$$27$$ −1.00000 −0.192450
$$28$$ −1.56155 −0.295106
$$29$$ −1.12311 −0.208555 −0.104278 0.994548i $$-0.533253\pi$$
−0.104278 + 0.994548i $$0.533253\pi$$
$$30$$ −1.00000 −0.182574
$$31$$ 1.00000 0.179605
$$32$$ 1.00000 0.176777
$$33$$ 1.56155 0.271831
$$34$$ 5.12311 0.878605
$$35$$ −1.56155 −0.263951
$$36$$ 1.00000 0.166667
$$37$$ 5.12311 0.842233 0.421117 0.907006i $$-0.361638\pi$$
0.421117 + 0.907006i $$0.361638\pi$$
$$38$$ 4.68466 0.759952
$$39$$ −2.00000 −0.320256
$$40$$ 1.00000 0.158114
$$41$$ −1.12311 −0.175400 −0.0876998 0.996147i $$-0.527952\pi$$
−0.0876998 + 0.996147i $$0.527952\pi$$
$$42$$ 1.56155 0.240953
$$43$$ −7.80776 −1.19067 −0.595336 0.803477i $$-0.702981\pi$$
−0.595336 + 0.803477i $$0.702981\pi$$
$$44$$ −1.56155 −0.235413
$$45$$ 1.00000 0.149071
$$46$$ 5.56155 0.820006
$$47$$ −3.12311 −0.455552 −0.227776 0.973714i $$-0.573145\pi$$
−0.227776 + 0.973714i $$0.573145\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ −4.56155 −0.651650
$$50$$ 1.00000 0.141421
$$51$$ −5.12311 −0.717378
$$52$$ 2.00000 0.277350
$$53$$ 11.5616 1.58810 0.794051 0.607852i $$-0.207968\pi$$
0.794051 + 0.607852i $$0.207968\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ −1.56155 −0.210560
$$56$$ −1.56155 −0.208671
$$57$$ −4.68466 −0.620498
$$58$$ −1.12311 −0.147471
$$59$$ −4.87689 −0.634918 −0.317459 0.948272i $$-0.602830\pi$$
−0.317459 + 0.948272i $$0.602830\pi$$
$$60$$ −1.00000 −0.129099
$$61$$ 6.00000 0.768221 0.384111 0.923287i $$-0.374508\pi$$
0.384111 + 0.923287i $$0.374508\pi$$
$$62$$ 1.00000 0.127000
$$63$$ −1.56155 −0.196737
$$64$$ 1.00000 0.125000
$$65$$ 2.00000 0.248069
$$66$$ 1.56155 0.192214
$$67$$ 9.36932 1.14464 0.572322 0.820029i $$-0.306043\pi$$
0.572322 + 0.820029i $$0.306043\pi$$
$$68$$ 5.12311 0.621268
$$69$$ −5.56155 −0.669532
$$70$$ −1.56155 −0.186641
$$71$$ 4.68466 0.555967 0.277983 0.960586i $$-0.410334\pi$$
0.277983 + 0.960586i $$0.410334\pi$$
$$72$$ 1.00000 0.117851
$$73$$ 9.80776 1.14791 0.573956 0.818886i $$-0.305408\pi$$
0.573956 + 0.818886i $$0.305408\pi$$
$$74$$ 5.12311 0.595549
$$75$$ −1.00000 −0.115470
$$76$$ 4.68466 0.537367
$$77$$ 2.43845 0.277887
$$78$$ −2.00000 −0.226455
$$79$$ −16.6847 −1.87717 −0.938585 0.345047i $$-0.887863\pi$$
−0.938585 + 0.345047i $$0.887863\pi$$
$$80$$ 1.00000 0.111803
$$81$$ 1.00000 0.111111
$$82$$ −1.12311 −0.124026
$$83$$ −2.24621 −0.246554 −0.123277 0.992372i $$-0.539340\pi$$
−0.123277 + 0.992372i $$0.539340\pi$$
$$84$$ 1.56155 0.170379
$$85$$ 5.12311 0.555679
$$86$$ −7.80776 −0.841933
$$87$$ 1.12311 0.120410
$$88$$ −1.56155 −0.166462
$$89$$ −1.31534 −0.139426 −0.0697130 0.997567i $$-0.522208\pi$$
−0.0697130 + 0.997567i $$0.522208\pi$$
$$90$$ 1.00000 0.105409
$$91$$ −3.12311 −0.327390
$$92$$ 5.56155 0.579832
$$93$$ −1.00000 −0.103695
$$94$$ −3.12311 −0.322124
$$95$$ 4.68466 0.480636
$$96$$ −1.00000 −0.102062
$$97$$ −6.00000 −0.609208 −0.304604 0.952479i $$-0.598524\pi$$
−0.304604 + 0.952479i $$0.598524\pi$$
$$98$$ −4.56155 −0.460786
$$99$$ −1.56155 −0.156942
$$100$$ 1.00000 0.100000
$$101$$ 3.56155 0.354388 0.177194 0.984176i $$-0.443298\pi$$
0.177194 + 0.984176i $$0.443298\pi$$
$$102$$ −5.12311 −0.507263
$$103$$ −18.2462 −1.79785 −0.898926 0.438100i $$-0.855652\pi$$
−0.898926 + 0.438100i $$0.855652\pi$$
$$104$$ 2.00000 0.196116
$$105$$ 1.56155 0.152392
$$106$$ 11.5616 1.12296
$$107$$ 1.56155 0.150961 0.0754805 0.997147i $$-0.475951\pi$$
0.0754805 + 0.997147i $$0.475951\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ −19.3693 −1.85524 −0.927622 0.373520i $$-0.878151\pi$$
−0.927622 + 0.373520i $$0.878151\pi$$
$$110$$ −1.56155 −0.148888
$$111$$ −5.12311 −0.486264
$$112$$ −1.56155 −0.147553
$$113$$ 10.6847 1.00513 0.502564 0.864540i $$-0.332390\pi$$
0.502564 + 0.864540i $$0.332390\pi$$
$$114$$ −4.68466 −0.438758
$$115$$ 5.56155 0.518617
$$116$$ −1.12311 −0.104278
$$117$$ 2.00000 0.184900
$$118$$ −4.87689 −0.448955
$$119$$ −8.00000 −0.733359
$$120$$ −1.00000 −0.0912871
$$121$$ −8.56155 −0.778323
$$122$$ 6.00000 0.543214
$$123$$ 1.12311 0.101267
$$124$$ 1.00000 0.0898027
$$125$$ 1.00000 0.0894427
$$126$$ −1.56155 −0.139114
$$127$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 7.80776 0.687435
$$130$$ 2.00000 0.175412
$$131$$ 9.36932 0.818601 0.409301 0.912400i $$-0.365773\pi$$
0.409301 + 0.912400i $$0.365773\pi$$
$$132$$ 1.56155 0.135916
$$133$$ −7.31534 −0.634321
$$134$$ 9.36932 0.809386
$$135$$ −1.00000 −0.0860663
$$136$$ 5.12311 0.439303
$$137$$ 3.75379 0.320708 0.160354 0.987060i $$-0.448736\pi$$
0.160354 + 0.987060i $$0.448736\pi$$
$$138$$ −5.56155 −0.473431
$$139$$ −8.87689 −0.752928 −0.376464 0.926431i $$-0.622860\pi$$
−0.376464 + 0.926431i $$0.622860\pi$$
$$140$$ −1.56155 −0.131975
$$141$$ 3.12311 0.263013
