# Properties

 Label 4400.2.a.bn.1.2 Level $4400$ Weight $2$ Character 4400.1 Self dual yes Analytic conductor $35.134$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4400 = 2^{4} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4400.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$35.1341768894$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 55) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 4400.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.82843 q^{3} -2.00000 q^{7} +5.00000 q^{9} +O(q^{10})$$ $$q+2.82843 q^{3} -2.00000 q^{7} +5.00000 q^{9} -1.00000 q^{11} +6.82843 q^{13} -1.17157 q^{17} -5.65685 q^{21} +2.82843 q^{23} +5.65685 q^{27} +7.65685 q^{29} -2.82843 q^{33} -3.65685 q^{37} +19.3137 q^{39} +6.00000 q^{41} -6.00000 q^{43} -2.82843 q^{47} -3.00000 q^{49} -3.31371 q^{51} -0.343146 q^{53} +9.65685 q^{59} +13.3137 q^{61} -10.0000 q^{63} -4.48528 q^{67} +8.00000 q^{69} +11.3137 q^{71} +6.82843 q^{73} +2.00000 q^{77} -4.00000 q^{79} +1.00000 q^{81} -6.00000 q^{83} +21.6569 q^{87} +9.31371 q^{89} -13.6569 q^{91} +7.65685 q^{97} -5.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{7} + 10 q^{9}+O(q^{10})$$ 2 * q - 4 * q^7 + 10 * q^9 $$2 q - 4 q^{7} + 10 q^{9} - 2 q^{11} + 8 q^{13} - 8 q^{17} + 4 q^{29} + 4 q^{37} + 16 q^{39} + 12 q^{41} - 12 q^{43} - 6 q^{49} + 16 q^{51} - 12 q^{53} + 8 q^{59} + 4 q^{61} - 20 q^{63} + 8 q^{67} + 16 q^{69} + 8 q^{73} + 4 q^{77} - 8 q^{79} + 2 q^{81} - 12 q^{83} + 32 q^{87} - 4 q^{89} - 16 q^{91} + 4 q^{97} - 10 q^{99}+O(q^{100})$$ 2 * q - 4 * q^7 + 10 * q^9 - 2 * q^11 + 8 * q^13 - 8 * q^17 + 4 * q^29 + 4 * q^37 + 16 * q^39 + 12 * q^41 - 12 * q^43 - 6 * q^49 + 16 * q^51 - 12 * q^53 + 8 * q^59 + 4 * q^61 - 20 * q^63 + 8 * q^67 + 16 * q^69 + 8 * q^73 + 4 * q^77 - 8 * q^79 + 2 * q^81 - 12 * q^83 + 32 * q^87 - 4 * q^89 - 16 * q^91 + 4 * q^97 - 10 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 2.82843 1.63299 0.816497 0.577350i $$-0.195913\pi$$
0.816497 + 0.577350i $$0.195913\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −2.00000 −0.755929 −0.377964 0.925820i $$-0.623376\pi$$
−0.377964 + 0.925820i $$0.623376\pi$$
$$8$$ 0 0
$$9$$ 5.00000 1.66667
$$10$$ 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 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −1.17157 −0.284148 −0.142074 0.989856i $$-0.545377\pi$$
−0.142074 + 0.989856i $$0.545377\pi$$
$$18$$ 0 0
$$19$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$20$$ 0 0
$$21$$ −5.65685 −1.23443
$$22$$ 0 0
$$23$$ 2.82843 0.589768 0.294884 0.955533i $$-0.404719\pi$$
0.294884 + 0.955533i $$0.404719\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 5.65685 1.08866
$$28$$ 0 0
$$29$$ 7.65685 1.42184 0.710921 0.703272i $$-0.248278\pi$$
0.710921 + 0.703272i $$0.248278\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ 0 0
$$33$$ −2.82843 −0.492366
$$34$$ 0 0
$$35$$ 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
$$39$$ 19.3137 3.09267
$$40$$ 0 0
$$41$$ 6.00000 0.937043 0.468521 0.883452i $$-0.344787\pi$$
0.468521 + 0.883452i $$0.344787\pi$$
$$42$$ 0 0
$$43$$ −6.00000 −0.914991 −0.457496 0.889212i $$-0.651253\pi$$
−0.457496 + 0.889212i $$0.651253\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −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
$$51$$ −3.31371 −0.464012
$$52$$ 0 0
$$53$$ −0.343146 −0.0471347 −0.0235673 0.999722i $$-0.507502\pi$$
