# Properties

 Label 4650.2.a.ce.1.2 Level $4650$ Weight $2$ Character 4650.1 Self dual yes Analytic conductor $37.130$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4650.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$37.1304369399$$ Analytic rank: $$1$$ 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: no (minimal twist has level 930) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-1.56155$$ of defining polynomial Character $$\chi$$ $$=$$ 4650.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.56155 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.56155 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.56155 q^{11} +1.00000 q^{12} -2.00000 q^{13} -1.56155 q^{14} +1.00000 q^{16} -5.12311 q^{17} -1.00000 q^{18} +4.68466 q^{19} +1.56155 q^{21} +1.56155 q^{22} -5.56155 q^{23} -1.00000 q^{24} +2.00000 q^{26} +1.00000 q^{27} +1.56155 q^{28} -1.12311 q^{29} +1.00000 q^{31} -1.00000 q^{32} -1.56155 q^{33} +5.12311 q^{34} +1.00000 q^{36} -5.12311 q^{37} -4.68466 q^{38} -2.00000 q^{39} -1.12311 q^{41} -1.56155 q^{42} +7.80776 q^{43} -1.56155 q^{44} +5.56155 q^{46} +3.12311 q^{47} +1.00000 q^{48} -4.56155 q^{49} -5.12311 q^{51} -2.00000 q^{52} -11.5616 q^{53} -1.00000 q^{54} -1.56155 q^{56} +4.68466 q^{57} +1.12311 q^{58} -4.87689 q^{59} +6.00000 q^{61} -1.00000 q^{62} +1.56155 q^{63} +1.00000 q^{64} +1.56155 q^{66} -9.36932 q^{67} -5.12311 q^{68} -5.56155 q^{69} +4.68466 q^{71} -1.00000 q^{72} -9.80776 q^{73} +5.12311 q^{74} +4.68466 q^{76} -2.43845 q^{77} +2.00000 q^{78} -16.6847 q^{79} +1.00000 q^{81} +1.12311 q^{82} +2.24621 q^{83} +1.56155 q^{84} -7.80776 q^{86} -1.12311 q^{87} +1.56155 q^{88} -1.31534 q^{89} -3.12311 q^{91} -5.56155 q^{92} +1.00000 q^{93} -3.12311 q^{94} -1.00000 q^{96} +6.00000 q^{97} +4.56155 q^{98} -1.56155 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 - q^7 - 2 * q^8 + 2 * q^9 $$2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - q^{7} - 2 q^{8} + 2 q^{9} + q^{11} + 2 q^{12} - 4 q^{13} + q^{14} + 2 q^{16} - 2 q^{17} - 2 q^{18} - 3 q^{19} - q^{21} - q^{22} - 7 q^{23} - 2 q^{24} + 4 q^{26} + 2 q^{27} - q^{28} + 6 q^{29} + 2 q^{31} - 2 q^{32} + q^{33} + 2 q^{34} + 2 q^{36} - 2 q^{37} + 3 q^{38} - 4 q^{39} + 6 q^{41} + q^{42} - 5 q^{43} + q^{44} + 7 q^{46} - 2 q^{47} + 2 q^{48} - 5 q^{49} - 2 q^{51} - 4 q^{52} - 19 q^{53} - 2 q^{54} + q^{56} - 3 q^{57} - 6 q^{58} - 18 q^{59} + 12 q^{61} - 2 q^{62} - q^{63} + 2 q^{64} - q^{66} + 6 q^{67} - 2 q^{68} - 7 q^{69} - 3 q^{71} - 2 q^{72} + q^{73} + 2 q^{74} - 3 q^{76} - 9 q^{77} + 4 q^{78} - 21 q^{79} + 2 q^{81} - 6 q^{82} - 12 q^{83} - q^{84} + 5 q^{86} + 6 q^{87} - q^{88} - 15 q^{89} + 2 q^{91} - 7 q^{92} + 2 q^{93} + 2 q^{94} - 2 q^{96} + 12 q^{97} + 5 q^{98} + q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 - q^7 - 2 * q^8 + 2 * q^9 + q^11 + 2 * q^12 - 4 * q^13 + q^14 + 2 * q^16 - 2 * q^17 - 2 * q^18 - 3 * q^19 - q^21 - q^22 - 7 * q^23 - 2 * q^24 + 4 * q^26 + 2 * q^27 - q^28 + 6 * q^29 + 2 * q^31 - 2 * q^32 + q^33 + 2 * q^34 + 2 * q^36 - 2 * q^37 + 3 * q^38 - 4 * q^39 + 6 * q^41 + q^42 - 5 * q^43 + q^44 + 7 * q^46 - 2 * q^47 + 2 * q^48 - 5 * q^49 - 2 * q^51 - 4 * q^52 - 19 * q^53 - 2 * q^54 + q^56 - 3 * q^57 - 6 * q^58 - 18 * q^59 + 12 * q^61 - 2 * q^62 - q^63 + 2 * q^64 - q^66 + 6 * q^67 - 2 * q^68 - 7 * q^69 - 3 * q^71 - 2 * q^72 + q^73 + 2 * q^74 - 3 * q^76 - 9 * q^77 + 4 * q^78 - 21 * q^79 + 2 * q^81 - 6 * q^82 - 12 * q^83 - q^84 + 5 * q^86 + 6 * q^87 - q^88 - 15 * q^89 + 2 * q^91 - 7 * q^92 + 2 * q^93 + 2 * q^94 - 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
$$5$$ 0 0
$$6$$ −1.00000 −0.408248
$$7$$ 1.56155 0.590211 0.295106 0.955465i $$-0.404645\pi$$
0.295106 + 0.955465i $$0.404645\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −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 $$-0.589456\pi$$
−0.277350 + 0.960769i $$0.589456\pi$$
$$14$$ −1.56155 −0.417343
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −5.12311 −1.24254 −0.621268 0.783598i $$-0.713382\pi$$