$$142$$ 4.68466 0.393128
$$143$$ −3.12311 −0.261167
$$144$$ 1.00000 0.0833333
$$145$$ −1.12311 −0.0932688
$$146$$ 9.80776 0.811696
$$147$$ 4.56155 0.376231
$$148$$ 5.12311 0.421117
$$149$$ −20.0540 −1.64289 −0.821443 0.570291i $$-0.806830\pi$$
−0.821443 + 0.570291i $$0.806830\pi$$
$$150$$ −1.00000 −0.0816497
$$151$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$152$$ 4.68466 0.379976
$$153$$ 5.12311 0.414179
$$154$$ 2.43845 0.196496
$$155$$ 1.00000 0.0803219
$$156$$ −2.00000 −0.160128
$$157$$ −20.0540 −1.60048 −0.800241 0.599679i $$-0.795295\pi$$
−0.800241 + 0.599679i $$0.795295\pi$$
$$158$$ −16.6847 −1.32736
$$159$$ −11.5616 −0.916891
$$160$$ 1.00000 0.0790569
$$161$$ −8.68466 −0.684447
$$162$$ 1.00000 0.0785674
$$163$$ 9.36932 0.733862 0.366931 0.930248i $$-0.380409\pi$$
0.366931 + 0.930248i $$0.380409\pi$$
$$164$$ −1.12311 −0.0876998
$$165$$ 1.56155 0.121567
$$166$$ −2.24621 −0.174340
$$167$$ −2.43845 −0.188693 −0.0943464 0.995539i $$-0.530076\pi$$
−0.0943464 + 0.995539i $$0.530076\pi$$
$$168$$ 1.56155 0.120476
$$169$$ −9.00000 −0.692308
$$170$$ 5.12311 0.392924
$$171$$ 4.68466 0.358245
$$172$$ −7.80776 −0.595336
$$173$$ −8.24621 −0.626948 −0.313474 0.949597i $$-0.601493\pi$$
−0.313474 + 0.949597i $$0.601493\pi$$
$$174$$ 1.12311 0.0851424
$$175$$ −1.56155 −0.118042
$$176$$ −1.56155 −0.117706
$$177$$ 4.87689 0.366570
$$178$$ −1.31534 −0.0985890
$$179$$ −12.0000 −0.896922 −0.448461 0.893802i $$-0.648028\pi$$
−0.448461 + 0.893802i $$0.648028\pi$$
$$180$$ 1.00000 0.0745356
$$181$$ 24.0540 1.78792 0.893959 0.448149i $$-0.147917\pi$$
0.893959 + 0.448149i $$0.147917\pi$$
$$182$$ −3.12311 −0.231500
$$183$$ −6.00000 −0.443533
$$184$$ 5.56155 0.410003
$$185$$ 5.12311 0.376658
$$186$$ −1.00000 −0.0733236
$$187$$ −8.00000 −0.585018
$$188$$ −3.12311 −0.227776
$$189$$ 1.56155 0.113586
$$190$$ 4.68466 0.339861
$$191$$ 2.24621 0.162530 0.0812651 0.996693i $$-0.474104\pi$$
0.0812651 + 0.996693i $$0.474104\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ −18.4924 −1.33111 −0.665557 0.746347i $$-0.731806\pi$$
−0.665557 + 0.746347i $$0.731806\pi$$
$$194$$ −6.00000 −0.430775
$$195$$ −2.00000 −0.143223
$$196$$ −4.56155 −0.325825
$$197$$ 6.00000 0.427482 0.213741 0.976890i $$-0.431435\pi$$
0.213741 + 0.976890i $$0.431435\pi$$
$$198$$ −1.56155 −0.110975
$$199$$ 3.80776 0.269925 0.134963 0.990851i $$-0.456909\pi$$
0.134963 + 0.990851i $$0.456909\pi$$
$$200$$ 1.00000 0.0707107
$$201$$ −9.36932 −0.660861
$$202$$ 3.56155 0.250590
$$203$$ 1.75379 0.123092
$$204$$ −5.12311 −0.358689
$$205$$ −1.12311 −0.0784411
$$206$$ −18.2462 −1.27127
$$207$$ 5.56155 0.386555
$$208$$ 2.00000 0.138675
$$209$$ −7.31534 −0.506013
$$210$$ 1.56155 0.107757
$$211$$ −11.3153 −0.778980 −0.389490 0.921031i $$-0.627349\pi$$
−0.389490 + 0.921031i $$0.627349\pi$$
$$212$$ 11.5616 0.794051
$$213$$ −4.68466 −0.320988
$$214$$ 1.56155 0.106746
$$215$$ −7.80776 −0.532485
$$216$$ −1.00000 −0.0680414
$$217$$ −1.56155 −0.106005
$$218$$ −19.3693 −1.31186
$$219$$ −9.80776 −0.662747
$$220$$ −1.56155 −0.105280
$$221$$ 10.2462 0.689235
$$222$$ −5.12311 −0.343840
$$223$$ −12.8769 −0.862301 −0.431150 0.902280i $$-0.641892\pi$$
−0.431150 + 0.902280i $$0.641892\pi$$
$$224$$ −1.56155 −0.104336
$$225$$ 1.00000 0.0666667
$$226$$ 10.6847 0.710733
$$227$$ −4.68466 −0.310932 −0.155466 0.987841i $$-0.549688\pi$$
−0.155466 + 0.987841i $$0.549688\pi$$
$$228$$ −4.68466 −0.310249
$$229$$ −12.4384 −0.821956 −0.410978 0.911645i $$-0.634813\pi$$
−0.410978 + 0.911645i $$0.634813\pi$$
$$230$$ 5.56155 0.366718
$$231$$ −2.43845 −0.160438
$$232$$ −1.12311 −0.0737355
$$233$$ 20.0540 1.31378 0.656890 0.753987i $$-0.271872\pi$$
0.656890 + 0.753987i $$0.271872\pi$$
$$234$$ 2.00000 0.130744
$$235$$ −3.12311 −0.203729
$$236$$ −4.87689 −0.317459
$$237$$ 16.6847 1.08379
$$238$$ −8.00000 −0.518563
$$239$$ 24.0000 1.55243 0.776215 0.630468i $$-0.217137\pi$$
0.776215 + 0.630468i $$0.217137\pi$$
$$240$$ −1.00000 −0.0645497
$$241$$ −4.24621 −0.273523 −0.136761 0.990604i $$-0.543669\pi$$
−0.136761 + 0.990604i $$0.543669\pi$$
$$242$$ −8.56155 −0.550357
$$243$$ −1.00000 −0.0641500
$$244$$ 6.00000 0.384111
$$245$$ −4.56155 −0.291427
$$246$$ 1.12311 0.0716066
$$247$$ 9.36932 0.596155
$$248$$ 1.00000 0.0635001
$$249$$ 2.24621 0.142348
$$250$$ 1.00000 0.0632456
$$251$$ −16.4924 −1.04099 −0.520496 0.853864i $$-0.674253\pi$$
−0.520496 + 0.853864i $$0.674253\pi$$
$$252$$ −1.56155 −0.0983686
$$253$$ −8.68466 −0.546000
$$254$$ 0 0
$$255$$ −5.12311 −0.320821
$$256$$ 1.00000 0.0625000
$$257$$ 10.6847 0.666491 0.333245 0.942840i $$-0.391856\pi$$
0.333245 + 0.942840i $$0.391856\pi$$
$$258$$ 7.80776 0.486090
$$259$$ −8.00000 −0.497096
$$260$$ 2.00000 0.124035