−0.0235673 + 0.999722i $$0.507502\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 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
$$63$$ −10.0000 −1.25988
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −4.48528 −0.547964 −0.273982 0.961735i $$-0.588341\pi$$
−0.273982 + 0.961735i $$0.588341\pi$$
$$68$$ 0 0
$$69$$ 8.00000 0.963087
$$70$$ 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 $$0.369148\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 2.00000 0.227921
$$78$$ 0 0
$$79$$ −4.00000 −0.450035 −0.225018 0.974355i $$-0.572244\pi$$
−0.225018 + 0.974355i $$0.572244\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −6.00000 −0.658586 −0.329293 0.944228i $$-0.606810\pi$$
−0.329293 + 0.944228i $$0.606810\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 21.6569 2.32186
$$88$$ 0 0
$$89$$ 9.31371 0.987251 0.493626 0.869675i $$-0.335671\pi$$
0.493626 + 0.869675i $$0.335671\pi$$
$$90$$ 0 0
$$91$$ −13.6569 −1.43163
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 7.65685 0.777436 0.388718 0.921357i $$-0.372918\pi$$
0.388718 + 0.921357i $$0.372918\pi$$
$$98$$ 0 0
$$99$$ −5.00000 −0.502519
$$100$$ 0 0
$$101$$ −13.3137 −1.32476 −0.662382 0.749166i $$-0.730454\pi$$
−0.662382 + 0.749166i $$0.730454\pi$$
$$102$$ 0 0
$$103$$ 1.17157 0.115439 0.0577193 0.998333i $$-0.481617\pi$$
0.0577193 + 0.998333i $$0.481617\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −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
$$111$$ −10.3431 −0.981728
$$112$$ 0 0
$$113$$ −8.34315 −0.784857 −0.392429 0.919782i $$-0.628365\pi$$
−0.392429 + 0.919782i $$0.628365\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 34.1421 3.15644
$$118$$ 0 0
$$119$$ 2.34315 0.214796
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 0 0
$$123$$ 16.9706 1.53018
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 15.6569 1.38932 0.694661 0.719338i $$-0.255555\pi$$
0.694661 + 0.719338i $$0.255555\pi$$
$$128$$ 0 0
$$129$$ −16.9706 −1.49417
$$130$$ 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
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −22.9706 −1.96251 −0.981254 0.192720i $$-0.938269\pi$$
−0.981254 + 0.192720i $$0.938269\pi$$
$$138$$ 0 0
$$139$$ 4.00000 0.339276 0.169638 0.985506i $$-0.445740\pi$$
0.169638 + 0.985506i $$0.445740\pi$$
$$140$$ 0 0
$$141$$ −8.00000 −0.673722
$$142$$ 0 0
$$143$$ −6.82843 −0.571022
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ −8.48528 −0.699854
$$148$$ 0 0
$$149$$ 11.6569 0.954967 0.477483 0.878641i $$-0.341549\pi$$
0.477483 + 0.878641i $$0.341549\pi$$
$$150$$ 0 0
$$151$$ 12.0000 0.976546 0.488273 0.872691i $$-0.337627\pi$$
0.488273 + 0.872691i $$0.337627\pi$$
$$152$$ 0 0
$$153$$ −5.85786 −0.473580
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 14.0000 1.11732 0.558661 0.829396i $$-0.311315\pi$$
0.558661 + 0.829396i $$0.311315\pi$$
$$158$$ 0 0
$$159$$ −0.970563 −0.0769706
$$160$$ 0 0
$$161$$ −5.65685 −0.445823
$$162$$ 0 0
$$163$$ −0.485281 −0.0380102 −0.0190051 0.999819i $$-0.506050\pi$$
−0.0190051 + 0.999819i $$0.506050\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 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
$$172$$ 0 0
$$173$$ −6.14214 −0.466978 −0.233489 0.972359i $$-0.575014\pi$$
−0.233489 + 0.972359i $$0.575014\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 27.3137 2.05302