−0.621268 + 0.783598i $$0.713382\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ 4.68466 1.07473 0.537367 0.843348i $$-0.319419\pi$$
0.537367 + 0.843348i $$0.319419\pi$$
$$20$$ 0 0
$$21$$ 1.56155 0.340759
$$22$$ 1.56155 0.332924
$$23$$ −5.56155 −1.15966 −0.579832 0.814736i $$-0.696882\pi$$
−0.579832 + 0.814736i $$0.696882\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ 0 0
$$26$$ 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 $$0.533253\pi$$
$$30$$ 0 0
$$31$$ 1.00000 0.179605
$$32$$ −1.00000 −0.176777
$$33$$ −1.56155 −0.271831
$$34$$ 5.12311 0.878605
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ −5.12311 −0.842233 −0.421117 0.907006i $$-0.638362\pi$$
−0.421117 + 0.907006i $$0.638362\pi$$
$$38$$ −4.68466 −0.759952
$$39$$ −2.00000 −0.320256
$$40$$ 0 0
$$41$$ −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 $$-0.297019\pi$$
0.595336 + 0.803477i $$0.297019\pi$$
$$44$$ −1.56155 −0.235413
$$45$$ 0 0
$$46$$ 5.56155 0.820006
$$47$$ 3.12311 0.455552 0.227776 0.973714i $$-0.426855\pi$$
0.227776 + 0.973714i $$0.426855\pi$$
$$48$$ 1.00000 0.144338
$$49$$ −4.56155 −0.651650
$$50$$ 0 0
$$51$$ −5.12311 −0.717378
$$52$$ −2.00000 −0.277350
$$53$$ −11.5616 −1.58810 −0.794051 0.607852i $$-0.792032\pi$$
−0.794051 + 0.607852i $$0.792032\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 0 0
$$56$$ −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 $$0.602830\pi$$
$$60$$ 0 0
$$61$$ 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
$$65$$ 0 0
$$66$$ 1.56155 0.192214
$$67$$ −9.36932 −1.14464 −0.572322 0.820029i $$-0.693957\pi$$
−0.572322 + 0.820029i $$0.693957\pi$$
$$68$$ −5.12311 −0.621268
$$69$$ −5.56155 −0.669532
$$70$$ 0 0
$$71$$ 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 $$-0.694592\pi$$
−0.573956 + 0.818886i $$0.694592\pi$$
$$74$$ 5.12311 0.595549
$$75$$ 0 0
$$76$$ 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 $$0.887863\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 1.12311 0.124026
$$83$$ 2.24621 0.246554 0.123277 0.992372i $$-0.460660\pi$$
0.123277 + 0.992372i $$0.460660\pi$$
$$84$$ 1.56155 0.170379
$$85$$ 0 0
$$86$$ −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 $$0.522208\pi$$
$$90$$ 0 0
$$91$$ −3.12311 −0.327390
$$92$$ −5.56155 −0.579832
$$93$$ 1.00000 0.103695
$$94$$ −3.12311 −0.322124
$$95$$ 0 0
$$96$$ −1.00000 −0.102062
$$97$$ 6.00000 0.609208 0.304604 0.952479i $$-0.401476\pi$$
0.304604 + 0.952479i $$0.401476\pi$$
$$98$$ 4.56155 0.460786
$$99$$ −1.56155 −0.156942
$$100$$ 0 0
$$101$$ 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 $$-0.144348\pi$$
0.898926 + 0.438100i $$0.144348\pi$$
$$104$$ 2.00000 0.196116
$$105$$ 0 0
$$106$$ 11.5616 1.12296
$$107$$ −1.56155 −0.150961 −0.0754805 0.997147i $$-0.524049\pi$$
−0.0754805 + 0.997147i $$0.524049\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ −19.3693 −1.85524 −0.927622 0.373520i $$-0.878151\pi$$
−0.927622 + 0.373520i $$0.878151\pi$$
$$110$$ 0 0
$$111$$ −5.12311 −0.486264
$$112$$ 1.56155 0.147553
$$113$$ −10.6847 −1.00513 −0.502564 0.864540i $$-0.667610\pi$$
−0.502564 + 0.864540i $$0.667610\pi$$
$$114$$ −4.68466 −0.438758
$$115$$ 0 0
$$116$$ −1.12311 −0.104278
$$117$$ −2.00000 −0.184900
$$118$$ 4.87689 0.448955
$$119$$ −8.00000 −0.733359
$$120$$ 0 0
$$121$$ −8.56155 −0.778323
$$122$$ −6.00000 −0.543214
$$123$$ −1.12311 −0.101267
$$124$$ 1.00000 0.0898027
$$125$$ 0 0
$$126$$ −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
$$130$$ 0 0
$$131$$ 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
$$135$$ 0 0
$$136$$ 5.12311 0.439303
$$137$$ −3.75379 −0.320708 −0.160354 0.987060i $$-0.551264\pi$$
−0.160354 + 0.987060i $$0.551264\pi$$
$$138$$ 5.56155 0.473431
$$139$$ −8.87689 −0.752928 −0.376464 0.926431i $$-0.622860\pi$$
−0.376464 + 0.926431i $$0.622860\pi$$
$$140$$ 0 0
$$141$$ 3.12311 0.263013
$$142$$ −4.68466 −0.393128
$$143$$ 3.12311 0.261167