$$261$$ −1.12311 −0.0695185
$$262$$ 9.36932 0.578838
$$263$$ 12.4924 0.770316 0.385158 0.922851i $$-0.374147\pi$$
0.385158 + 0.922851i $$0.374147\pi$$
$$264$$ 1.56155 0.0961069
$$265$$ 11.5616 0.710221
$$266$$ −7.31534 −0.448532
$$267$$ 1.31534 0.0804976
$$268$$ 9.36932 0.572322
$$269$$ 16.2462 0.990549 0.495274 0.868737i $$-0.335067\pi$$
0.495274 + 0.868737i $$0.335067\pi$$
$$270$$ −1.00000 −0.0608581
$$271$$ −10.0540 −0.610736 −0.305368 0.952234i $$-0.598779\pi$$
−0.305368 + 0.952234i $$0.598779\pi$$
$$272$$ 5.12311 0.310634
$$273$$ 3.12311 0.189019
$$274$$ 3.75379 0.226775
$$275$$ −1.56155 −0.0941652
$$276$$ −5.56155 −0.334766
$$277$$ 19.3693 1.16379 0.581895 0.813264i $$-0.302312\pi$$
0.581895 + 0.813264i $$0.302312\pi$$
$$278$$ −8.87689 −0.532401
$$279$$ 1.00000 0.0598684
$$280$$ −1.56155 −0.0933206
$$281$$ 13.1231 0.782859 0.391429 0.920208i $$-0.371981\pi$$
0.391429 + 0.920208i $$0.371981\pi$$
$$282$$ 3.12311 0.185978
$$283$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$284$$ 4.68466 0.277983
$$285$$ −4.68466 −0.277495
$$286$$ −3.12311 −0.184673
$$287$$ 1.75379 0.103523
$$288$$ 1.00000 0.0589256
$$289$$ 9.24621 0.543895
$$290$$ −1.12311 −0.0659510
$$291$$ 6.00000 0.351726
$$292$$ 9.80776 0.573956
$$293$$ −6.49242 −0.379291 −0.189646 0.981853i $$-0.560734\pi$$
−0.189646 + 0.981853i $$0.560734\pi$$
$$294$$ 4.56155 0.266035
$$295$$ −4.87689 −0.283944
$$296$$ 5.12311 0.297774
$$297$$ 1.56155 0.0906105
$$298$$ −20.0540 −1.16170
$$299$$ 11.1231 0.643266
$$300$$ −1.00000 −0.0577350
$$301$$ 12.1922 0.702749
$$302$$ 0 0
$$303$$ −3.56155 −0.204606
$$304$$ 4.68466 0.268684
$$305$$ 6.00000 0.343559
$$306$$ 5.12311 0.292868
$$307$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$308$$ 2.43845 0.138943
$$309$$ 18.2462 1.03799
$$310$$ 1.00000 0.0567962
$$311$$ −18.2462 −1.03465 −0.517324 0.855790i $$-0.673072\pi$$
−0.517324 + 0.855790i $$0.673072\pi$$
$$312$$ −2.00000 −0.113228
$$313$$ −10.0000 −0.565233 −0.282617 0.959233i $$-0.591202\pi$$
−0.282617 + 0.959233i $$0.591202\pi$$
$$314$$ −20.0540 −1.13171
$$315$$ −1.56155 −0.0879835
$$316$$ −16.6847 −0.938585
$$317$$ 25.1231 1.41105 0.705527 0.708683i $$-0.250710\pi$$
0.705527 + 0.708683i $$0.250710\pi$$
$$318$$ −11.5616 −0.648340
$$319$$ 1.75379 0.0981933
$$320$$ 1.00000 0.0559017
$$321$$ −1.56155 −0.0871574
$$322$$ −8.68466 −0.483977
$$323$$ 24.0000 1.33540
$$324$$ 1.00000 0.0555556
$$325$$ 2.00000 0.110940
$$326$$ 9.36932 0.518918
$$327$$ 19.3693 1.07113
$$328$$ −1.12311 −0.0620131
$$329$$ 4.87689 0.268872
$$330$$ 1.56155 0.0859607
$$331$$ 10.2462 0.563183 0.281591 0.959534i $$-0.409138\pi$$
0.281591 + 0.959534i $$0.409138\pi$$
$$332$$ −2.24621 −0.123277
$$333$$ 5.12311 0.280744
$$334$$ −2.43845 −0.133426
$$335$$ 9.36932 0.511900
$$336$$ 1.56155 0.0851897
$$337$$ −10.0000 −0.544735 −0.272367 0.962193i $$-0.587807\pi$$
−0.272367 + 0.962193i $$0.587807\pi$$
$$338$$ −9.00000 −0.489535
$$339$$ −10.6847 −0.580311
$$340$$ 5.12311 0.277839
$$341$$ −1.56155 −0.0845628
$$342$$ 4.68466 0.253317
$$343$$ 18.0540 0.974823
$$344$$ −7.80776 −0.420966
$$345$$ −5.56155 −0.299424
$$346$$ −8.24621 −0.443319
$$347$$ −2.24621 −0.120583 −0.0602915 0.998181i $$-0.519203\pi$$
−0.0602915 + 0.998181i $$0.519203\pi$$
$$348$$ 1.12311 0.0602048
$$349$$ −19.3693 −1.03682 −0.518408 0.855133i $$-0.673475\pi$$
−0.518408 + 0.855133i $$0.673475\pi$$
$$350$$ −1.56155 −0.0834685
$$351$$ −2.00000 −0.106752
$$352$$ −1.56155 −0.0832310
$$353$$ 16.2462 0.864699 0.432349 0.901706i $$-0.357685\pi$$
0.432349 + 0.901706i $$0.357685\pi$$
$$354$$ 4.87689 0.259204
$$355$$ 4.68466 0.248636
$$356$$ −1.31534 −0.0697130
$$357$$ 8.00000 0.423405
$$358$$ −12.0000 −0.634220
$$359$$ −19.3153 −1.01942 −0.509712 0.860345i $$-0.670248\pi$$
−0.509712 + 0.860345i $$0.670248\pi$$
$$360$$ 1.00000 0.0527046
$$361$$ 2.94602 0.155054
$$362$$ 24.0540 1.26425
$$363$$ 8.56155 0.449365
$$364$$ −3.12311 −0.163695
$$365$$ 9.80776 0.513362
$$366$$ −6.00000 −0.313625
$$367$$ −9.36932 −0.489074 −0.244537 0.969640i $$-0.578636\pi$$
−0.244537 + 0.969640i $$0.578636\pi$$
$$368$$ 5.56155 0.289916
$$369$$ −1.12311 −0.0584665
$$370$$ 5.12311 0.266338
$$371$$ −18.0540 −0.937316
$$372$$ −1.00000 −0.0518476
$$373$$ 6.68466 0.346118 0.173059 0.984911i $$-0.444635\pi$$
0.173059 + 0.984911i $$0.444635\pi$$
$$374$$ −8.00000 −0.413670
$$375$$ −1.00000 −0.0516398
$$376$$ −3.12311 −0.161062
$$377$$ −2.24621 −0.115686
$$378$$ 1.56155 0.0803176
$$379$$ 30.0540 1.54377 0.771885 0.635763i $$-0.219314\pi$$
0.771885 + 0.635763i $$0.219314\pi$$
$$380$$ 4.68466 0.240318
$$381$$ 0 0
$$382$$ 2.24621 0.114926
$$383$$ 6.24621 0.319166 0.159583 0.987184i $$-0.448985\pi$$
0.159583 + 0.987184i $$0.448985\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 2.43845 0.124275