$$178$$ 0 0
$$179$$ 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 $$0.515547\pi$$
$$182$$ 0 0
$$183$$ 37.6569 2.78367
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 1.17157 0.0856739
$$188$$ 0 0
$$189$$ −11.3137 −0.822951
$$190$$ 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 $$0.420962\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 5.17157 0.368459 0.184230 0.982883i $$-0.441021\pi$$
0.184230 + 0.982883i $$0.441021\pi$$
$$198$$ 0 0
$$199$$ −21.6569 −1.53521 −0.767607 0.640921i $$-0.778553\pi$$
−0.767607 + 0.640921i $$0.778553\pi$$
$$200$$ 0 0
$$201$$ −12.6863 −0.894822
$$202$$ 0 0
$$203$$ −15.3137 −1.07481
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 14.1421 0.982946
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 16.0000 1.10149 0.550743 0.834675i $$-0.314345\pi$$
0.550743 + 0.834675i $$0.314345\pi$$
$$212$$ 0 0
$$213$$ 32.0000 2.19260
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 19.3137 1.30510
$$220$$ 0 0
$$221$$ −8.00000 −0.538138
$$222$$ 0 0
$$223$$ −5.17157 −0.346314 −0.173157 0.984894i $$-0.555397\pi$$
−0.173157 + 0.984894i $$0.555397\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 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
$$231$$ 5.65685 0.372194
$$232$$ 0 0
$$233$$ −22.1421 −1.45058 −0.725290 0.688444i $$-0.758294\pi$$
−0.725290 + 0.688444i $$0.758294\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −11.3137 −0.734904
$$238$$ 0 0
$$239$$ 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 0
$$243$$ −14.1421 −0.907218
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ −16.9706 −1.07547
$$250$$ 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
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −13.3137 −0.830486 −0.415243 0.909710i $$-0.636304\pi$$
−0.415243 + 0.909710i $$0.636304\pi$$
$$258$$ 0 0
$$259$$ 7.31371 0.454452
$$260$$ 0 0
$$261$$ 38.2843 2.36974
$$262$$ 0 0
$$263$$ 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
$$267$$ 26.3431 1.61217
$$268$$ 0 0
$$269$$ −5.31371 −0.323983 −0.161991 0.986792i $$-0.551792\pi$$
−0.161991 + 0.986792i $$0.551792\pi$$
$$270$$ 0 0
$$271$$ 15.3137 0.930242 0.465121 0.885247i $$-0.346011\pi$$
0.465121 + 0.885247i $$0.346011\pi$$
$$272$$ 0 0
$$273$$ −38.6274 −2.33784
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −1.17157 −0.0703930 −0.0351965 0.999380i $$-0.511206\pi$$
−0.0351965 + 0.999380i $$0.511206\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −5.31371 −0.316989 −0.158495 0.987360i $$-0.550664\pi$$
−0.158495 + 0.987360i $$0.550664\pi$$
$$282$$ 0 0
$$283$$ −12.6274 −0.750622 −0.375311 0.926899i $$-0.622464\pi$$
−0.375311 + 0.926899i $$0.622464\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −12.0000 −0.708338
$$288$$ 0 0
$$289$$ −15.6274 −0.919260
$$290$$ 0 0
$$291$$ 21.6569 1.26955
$$292$$ 0 0
$$293$$ 14.8284 0.866286 0.433143 0.901325i $$-0.357405\pi$$
0.433143 + 0.901325i $$0.357405\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −5.65685 −0.328244
$$298$$ 0 0
$$299$$ 19.3137 1.11694
$$300$$ 0 0
$$301$$ 12.0000 0.691669
$$302$$ 0 0
$$303$$ −37.6569 −2.16333
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −27.6569 −1.57846 −0.789230 0.614098i $$-0.789520\pi$$
−0.789230 + 0.614098i $$0.789520\pi$$
$$308$$ 0 0
$$309$$ 3.31371 0.188510