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ 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 $$0.806830\pi$$
$$150$$ 0 0
$$151$$ 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
$$155$$ 0 0
$$156$$ −2.00000 −0.160128
$$157$$ 20.0540 1.60048 0.800241 0.599679i $$-0.204705\pi$$
0.800241 + 0.599679i $$0.204705\pi$$
$$158$$ 16.6847 1.32736
$$159$$ −11.5616 −0.916891
$$160$$ 0 0
$$161$$ −8.68466 −0.684447
$$162$$ −1.00000 −0.0785674
$$163$$ −9.36932 −0.733862 −0.366931 0.930248i $$-0.619591\pi$$
−0.366931 + 0.930248i $$0.619591\pi$$
$$164$$ −1.12311 −0.0876998
$$165$$ 0 0
$$166$$ −2.24621 −0.174340
$$167$$ 2.43845 0.188693 0.0943464 0.995539i $$-0.469924\pi$$
0.0943464 + 0.995539i $$0.469924\pi$$
$$168$$ −1.56155 −0.120476
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ 4.68466 0.358245
$$172$$ 7.80776 0.595336
$$173$$ 8.24621 0.626948 0.313474 0.949597i $$-0.398507\pi$$
0.313474 + 0.949597i $$0.398507\pi$$
$$174$$ 1.12311 0.0851424
$$175$$ 0 0
$$176$$ −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 $$0.648028\pi$$
$$180$$ 0 0
$$181$$ 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
$$185$$ 0 0
$$186$$ −1.00000 −0.0733236
$$187$$ 8.00000 0.585018
$$188$$ 3.12311 0.227776
$$189$$ 1.56155 0.113586
$$190$$ 0 0
$$191$$ 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 $$-0.268194\pi$$
0.665557 + 0.746347i $$0.268194\pi$$
$$194$$ −6.00000 −0.430775
$$195$$ 0 0
$$196$$ −4.56155 −0.325825
$$197$$ −6.00000 −0.427482 −0.213741 0.976890i $$-0.568565\pi$$
−0.213741 + 0.976890i $$0.568565\pi$$
$$198$$ 1.56155 0.110975
$$199$$ 3.80776 0.269925 0.134963 0.990851i $$-0.456909\pi$$
0.134963 + 0.990851i $$0.456909\pi$$
$$200$$ 0 0
$$201$$ −9.36932 −0.660861
$$202$$ −3.56155 −0.250590
$$203$$ −1.75379 −0.123092
$$204$$ −5.12311 −0.358689
$$205$$ 0 0
$$206$$ −18.2462 −1.27127
$$207$$ −5.56155 −0.386555
$$208$$ −2.00000 −0.138675
$$209$$ −7.31534 −0.506013
$$210$$ 0 0
$$211$$ −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
$$215$$ 0 0
$$216$$ −1.00000 −0.0680414
$$217$$ 1.56155 0.106005
$$218$$ 19.3693 1.31186
$$219$$ −9.80776 −0.662747
$$220$$ 0 0
$$221$$ 10.2462 0.689235
$$222$$ 5.12311 0.343840
$$223$$ 12.8769 0.862301 0.431150 0.902280i $$-0.358108\pi$$
0.431150 + 0.902280i $$0.358108\pi$$
$$224$$ −1.56155 −0.104336
$$225$$ 0 0
$$226$$ 10.6847 0.710733
$$227$$ 4.68466 0.310932 0.155466 0.987841i $$-0.450312\pi$$
0.155466 + 0.987841i $$0.450312\pi$$
$$228$$ 4.68466 0.310249
$$229$$ −12.4384 −0.821956 −0.410978 0.911645i $$-0.634813\pi$$
−0.410978 + 0.911645i $$0.634813\pi$$
$$230$$ 0 0
$$231$$ −2.43845 −0.160438
$$232$$ 1.12311 0.0737355
$$233$$ −20.0540 −1.31378 −0.656890 0.753987i $$-0.728128\pi$$
−0.656890 + 0.753987i $$0.728128\pi$$
$$234$$ 2.00000 0.130744
$$235$$ 0 0
$$236$$ −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 $$0.217137\pi$$
$$240$$ 0 0
$$241$$ −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
$$245$$ 0 0
$$246$$ 1.12311 0.0716066
$$247$$ −9.36932 −0.596155
$$248$$ −1.00000 −0.0635001
$$249$$ 2.24621 0.142348
$$250$$ 0 0
$$251$$ −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
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −10.6847 −0.666491 −0.333245 0.942840i $$-0.608144\pi$$
−0.333245 + 0.942840i $$0.608144\pi$$
$$258$$ −7.80776 −0.486090
$$259$$ −8.00000 −0.497096
$$260$$ 0 0
$$261$$ −1.12311 −0.0695185
$$262$$ −9.36932 −0.578838
$$263$$ −12.4924 −0.770316 −0.385158 0.922851i $$-0.625853\pi$$
−0.385158 + 0.922851i $$0.625853\pi$$
$$264$$ 1.56155 0.0961069
$$265$$ 0 0
$$266$$ −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 $$0.335067\pi$$
$$270$$ 0 0
$$271$$ −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
$$275$$ 0 0
$$276$$ −5.56155 −0.334766
$$277$$ −19.3693 −1.16379 −0.581895 0.813264i $$-0.697688\pi$$
−0.581895 + 0.813264i $$0.697688\pi$$
$$278$$ 8.87689 0.532401
$$279$$ 1.00000 0.0598684
$$280$$ 0 0
$$281$$ 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
$$285$$ 0 0
$$286$$ −3.12311 −0.184673
$$287$$ −1.75379 −0.103523