$$386$$ −18.4924 −0.941240
$$387$$ −7.80776 −0.396891
$$388$$ −6.00000 −0.304604
$$389$$ 24.2462 1.22933 0.614666 0.788788i $$-0.289291\pi$$
0.614666 + 0.788788i $$0.289291\pi$$
$$390$$ −2.00000 −0.101274
$$391$$ 28.4924 1.44092
$$392$$ −4.56155 −0.230393
$$393$$ −9.36932 −0.472620
$$394$$ 6.00000 0.302276
$$395$$ −16.6847 −0.839496
$$396$$ −1.56155 −0.0784710
$$397$$ −28.0540 −1.40799 −0.703994 0.710206i $$-0.748602\pi$$
−0.703994 + 0.710206i $$0.748602\pi$$
$$398$$ 3.80776 0.190866
$$399$$ 7.31534 0.366225
$$400$$ 1.00000 0.0500000
$$401$$ −1.31534 −0.0656850 −0.0328425 0.999461i $$-0.510456\pi$$
−0.0328425 + 0.999461i $$0.510456\pi$$
$$402$$ −9.36932 −0.467299
$$403$$ 2.00000 0.0996271
$$404$$ 3.56155 0.177194
$$405$$ 1.00000 0.0496904
$$406$$ 1.75379 0.0870391
$$407$$ −8.00000 −0.396545
$$408$$ −5.12311 −0.253632
$$409$$ −12.6307 −0.624547 −0.312274 0.949992i $$-0.601091\pi$$
−0.312274 + 0.949992i $$0.601091\pi$$
$$410$$ −1.12311 −0.0554662
$$411$$ −3.75379 −0.185161
$$412$$ −18.2462 −0.898926
$$413$$ 7.61553 0.374736
$$414$$ 5.56155 0.273335
$$415$$ −2.24621 −0.110262
$$416$$ 2.00000 0.0980581
$$417$$ 8.87689 0.434703
$$418$$ −7.31534 −0.357805
$$419$$ 22.2462 1.08680 0.543399 0.839474i $$-0.317137\pi$$
0.543399 + 0.839474i $$0.317137\pi$$
$$420$$ 1.56155 0.0761960
$$421$$ 18.4924 0.901266 0.450633 0.892709i $$-0.351198\pi$$
0.450633 + 0.892709i $$0.351198\pi$$
$$422$$ −11.3153 −0.550822
$$423$$ −3.12311 −0.151851
$$424$$ 11.5616 0.561479
$$425$$ 5.12311 0.248507
$$426$$ −4.68466 −0.226972
$$427$$ −9.36932 −0.453413
$$428$$ 1.56155 0.0754805
$$429$$ 3.12311 0.150785
$$430$$ −7.80776 −0.376524
$$431$$ −13.7538 −0.662497 −0.331248 0.943544i $$-0.607470\pi$$
−0.331248 + 0.943544i $$0.607470\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ −18.3002 −0.879451 −0.439725 0.898132i $$-0.644924\pi$$
−0.439725 + 0.898132i $$0.644924\pi$$
$$434$$ −1.56155 −0.0749569
$$435$$ 1.12311 0.0538488
$$436$$ −19.3693 −0.927622
$$437$$ 26.0540 1.24633
$$438$$ −9.80776 −0.468633
$$439$$ −31.6155 −1.50893 −0.754463 0.656342i $$-0.772103\pi$$
−0.754463 + 0.656342i $$0.772103\pi$$
$$440$$ −1.56155 −0.0744441
$$441$$ −4.56155 −0.217217
$$442$$ 10.2462 0.487363
$$443$$ −14.0540 −0.667725 −0.333862 0.942622i $$-0.608352\pi$$
−0.333862 + 0.942622i $$0.608352\pi$$
$$444$$ −5.12311 −0.243132
$$445$$ −1.31534 −0.0623532
$$446$$ −12.8769 −0.609739
$$447$$ 20.0540 0.948520
$$448$$ −1.56155 −0.0737764
$$449$$ −22.4924 −1.06148 −0.530742 0.847534i $$-0.678087\pi$$
−0.530742 + 0.847534i $$0.678087\pi$$
$$450$$ 1.00000 0.0471405
$$451$$ 1.75379 0.0825827
$$452$$ 10.6847 0.502564
$$453$$ 0 0
$$454$$ −4.68466 −0.219862
$$455$$ −3.12311 −0.146413
$$456$$ −4.68466 −0.219379
$$457$$ 24.7386 1.15722 0.578612 0.815603i $$-0.303594\pi$$
0.578612 + 0.815603i $$0.303594\pi$$
$$458$$ −12.4384 −0.581210
$$459$$ −5.12311 −0.239126
$$460$$ 5.56155 0.259309
$$461$$ −15.3693 −0.715820 −0.357910 0.933756i $$-0.616511\pi$$
−0.357910 + 0.933756i $$0.616511\pi$$
$$462$$ −2.43845 −0.113447
$$463$$ −18.7386 −0.870858 −0.435429 0.900223i $$-0.643403\pi$$
−0.435429 + 0.900223i $$0.643403\pi$$
$$464$$ −1.12311 −0.0521389
$$465$$ −1.00000 −0.0463739
$$466$$ 20.0540 0.928982
$$467$$ −26.2462 −1.21453 −0.607265 0.794499i $$-0.707733\pi$$
−0.607265 + 0.794499i $$0.707733\pi$$
$$468$$ 2.00000 0.0924500
$$469$$ −14.6307 −0.675582
$$470$$ −3.12311 −0.144058
$$471$$ 20.0540 0.924038
$$472$$ −4.87689 −0.224477
$$473$$ 12.1922 0.560600
$$474$$ 16.6847 0.766352
$$475$$ 4.68466 0.214947
$$476$$ −8.00000 −0.366679
$$477$$ 11.5616 0.529367
$$478$$ 24.0000 1.09773
$$479$$ 6.43845 0.294180 0.147090 0.989123i $$-0.453009\pi$$
0.147090 + 0.989123i $$0.453009\pi$$
$$480$$ −1.00000 −0.0456435
$$481$$ 10.2462 0.467187
$$482$$ −4.24621 −0.193410
$$483$$ 8.68466 0.395166
$$484$$ −8.56155 −0.389161
$$485$$ −6.00000 −0.272446
$$486$$ −1.00000 −0.0453609
$$487$$ 24.0000 1.08754 0.543772 0.839233i $$-0.316996\pi$$
0.543772 + 0.839233i $$0.316996\pi$$
$$488$$ 6.00000 0.271607
$$489$$ −9.36932 −0.423695
$$490$$ −4.56155 −0.206070
$$491$$ 2.93087 0.132268 0.0661341 0.997811i $$-0.478933\pi$$
0.0661341 + 0.997811i $$0.478933\pi$$
$$492$$ 1.12311 0.0506335
$$493$$ −5.75379 −0.259138
$$494$$ 9.36932 0.421545
$$495$$ −1.56155 −0.0701866
$$496$$ 1.00000 0.0449013
$$497$$ −7.31534 −0.328138
$$498$$ 2.24621 0.100655
$$499$$ 20.0000 0.895323 0.447661 0.894203i $$-0.352257\pi$$
0.447661 + 0.894203i $$0.352257\pi$$
$$500$$ 1.00000 0.0447214
$$501$$ 2.43845 0.108942
$$502$$ −16.4924 −0.736093
$$503$$ 9.75379 0.434900 0.217450 0.976071i $$-0.430226\pi$$
0.217450 + 0.976071i $$0.430226\pi$$
$$504$$ −1.56155 −0.0695571