$$310$$ 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 $$0.705773\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −21.3137 −1.19710 −0.598549 0.801087i $$-0.704256\pi$$
−0.598549 + 0.801087i $$0.704256\pi$$
$$318$$ 0 0
$$319$$ −7.65685 −0.428702
$$320$$ 0 0
$$321$$ −10.3431 −0.577298
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 10.3431 0.571977
$$328$$ 0 0
$$329$$ 5.65685 0.311872
$$330$$ 0 0
$$331$$ −15.3137 −0.841718 −0.420859 0.907126i $$-0.638271\pi$$
−0.420859 + 0.907126i $$0.638271\pi$$
$$332$$ 0 0
$$333$$ −18.2843 −1.00197
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 3.51472 0.191459 0.0957295 0.995407i $$-0.469482\pi$$
0.0957295 + 0.995407i $$0.469482\pi$$
$$338$$ 0 0
$$339$$ −23.5980 −1.28167
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 20.0000 1.07990
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −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
$$351$$ 38.6274 2.06178
$$352$$ 0 0
$$353$$ 1.31371 0.0699216 0.0349608 0.999389i $$-0.488869\pi$$
0.0349608 + 0.999389i $$0.488869\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 6.62742 0.350760
$$358$$ 0 0
$$359$$ −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 0
$$363$$ 2.82843 0.148454
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 8.48528 0.442928 0.221464 0.975169i $$-0.428916\pi$$
0.221464 + 0.975169i $$0.428916\pi$$
$$368$$ 0 0
$$369$$ 30.0000 1.56174
$$370$$ 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 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 52.2843 2.69278
$$378$$ 0 0
$$379$$ −22.3431 −1.14769 −0.573845 0.818964i $$-0.694549\pi$$
−0.573845 + 0.818964i $$0.694549\pi$$
$$380$$ 0 0
$$381$$ 44.2843 2.26875
$$382$$ 0 0
$$383$$ −34.1421 −1.74458 −0.872291 0.488987i $$-0.837366\pi$$
−0.872291 + 0.488987i $$0.837366\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −30.0000 −1.52499
$$388$$ 0 0
$$389$$ −24.6274 −1.24866 −0.624330 0.781161i $$-0.714628\pi$$
−0.624330 + 0.781161i $$0.714628\pi$$
$$390$$ 0 0
$$391$$ −3.31371 −0.167581
$$392$$ 0 0
$$393$$ −32.0000 −1.61419
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −13.3137 −0.668196 −0.334098 0.942538i $$-0.608432\pi$$
−0.334098 + 0.942538i $$0.608432\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 17.3137 0.864605 0.432303 0.901729i $$-0.357701\pi$$
0.432303 + 0.901729i $$0.357701\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 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
$$411$$ −64.9706 −3.20476
$$412$$ 0 0
$$413$$ −19.3137 −0.950365
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 11.3137 0.554035
$$418$$ 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 $$0.546708\pi$$
$$422$$ 0 0
$$423$$ −14.1421 −0.687614
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −26.6274 −1.28859
$$428$$ 0 0
$$429$$ −19.3137 −0.932475
$$430$$ 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
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 16.0000 0.763638 0.381819 0.924237i $$-0.375298\pi$$
0.381819 + 0.924237i $$0.375298\pi$$
$$440$$ 0 0
$$441$$ −15.0000 −0.714286
$$442$$ 0 0
$$443$$ −21.1716 −1.00589 −0.502946 0.864318i $$-0.667751\pi$$
−0.502946 + 0.864318i $$0.667751\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 32.9706 1.55945
$$448$$ 0 0
$$449$$ −16.6274 −0.784696 −0.392348 0.919817i $$-0.628337\pi$$
−0.392348 + 0.919817i $$0.628337\pi$$
$$450$$ 0 0
$$451$$ −6.00000 −0.282529
$$452$$ 0 0