$$288$$ −1.00000 −0.0589256
$$289$$ 9.24621 0.543895
$$290$$ 0 0
$$291$$ 6.00000 0.351726
$$292$$ −9.80776 −0.573956
$$293$$ 6.49242 0.379291 0.189646 0.981853i $$-0.439266\pi$$
0.189646 + 0.981853i $$0.439266\pi$$
$$294$$ 4.56155 0.266035
$$295$$ 0 0
$$296$$ 5.12311 0.297774
$$297$$ −1.56155 −0.0906105
$$298$$ 20.0540 1.16170
$$299$$ 11.1231 0.643266
$$300$$ 0 0
$$301$$ 12.1922 0.702749
$$302$$ 0 0
$$303$$ 3.56155 0.204606
$$304$$ 4.68466 0.268684
$$305$$ 0 0
$$306$$ 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
$$310$$ 0 0
$$311$$ −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 $$-0.408798\pi$$
0.282617 + 0.959233i $$0.408798\pi$$
$$314$$ −20.0540 −1.13171
$$315$$ 0 0
$$316$$ −16.6847 −0.938585
$$317$$ −25.1231 −1.41105 −0.705527 0.708683i $$-0.749290\pi$$
−0.705527 + 0.708683i $$0.749290\pi$$
$$318$$ 11.5616 0.648340
$$319$$ 1.75379 0.0981933
$$320$$ 0 0
$$321$$ −1.56155 −0.0871574
$$322$$ 8.68466 0.483977
$$323$$ −24.0000 −1.33540
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ 9.36932 0.518918
$$327$$ −19.3693 −1.07113
$$328$$ 1.12311 0.0620131
$$329$$ 4.87689 0.268872
$$330$$ 0 0
$$331$$ 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
$$335$$ 0 0
$$336$$ 1.56155 0.0851897
$$337$$ 10.0000 0.544735 0.272367 0.962193i $$-0.412193\pi$$
0.272367 + 0.962193i $$0.412193\pi$$
$$338$$ 9.00000 0.489535
$$339$$ −10.6847 −0.580311
$$340$$ 0 0
$$341$$ −1.56155 −0.0845628
$$342$$ −4.68466 −0.253317
$$343$$ −18.0540 −0.974823
$$344$$ −7.80776 −0.420966
$$345$$ 0 0
$$346$$ −8.24621 −0.443319
$$347$$ 2.24621 0.120583 0.0602915 0.998181i $$-0.480797\pi$$
0.0602915 + 0.998181i $$0.480797\pi$$
$$348$$ −1.12311 −0.0602048
$$349$$ −19.3693 −1.03682 −0.518408 0.855133i $$-0.673475\pi$$
−0.518408 + 0.855133i $$0.673475\pi$$
$$350$$ 0 0
$$351$$ −2.00000 −0.106752
$$352$$ 1.56155 0.0832310
$$353$$ −16.2462 −0.864699 −0.432349 0.901706i $$-0.642315\pi$$
−0.432349 + 0.901706i $$0.642315\pi$$
$$354$$ 4.87689 0.259204
$$355$$ 0 0
$$356$$ −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 $$0.670248\pi$$
$$360$$ 0 0
$$361$$ 2.94602 0.155054
$$362$$ −24.0540 −1.26425
$$363$$ −8.56155 −0.449365
$$364$$ −3.12311 −0.163695
$$365$$ 0 0
$$366$$ −6.00000 −0.313625
$$367$$ 9.36932 0.489074 0.244537 0.969640i $$-0.421364\pi$$
0.244537 + 0.969640i $$0.421364\pi$$
$$368$$ −5.56155 −0.289916
$$369$$ −1.12311 −0.0584665
$$370$$ 0 0
$$371$$ −18.0540 −0.937316
$$372$$ 1.00000 0.0518476
$$373$$ −6.68466 −0.346118 −0.173059 0.984911i $$-0.555365\pi$$
−0.173059 + 0.984911i $$0.555365\pi$$
$$374$$ −8.00000 −0.413670
$$375$$ 0 0
$$376$$ −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 $$0.219314\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ −2.24621 −0.114926
$$383$$ −6.24621 −0.319166 −0.159583 0.987184i $$-0.551015\pi$$
−0.159583 + 0.987184i $$0.551015\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ −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 $$0.289291\pi$$
$$390$$ 0 0
$$391$$ 28.4924 1.44092
$$392$$ 4.56155 0.230393
$$393$$ 9.36932 0.472620
$$394$$ 6.00000 0.302276
$$395$$ 0 0
$$396$$ −1.56155 −0.0784710
$$397$$ 28.0540 1.40799 0.703994 0.710206i $$-0.251398\pi$$
0.703994 + 0.710206i $$0.251398\pi$$
$$398$$ −3.80776 −0.190866
$$399$$ 7.31534 0.366225
$$400$$ 0 0
$$401$$ −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
$$405$$ 0 0
$$406$$ 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 $$0.601091\pi$$
$$410$$ 0 0
$$411$$ −3.75379 −0.185161
$$412$$ 18.2462 0.898926
$$413$$ −7.61553 −0.374736
$$414$$ 5.56155 0.273335
$$415$$ 0 0
$$416$$ 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 $$0.317137\pi$$
$$420$$ 0 0
$$421$$ 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
$$425$$ 0 0
$$426$$ −4.68466 −0.226972
$$427$$ 9.36932 0.453413
$$428$$ −1.56155 −0.0754805
$$429$$ 3.12311 0.150785
$$430$$ 0 0
$$431$$ −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 $$-0.355076\pi$$
0.439725 + 0.898132i $$0.355076\pi$$
$$434$$ −1.56155 −0.0749569
