# Properties

 Label 2475.2.a.x.1.1 Level $2475$ Weight $2$ Character 2475.1 Self dual yes Analytic conductor $19.763$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2475 = 3^{2} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2475.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$19.7629745003$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 55) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 2475.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-0.414214 q^{2} -1.82843 q^{4} +2.00000 q^{7} +1.58579 q^{8} +O(q^{10})$$ $$q-0.414214 q^{2} -1.82843 q^{4} +2.00000 q^{7} +1.58579 q^{8} -1.00000 q^{11} +6.82843 q^{13} -0.828427 q^{14} +3.00000 q^{16} +1.17157 q^{17} +0.414214 q^{22} +2.82843 q^{23} -2.82843 q^{26} -3.65685 q^{28} -7.65685 q^{29} -4.41421 q^{32} -0.485281 q^{34} -3.65685 q^{37} -6.00000 q^{41} +6.00000 q^{43} +1.82843 q^{44} -1.17157 q^{46} -2.82843 q^{47} -3.00000 q^{49} -12.4853 q^{52} +0.343146 q^{53} +3.17157 q^{56} +3.17157 q^{58} +9.65685 q^{59} +13.3137 q^{61} -4.17157 q^{64} +4.48528 q^{67} -2.14214 q^{68} +11.3137 q^{71} +6.82843 q^{73} +1.51472 q^{74} -2.00000 q^{77} +4.00000 q^{79} +2.48528 q^{82} -6.00000 q^{83} -2.48528 q^{86} -1.58579 q^{88} -9.31371 q^{89} +13.6569 q^{91} -5.17157 q^{92} +1.17157 q^{94} +7.65685 q^{97} +1.24264 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{4} + 4 q^{7} + 6 q^{8}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^4 + 4 * q^7 + 6 * q^8 $$2 q + 2 q^{2} + 2 q^{4} + 4 q^{7} + 6 q^{8} - 2 q^{11} + 8 q^{13} + 4 q^{14} + 6 q^{16} + 8 q^{17} - 2 q^{22} + 4 q^{28} - 4 q^{29} - 6 q^{32} + 16 q^{34} + 4 q^{37} - 12 q^{41} + 12 q^{43} - 2 q^{44} - 8 q^{46} - 6 q^{49} - 8 q^{52} + 12 q^{53} + 12 q^{56} + 12 q^{58} + 8 q^{59} + 4 q^{61} - 14 q^{64} - 8 q^{67} + 24 q^{68} + 8 q^{73} + 20 q^{74} - 4 q^{77} + 8 q^{79} - 12 q^{82} - 12 q^{83} + 12 q^{86} - 6 q^{88} + 4 q^{89} + 16 q^{91} - 16 q^{92} + 8 q^{94} + 4 q^{97} - 6 q^{98}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^4 + 4 * q^7 + 6 * q^8 - 2 * q^11 + 8 * q^13 + 4 * q^14 + 6 * q^16 + 8 * q^17 - 2 * q^22 + 4 * q^28 - 4 * q^29 - 6 * q^32 + 16 * q^34 + 4 * q^37 - 12 * q^41 + 12 * q^43 - 2 * q^44 - 8 * q^46 - 6 * q^49 - 8 * q^52 + 12 * q^53 + 12 * q^56 + 12 * q^58 + 8 * q^59 + 4 * q^61 - 14 * q^64 - 8 * q^67 + 24 * q^68 + 8 * q^73 + 20 * q^74 - 4 * q^77 + 8 * q^79 - 12 * q^82 - 12 * q^83 + 12 * q^86 - 6 * q^88 + 4 * q^89 + 16 * q^91 - 16 * q^92 + 8 * q^94 + 4 * q^97 - 6 * q^98

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −0.414214 −0.292893 −0.146447 0.989219i $$-0.546784\pi$$
−0.146447 + 0.989219i $$0.546784\pi$$
$$3$$ 0 0
$$4$$ −1.82843 −0.914214
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 2.00000 0.755929 0.377964 0.925820i $$-0.376624\pi$$
0.377964 + 0.925820i $$0.376624\pi$$
$$8$$ 1.58579 0.560660
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −1.00000 −0.301511
$$12$$ 0 0
$$13$$ 6.82843 1.89386 0.946932 0.321433i $$-0.104164\pi$$
0.946932 + 0.321433i $$0.104164\pi$$
$$14$$ −0.828427 −0.221406
$$15$$ 0 0
$$16$$ 3.00000 0.750000
$$17$$ 1.17157 0.284148 0.142074 0.989856i $$-0.454623\pi$$
0.142074 + 0.989856i $$0.454623\pi$$
$$18$$ 0 0
$$19$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0.414214 0.0883106
$$23$$ 2.82843 0.589768 0.294884 0.955533i $$-0.404719\pi$$
0.294884 + 0.955533i $$0.404719\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −2.82843 −0.554700
$$27$$ 0 0
$$28$$ −3.65685 −0.691080
$$29$$ −7.65685 −1.42184 −0.710921 0.703272i $$-0.751722\pi$$
−0.710921 + 0.703272i $$0.751722\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ −4.41421 −0.780330
$$33$$ 0 0
$$34$$ −0.485281 −0.0832251
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −3.65685 −0.601183 −0.300592 0.953753i $$-0.597184\pi$$
−0.300592 + 0.953753i $$0.597184\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ 0 0
$$43$$ 6.00000 0.914991 0.457496 0.889212i $$-0.348747\pi$$
0.457496 + 0.889212i $$0.348747\pi$$
$$44$$ 1.82843 0.275646
$$45$$ 0 0
$$46$$ −1.17157 −0.172739
$$47$$ −2.82843 −0.412568 −0.206284 0.978492i $$-0.566137\pi$$
−0.206284 + 0.978492i $$0.566137\pi$$
$$48$$ 0 0
$$49$$ −3.00000 −0.428571
$$50$$ 0 0
$$51$$ 0 0
$$52$$ −12.4853 −1.73140
$$53$$ 0.343146 0.0471347 0.0235673 0.999722i $$-0.492498\pi$$
0.0235673 + 0.999722i $$0.492498\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 3.17157 0.423819
$$57$$ 0 0
$$58$$ 3.17157 0.416448
$$59$$ 9.65685 1.25722 0.628608 0.777723i $$-0.283625\pi$$
0.628608 + 0.777723i $$0.283625\pi$$
$$60$$ 0 0
$$61$$ 13.3137 1.70465 0.852323 0.523016i $$-0.175193\pi$$
0.852323 + 0.523016i $$0.175193\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ −4.17157 −0.521447
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 4.48528 0.547964 0.273982 0.961735i $$-0.411659\pi$$