$$453$$ 33.9411 1.59469
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 16.4853 0.771149 0.385574 0.922677i $$-0.374003\pi$$
0.385574 + 0.922677i $$0.374003\pi$$
$$458$$ 0 0
$$459$$ −6.62742 −0.309341
$$460$$ 0 0
$$461$$ −32.6274 −1.51961 −0.759805 0.650151i $$-0.774706\pi$$
−0.759805 + 0.650151i $$0.774706\pi$$
$$462$$ 0 0
$$463$$ 22.1421 1.02903 0.514516 0.857481i $$-0.327972\pi$$
0.514516 + 0.857481i $$0.327972\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −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
$$471$$ 39.5980 1.82458
$$472$$ 0 0
$$473$$ 6.00000 0.275880
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −1.71573 −0.0785578
$$478$$ 0 0
$$479$$ 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
$$482$$ 0 0
$$483$$ −16.0000 −0.728025
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −7.51472 −0.340524 −0.170262 0.985399i $$-0.554461\pi$$
−0.170262 + 0.985399i $$0.554461\pi$$
$$488$$ 0 0
$$489$$ −1.37258 −0.0620703
$$490$$ 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 $$-0.488193\pi$$
0.0370855 + 0.999312i $$0.488193\pi$$
$$500$$ 0 0
$$501$$ 31.0294 1.38629
$$502$$ 0 0
$$503$$ −28.6274 −1.27643 −0.638217 0.769857i $$-0.720328\pi$$
−0.638217 + 0.769857i $$0.720328\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 95.1127 4.22410
$$508$$ 0 0
$$509$$ 9.31371 0.412823 0.206411 0.978465i $$-0.433821\pi$$
0.206411 + 0.978465i $$0.433821\pi$$
$$510$$ 0 0
$$511$$ −13.6569 −0.604144
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 2.82843 0.124394
$$518$$ 0 0
$$519$$ −17.3726 −0.762572
$$520$$ 0 0
$$521$$ 2.68629 0.117689 0.0588443 0.998267i $$-0.481258\pi$$
0.0588443 + 0.998267i $$0.481258\pi$$
$$522$$ 0 0
$$523$$ 37.5980 1.64404 0.822022 0.569455i $$-0.192846\pi$$
0.822022 + 0.569455i $$0.192846\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −15.0000 −0.652174
$$530$$ 0 0
$$531$$ 48.2843 2.09536
$$532$$ 0 0
$$533$$ 40.9706 1.77463
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 4.68629 0.202228
$$538$$ 0 0
$$539$$ 3.00000 0.129219
$$540$$ 0 0
$$541$$ 6.00000 0.257960 0.128980 0.991647i $$-0.458830\pi$$
0.128980 + 0.991647i $$0.458830\pi$$
$$542$$ 0 0
$$543$$ −3.71573 −0.159457
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −34.0000 −1.45374 −0.726868 0.686778i $$-0.759025\pi$$
−0.726868 + 0.686778i $$0.759025\pi$$
$$548$$ 0 0
$$549$$ 66.5685 2.84108
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 8.00000 0.340195
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −38.1421 −1.61613 −0.808067 0.589090i $$-0.799486\pi$$
−0.808067 + 0.589090i $$0.799486\pi$$
$$558$$ 0 0
$$559$$ −40.9706 −1.73287
$$560$$ 0 0
$$561$$ 3.31371 0.139905
$$562$$ 0 0
$$563$$ 11.6569 0.491278 0.245639 0.969361i $$-0.421002\pi$$
0.245639 + 0.969361i $$0.421002\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ −2.00000 −0.0839921
$$568$$ 0 0
$$569$$ 20.3431 0.852829 0.426415 0.904528i $$-0.359776\pi$$
0.426415 + 0.904528i $$0.359776\pi$$
$$570$$ 0 0
$$571$$ −45.9411 −1.92258 −0.961288 0.275545i $$-0.911142\pi$$
−0.961288 + 0.275545i $$0.911142\pi$$
$$572$$ 0 0
$$573$$ 54.6274 2.28209
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −6.97056 −0.290188 −0.145094 0.989418i $$-0.546349\pi$$
−0.145094 + 0.989418i $$0.546349\pi$$
$$578$$ 0 0
$$579$$ 19.3137 0.802650
$$580$$ 0 0
$$581$$ 12.0000 0.497844
$$582$$ 0 0
$$583$$ 0.343146 0.0142116