$$505$$ 3.56155 0.158487
$$506$$ −8.68466 −0.386080
$$507$$ 9.00000 0.399704
$$508$$ 0 0
$$509$$ −21.6155 −0.958091 −0.479046 0.877790i $$-0.659017\pi$$
−0.479046 + 0.877790i $$0.659017\pi$$
$$510$$ −5.12311 −0.226855
$$511$$ −15.3153 −0.677511
$$512$$ 1.00000 0.0441942
$$513$$ −4.68466 −0.206833
$$514$$ 10.6847 0.471280
$$515$$ −18.2462 −0.804024
$$516$$ 7.80776 0.343718
$$517$$ 4.87689 0.214486
$$518$$ −8.00000 −0.351500
$$519$$ 8.24621 0.361968
$$520$$ 2.00000 0.0877058
$$521$$ −15.7538 −0.690186 −0.345093 0.938568i $$-0.612153\pi$$
−0.345093 + 0.938568i $$0.612153\pi$$
$$522$$ −1.12311 −0.0491570
$$523$$ 39.4233 1.72386 0.861930 0.507027i $$-0.169256\pi$$
0.861930 + 0.507027i $$0.169256\pi$$
$$524$$ 9.36932 0.409301
$$525$$ 1.56155 0.0681518
$$526$$ 12.4924 0.544696
$$527$$ 5.12311 0.223166
$$528$$ 1.56155 0.0679579
$$529$$ 7.93087 0.344820
$$530$$ 11.5616 0.502202
$$531$$ −4.87689 −0.211639
$$532$$ −7.31534 −0.317160
$$533$$ −2.24621 −0.0972942
$$534$$ 1.31534 0.0569204
$$535$$ 1.56155 0.0675118
$$536$$ 9.36932 0.404693
$$537$$ 12.0000 0.517838
$$538$$ 16.2462 0.700424
$$539$$ 7.12311 0.306814
$$540$$ −1.00000 −0.0430331
$$541$$ 36.2462 1.55835 0.779173 0.626809i $$-0.215639\pi$$
0.779173 + 0.626809i $$0.215639\pi$$
$$542$$ −10.0540 −0.431855
$$543$$ −24.0540 −1.03225
$$544$$ 5.12311 0.219651
$$545$$ −19.3693 −0.829690
$$546$$ 3.12311 0.133657
$$547$$ 35.1231 1.50176 0.750878 0.660441i $$-0.229631\pi$$
0.750878 + 0.660441i $$0.229631\pi$$
$$548$$ 3.75379 0.160354
$$549$$ 6.00000 0.256074
$$550$$ −1.56155 −0.0665848
$$551$$ −5.26137 −0.224142
$$552$$ −5.56155 −0.236715
$$553$$ 26.0540 1.10793
$$554$$ 19.3693 0.822923
$$555$$ −5.12311 −0.217464
$$556$$ −8.87689 −0.376464
$$557$$ −6.19224 −0.262373 −0.131187 0.991358i $$-0.541879\pi$$
−0.131187 + 0.991358i $$0.541879\pi$$
$$558$$ 1.00000 0.0423334
$$559$$ −15.6155 −0.660466
$$560$$ −1.56155 −0.0659877
$$561$$ 8.00000 0.337760
$$562$$ 13.1231 0.553565
$$563$$ −16.4924 −0.695073 −0.347536 0.937667i $$-0.612982\pi$$
−0.347536 + 0.937667i $$0.612982\pi$$
$$564$$ 3.12311 0.131506
$$565$$ 10.6847 0.449507
$$566$$ 0 0
$$567$$ −1.56155 −0.0655791
$$568$$ 4.68466 0.196564
$$569$$ 10.1922 0.427281 0.213640 0.976912i $$-0.431468\pi$$
0.213640 + 0.976912i $$0.431468\pi$$
$$570$$ −4.68466 −0.196219
$$571$$ 32.8769 1.37586 0.687928 0.725779i $$-0.258521\pi$$
0.687928 + 0.725779i $$0.258521\pi$$
$$572$$ −3.12311 −0.130584
$$573$$ −2.24621 −0.0938368
$$574$$ 1.75379 0.0732017
$$575$$ 5.56155 0.231933
$$576$$ 1.00000 0.0416667
$$577$$ 6.87689 0.286289 0.143144 0.989702i $$-0.454279\pi$$
0.143144 + 0.989702i $$0.454279\pi$$
$$578$$ 9.24621 0.384592
$$579$$ 18.4924 0.768519
$$580$$ −1.12311 −0.0466344
$$581$$ 3.50758 0.145519
$$582$$ 6.00000 0.248708
$$583$$ −18.0540 −0.747719
$$584$$ 9.80776 0.405848
$$585$$ 2.00000 0.0826898
$$586$$ −6.49242 −0.268200
$$587$$ 32.4924 1.34111 0.670553 0.741862i $$-0.266057\pi$$
0.670553 + 0.741862i $$0.266057\pi$$
$$588$$ 4.56155 0.188115
$$589$$ 4.68466 0.193028
$$590$$ −4.87689 −0.200779
$$591$$ −6.00000 −0.246807
$$592$$ 5.12311 0.210558
$$593$$ −30.0000 −1.23195 −0.615976 0.787765i $$-0.711238\pi$$
−0.615976 + 0.787765i $$0.711238\pi$$
$$594$$ 1.56155 0.0640713
$$595$$ −8.00000 −0.327968
$$596$$ −20.0540 −0.821443
$$597$$ −3.80776 −0.155841
$$598$$ 11.1231 0.454858
$$599$$ −23.4233 −0.957050 −0.478525 0.878074i $$-0.658828\pi$$
−0.478525 + 0.878074i $$0.658828\pi$$
$$600$$ −1.00000 −0.0408248
$$601$$ 2.00000 0.0815817 0.0407909 0.999168i $$-0.487012\pi$$
0.0407909 + 0.999168i $$0.487012\pi$$
$$602$$ 12.1922 0.496918
$$603$$ 9.36932 0.381548
$$604$$ 0 0
$$605$$ −8.56155 −0.348077
$$606$$ −3.56155 −0.144678
$$607$$ −30.0540 −1.21985 −0.609927 0.792458i $$-0.708801\pi$$
−0.609927 + 0.792458i $$0.708801\pi$$
$$608$$ 4.68466 0.189988
$$609$$ −1.75379 −0.0710671
$$610$$ 6.00000 0.242933
$$611$$ −6.24621 −0.252695
$$612$$ 5.12311 0.207089
$$613$$ 2.00000 0.0807792 0.0403896 0.999184i $$-0.487140\pi$$
0.0403896 + 0.999184i $$0.487140\pi$$
$$614$$ 0 0
$$615$$ 1.12311 0.0452880
$$616$$ 2.43845 0.0982478
$$617$$ −19.5616 −0.787518 −0.393759 0.919214i $$-0.628826\pi$$
−0.393759 + 0.919214i $$0.628826\pi$$
$$618$$ 18.2462 0.733970
$$619$$ −13.3693 −0.537358 −0.268679 0.963230i $$-0.586587\pi$$
−0.268679 + 0.963230i $$0.586587\pi$$
$$620$$ 1.00000 0.0401610
$$621$$ −5.56155 −0.223177
$$622$$ −18.2462 −0.731606
$$623$$ 2.05398 0.0822908
$$624$$ −2.00000 −0.0800641
$$625$$ 1.00000 0.0400000
$$626$$ −10.0000 −0.399680
$$627$$ 7.31534 0.292147
$$628$$ −20.0540 −0.800241
$$629$$ 26.2462 1.04650
$$630$$ −1.56155 −0.0622138
$$631$$ −24.3002 −0.967375 −0.483688 0.875241i $$-0.660703\pi$$
−0.483688 + 0.875241i $$0.660703\pi$$