0.273982 + 0.961735i $$0.411659\pi$$
$$68$$ −2.14214 −0.259772
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 11.3137 1.34269 0.671345 0.741145i $$-0.265717\pi$$
0.671345 + 0.741145i $$0.265717\pi$$
$$72$$ 0 0
$$73$$ 6.82843 0.799207 0.399603 0.916688i $$-0.369148\pi$$
0.399603 + 0.916688i $$0.369148\pi$$
$$74$$ 1.51472 0.176082
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −2.00000 −0.227921
$$78$$ 0 0
$$79$$ 4.00000 0.450035 0.225018 0.974355i $$-0.427756\pi$$
0.225018 + 0.974355i $$0.427756\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 2.48528 0.274453
$$83$$ −6.00000 −0.658586 −0.329293 0.944228i $$-0.606810\pi$$
−0.329293 + 0.944228i $$0.606810\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −2.48528 −0.267995
$$87$$ 0 0
$$88$$ −1.58579 −0.169045
$$89$$ −9.31371 −0.987251 −0.493626 0.869675i $$-0.664329\pi$$
−0.493626 + 0.869675i $$0.664329\pi$$
$$90$$ 0 0
$$91$$ 13.6569 1.43163
$$92$$ −5.17157 −0.539174
$$93$$ 0 0
$$94$$ 1.17157 0.120839
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 7.65685 0.777436 0.388718 0.921357i $$-0.372918\pi$$
0.388718 + 0.921357i $$0.372918\pi$$
$$98$$ 1.24264 0.125526
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 13.3137 1.32476 0.662382 0.749166i $$-0.269546\pi$$
0.662382 + 0.749166i $$0.269546\pi$$
$$102$$ 0 0
$$103$$ −1.17157 −0.115439 −0.0577193 0.998333i $$-0.518383\pi$$
−0.0577193 + 0.998333i $$0.518383\pi$$
$$104$$ 10.8284 1.06181
$$105$$ 0 0
$$106$$ −0.142136 −0.0138054
$$107$$ −3.65685 −0.353521 −0.176761 0.984254i $$-0.556562\pi$$
−0.176761 + 0.984254i $$0.556562\pi$$
$$108$$ 0 0
$$109$$ 3.65685 0.350263 0.175132 0.984545i $$-0.443965\pi$$
0.175132 + 0.984545i $$0.443965\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 6.00000 0.566947
$$113$$ 8.34315 0.784857 0.392429 0.919782i $$-0.371635\pi$$
0.392429 + 0.919782i $$0.371635\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 14.0000 1.29987
$$117$$ 0 0
$$118$$ −4.00000 −0.368230
$$119$$ 2.34315 0.214796
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ −5.51472 −0.499279
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −15.6569 −1.38932 −0.694661 0.719338i $$-0.744445\pi$$
−0.694661 + 0.719338i $$0.744445\pi$$
$$128$$ 10.5563 0.933058
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −11.3137 −0.988483 −0.494242 0.869325i $$-0.664554\pi$$
−0.494242 + 0.869325i $$0.664554\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ −1.85786 −0.160495
$$135$$ 0 0
$$136$$ 1.85786 0.159311
$$137$$ 22.9706 1.96251 0.981254 0.192720i $$-0.0617309\pi$$
0.981254 + 0.192720i $$0.0617309\pi$$
$$138$$ 0 0
$$139$$ −4.00000 −0.339276 −0.169638 0.985506i $$-0.554260\pi$$
−0.169638 + 0.985506i $$0.554260\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −4.68629 −0.393265
$$143$$ −6.82843 −0.571022
$$144$$ 0 0
$$145$$ 0 0
$$146$$ −2.82843 −0.234082
$$147$$ 0 0
$$148$$ 6.68629 0.549610
$$149$$ −11.6569 −0.954967 −0.477483 0.878641i $$-0.658451\pi$$
−0.477483 + 0.878641i $$0.658451\pi$$
$$150$$ 0 0
$$151$$ −12.0000 −0.976546 −0.488273 0.872691i $$-0.662373\pi$$
−0.488273 + 0.872691i $$0.662373\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0.828427 0.0667566
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 14.0000 1.11732 0.558661 0.829396i $$-0.311315\pi$$
0.558661 + 0.829396i $$0.311315\pi$$
$$158$$ −1.65685 −0.131812
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 5.65685 0.445823
$$162$$ 0 0
$$163$$ 0.485281 0.0380102 0.0190051 0.999819i $$-0.493950\pi$$
0.0190051 + 0.999819i $$0.493950\pi$$
$$164$$ 10.9706 0.856657
$$165$$ 0 0
$$166$$ 2.48528 0.192895
$$167$$ 10.9706 0.848928 0.424464 0.905445i $$-0.360463\pi$$
0.424464 + 0.905445i $$0.360463\pi$$
$$168$$ 0 0
$$169$$ 33.6274 2.58672
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −10.9706 −0.836498
$$173$$ 6.14214 0.466978 0.233489 0.972359i $$-0.424986\pi$$
0.233489 + 0.972359i $$0.424986\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −3.00000 −0.226134
$$177$$ 0 0
$$178$$ 3.85786 0.289159
$$179$$ 1.65685 0.123839 0.0619196 0.998081i $$-0.480278\pi$$
0.0619196 + 0.998081i $$0.480278\pi$$
$$180$$ 0 0
$$181$$ −1.31371 −0.0976472 −0.0488236 0.998807i $$-0.515547\pi$$
−0.0488236 + 0.998807i $$0.515547\pi$$
$$182$$ −5.65685 −0.419314
$$183$$ 0 0
$$184$$ 4.48528 0.330659
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −1.17157 −0.0856739
$$188$$ 5.17157 0.377176
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 19.3137 1.39749 0.698745 0.715370i $$-0.253742\pi$$
0.698745 + 0.715370i $$0.253742\pi$$
$$192$$ 0 0
$$193$$ 6.82843 0.491521 0.245760 0.969331i $$-0.420962\pi$$
0.245760 + 0.969331i $$0.420962\pi$$
$$194$$ −3.17157 −0.227706
$$195$$ 0 0