$$435$$ 0 0
$$436$$ −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 $$0.772103\pi$$
$$440$$ 0 0
$$441$$ −4.56155 −0.217217
$$442$$ −10.2462 −0.487363
$$443$$ 14.0540 0.667725 0.333862 0.942622i $$-0.391648\pi$$
0.333862 + 0.942622i $$0.391648\pi$$
$$444$$ −5.12311 −0.243132
$$445$$ 0 0
$$446$$ −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 $$0.678087\pi$$
$$450$$ 0 0
$$451$$ 1.75379 0.0825827
$$452$$ −10.6847 −0.502564
$$453$$ 0 0
$$454$$ −4.68466 −0.219862
$$455$$ 0 0
$$456$$ −4.68466 −0.219379
$$457$$ −24.7386 −1.15722 −0.578612 0.815603i $$-0.696406\pi$$
−0.578612 + 0.815603i $$0.696406\pi$$
$$458$$ 12.4384 0.581210
$$459$$ −5.12311 −0.239126
$$460$$ 0 0
$$461$$ −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 $$-0.356597\pi$$
0.435429 + 0.900223i $$0.356597\pi$$
$$464$$ −1.12311 −0.0521389
$$465$$ 0 0
$$466$$ 20.0540 0.928982
$$467$$ 26.2462 1.21453 0.607265 0.794499i $$-0.292267\pi$$
0.607265 + 0.794499i $$0.292267\pi$$
$$468$$ −2.00000 −0.0924500
$$469$$ −14.6307 −0.675582
$$470$$ 0 0
$$471$$ 20.0540 0.924038
$$472$$ 4.87689 0.224477
$$473$$ −12.1922 −0.560600
$$474$$ 16.6847 0.766352
$$475$$ 0 0
$$476$$ −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 $$0.453009\pi$$
$$480$$ 0 0
$$481$$ 10.2462 0.467187
$$482$$ 4.24621 0.193410
$$483$$ −8.68466 −0.395166
$$484$$ −8.56155 −0.389161
$$485$$ 0 0
$$486$$ −1.00000 −0.0453609
$$487$$ −24.0000 −1.08754 −0.543772 0.839233i $$-0.683004\pi$$
−0.543772 + 0.839233i $$0.683004\pi$$
$$488$$ −6.00000 −0.271607
$$489$$ −9.36932 −0.423695
$$490$$ 0 0
$$491$$ 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
$$495$$ 0 0
$$496$$ 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 $$0.352257\pi$$
$$500$$ 0 0
$$501$$ 2.43845 0.108942
$$502$$ 16.4924 0.736093
$$503$$ −9.75379 −0.434900 −0.217450 0.976071i $$-0.569774\pi$$
−0.217450 + 0.976071i $$0.569774\pi$$
$$504$$ −1.56155 −0.0695571
$$505$$ 0 0
$$506$$ −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 $$0.659017\pi$$
$$510$$ 0 0
$$511$$ −15.3153 −0.677511
$$512$$ −1.00000 −0.0441942
$$513$$ 4.68466 0.206833
$$514$$ 10.6847 0.471280
$$515$$ 0 0
$$516$$ 7.80776 0.343718
$$517$$ −4.87689 −0.214486
$$518$$ 8.00000 0.351500
$$519$$ 8.24621 0.361968
$$520$$ 0 0
$$521$$ −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 $$-0.830744\pi$$
−0.861930 + 0.507027i $$0.830744\pi$$
$$524$$ 9.36932 0.409301
$$525$$ 0 0
$$526$$ 12.4924 0.544696
$$527$$ −5.12311 −0.223166
$$528$$ −1.56155 −0.0679579
$$529$$ 7.93087 0.344820
$$530$$ 0 0
$$531$$ −4.87689 −0.211639
$$532$$ 7.31534 0.317160
$$533$$ 2.24621 0.0972942
$$534$$ 1.31534 0.0569204
$$535$$ 0 0
$$536$$ 9.36932 0.404693
$$537$$ −12.0000 −0.517838
$$538$$ −16.2462 −0.700424
$$539$$ 7.12311 0.306814
$$540$$ 0 0
$$541$$ 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
$$545$$ 0 0
$$546$$ 3.12311 0.133657
$$547$$ −35.1231 −1.50176 −0.750878 0.660441i $$-0.770369\pi$$
−0.750878 + 0.660441i $$0.770369\pi$$
$$548$$ −3.75379 −0.160354
$$549$$ 6.00000 0.256074
$$550$$ 0 0
$$551$$ −5.26137 −0.224142
$$552$$ 5.56155 0.236715
$$553$$ −26.0540 −1.10793
$$554$$ 19.3693 0.822923
$$555$$ 0 0
$$556$$ −8.87689 −0.376464
$$557$$ 6.19224 0.262373 0.131187 0.991358i $$-0.458121\pi$$
0.131187 + 0.991358i $$0.458121\pi$$
$$558$$ −1.00000 −0.0423334
$$559$$ −15.6155 −0.660466
$$560$$ 0 0
$$561$$ 8.00000 0.337760
$$562$$ −13.1231 −0.553565
$$563$$ 16.4924 0.695073 0.347536 0.937667i $$-0.387018\pi$$
0.347536 + 0.937667i $$0.387018\pi$$
$$564$$ 3.12311 0.131506
$$565$$ 0 0
$$566$$ 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 $$0.431468\pi$$
$$570$$ 0 0
$$571$$ 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
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ −6.87689 −0.286289 −0.143144 0.989702i $$-0.545721\pi$$
−0.143144 + 0.989702i $$0.545721\pi$$
$$578$$ −9.24621 −0.384592
$$579$$ 18.4924 0.768519
$$580$$ 0 0
$$581$$ 3.50758 0.145519
$$582$$ −6.00000 −0.248708
$$583$$ 18.0540 0.747719