$$632$$ −16.6847 −0.663680
$$633$$ 11.3153 0.449744
$$634$$ 25.1231 0.997766
$$635$$ 0 0
$$636$$ −11.5616 −0.458445
$$637$$ −9.12311 −0.361471
$$638$$ 1.75379 0.0694332
$$639$$ 4.68466 0.185322
$$640$$ 1.00000 0.0395285
$$641$$ −38.4924 −1.52036 −0.760180 0.649713i $$-0.774889\pi$$
−0.760180 + 0.649713i $$0.774889\pi$$
$$642$$ −1.56155 −0.0616296
$$643$$ 30.0540 1.18521 0.592607 0.805492i $$-0.298099\pi$$
0.592607 + 0.805492i $$0.298099\pi$$
$$644$$ −8.68466 −0.342223
$$645$$ 7.80776 0.307430
$$646$$ 24.0000 0.944267
$$647$$ −17.0691 −0.671057 −0.335528 0.942030i $$-0.608915\pi$$
−0.335528 + 0.942030i $$0.608915\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ 7.61553 0.298936
$$650$$ 2.00000 0.0784465
$$651$$ 1.56155 0.0612021
$$652$$ 9.36932 0.366931
$$653$$ 23.7538 0.929558 0.464779 0.885427i $$-0.346134\pi$$
0.464779 + 0.885427i $$0.346134\pi$$
$$654$$ 19.3693 0.757400
$$655$$ 9.36932 0.366090
$$656$$ −1.12311 −0.0438499
$$657$$ 9.80776 0.382637
$$658$$ 4.87689 0.190121
$$659$$ 18.7386 0.729954 0.364977 0.931017i $$-0.381077\pi$$
0.364977 + 0.931017i $$0.381077\pi$$
$$660$$ 1.56155 0.0607834
$$661$$ −6.49242 −0.252526 −0.126263 0.991997i $$-0.540298\pi$$
−0.126263 + 0.991997i $$0.540298\pi$$
$$662$$ 10.2462 0.398230
$$663$$ −10.2462 −0.397930
$$664$$ −2.24621 −0.0871699
$$665$$ −7.31534 −0.283677
$$666$$ 5.12311 0.198516
$$667$$ −6.24621 −0.241854
$$668$$ −2.43845 −0.0943464
$$669$$ 12.8769 0.497849
$$670$$ 9.36932 0.361968
$$671$$ −9.36932 −0.361698
$$672$$ 1.56155 0.0602382
$$673$$ 28.2462 1.08881 0.544406 0.838822i $$-0.316755\pi$$
0.544406 + 0.838822i $$0.316755\pi$$
$$674$$ −10.0000 −0.385186
$$675$$ −1.00000 −0.0384900
$$676$$ −9.00000 −0.346154
$$677$$ −29.4233 −1.13083 −0.565414 0.824807i $$-0.691284\pi$$
−0.565414 + 0.824807i $$0.691284\pi$$
$$678$$ −10.6847 −0.410342
$$679$$ 9.36932 0.359561
$$680$$ 5.12311 0.196462
$$681$$ 4.68466 0.179517
$$682$$ −1.56155 −0.0597949
$$683$$ 19.3153 0.739081 0.369541 0.929215i $$-0.379515\pi$$
0.369541 + 0.929215i $$0.379515\pi$$
$$684$$ 4.68466 0.179122
$$685$$ 3.75379 0.143425
$$686$$ 18.0540 0.689304
$$687$$ 12.4384 0.474556
$$688$$ −7.80776 −0.297668
$$689$$ 23.1231 0.880920
$$690$$ −5.56155 −0.211725
$$691$$ −22.0540 −0.838973 −0.419486 0.907762i $$-0.637790\pi$$
−0.419486 + 0.907762i $$0.637790\pi$$
$$692$$ −8.24621 −0.313474
$$693$$ 2.43845 0.0926289
$$694$$ −2.24621 −0.0852650
$$695$$ −8.87689 −0.336720
$$696$$ 1.12311 0.0425712
$$697$$ −5.75379 −0.217940
$$698$$ −19.3693 −0.733139
$$699$$ −20.0540 −0.758511
$$700$$ −1.56155 −0.0590211
$$701$$ 25.8078 0.974746 0.487373 0.873194i $$-0.337955\pi$$
0.487373 + 0.873194i $$0.337955\pi$$
$$702$$ −2.00000 −0.0754851
$$703$$ 24.0000 0.905177
$$704$$ −1.56155 −0.0588532
$$705$$ 3.12311 0.117623
$$706$$ 16.2462 0.611434
$$707$$ −5.56155 −0.209164
$$708$$ 4.87689 0.183285
$$709$$ −20.0540 −0.753143 −0.376571 0.926388i $$-0.622897\pi$$
−0.376571 + 0.926388i $$0.622897\pi$$
$$710$$ 4.68466 0.175812
$$711$$ −16.6847 −0.625724
$$712$$ −1.31534 −0.0492945
$$713$$ 5.56155 0.208282
$$714$$ 8.00000 0.299392
$$715$$ −3.12311 −0.116798
$$716$$ −12.0000 −0.448461
$$717$$ −24.0000 −0.896296
$$718$$ −19.3153 −0.720842
$$719$$ −43.1231 −1.60822 −0.804110 0.594480i $$-0.797358\pi$$
−0.804110 + 0.594480i $$0.797358\pi$$
$$720$$ 1.00000 0.0372678
$$721$$ 28.4924 1.06111
$$722$$ 2.94602 0.109640
$$723$$ 4.24621 0.157918
$$724$$ 24.0540 0.893959
$$725$$ −1.12311 −0.0417111
$$726$$ 8.56155 0.317749
$$727$$ −14.0540 −0.521233 −0.260617 0.965442i $$-0.583926\pi$$
−0.260617 + 0.965442i $$0.583926\pi$$
$$728$$ −3.12311 −0.115750
$$729$$ 1.00000 0.0370370
$$730$$ 9.80776 0.363002
$$731$$ −40.0000 −1.47945
$$732$$ −6.00000 −0.221766
$$733$$ 26.4924 0.978520 0.489260 0.872138i $$-0.337267\pi$$
0.489260 + 0.872138i $$0.337267\pi$$
$$734$$ −9.36932 −0.345828
$$735$$ 4.56155 0.168255
$$736$$ 5.56155 0.205002
$$737$$ −14.6307 −0.538928
$$738$$ −1.12311 −0.0413421
$$739$$ −12.9848 −0.477655 −0.238828 0.971062i $$-0.576763\pi$$
−0.238828 + 0.971062i $$0.576763\pi$$
$$740$$ 5.12311 0.188329
$$741$$ −9.36932 −0.344190
$$742$$ −18.0540 −0.662782
$$743$$ 11.4233 0.419080 0.209540 0.977800i $$-0.432803\pi$$
0.209540 + 0.977800i $$0.432803\pi$$
$$744$$ −1.00000 −0.0366618
$$745$$ −20.0540 −0.734721
$$746$$ 6.68466 0.244743
$$747$$ −2.24621 −0.0821846
$$748$$ −8.00000 −0.292509
$$749$$ −2.43845 −0.0890989
$$750$$ −1.00000 −0.0365148
$$751$$ 36.8769 1.34566 0.672828 0.739798i $$-0.265079\pi$$
0.672828 + 0.739798i $$0.265079\pi$$
$$752$$ −3.12311 −0.113888
$$753$$ 16.4924 0.601017
$$754$$ −2.24621 −0.0818022
$$755$$ 0 0
$$756$$ 1.56155 0.0567931
$$757$$ 10.0000 0.363456 0.181728 0.983349i $$-0.441831\pi$$