$$196$$ 5.48528 0.391806
$$197$$ −5.17157 −0.368459 −0.184230 0.982883i $$-0.558979\pi$$
−0.184230 + 0.982883i $$0.558979\pi$$
$$198$$ 0 0
$$199$$ 21.6569 1.53521 0.767607 0.640921i $$-0.221447\pi$$
0.767607 + 0.640921i $$0.221447\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ −5.51472 −0.388014
$$203$$ −15.3137 −1.07481
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0.485281 0.0338112
$$207$$ 0 0
$$208$$ 20.4853 1.42040
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −16.0000 −1.10149 −0.550743 0.834675i $$-0.685655\pi$$
−0.550743 + 0.834675i $$0.685655\pi$$
$$212$$ −0.627417 −0.0430912
$$213$$ 0 0
$$214$$ 1.51472 0.103544
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 0 0
$$218$$ −1.51472 −0.102590
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 8.00000 0.538138
$$222$$ 0 0
$$223$$ 5.17157 0.346314 0.173157 0.984894i $$-0.444603\pi$$
0.173157 + 0.984894i $$0.444603\pi$$
$$224$$ −8.82843 −0.589874
$$225$$ 0 0
$$226$$ −3.45584 −0.229879
$$227$$ 2.68629 0.178295 0.0891477 0.996018i $$-0.471586\pi$$
0.0891477 + 0.996018i $$0.471586\pi$$
$$228$$ 0 0
$$229$$ −21.3137 −1.40845 −0.704225 0.709977i $$-0.748705\pi$$
−0.704225 + 0.709977i $$0.748705\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −12.1421 −0.797170
$$233$$ 22.1421 1.45058 0.725290 0.688444i $$-0.241706\pi$$
0.725290 + 0.688444i $$0.241706\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −17.6569 −1.14936
$$237$$ 0 0
$$238$$ −0.970563 −0.0629122
$$239$$ 0.686292 0.0443925 0.0221963 0.999754i $$-0.492934\pi$$
0.0221963 + 0.999754i $$0.492934\pi$$
$$240$$ 0 0
$$241$$ 6.00000 0.386494 0.193247 0.981150i $$-0.438098\pi$$
0.193247 + 0.981150i $$0.438098\pi$$
$$242$$ −0.414214 −0.0266267
$$243$$ 0 0
$$244$$ −24.3431 −1.55841
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −12.0000 −0.757433 −0.378717 0.925513i $$-0.623635\pi$$
−0.378717 + 0.925513i $$0.623635\pi$$
$$252$$ 0 0
$$253$$ −2.82843 −0.177822
$$254$$ 6.48528 0.406923
$$255$$ 0 0
$$256$$ 3.97056 0.248160
$$257$$ 13.3137 0.830486 0.415243 0.909710i $$-0.363696\pi$$
0.415243 + 0.909710i $$0.363696\pi$$
$$258$$ 0 0
$$259$$ −7.31371 −0.454452
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 4.68629 0.289520
$$263$$ 22.9706 1.41643 0.708213 0.705999i $$-0.249502\pi$$
0.708213 + 0.705999i $$0.249502\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ −8.20101 −0.500956
$$269$$ 5.31371 0.323983 0.161991 0.986792i $$-0.448208\pi$$
0.161991 + 0.986792i $$0.448208\pi$$
$$270$$ 0 0
$$271$$ −15.3137 −0.930242 −0.465121 0.885247i $$-0.653989\pi$$
−0.465121 + 0.885247i $$0.653989\pi$$
$$272$$ 3.51472 0.213111
$$273$$ 0 0
$$274$$ −9.51472 −0.574805
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −1.17157 −0.0703930 −0.0351965 0.999380i $$-0.511206\pi$$
−0.0351965 + 0.999380i $$0.511206\pi$$
$$278$$ 1.65685 0.0993715
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 5.31371 0.316989 0.158495 0.987360i $$-0.449336\pi$$
0.158495 + 0.987360i $$0.449336\pi$$
$$282$$ 0 0
$$283$$ 12.6274 0.750622 0.375311 0.926899i $$-0.377536\pi$$
0.375311 + 0.926899i $$0.377536\pi$$
$$284$$ −20.6863 −1.22751
$$285$$ 0 0
$$286$$ 2.82843 0.167248
$$287$$ −12.0000 −0.708338
$$288$$ 0 0
$$289$$ −15.6274 −0.919260
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −12.4853 −0.730646
$$293$$ −14.8284 −0.866286 −0.433143 0.901325i $$-0.642595\pi$$
−0.433143 + 0.901325i $$0.642595\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −5.79899 −0.337059
$$297$$ 0 0
$$298$$ 4.82843 0.279703
$$299$$ 19.3137 1.11694
$$300$$ 0 0
$$301$$ 12.0000 0.691669
$$302$$ 4.97056 0.286024
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 27.6569 1.57846 0.789230 0.614098i $$-0.210480\pi$$
0.789230 + 0.614098i $$0.210480\pi$$
$$308$$ 3.65685 0.208369
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −27.3137 −1.54882 −0.774409 0.632685i $$-0.781953\pi$$
−0.774409 + 0.632685i $$0.781953\pi$$
$$312$$ 0 0
$$313$$ −21.3137 −1.20472 −0.602361 0.798224i $$-0.705773\pi$$
−0.602361 + 0.798224i $$0.705773\pi$$
$$314$$ −5.79899 −0.327256
$$315$$ 0 0
$$316$$ −7.31371 −0.411428
$$317$$ 21.3137 1.19710 0.598549 0.801087i $$-0.295744\pi$$
0.598549 + 0.801087i $$0.295744\pi$$
$$318$$ 0 0
$$319$$ 7.65685 0.428702
$$320$$ 0 0
$$321$$ 0 0
$$322$$ −2.34315 −0.130578
$$323$$ 0 0
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −0.201010 −0.0111329
$$327$$ 0 0
$$328$$ −9.51472 −0.525362
$$329$$ −5.65685 −0.311872
$$330$$ 0 0
$$331$$ 15.3137 0.841718 0.420859 0.907126i $$-0.361729\pi$$
0.420859 + 0.907126i $$0.361729\pi$$
$$332$$ 10.9706 0.602088
$$333$$ 0 0
$$334$$ −4.54416 −0.248645
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 3.51472 0.191459 0.0957295 0.995407i $$-0.469482\pi$$
0.0957295 + 0.995407i $$0.469482\pi$$