$$584$$ 9.80776 0.405848
$$585$$ 0 0
$$586$$ −6.49242 −0.268200
$$587$$ −32.4924 −1.34111 −0.670553 0.741862i $$-0.733943\pi$$
−0.670553 + 0.741862i $$0.733943\pi$$
$$588$$ −4.56155 −0.188115
$$589$$ 4.68466 0.193028
$$590$$ 0 0
$$591$$ −6.00000 −0.246807
$$592$$ −5.12311 −0.210558
$$593$$ 30.0000 1.23195 0.615976 0.787765i $$-0.288762\pi$$
0.615976 + 0.787765i $$0.288762\pi$$
$$594$$ 1.56155 0.0640713
$$595$$ 0 0
$$596$$ −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 $$0.658828\pi$$
$$600$$ 0 0
$$601$$ 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
$$605$$ 0 0
$$606$$ −3.56155 −0.144678
$$607$$ 30.0540 1.21985 0.609927 0.792458i $$-0.291199\pi$$
0.609927 + 0.792458i $$0.291199\pi$$
$$608$$ −4.68466 −0.189988
$$609$$ −1.75379 −0.0710671
$$610$$ 0 0
$$611$$ −6.24621 −0.252695
$$612$$ −5.12311 −0.207089
$$613$$ −2.00000 −0.0807792 −0.0403896 0.999184i $$-0.512860\pi$$
−0.0403896 + 0.999184i $$0.512860\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 2.43845 0.0982478
$$617$$ 19.5616 0.787518 0.393759 0.919214i $$-0.371174\pi$$
0.393759 + 0.919214i $$0.371174\pi$$
$$618$$ −18.2462 −0.733970
$$619$$ −13.3693 −0.537358 −0.268679 0.963230i $$-0.586587\pi$$
−0.268679 + 0.963230i $$0.586587\pi$$
$$620$$ 0 0
$$621$$ −5.56155 −0.223177
$$622$$ 18.2462 0.731606
$$623$$ −2.05398 −0.0822908
$$624$$ −2.00000 −0.0800641
$$625$$ 0 0
$$626$$ −10.0000 −0.399680
$$627$$ −7.31534 −0.292147
$$628$$ 20.0540 0.800241
$$629$$ 26.2462 1.04650
$$630$$ 0 0
$$631$$ −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
$$640$$ 0 0
$$641$$ −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 $$-0.701901\pi$$
−0.592607 + 0.805492i $$0.701901\pi$$
$$644$$ −8.68466 −0.342223
$$645$$ 0 0
$$646$$ 24.0000 0.944267
$$647$$ 17.0691 0.671057 0.335528 0.942030i $$-0.391085\pi$$
0.335528 + 0.942030i $$0.391085\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ 7.61553 0.298936
$$650$$ 0 0
$$651$$ 1.56155 0.0612021
$$652$$ −9.36932 −0.366931
$$653$$ −23.7538 −0.929558 −0.464779 0.885427i $$-0.653866\pi$$
−0.464779 + 0.885427i $$0.653866\pi$$
$$654$$ 19.3693 0.757400
$$655$$ 0 0
$$656$$ −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 $$0.381077\pi$$
$$660$$ 0 0
$$661$$ −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
$$665$$ 0 0
$$666$$ 5.12311 0.198516
$$667$$ 6.24621 0.241854
$$668$$ 2.43845 0.0943464
$$669$$ 12.8769 0.497849
$$670$$ 0 0
$$671$$ −9.36932 −0.361698
$$672$$ −1.56155 −0.0602382
$$673$$ −28.2462 −1.08881 −0.544406 0.838822i $$-0.683245\pi$$
−0.544406 + 0.838822i $$0.683245\pi$$
$$674$$ −10.0000 −0.385186
$$675$$ 0 0
$$676$$ −9.00000 −0.346154
$$677$$ 29.4233 1.13083 0.565414 0.824807i $$-0.308716\pi$$
0.565414 + 0.824807i $$0.308716\pi$$
$$678$$ 10.6847 0.410342
$$679$$ 9.36932 0.359561
$$680$$ 0 0
$$681$$ 4.68466 0.179517
$$682$$ 1.56155 0.0597949
$$683$$ −19.3153 −0.739081 −0.369541 0.929215i $$-0.620485\pi$$
−0.369541 + 0.929215i $$0.620485\pi$$
$$684$$ 4.68466 0.179122
$$685$$ 0 0
$$686$$ 18.0540 0.689304
$$687$$ −12.4384 −0.474556
$$688$$ 7.80776 0.297668
$$689$$ 23.1231 0.880920
$$690$$ 0 0
$$691$$ −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
$$695$$ 0 0
$$696$$ 1.12311 0.0425712
$$697$$ 5.75379 0.217940
$$698$$ 19.3693 0.733139
$$699$$ −20.0540 −0.758511
$$700$$ 0 0
$$701$$ 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
$$705$$ 0 0
$$706$$ 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 $$0.622897\pi$$
$$710$$ 0 0
$$711$$ −16.6847 −0.625724
$$712$$ 1.31534 0.0492945
$$713$$ −5.56155 −0.208282
$$714$$ 8.00000 0.299392
$$715$$ 0 0
$$716$$ −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 $$0.797358\pi$$
$$720$$ 0 0
$$721$$ 28.4924 1.06111
$$722$$ −2.94602 −0.109640
$$723$$ −4.24621 −0.157918
$$724$$ 24.0540 0.893959
$$725$$ 0 0
$$726$$ 8.56155 0.317749
$$727$$ 14.0540 0.521233 0.260617 0.965442i $$-0.416074\pi$$
0.260617 + 0.965442i $$0.416074\pi$$
$$728$$ 3.12311 0.115750
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −40.0000 −1.47945