0.181728 + 0.983349i $$0.441831\pi$$
$$758$$ 30.0540 1.09161
$$759$$ 8.68466 0.315233
$$760$$ 4.68466 0.169930
$$761$$ 41.4233 1.50159 0.750797 0.660533i $$-0.229670\pi$$
0.750797 + 0.660533i $$0.229670\pi$$
$$762$$ 0 0
$$763$$ 30.2462 1.09499
$$764$$ 2.24621 0.0812651
$$765$$ 5.12311 0.185226
$$766$$ 6.24621 0.225685
$$767$$ −9.75379 −0.352189
$$768$$ −1.00000 −0.0360844
$$769$$ −16.4384 −0.592786 −0.296393 0.955066i $$-0.595784\pi$$
−0.296393 + 0.955066i $$0.595784\pi$$
$$770$$ 2.43845 0.0878755
$$771$$ −10.6847 −0.384799
$$772$$ −18.4924 −0.665557
$$773$$ −15.5616 −0.559710 −0.279855 0.960042i $$-0.590286\pi$$
−0.279855 + 0.960042i $$0.590286\pi$$
$$774$$ −7.80776 −0.280644
$$775$$ 1.00000 0.0359211
$$776$$ −6.00000 −0.215387
$$777$$ 8.00000 0.286998
$$778$$ 24.2462 0.869269
$$779$$ −5.26137 −0.188508
$$780$$ −2.00000 −0.0716115
$$781$$ −7.31534 −0.261764
$$782$$ 28.4924 1.01889
$$783$$ 1.12311 0.0401365
$$784$$ −4.56155 −0.162913
$$785$$ −20.0540 −0.715757
$$786$$ −9.36932 −0.334192
$$787$$ 10.9309 0.389643 0.194822 0.980839i $$-0.437587\pi$$
0.194822 + 0.980839i $$0.437587\pi$$
$$788$$ 6.00000 0.213741
$$789$$ −12.4924 −0.444742
$$790$$ −16.6847 −0.593614
$$791$$ −16.6847 −0.593238
$$792$$ −1.56155 −0.0554874
$$793$$ 12.0000 0.426132
$$794$$ −28.0540 −0.995598
$$795$$ −11.5616 −0.410046
$$796$$ 3.80776 0.134963
$$797$$ 38.9848 1.38091 0.690457 0.723373i $$-0.257409\pi$$
0.690457 + 0.723373i $$0.257409\pi$$
$$798$$ 7.31534 0.258960
$$799$$ −16.0000 −0.566039
$$800$$ 1.00000 0.0353553
$$801$$ −1.31534 −0.0464753
$$802$$ −1.31534 −0.0464463
$$803$$ −15.3153 −0.540467
$$804$$ −9.36932 −0.330430
$$805$$ −8.68466 −0.306094
$$806$$ 2.00000 0.0704470
$$807$$ −16.2462 −0.571894
$$808$$ 3.56155 0.125295
$$809$$ −25.3153 −0.890040 −0.445020 0.895521i $$-0.646803\pi$$
−0.445020 + 0.895521i $$0.646803\pi$$
$$810$$ 1.00000 0.0351364
$$811$$ 53.6695 1.88459 0.942296 0.334782i $$-0.108663\pi$$
0.942296 + 0.334782i $$0.108663\pi$$
$$812$$ 1.75379 0.0615459
$$813$$ 10.0540 0.352608
$$814$$ −8.00000 −0.280400
$$815$$ 9.36932 0.328193
$$816$$ −5.12311 −0.179345
$$817$$ −36.5767 −1.27966
$$818$$ −12.6307 −0.441621
$$819$$ −3.12311 −0.109130
$$820$$ −1.12311 −0.0392205
$$821$$ −21.6155 −0.754387 −0.377194 0.926134i $$-0.623111\pi$$
−0.377194 + 0.926134i $$0.623111\pi$$
$$822$$ −3.75379 −0.130928
$$823$$ 15.6155 0.544323 0.272162 0.962252i $$-0.412261\pi$$
0.272162 + 0.962252i $$0.412261\pi$$
$$824$$ −18.2462 −0.635637
$$825$$ 1.56155 0.0543663
$$826$$ 7.61553 0.264978
$$827$$ 18.2462 0.634483 0.317241 0.948345i $$-0.397243\pi$$
0.317241 + 0.948345i $$0.397243\pi$$
$$828$$ 5.56155 0.193277
$$829$$ −37.4233 −1.29976 −0.649882 0.760035i $$-0.725182\pi$$
−0.649882 + 0.760035i $$0.725182\pi$$
$$830$$ −2.24621 −0.0779671
$$831$$ −19.3693 −0.671914
$$832$$ 2.00000 0.0693375
$$833$$ −23.3693 −0.809699
$$834$$ 8.87689 0.307382
$$835$$ −2.43845 −0.0843859
$$836$$ −7.31534 −0.253006
$$837$$ −1.00000 −0.0345651
$$838$$ 22.2462 0.768483
$$839$$ 34.9309 1.20595 0.602974 0.797761i $$-0.293982\pi$$
0.602974 + 0.797761i $$0.293982\pi$$
$$840$$ 1.56155 0.0538787
$$841$$ −27.7386 −0.956505
$$842$$ 18.4924 0.637291
$$843$$ −13.1231 −0.451984
$$844$$ −11.3153 −0.389490
$$845$$ −9.00000 −0.309609
$$846$$ −3.12311 −0.107375
$$847$$ 13.3693 0.459375
$$848$$ 11.5616 0.397025
$$849$$ 0 0
$$850$$ 5.12311 0.175721
$$851$$ 28.4924 0.976708
$$852$$ −4.68466 −0.160494
$$853$$ −38.7926 −1.32823 −0.664117 0.747629i $$-0.731192\pi$$
−0.664117 + 0.747629i $$0.731192\pi$$
$$854$$ −9.36932 −0.320611
$$855$$ 4.68466 0.160212
$$856$$ 1.56155 0.0533728
$$857$$ −53.2311 −1.81834 −0.909169 0.416427i $$-0.863282\pi$$
−0.909169 + 0.416427i $$0.863282\pi$$
$$858$$ 3.12311 0.106621
$$859$$ −22.7386 −0.775832 −0.387916 0.921695i $$-0.626805\pi$$
−0.387916 + 0.921695i $$0.626805\pi$$
$$860$$ −7.80776 −0.266243
$$861$$ −1.75379 −0.0597690
$$862$$ −13.7538 −0.468456
$$863$$ −30.5464 −1.03981 −0.519906 0.854224i $$-0.674033\pi$$
−0.519906 + 0.854224i $$0.674033\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ −8.24621 −0.280380
$$866$$ −18.3002 −0.621866
$$867$$ −9.24621 −0.314018
$$868$$ −1.56155 −0.0530026
$$869$$ 26.0540 0.883821
$$870$$ 1.12311 0.0380768
$$871$$ 18.7386 0.634934
$$872$$ −19.3693 −0.655928
$$873$$ −6.00000 −0.203069
$$874$$ 26.0540 0.881289
$$875$$ −1.56155 −0.0527901
$$876$$ −9.80776 −0.331374
$$877$$ −50.0000 −1.68838 −0.844190 0.536044i $$-0.819918\pi$$
−0.844190 + 0.536044i $$0.819918\pi$$
$$878$$ −31.6155 −1.06697
$$879$$ 6.49242 0.218984
$$880$$ −1.56155 −0.0526399
$$881$$ 42.4924 1.43161 0.715803 0.698302i $$-0.246061\pi$$
0.715803 + 0.698302i $$0.246061\pi$$
$$882$$ −4.56155 −0.153595