$$338$$ −13.9289 −0.757634
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ −20.0000 −1.07990
$$344$$ 9.51472 0.512999
$$345$$ 0 0
$$346$$ −2.54416 −0.136775
$$347$$ −22.9706 −1.23312 −0.616562 0.787306i $$-0.711475\pi$$
−0.616562 + 0.787306i $$0.711475\pi$$
$$348$$ 0 0
$$349$$ −6.97056 −0.373126 −0.186563 0.982443i $$-0.559735\pi$$
−0.186563 + 0.982443i $$0.559735\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 4.41421 0.235278
$$353$$ −1.31371 −0.0699216 −0.0349608 0.999389i $$-0.511131\pi$$
−0.0349608 + 0.999389i $$0.511131\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 17.0294 0.902558
$$357$$ 0 0
$$358$$ −0.686292 −0.0362716
$$359$$ −23.3137 −1.23045 −0.615225 0.788351i $$-0.710935\pi$$
−0.615225 + 0.788351i $$0.710935\pi$$
$$360$$ 0 0
$$361$$ −19.0000 −1.00000
$$362$$ 0.544156 0.0286002
$$363$$ 0 0
$$364$$ −24.9706 −1.30881
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −8.48528 −0.442928 −0.221464 0.975169i $$-0.571084\pi$$
−0.221464 + 0.975169i $$0.571084\pi$$
$$368$$ 8.48528 0.442326
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0.686292 0.0356305
$$372$$ 0 0
$$373$$ 3.79899 0.196704 0.0983521 0.995152i $$-0.468643\pi$$
0.0983521 + 0.995152i $$0.468643\pi$$
$$374$$ 0.485281 0.0250933
$$375$$ 0 0
$$376$$ −4.48528 −0.231311
$$377$$ −52.2843 −2.69278
$$378$$ 0 0
$$379$$ 22.3431 1.14769 0.573845 0.818964i $$-0.305451\pi$$
0.573845 + 0.818964i $$0.305451\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ −8.00000 −0.409316
$$383$$ −34.1421 −1.74458 −0.872291 0.488987i $$-0.837366\pi$$
−0.872291 + 0.488987i $$0.837366\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −2.82843 −0.143963
$$387$$ 0 0
$$388$$ −14.0000 −0.710742
$$389$$ 24.6274 1.24866 0.624330 0.781161i $$-0.285372\pi$$
0.624330 + 0.781161i $$0.285372\pi$$
$$390$$ 0 0
$$391$$ 3.31371 0.167581
$$392$$ −4.75736 −0.240283
$$393$$ 0 0
$$394$$ 2.14214 0.107919
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −13.3137 −0.668196 −0.334098 0.942538i $$-0.608432\pi$$
−0.334098 + 0.942538i $$0.608432\pi$$
$$398$$ −8.97056 −0.449654
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −17.3137 −0.864605 −0.432303 0.901729i $$-0.642299\pi$$
−0.432303 + 0.901729i $$0.642299\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ −24.3431 −1.21112
$$405$$ 0 0
$$406$$ 6.34315 0.314805
$$407$$ 3.65685 0.181264
$$408$$ 0 0
$$409$$ 34.9706 1.72918 0.864592 0.502475i $$-0.167577\pi$$
0.864592 + 0.502475i $$0.167577\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 2.14214 0.105535
$$413$$ 19.3137 0.950365
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −30.1421 −1.47784
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 14.3431 0.700709 0.350354 0.936617i $$-0.386061\pi$$
0.350354 + 0.936617i $$0.386061\pi$$
$$420$$ 0 0
$$421$$ −6.00000 −0.292422 −0.146211 0.989253i $$-0.546708\pi$$
−0.146211 + 0.989253i $$0.546708\pi$$
$$422$$ 6.62742 0.322618
$$423$$ 0 0
$$424$$ 0.544156 0.0264265
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 26.6274 1.28859
$$428$$ 6.68629 0.323194
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −11.3137 −0.544962 −0.272481 0.962161i $$-0.587844\pi$$
−0.272481 + 0.962161i $$0.587844\pi$$
$$432$$ 0 0
$$433$$ −3.65685 −0.175737 −0.0878686 0.996132i $$-0.528006\pi$$
−0.0878686 + 0.996132i $$0.528006\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −6.68629 −0.320215
$$437$$ 0 0
$$438$$ 0 0
$$439$$ −16.0000 −0.763638 −0.381819 0.924237i $$-0.624702\pi$$
−0.381819 + 0.924237i $$0.624702\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −3.31371 −0.157617
$$443$$ −21.1716 −1.00589 −0.502946 0.864318i $$-0.667751\pi$$
−0.502946 + 0.864318i $$0.667751\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −2.14214 −0.101433
$$447$$ 0 0
$$448$$ −8.34315 −0.394177
$$449$$ 16.6274 0.784696 0.392348 0.919817i $$-0.371663\pi$$
0.392348 + 0.919817i $$0.371663\pi$$
$$450$$ 0 0
$$451$$ 6.00000 0.282529
$$452$$ −15.2548 −0.717527
$$453$$ 0 0
$$454$$ −1.11270 −0.0522215
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 16.4853 0.771149 0.385574 0.922677i $$-0.374003\pi$$
0.385574 + 0.922677i $$0.374003\pi$$
$$458$$ 8.82843 0.412525
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 32.6274 1.51961 0.759805 0.650151i $$-0.225294\pi$$
0.759805 + 0.650151i $$0.225294\pi$$
$$462$$ 0 0
$$463$$ −22.1421 −1.02903 −0.514516 0.857481i $$-0.672028\pi$$
−0.514516 + 0.857481i $$0.672028\pi$$
$$464$$ −22.9706 −1.06638
$$465$$ 0 0
$$466$$ −9.17157 −0.424865
$$467$$ −9.17157 −0.424410 −0.212205 0.977225i $$-0.568064\pi$$
−0.212205 + 0.977225i $$0.568064\pi$$
$$468$$ 0 0
$$469$$ 8.97056 0.414222
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 15.3137 0.704871
$$473$$ −6.00000 −0.275880
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −4.28427 −0.196369