$$732$$ 6.00000 0.221766
$$733$$ −26.4924 −0.978520 −0.489260 0.872138i $$-0.662733\pi$$
−0.489260 + 0.872138i $$0.662733\pi$$
$$734$$ −9.36932 −0.345828
$$735$$ 0 0
$$736$$ 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 $$0.576763\pi$$
$$740$$ 0 0
$$741$$ −9.36932 −0.344190
$$742$$ 18.0540 0.662782
$$743$$ −11.4233 −0.419080 −0.209540 0.977800i $$-0.567197\pi$$
−0.209540 + 0.977800i $$0.567197\pi$$
$$744$$ −1.00000 −0.0366618
$$745$$ 0 0
$$746$$ 6.68466 0.244743
$$747$$ 2.24621 0.0821846
$$748$$ 8.00000 0.292509
$$749$$ −2.43845 −0.0890989
$$750$$ 0 0
$$751$$ 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 $$-0.558169\pi$$
−0.181728 + 0.983349i $$0.558169\pi$$
$$758$$ −30.0540 −1.09161
$$759$$ 8.68466 0.315233
$$760$$ 0 0
$$761$$ 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
$$765$$ 0 0
$$766$$ 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 $$0.595784\pi$$
$$770$$ 0 0
$$771$$ −10.6847 −0.384799
$$772$$ 18.4924 0.665557
$$773$$ 15.5616 0.559710 0.279855 0.960042i $$-0.409714\pi$$
0.279855 + 0.960042i $$0.409714\pi$$
$$774$$ −7.80776 −0.280644
$$775$$ 0 0
$$776$$ −6.00000 −0.215387
$$777$$ −8.00000 −0.286998
$$778$$ −24.2462 −0.869269
$$779$$ −5.26137 −0.188508
$$780$$ 0 0
$$781$$ −7.31534 −0.261764
$$782$$ −28.4924 −1.01889
$$783$$ −1.12311 −0.0401365
$$784$$ −4.56155 −0.162913
$$785$$ 0 0
$$786$$ −9.36932 −0.334192
$$787$$ −10.9309 −0.389643 −0.194822 0.980839i $$-0.562413\pi$$
−0.194822 + 0.980839i $$0.562413\pi$$
$$788$$ −6.00000 −0.213741
$$789$$ −12.4924 −0.444742
$$790$$ 0 0
$$791$$ −16.6847 −0.593238
$$792$$ 1.56155 0.0554874
$$793$$ −12.0000 −0.426132
$$794$$ −28.0540 −0.995598
$$795$$ 0 0
$$796$$ 3.80776 0.134963
$$797$$ −38.9848 −1.38091 −0.690457 0.723373i $$-0.742591\pi$$
−0.690457 + 0.723373i $$0.742591\pi$$
$$798$$ −7.31534 −0.258960
$$799$$ −16.0000 −0.566039
$$800$$ 0 0
$$801$$ −1.31534 −0.0464753
$$802$$ 1.31534 0.0464463
$$803$$ 15.3153 0.540467
$$804$$ −9.36932 −0.330430
$$805$$ 0 0
$$806$$ 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 $$0.646803\pi$$
$$810$$ 0 0
$$811$$ 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
$$815$$ 0 0
$$816$$ −5.12311 −0.179345
$$817$$ 36.5767 1.27966
$$818$$ 12.6307 0.441621
$$819$$ −3.12311 −0.109130
$$820$$ 0 0
$$821$$ −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 $$-0.587739\pi$$
−0.272162 + 0.962252i $$0.587739\pi$$
$$824$$ −18.2462 −0.635637
$$825$$ 0 0
$$826$$ 7.61553 0.264978
$$827$$ −18.2462 −0.634483 −0.317241 0.948345i $$-0.602757\pi$$
−0.317241 + 0.948345i $$0.602757\pi$$
$$828$$ −5.56155 −0.193277
$$829$$ −37.4233 −1.29976 −0.649882 0.760035i $$-0.725182\pi$$
−0.649882 + 0.760035i $$0.725182\pi$$
$$830$$ 0 0
$$831$$ −19.3693 −0.671914
$$832$$ −2.00000 −0.0693375
$$833$$ 23.3693 0.809699
$$834$$ 8.87689 0.307382
$$835$$ 0 0
$$836$$ −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 $$0.293982\pi$$
$$840$$ 0 0
$$841$$ −27.7386 −0.956505
$$842$$ −18.4924 −0.637291
$$843$$ 13.1231 0.451984
$$844$$ −11.3153 −0.389490
$$845$$ 0 0
$$846$$ −3.12311 −0.107375
$$847$$ −13.3693 −0.459375
$$848$$ −11.5616 −0.397025
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 28.4924 0.976708
$$852$$ 4.68466 0.160494
$$853$$ 38.7926 1.32823 0.664117 0.747629i $$-0.268808\pi$$
0.664117 + 0.747629i $$0.268808\pi$$
$$854$$ −9.36932 −0.320611
$$855$$ 0 0
$$856$$ 1.56155 0.0533728
$$857$$ 53.2311 1.81834 0.909169 0.416427i $$-0.136718\pi$$
0.909169 + 0.416427i $$0.136718\pi$$
$$858$$ −3.12311 −0.106621
$$859$$ −22.7386 −0.775832 −0.387916 0.921695i $$-0.626805\pi$$
−0.387916 + 0.921695i $$0.626805\pi$$
$$860$$ 0 0
$$861$$ −1.75379 −0.0597690
$$862$$ 13.7538 0.468456
$$863$$ 30.5464 1.03981 0.519906 0.854224i $$-0.325967\pi$$
0.519906 + 0.854224i $$0.325967\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 0 0
$$866$$ −18.3002 −0.621866
$$867$$ 9.24621 0.314018
$$868$$ 1.56155 0.0530026
$$869$$ 26.0540 0.883821
$$870$$ 0 0
$$871$$ 18.7386 0.634934
$$872$$ 19.3693 0.655928