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 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
$$591$$ 14.6274 0.601692
$$592$$ 0 0
$$593$$ −20.4853 −0.841230 −0.420615 0.907239i $$-0.638186\pi$$
−0.420615 + 0.907239i $$0.638186\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −61.2548 −2.50699
$$598$$ 0 0
$$599$$ −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 $$0.853682\pi$$
$$602$$ 0 0
$$603$$ −22.4264 −0.913274
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −18.2843 −0.742136 −0.371068 0.928606i $$-0.621008\pi$$
−0.371068 + 0.928606i $$0.621008\pi$$
$$608$$ 0 0
$$609$$ −43.3137 −1.75516
$$610$$ 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 $$0.671865\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −11.6569 −0.469287 −0.234644 0.972081i $$-0.575392\pi$$
−0.234644 + 0.972081i $$0.575392\pi$$
$$618$$ 0 0
$$619$$ 25.6569 1.03124 0.515618 0.856819i $$-0.327562\pi$$
0.515618 + 0.856819i $$0.327562\pi$$
$$620$$ 0 0
$$621$$ 16.0000 0.642058
$$622$$ 0 0
$$623$$ −18.6274 −0.746292
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 4.28427 0.170825
$$630$$ 0 0
$$631$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$632$$ 0 0
$$633$$ 45.2548 1.79872
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −20.4853 −0.811656
$$638$$ 0 0
$$639$$ 56.5685 2.23782
$$640$$ 0 0
$$641$$ 30.0000 1.18493 0.592464 0.805597i $$-0.298155\pi$$
0.592464 + 0.805597i $$0.298155\pi$$
$$642$$ 0 0
$$643$$ 49.4558 1.95035 0.975174 0.221440i $$-0.0710756\pi$$
0.975174 + 0.221440i $$0.0710756\pi$$
$$644$$ 0 0
$$645$$ 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 0
$$653$$ −0.343146 −0.0134283 −0.00671417 0.999977i $$-0.502137\pi$$
−0.00671417 + 0.999977i $$0.502137\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 34.1421 1.33201
$$658$$ 0 0
$$659$$ 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 $$0.503884\pi$$
$$662$$ 0 0
$$663$$ −22.6274 −0.878776
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 21.6569 0.838557
$$668$$ 0 0
$$669$$ −14.6274 −0.565529
$$670$$ 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 $$0.527551\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −17.1716 −0.659957 −0.329979 0.943988i $$-0.607042\pi$$
−0.329979 + 0.943988i $$0.607042\pi$$
$$678$$ 0 0
$$679$$ −15.3137 −0.587686
$$680$$ 0 0
$$681$$ 7.59798 0.291155
$$682$$ 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
$$686$$ 0 0
$$687$$ −60.2843 −2.29999
$$688$$ 0 0
$$689$$ −2.34315 −0.0892667
$$690$$ 0 0
$$691$$ 16.6863 0.634776 0.317388 0.948296i $$-0.397194\pi$$
0.317388 + 0.948296i $$0.397194\pi$$
$$692$$ 0 0
$$693$$ 10.0000 0.379869
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −7.02944 −0.266259
$$698$$ 0 0
$$699$$ −62.6274 −2.36879
$$700$$ 0 0
$$701$$ 32.6274 1.23232 0.616160 0.787621i $$-0.288687\pi$$
0.616160 + 0.787621i $$0.288687\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 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
$$711$$ −20.0000 −0.750059
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 1.94113 0.0724927
$$718$$ 0 0
$$719$$ −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
$$722$$ 0 0
$$723$$ 16.9706 0.631142
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −36.4853 −1.35316 −0.676582 0.736367i $$-0.736540\pi$$
−0.676582 + 0.736367i $$0.736540\pi$$
$$728$$ 0 0
$$729$$ −43.0000 −1.59259