$$477$$ 0 0
$$478$$ −0.284271 −0.0130023
$$479$$ 36.0000 1.64488 0.822441 0.568850i $$-0.192612\pi$$
0.822441 + 0.568850i $$0.192612\pi$$
$$480$$ 0 0
$$481$$ −24.9706 −1.13856
$$482$$ −2.48528 −0.113201
$$483$$ 0 0
$$484$$ −1.82843 −0.0831103
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 7.51472 0.340524 0.170262 0.985399i $$-0.445539\pi$$
0.170262 + 0.985399i $$0.445539\pi$$
$$488$$ 21.1127 0.955727
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 23.3137 1.05213 0.526066 0.850443i $$-0.323666\pi$$
0.526066 + 0.850443i $$0.323666\pi$$
$$492$$ 0 0
$$493$$ −8.97056 −0.404014
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 22.6274 1.01498
$$498$$ 0 0
$$499$$ −1.65685 −0.0741710 −0.0370855 0.999312i $$-0.511807\pi$$
−0.0370855 + 0.999312i $$0.511807\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 4.97056 0.221847
$$503$$ −28.6274 −1.27643 −0.638217 0.769857i $$-0.720328\pi$$
−0.638217 + 0.769857i $$0.720328\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 1.17157 0.0520828
$$507$$ 0 0
$$508$$ 28.6274 1.27014
$$509$$ −9.31371 −0.412823 −0.206411 0.978465i $$-0.566179\pi$$
−0.206411 + 0.978465i $$0.566179\pi$$
$$510$$ 0 0
$$511$$ 13.6569 0.604144
$$512$$ −22.7574 −1.00574
$$513$$ 0 0
$$514$$ −5.51472 −0.243244
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 2.82843 0.124394
$$518$$ 3.02944 0.133106
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −2.68629 −0.117689 −0.0588443 0.998267i $$-0.518742\pi$$
−0.0588443 + 0.998267i $$0.518742\pi$$
$$522$$ 0 0
$$523$$ −37.5980 −1.64404 −0.822022 0.569455i $$-0.807154\pi$$
−0.822022 + 0.569455i $$0.807154\pi$$
$$524$$ 20.6863 0.903685
$$525$$ 0 0
$$526$$ −9.51472 −0.414861
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −15.0000 −0.652174
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −40.9706 −1.77463
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 7.11270 0.307222
$$537$$ 0 0
$$538$$ −2.20101 −0.0948923
$$539$$ 3.00000 0.129219
$$540$$ 0 0
$$541$$ 6.00000 0.257960 0.128980 0.991647i $$-0.458830\pi$$
0.128980 + 0.991647i $$0.458830\pi$$
$$542$$ 6.34315 0.272461
$$543$$ 0 0
$$544$$ −5.17157 −0.221729
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 34.0000 1.45374 0.726868 0.686778i $$-0.240975\pi$$
0.726868 + 0.686778i $$0.240975\pi$$
$$548$$ −42.0000 −1.79415
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 8.00000 0.340195
$$554$$ 0.485281 0.0206176
$$555$$ 0 0
$$556$$ 7.31371 0.310170
$$557$$ 38.1421 1.61613 0.808067 0.589090i $$-0.200514\pi$$
0.808067 + 0.589090i $$0.200514\pi$$
$$558$$ 0 0
$$559$$ 40.9706 1.73287
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −2.20101 −0.0928440
$$563$$ 11.6569 0.491278 0.245639 0.969361i $$-0.421002\pi$$
0.245639 + 0.969361i $$0.421002\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −5.23045 −0.219852
$$567$$ 0 0
$$568$$ 17.9411 0.752793
$$569$$ −20.3431 −0.852829 −0.426415 0.904528i $$-0.640224\pi$$
−0.426415 + 0.904528i $$0.640224\pi$$
$$570$$ 0 0
$$571$$ 45.9411 1.92258 0.961288 0.275545i $$-0.0888584\pi$$
0.961288 + 0.275545i $$0.0888584\pi$$
$$572$$ 12.4853 0.522036
$$573$$ 0 0
$$574$$ 4.97056 0.207467
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −6.97056 −0.290188 −0.145094 0.989418i $$-0.546349\pi$$
−0.145094 + 0.989418i $$0.546349\pi$$
$$578$$ 6.47309 0.269245
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −12.0000 −0.497844
$$582$$ 0 0
$$583$$ −0.343146 −0.0142116
$$584$$ 10.8284 0.448084
$$585$$ 0 0
$$586$$ 6.14214 0.253729
$$587$$ 26.1421 1.07900 0.539501 0.841985i $$-0.318613\pi$$
0.539501 + 0.841985i $$0.318613\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 0 0
$$592$$ −10.9706 −0.450887
$$593$$ 20.4853 0.841230 0.420615 0.907239i $$-0.361814\pi$$
0.420615 + 0.907239i $$0.361814\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 21.3137 0.873044
$$597$$ 0 0
$$598$$ −8.00000 −0.327144
$$599$$ −5.65685 −0.231133 −0.115566 0.993300i $$-0.536868\pi$$
−0.115566 + 0.993300i $$0.536868\pi$$
$$600$$ 0 0
$$601$$ −43.9411 −1.79240 −0.896198 0.443654i $$-0.853682\pi$$
−0.896198 + 0.443654i $$0.853682\pi$$
$$602$$ −4.97056 −0.202585
$$603$$ 0 0
$$604$$ 21.9411 0.892772
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 18.2843 0.742136 0.371068 0.928606i $$-0.378992\pi$$
0.371068 + 0.928606i $$0.378992\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −19.3137 −0.781349
$$612$$ 0 0
$$613$$ −25.4558 −1.02815 −0.514076 0.857745i $$-0.671865\pi$$
−0.514076 + 0.857745i $$0.671865\pi$$
$$614$$ −11.4558 −0.462320
$$615$$ 0 0
$$616$$ −3.17157 −0.127786
$$617$$ 11.6569 0.469287 0.234644 0.972081i $$-0.424608\pi$$
0.234644 + 0.972081i $$0.424608\pi$$
$$618$$ 0 0
$$619$$ −25.6569 −1.03124 −0.515618 0.856819i $$-0.672438\pi$$