$$873$$ 6.00000 0.203069
$$874$$ 26.0540 0.881289
$$875$$ 0 0
$$876$$ −9.80776 −0.331374
$$877$$ 50.0000 1.68838 0.844190 0.536044i $$-0.180082\pi$$
0.844190 + 0.536044i $$0.180082\pi$$
$$878$$ 31.6155 1.06697
$$879$$ 6.49242 0.218984
$$880$$ 0 0
$$881$$ 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 $$-0.322665\pi$$
0.528739 + 0.848784i $$0.322665\pi$$
$$884$$ 10.2462 0.344617
$$885$$ 0 0
$$886$$ −14.0540 −0.472153
$$887$$ 57.3693 1.92627 0.963137 0.269013i $$-0.0866974\pi$$
0.963137 + 0.269013i $$0.0866974\pi$$
$$888$$ 5.12311 0.171920
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −1.56155 −0.0523140
$$892$$ 12.8769 0.431150
$$893$$ 14.6307 0.489597
$$894$$ 20.0540 0.670705
$$895$$ 0 0
$$896$$ −1.56155 −0.0521678
$$897$$ 11.1231 0.371390
$$898$$ 22.4924 0.750582
$$899$$ −1.12311 −0.0374577
$$900$$ 0 0
$$901$$ 59.2311 1.97327
$$902$$ −1.75379 −0.0583948
$$903$$ 12.1922 0.405732
$$904$$ 10.6847 0.355366
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −24.0000 −0.796907 −0.398453 0.917189i $$-0.630453\pi$$
−0.398453 + 0.917189i $$0.630453\pi$$
$$908$$ 4.68466 0.155466
$$909$$ 3.56155 0.118129
$$910$$ 0 0
$$911$$ −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
$$915$$ 0 0
$$916$$ −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 $$0.273343\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 15.3693 0.506161
$$923$$ −9.36932 −0.308395
$$924$$ −2.43845 −0.0802190
$$925$$ 0 0
$$926$$ −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 $$0.439261\pi$$
$$930$$ 0 0
$$931$$ −21.3693 −0.700351
$$932$$ −20.0540 −0.656890
$$933$$ −18.2462 −0.597354
$$934$$ −26.2462 −0.858802
$$935$$ 0 0
$$936$$ 2.00000 0.0653720
$$937$$ −16.6307 −0.543301 −0.271650 0.962396i $$-0.587569\pi$$
−0.271650 + 0.962396i $$0.587569\pi$$
$$938$$ 14.6307 0.477709
$$939$$ 10.0000 0.326338
$$940$$ 0 0
$$941$$ 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
$$945$$ 0 0
$$946$$ 12.1922 0.396404
$$947$$ 0.876894 0.0284952 0.0142476 0.999898i $$-0.495465\pi$$
0.0142476 + 0.999898i $$0.495465\pi$$
$$948$$ −16.6847 −0.541893
$$949$$ 19.6155 0.636747
$$950$$ 0 0
$$951$$ −25.1231 −0.814673
$$952$$ 8.00000 0.259281
$$953$$ 58.1080 1.88230 0.941151 0.337988i $$-0.109746\pi$$
0.941151 + 0.337988i $$0.109746\pi$$
$$954$$ 11.5616 0.374319
$$955$$ 0 0
$$956$$ 24.0000 0.776215
$$957$$ 1.75379 0.0566919
$$958$$ −6.43845 −0.208017
$$959$$ −5.86174 −0.189285
$$960$$ 0 0
$$961$$ 1.00000 0.0322581
$$962$$ −10.2462 −0.330351
$$963$$ −1.56155 −0.0503203
$$964$$ −4.24621 −0.136761
$$965$$ 0 0
$$966$$ 8.68466 0.279424
$$967$$ 25.7538 0.828186 0.414093 0.910235i $$-0.364099\pi$$
0.414093 + 0.910235i $$0.364099\pi$$
$$968$$ 8.56155 0.275179
$$969$$ −24.0000 −0.770991
$$970$$ 0 0
$$971$$ −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
$$975$$ 0 0
$$976$$ 6.00000 0.192055
$$977$$ 22.9848 0.735350 0.367675 0.929954i $$-0.380154\pi$$
0.367675 + 0.929954i $$0.380154\pi$$
$$978$$ 9.36932 0.299598
$$979$$ 2.05398 0.0656453
$$980$$ 0 0
$$981$$ −19.3693 −0.618415
$$982$$ −2.93087 −0.0935278
$$983$$ 24.9848 0.796893 0.398446 0.917192i $$-0.369549\pi$$
0.398446 + 0.917192i $$0.369549\pi$$
$$984$$ 1.12311 0.0358033
$$985$$ 0 0
$$986$$ −5.75379 −0.183238
$$987$$ 4.87689 0.155233
$$988$$ −9.36932 −0.298078
$$989$$ −43.4233 −1.38078
$$990$$ 0 0
$$991$$ 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
$$995$$ 0 0
$$996$$ 2.24621 0.0711739
$$997$$ 62.4924 1.97915 0.989577 0.144002i $$-0.0459971\pi$$
0.989577 + 0.144002i $$0.0459971\pi$$
$$998$$ −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 4650.2.a.ce.1.2 2
5.2 odd 4 4650.2.d.bd.3349.2 4
5.3 odd 4 4650.2.d.bd.3349.3 4
5.4 even 2 930.2.a.p.1.1 2
15.14 odd 2 2790.2.a.be.1.1 2
20.19 odd 2 7440.2.a.bl.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.a.p.1.1 2 5.4 even 2
2790.2.a.be.1.1 2 15.14 odd 2
4650.2.a.ce.1.2 2 1.1 even 1 trivial
4650.2.d.bd.3349.2 4 5.2 odd 4
4650.2.d.bd.3349.3 4 5.3 odd 4
7440.2.a.bl.1.2 2 20.19 odd 2