$$730$$ 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 $$0.712000\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 4.48528 0.165217
$$738$$ 0 0
$$739$$ 37.9411 1.39569 0.697843 0.716250i $$-0.254143\pi$$
0.697843 + 0.716250i $$0.254143\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 29.5980 1.08584 0.542922 0.839783i $$-0.317318\pi$$
0.542922 + 0.839783i $$0.317318\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ −30.0000 −1.09764
$$748$$ 0 0
$$749$$ 7.31371 0.267237
$$750$$ 0 0
$$751$$ 16.0000 0.583848 0.291924 0.956441i $$-0.405705\pi$$
0.291924 + 0.956441i $$0.405705\pi$$
$$752$$ 0 0
$$753$$ −33.9411 −1.23688
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 9.31371 0.338512 0.169256 0.985572i $$-0.445863\pi$$
0.169256 + 0.985572i $$0.445863\pi$$
$$758$$ 0 0
$$759$$ −8.00000 −0.290382
$$760$$ 0 0
$$761$$ −30.0000 −1.08750 −0.543750 0.839248i $$-0.682996\pi$$
−0.543750 + 0.839248i $$0.682996\pi$$
$$762$$ 0 0
$$763$$ −7.31371 −0.264774
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 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
$$771$$ −37.6569 −1.35618
$$772$$ 0 0
$$773$$ 30.2843 1.08925 0.544625 0.838680i $$-0.316672\pi$$
0.544625 + 0.838680i $$0.316672\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 20.6863 0.742117
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ −11.3137 −0.404836
$$782$$ 0 0
$$783$$ 43.3137 1.54791
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 18.9706 0.676228 0.338114 0.941105i $$-0.390211\pi$$
0.338114 + 0.941105i $$0.390211\pi$$
$$788$$ 0 0
$$789$$ 64.9706 2.31301
$$790$$ 0 0
$$791$$ 16.6863 0.593296
$$792$$ 0 0
$$793$$ 90.9117 3.22837
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 12.6274 0.447286 0.223643 0.974671i $$-0.428205\pi$$
0.223643 + 0.974671i $$0.428205\pi$$
$$798$$ 0 0
$$799$$ 3.31371 0.117231
$$800$$ 0 0
$$801$$ 46.5685 1.64542
$$802$$ 0 0
$$803$$ −6.82843 −0.240970
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −15.0294 −0.529061
$$808$$ 0 0
$$809$$ 22.9706 0.807602 0.403801 0.914847i $$-0.367689\pi$$
0.403801 + 0.914847i $$0.367689\pi$$
$$810$$ 0 0
$$811$$ 13.9411 0.489539 0.244770 0.969581i $$-0.421288\pi$$
0.244770 + 0.969581i $$0.421288\pi$$
$$812$$ 0 0
$$813$$ 43.3137 1.51908
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 0 0
$$818$$ 0 0
$$819$$ −68.2843 −2.38605
$$820$$ 0 0
$$821$$ −18.6863 −0.652156 −0.326078 0.945343i $$-0.605727\pi$$
−0.326078 + 0.945343i $$0.605727\pi$$
$$822$$ 0 0
$$823$$ 36.4853 1.27180 0.635898 0.771773i $$-0.280630\pi$$
0.635898 + 0.771773i $$0.280630\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −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
$$831$$ −3.31371 −0.114951
$$832$$ 0 0
$$833$$ 3.51472 0.121778
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −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
$$842$$ 0 0
$$843$$ −15.0294 −0.517641
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −2.00000 −0.0687208
$$848$$ 0 0
$$849$$ −35.7157 −1.22576
$$850$$ 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 $$0.312283\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 48.7696 1.66594 0.832968 0.553321i $$-0.186640\pi$$
0.832968 + 0.553321i $$0.186640\pi$$
$$858$$ 0 0
$$859$$ 32.2843 1.10153 0.550763 0.834662i $$-0.314337\pi$$
0.550763 + 0.834662i $$0.314337\pi$$
$$860$$ 0 0
$$861$$ −33.9411 −1.15671