−0.515618 + 0.856819i $$0.672438\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 11.3137 0.453638
$$623$$ −18.6274 −0.746292
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 8.82843 0.352855
$$627$$ 0 0
$$628$$ −25.5980 −1.02147
$$629$$ −4.28427 −0.170825
$$630$$ 0 0
$$631$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$632$$ 6.34315 0.252317
$$633$$ 0 0
$$634$$ −8.82843 −0.350622
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −20.4853 −0.811656
$$638$$ −3.17157 −0.125564
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −30.0000 −1.18493 −0.592464 0.805597i $$-0.701845\pi$$
−0.592464 + 0.805597i $$0.701845\pi$$
$$642$$ 0 0
$$643$$ −49.4558 −1.95035 −0.975174 0.221440i $$-0.928924\pi$$
−0.975174 + 0.221440i $$0.928924\pi$$
$$644$$ −10.3431 −0.407577
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −35.1127 −1.38042 −0.690211 0.723608i $$-0.742482\pi$$
−0.690211 + 0.723608i $$0.742482\pi$$
$$648$$ 0 0
$$649$$ −9.65685 −0.379065
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −0.887302 −0.0347494
$$653$$ 0.343146 0.0134283 0.00671417 0.999977i $$-0.497863\pi$$
0.00671417 + 0.999977i $$0.497863\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −18.0000 −0.702782
$$657$$ 0 0
$$658$$ 2.34315 0.0913453
$$659$$ 21.9411 0.854705 0.427352 0.904085i $$-0.359446\pi$$
0.427352 + 0.904085i $$0.359446\pi$$
$$660$$ 0 0
$$661$$ −0.627417 −0.0244037 −0.0122018 0.999926i $$-0.503884\pi$$
−0.0122018 + 0.999926i $$0.503884\pi$$
$$662$$ −6.34315 −0.246533
$$663$$ 0 0
$$664$$ −9.51472 −0.369243
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −21.6569 −0.838557
$$668$$ −20.0589 −0.776101
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −13.3137 −0.513970
$$672$$ 0 0
$$673$$ −4.48528 −0.172895 −0.0864474 0.996256i $$-0.527551\pi$$
−0.0864474 + 0.996256i $$0.527551\pi$$
$$674$$ −1.45584 −0.0560770
$$675$$ 0 0
$$676$$ −61.4853 −2.36482
$$677$$ 17.1716 0.659957 0.329979 0.943988i $$-0.392958\pi$$
0.329979 + 0.943988i $$0.392958\pi$$
$$678$$ 0 0
$$679$$ 15.3137 0.587686
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 31.7990 1.21675 0.608377 0.793648i $$-0.291821\pi$$
0.608377 + 0.793648i $$0.291821\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 8.28427 0.316295
$$687$$ 0 0
$$688$$ 18.0000 0.686244
$$689$$ 2.34315 0.0892667
$$690$$ 0 0
$$691$$ −16.6863 −0.634776 −0.317388 0.948296i $$-0.602806\pi$$
−0.317388 + 0.948296i $$0.602806\pi$$
$$692$$ −11.2304 −0.426918
$$693$$ 0 0
$$694$$ 9.51472 0.361174
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −7.02944 −0.266259
$$698$$ 2.88730 0.109286
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −32.6274 −1.23232 −0.616160 0.787621i $$-0.711313\pi$$
−0.616160 + 0.787621i $$0.711313\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 4.17157 0.157222
$$705$$ 0 0
$$706$$ 0.544156 0.0204796
$$707$$ 26.6274 1.00143
$$708$$ 0 0
$$709$$ −20.6274 −0.774679 −0.387339 0.921937i $$-0.626606\pi$$
−0.387339 + 0.921937i $$0.626606\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ −14.7696 −0.553512
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −3.02944 −0.113215
$$717$$ 0 0
$$718$$ 9.65685 0.360391
$$719$$ −29.6569 −1.10601 −0.553007 0.833177i $$-0.686520\pi$$
−0.553007 + 0.833177i $$0.686520\pi$$
$$720$$ 0 0
$$721$$ −2.34315 −0.0872633
$$722$$ 7.87006 0.292893
$$723$$ 0 0
$$724$$ 2.40202 0.0892704
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 36.4853 1.35316 0.676582 0.736367i $$-0.263460\pi$$
0.676582 + 0.736367i $$0.263460\pi$$
$$728$$ 21.6569 0.802656
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 7.02944 0.259993
$$732$$ 0 0
$$733$$ −33.4558 −1.23572 −0.617860 0.786288i $$-0.712000\pi$$
−0.617860 + 0.786288i $$0.712000\pi$$
$$734$$ 3.51472 0.129731
$$735$$ 0 0
$$736$$ −12.4853 −0.460214
$$737$$ −4.48528 −0.165217
$$738$$ 0 0
$$739$$ −37.9411 −1.39569 −0.697843 0.716250i $$-0.745857\pi$$
−0.697843 + 0.716250i $$0.745857\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −0.284271 −0.0104359
$$743$$ 29.5980 1.08584 0.542922 0.839783i $$-0.317318\pi$$
0.542922 + 0.839783i $$0.317318\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −1.57359 −0.0576133
$$747$$ 0 0
$$748$$ 2.14214 0.0783242
$$749$$ −7.31371 −0.267237
$$750$$ 0 0
$$751$$ −16.0000 −0.583848 −0.291924 0.956441i $$-0.594295\pi$$
−0.291924 + 0.956441i $$0.594295\pi$$
$$752$$ −8.48528 −0.309426
$$753$$ 0 0
$$754$$ 21.6569 0.788696
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 9.31371 0.338512 0.169256 0.985572i $$-0.445863\pi$$
0.169256 + 0.985572i $$0.445863\pi$$
$$758$$ −9.25483 −0.336151
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 30.0000 1.08750 0.543750 0.839248i $$-0.317004\pi$$
0.543750 + 0.839248i $$0.317004\pi$$
$$762$$ 0 0
$$763$$ 7.31371 0.264774