$$883$$ −31.4233 −1.05748 −0.528739 0.848784i $$-0.677335\pi$$
−0.528739 + 0.848784i $$0.677335\pi$$
$$884$$ 10.2462 0.344617
$$885$$ 4.87689 0.163935
$$886$$ −14.0540 −0.472153
$$887$$ −57.3693 −1.92627 −0.963137 0.269013i $$-0.913303\pi$$
−0.963137 + 0.269013i $$0.913303\pi$$
$$888$$ −5.12311 −0.171920
$$889$$ 0 0
$$890$$ −1.31534 −0.0440903
$$891$$ −1.56155 −0.0523140
$$892$$ −12.8769 −0.431150
$$893$$ −14.6307 −0.489597
$$894$$ 20.0540 0.670705
$$895$$ −12.0000 −0.401116
$$896$$ −1.56155 −0.0521678
$$897$$ −11.1231 −0.371390
$$898$$ −22.4924 −0.750582
$$899$$ −1.12311 −0.0374577
$$900$$ 1.00000 0.0333333
$$901$$ 59.2311 1.97327
$$902$$ 1.75379 0.0583948
$$903$$ −12.1922 −0.405732
$$904$$ 10.6847 0.355366
$$905$$ 24.0540 0.799581
$$906$$ 0 0
$$907$$ 24.0000 0.796907 0.398453 0.917189i $$-0.369547\pi$$
0.398453 + 0.917189i $$0.369547\pi$$
$$908$$ −4.68466 −0.155466
$$909$$ 3.56155 0.118129
$$910$$ −3.12311 −0.103530
$$911$$ −44.4924 −1.47410 −0.737050 0.675838i $$-0.763782\pi$$
−0.737050 + 0.675838i $$0.763782\pi$$
$$912$$ −4.68466 −0.155125
$$913$$ 3.50758 0.116084
$$914$$ 24.7386 0.818281
$$915$$ −6.00000 −0.198354
$$916$$ −12.4384 −0.410978
$$917$$ −14.6307 −0.483148
$$918$$ −5.12311 −0.169088
$$919$$ 39.6155 1.30680 0.653398 0.757015i $$-0.273343\pi$$
0.653398 + 0.757015i $$0.273343\pi$$
$$920$$ 5.56155 0.183359
$$921$$ 0 0
$$922$$ −15.3693 −0.506161
$$923$$ 9.36932 0.308395
$$924$$ −2.43845 −0.0802190
$$925$$ 5.12311 0.168447
$$926$$ −18.7386 −0.615790
$$927$$ −18.2462 −0.599284
$$928$$ −1.12311 −0.0368677
$$929$$ 11.5616 0.379322 0.189661 0.981850i $$-0.439261\pi$$
0.189661 + 0.981850i $$0.439261\pi$$
$$930$$ −1.00000 −0.0327913
$$931$$ −21.3693 −0.700351
$$932$$ 20.0540 0.656890
$$933$$ 18.2462 0.597354
$$934$$ −26.2462 −0.858802
$$935$$ −8.00000 −0.261628
$$936$$ 2.00000 0.0653720
$$937$$ 16.6307 0.543301 0.271650 0.962396i $$-0.412431\pi$$
0.271650 + 0.962396i $$0.412431\pi$$
$$938$$ −14.6307 −0.477709
$$939$$ 10.0000 0.326338
$$940$$ −3.12311 −0.101864
$$941$$ 27.3693 0.892214 0.446107 0.894980i $$-0.352810\pi$$
0.446107 + 0.894980i $$0.352810\pi$$
$$942$$ 20.0540 0.653394
$$943$$ −6.24621 −0.203405
$$944$$ −4.87689 −0.158729
$$945$$ 1.56155 0.0507973
$$946$$ 12.1922 0.396404
$$947$$ −0.876894 −0.0284952 −0.0142476 0.999898i $$-0.504535\pi$$
−0.0142476 + 0.999898i $$0.504535\pi$$
$$948$$ 16.6847 0.541893
$$949$$ 19.6155 0.636747
$$950$$ 4.68466 0.151990
$$951$$ −25.1231 −0.814673
$$952$$ −8.00000 −0.259281
$$953$$ −58.1080 −1.88230 −0.941151 0.337988i $$-0.890254\pi$$
−0.941151 + 0.337988i $$0.890254\pi$$
$$954$$ 11.5616 0.374319
$$955$$ 2.24621 0.0726857
$$956$$ 24.0000 0.776215
$$957$$ −1.75379 −0.0566919
$$958$$ 6.43845 0.208017
$$959$$ −5.86174 −0.189285
$$960$$ −1.00000 −0.0322749
$$961$$ 1.00000 0.0322581
$$962$$ 10.2462 0.330351
$$963$$ 1.56155 0.0503203
$$964$$ −4.24621 −0.136761
$$965$$ −18.4924 −0.595292
$$966$$ 8.68466 0.279424
$$967$$ −25.7538 −0.828186 −0.414093 0.910235i $$-0.635901\pi$$
−0.414093 + 0.910235i $$0.635901\pi$$
$$968$$ −8.56155 −0.275179
$$969$$ −24.0000 −0.770991
$$970$$ −6.00000 −0.192648
$$971$$ −28.8769 −0.926704 −0.463352 0.886174i $$-0.653353\pi$$
−0.463352 + 0.886174i $$0.653353\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ 13.8617 0.444387
$$974$$ 24.0000 0.769010
$$975$$ −2.00000 −0.0640513
$$976$$ 6.00000 0.192055
$$977$$ −22.9848 −0.735350 −0.367675 0.929954i $$-0.619846\pi$$
−0.367675 + 0.929954i $$0.619846\pi$$
$$978$$ −9.36932 −0.299598
$$979$$ 2.05398 0.0656453
$$980$$ −4.56155 −0.145713
$$981$$ −19.3693 −0.618415
$$982$$ 2.93087 0.0935278
$$983$$ −24.9848 −0.796893 −0.398446 0.917192i $$-0.630451\pi$$
−0.398446 + 0.917192i $$0.630451\pi$$
$$984$$ 1.12311 0.0358033
$$985$$ 6.00000 0.191176
$$986$$ −5.75379 −0.183238
$$987$$ −4.87689 −0.155233
$$988$$ 9.36932 0.298078
$$989$$ −43.4233 −1.38078
$$990$$ −1.56155 −0.0496294
$$991$$ 23.3153 0.740636 0.370318 0.928905i $$-0.379249\pi$$
0.370318 + 0.928905i $$0.379249\pi$$
$$992$$ 1.00000 0.0317500
$$993$$ −10.2462 −0.325154
$$994$$ −7.31534 −0.232029
$$995$$ 3.80776 0.120714
$$996$$ 2.24621 0.0711739
$$997$$ −62.4924 −1.97915 −0.989577 0.144002i $$-0.954003\pi$$
−0.989577 + 0.144002i $$0.954003\pi$$
$$998$$ 20.0000 0.633089
$$999$$ −5.12311 −0.162088
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 930.2.a.p.1.1 2
3.2 odd 2 2790.2.a.be.1.1 2
4.3 odd 2 7440.2.a.bl.1.2 2
5.2 odd 4 4650.2.d.bd.3349.3 4
5.3 odd 4 4650.2.d.bd.3349.2 4
5.4 even 2 4650.2.a.ce.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.a.p.1.1 2 1.1 even 1 trivial
2790.2.a.be.1.1 2 3.2 odd 2
4650.2.a.ce.1.2 2 5.4 even 2
4650.2.d.bd.3349.2 4 5.3 odd 4
4650.2.d.bd.3349.3 4 5.2 odd 4
7440.2.a.bl.1.2 2 4.3 odd 2