$$764$$ −35.3137 −1.27761
$$765$$ 0 0
$$766$$ 14.1421 0.510976
$$767$$ 65.9411 2.38100
$$768$$ 0 0
$$769$$ 14.9706 0.539852 0.269926 0.962881i $$-0.413001\pi$$
0.269926 + 0.962881i $$0.413001\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −12.4853 −0.449355
$$773$$ −30.2843 −1.08925 −0.544625 0.838680i $$-0.683328\pi$$
−0.544625 + 0.838680i $$0.683328\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 12.1421 0.435877
$$777$$ 0 0
$$778$$ −10.2010 −0.365724
$$779$$ 0 0
$$780$$ 0 0
$$781$$ −11.3137 −0.404836
$$782$$ −1.37258 −0.0490835
$$783$$ 0 0
$$784$$ −9.00000 −0.321429
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −18.9706 −0.676228 −0.338114 0.941105i $$-0.609789\pi$$
−0.338114 + 0.941105i $$0.609789\pi$$
$$788$$ 9.45584 0.336850
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 16.6863 0.593296
$$792$$ 0 0
$$793$$ 90.9117 3.22837
$$794$$ 5.51472 0.195710
$$795$$ 0 0
$$796$$ −39.5980 −1.40351
$$797$$ −12.6274 −0.447286 −0.223643 0.974671i $$-0.571795\pi$$
−0.223643 + 0.974671i $$0.571795\pi$$
$$798$$ 0 0
$$799$$ −3.31371 −0.117231
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 7.17157 0.253237
$$803$$ −6.82843 −0.240970
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 21.1127 0.742742
$$809$$ −22.9706 −0.807602 −0.403801 0.914847i $$-0.632311\pi$$
−0.403801 + 0.914847i $$0.632311\pi$$
$$810$$ 0 0
$$811$$ −13.9411 −0.489539 −0.244770 0.969581i $$-0.578712\pi$$
−0.244770 + 0.969581i $$0.578712\pi$$
$$812$$ 28.0000 0.982607
$$813$$ 0 0
$$814$$ −1.51472 −0.0530909
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 0 0
$$818$$ −14.4853 −0.506466
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 18.6863 0.652156 0.326078 0.945343i $$-0.394273\pi$$
0.326078 + 0.945343i $$0.394273\pi$$
$$822$$ 0 0
$$823$$ −36.4853 −1.27180 −0.635898 0.771773i $$-0.719370\pi$$
−0.635898 + 0.771773i $$0.719370\pi$$
$$824$$ −1.85786 −0.0647218
$$825$$ 0 0
$$826$$ −8.00000 −0.278356
$$827$$ −34.2843 −1.19218 −0.596090 0.802917i $$-0.703280\pi$$
−0.596090 + 0.802917i $$0.703280\pi$$
$$828$$ 0 0
$$829$$ 18.0000 0.625166 0.312583 0.949890i $$-0.398806\pi$$
0.312583 + 0.949890i $$0.398806\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ −28.4853 −0.987549
$$833$$ −3.51472 −0.121778
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 0 0
$$838$$ −5.94113 −0.205233
$$839$$ −37.6569 −1.30006 −0.650029 0.759909i $$-0.725243\pi$$
−0.650029 + 0.759909i $$0.725243\pi$$
$$840$$ 0 0
$$841$$ 29.6274 1.02164
$$842$$ 2.48528 0.0856485
$$843$$ 0 0
$$844$$ 29.2548 1.00699
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 2.00000 0.0687208
$$848$$ 1.02944 0.0353510
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −10.3431 −0.354558
$$852$$ 0 0
$$853$$ 32.4853 1.11227 0.556137 0.831090i $$-0.312283\pi$$
0.556137 + 0.831090i $$0.312283\pi$$
$$854$$ −11.0294 −0.377420
$$855$$ 0 0
$$856$$ −5.79899 −0.198205
$$857$$ −48.7696 −1.66594 −0.832968 0.553321i $$-0.813360\pi$$
−0.832968 + 0.553321i $$0.813360\pi$$
$$858$$ 0 0
$$859$$ −32.2843 −1.10153 −0.550763 0.834662i $$-0.685663\pi$$
−0.550763 + 0.834662i $$0.685663\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 4.68629 0.159616
$$863$$ −14.8284 −0.504766 −0.252383 0.967627i $$-0.581214\pi$$
−0.252383 + 0.967627i $$0.581214\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 1.51472 0.0514722
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −4.00000 −0.135691
$$870$$ 0 0
$$871$$ 30.6274 1.03777
$$872$$ 5.79899 0.196379
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −1.45584 −0.0491604 −0.0245802 0.999698i $$-0.507825\pi$$
−0.0245802 + 0.999698i $$0.507825\pi$$
$$878$$ 6.62742 0.223664
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 52.6274 1.77306 0.886531 0.462668i $$-0.153108\pi$$
0.886531 + 0.462668i $$0.153108\pi$$
$$882$$ 0 0
$$883$$ −42.8284 −1.44129 −0.720646 0.693304i $$-0.756155\pi$$
−0.720646 + 0.693304i $$0.756155\pi$$
$$884$$ −14.6274 −0.491973
$$885$$ 0 0
$$886$$ 8.76955 0.294619
$$887$$ −18.2843 −0.613926 −0.306963 0.951721i $$-0.599313\pi$$
−0.306963 + 0.951721i $$0.599313\pi$$
$$888$$ 0 0
$$889$$ −31.3137 −1.05023
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −9.45584 −0.316605
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 21.1127 0.705326
$$897$$ 0 0
$$898$$ −6.88730 −0.229832
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 0.402020 0.0133932
$$902$$ −2.48528 −0.0827508
$$903$$ 0 0
$$904$$ 13.2304 0.440038
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 44.4853 1.47711 0.738555 0.674193i $$-0.235509\pi$$
0.738555 + 0.674193i $$0.235509\pi$$
$$908$$ −4.91169 −0.163000
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −57.9411 −1.91968 −0.959838 0.280556i $$-0.909481\pi$$
−0.959838 + 0.280556i $$0.909481\pi$$
$$912$$ 0 0
$$913$$ 6.00000 0.198571
$$914$$ −6.82843 −0.225864
$$915$$ 0 0
$$916$$ 38.9706 1.28762
$$917$$ −22.6274 −0.747223
$$918$$ 0 0
$$919$$ −32.0000 −1.05558 −0.527791 0.849374i $$-0.676980\pi$$
−0.527791 + 0.849374i $$0.676980\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −13.5147 −0.445084
$$923$$ 77.2548 2.54287
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 9.17157 0.301397
$$927$$ 0 0
$$928$$ 33.7990 1.10951
$$929$$ 17.3137 0.568044 0.284022 0.958818i $$-0.408331\pi$$
0.284022 + 0.958818i $$0.408331\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ −40.4853 −1.32614
$$933$$ 0 0
$$934$$ 3.79899 0.124307
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −49.4558 −1.61565 −0.807826 0.589421i $$-0.799356\pi$$
−0.807826 + 0.589421i $$0.799356\pi$$
$$938$$ −3.71573 −0.121323
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 29.3137 0.955600 0.477800 0.878469i $$-0.341434\pi$$
0.477800 + 0.878469i $$0.341434\pi$$
$$942$$ 0 0
$$943$$ −16.9706 −0.552638
$$944$$ 28.9706 0.942912
$$945$$ 0 0
$$946$$ 2.48528 0.0808035
$$947$$ 46.8284 1.52172 0.760860 0.648916i $$-0.224778\pi$$
0.760860 + 0.648916i $$0.224778\pi$$
$$948$$ 0 0
$$949$$ 46.6274 1.51359
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 3.71573 0.120427
$$953$$ 58.8284 1.90564 0.952820 0.303536i $$-0.0981674\pi$$
0.952820 + 0.303536i $$0.0981674\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −1.25483 −0.0405842
$$957$$ 0 0
$$958$$ −14.9117 −0.481775
$$959$$ 45.9411 1.48352
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ 10.3431 0.333476
$$963$$ 0 0
$$964$$ −10.9706 −0.353338
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −18.9706 −0.610052 −0.305026 0.952344i $$-0.598665\pi$$
−0.305026 + 0.952344i $$0.598665\pi$$
$$968$$ 1.58579 0.0509691
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 31.3137 1.00490 0.502452 0.864605i $$-0.332431\pi$$
0.502452 + 0.864605i $$0.332431\pi$$
$$972$$ 0 0
$$973$$ −8.00000 −0.256468
$$974$$ −3.11270 −0.0997373
$$975$$ 0 0
$$976$$ 39.9411 1.27848
$$977$$ 43.6569 1.39671 0.698353 0.715753i $$-0.253916\pi$$
0.698353 + 0.715753i $$0.253916\pi$$
$$978$$ 0 0
$$979$$ 9.31371 0.297667
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −9.65685 −0.308163
$$983$$ −50.1421 −1.59929 −0.799643 0.600476i $$-0.794978\pi$$
−0.799643 + 0.600476i $$0.794978\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 3.71573 0.118333
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 16.9706 0.539633
$$990$$ 0 0
$$991$$ 9.94113 0.315790 0.157895 0.987456i $$-0.449529\pi$$
0.157895 + 0.987456i $$0.449529\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ −9.37258 −0.297280
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −9.45584 −0.299470 −0.149735 0.988726i $$-0.547842\pi$$
−0.149735 + 0.988726i $$0.547842\pi$$
$$998$$ 0.686292 0.0217242
$$999$$ 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 2475.2.a.x.1.1 2
3.2 odd 2 275.2.a.c.1.2 2
5.2 odd 4 2475.2.c.l.199.2 4
5.3 odd 4 2475.2.c.l.199.3 4
5.4 even 2 495.2.a.b.1.2 2
12.11 even 2 4400.2.a.bn.1.2 2
15.2 even 4 275.2.b.d.199.3 4
15.8 even 4 275.2.b.d.199.2 4
15.14 odd 2 55.2.a.b.1.1 2
20.19 odd 2 7920.2.a.ch.1.1 2
33.32 even 2 3025.2.a.o.1.1 2
55.54 odd 2 5445.2.a.y.1.1 2
60.23 odd 4 4400.2.b.q.4049.4 4
60.47 odd 4 4400.2.b.q.4049.1 4
60.59 even 2 880.2.a.m.1.1 2
105.104 even 2 2695.2.a.f.1.1 2
120.29 odd 2 3520.2.a.bn.1.1 2
120.59 even 2 3520.2.a.bo.1.2 2
165.14 odd 10 605.2.g.f.251.1 8
165.29 even 10 605.2.g.l.511.2 8
165.59 odd 10 605.2.g.f.511.1 8
165.74 even 10 605.2.g.l.251.2 8
165.104 odd 10 605.2.g.f.366.2 8
165.119 odd 10 605.2.g.f.81.2 8
165.134 even 10 605.2.g.l.81.1 8
165.149 even 10 605.2.g.l.366.1 8
165.164 even 2 605.2.a.d.1.2 2
195.194 odd 2 9295.2.a.g.1.2 2
660.659 odd 2 9680.2.a.bn.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
55.2.a.b.1.1 2 15.14 odd 2
275.2.a.c.1.2 2 3.2 odd 2
275.2.b.d.199.2 4 15.8 even 4
275.2.b.d.199.3 4 15.2 even 4
495.2.a.b.1.2 2 5.4 even 2
605.2.a.d.1.2 2 165.164 even 2
605.2.g.f.81.2 8 165.119 odd 10
605.2.g.f.251.1 8 165.14 odd 10
605.2.g.f.366.2 8 165.104 odd 10
605.2.g.f.511.1 8 165.59 odd 10
605.2.g.l.81.1 8 165.134 even 10
605.2.g.l.251.2 8 165.74 even 10
605.2.g.l.366.1 8 165.149 even 10
605.2.g.l.511.2 8 165.29 even 10
880.2.a.m.1.1 2 60.59 even 2
2475.2.a.x.1.1 2 1.1 even 1 trivial
2475.2.c.l.199.2 4 5.2 odd 4
2475.2.c.l.199.3 4 5.3 odd 4
2695.2.a.f.1.1 2 105.104 even 2
3025.2.a.o.1.1 2 33.32 even 2
3520.2.a.bn.1.1 2 120.29 odd 2
3520.2.a.bo.1.2 2 120.59 even 2
4400.2.a.bn.1.2 2 12.11 even 2
4400.2.b.q.4049.1 4 60.47 odd 4
4400.2.b.q.4049.4 4 60.23 odd 4
5445.2.a.y.1.1 2 55.54 odd 2
7920.2.a.ch.1.1 2 20.19 odd 2
9295.2.a.g.1.2 2 195.194 odd 2
9680.2.a.bn.1.1 2 660.659 odd 2