$$862$$ 0 0
$$863$$ −14.8284 −0.504766 −0.252383 0.967627i $$-0.581214\pi$$
−0.252383 + 0.967627i $$0.581214\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ −44.2010 −1.50115
$$868$$ 0 0
$$869$$ 4.00000 0.135691
$$870$$ 0 0
$$871$$ −30.6274 −1.03777
$$872$$ 0 0
$$873$$ 38.2843 1.29573
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −1.45584 −0.0491604 −0.0245802 0.999698i $$-0.507825\pi$$
−0.0245802 + 0.999698i $$0.507825\pi$$
$$878$$ 0 0
$$879$$ 41.9411 1.41464
$$880$$ 0 0
$$881$$ −52.6274 −1.77306 −0.886531 0.462668i $$-0.846892\pi$$
−0.886531 + 0.462668i $$0.846892\pi$$
$$882$$ 0 0
$$883$$ 42.8284 1.44129 0.720646 0.693304i $$-0.243845\pi$$
0.720646 + 0.693304i $$0.243845\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −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
$$891$$ −1.00000 −0.0335013
$$892$$ 0 0
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 54.6274 1.82396
$$898$$ 0 0
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 0.402020 0.0133932
$$902$$ 0 0
$$903$$ 33.9411 1.12949
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −44.4853 −1.47711 −0.738555 0.674193i $$-0.764491\pi$$
−0.738555 + 0.674193i $$0.764491\pi$$
$$908$$ 0 0
$$909$$ −66.5685 −2.20794
$$910$$ 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
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 22.6274 0.747223
$$918$$ 0 0
$$919$$ 32.0000 1.05558 0.527791 0.849374i $$-0.323020\pi$$
0.527791 + 0.849374i $$0.323020\pi$$
$$920$$ 0 0
$$921$$ −78.2254 −2.57761
$$922$$ 0 0
$$923$$ 77.2548 2.54287
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 5.85786 0.192398
$$928$$ 0 0
$$929$$ −17.3137 −0.568044 −0.284022 0.958818i $$-0.591669\pi$$
−0.284022 + 0.958818i $$0.591669\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ −77.2548 −2.52921
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −49.4558 −1.61565 −0.807826 0.589421i $$-0.799356\pi$$
−0.807826 + 0.589421i $$0.799356\pi$$
$$938$$ 0 0
$$939$$ −60.2843 −1.96730
$$940$$ 0 0
$$941$$ −29.3137 −0.955600 −0.477800 0.878469i $$-0.658566\pi$$
−0.477800 + 0.878469i $$0.658566\pi$$
$$942$$ 0 0
$$943$$ 16.9706 0.552638
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 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
$$951$$ −60.2843 −1.95485
$$952$$ 0 0
$$953$$ −58.8284 −1.90564 −0.952820 0.303536i $$-0.901833\pi$$
−0.952820 + 0.303536i $$0.901833\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ −21.6569 −0.700067
$$958$$ 0 0
$$959$$ 45.9411 1.48352
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ 0 0
$$963$$ −18.2843 −0.589202
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 18.9706 0.610052 0.305026 0.952344i $$-0.401335\pi$$
0.305026 + 0.952344i $$0.401335\pi$$
$$968$$ 0 0
$$969$$ 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
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −43.6569 −1.39671 −0.698353 0.715753i $$-0.746084\pi$$
−0.698353 + 0.715753i $$0.746084\pi$$
$$978$$ 0 0
$$979$$ −9.31371 −0.297667
$$980$$ 0 0
$$981$$ 18.2843 0.583772
$$982$$ 0 0
$$983$$ −50.1421 −1.59929 −0.799643 0.600476i $$-0.794978\pi$$
−0.799643 + 0.600476i $$0.794978\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 16.0000 0.509286
$$988$$ 0 0
$$989$$ −16.9706 −0.539633
$$990$$ 0 0
$$991$$ −9.94113 −0.315790 −0.157895 0.987456i $$-0.550471\pi$$
−0.157895 + 0.987456i $$0.550471\pi$$
$$992$$ 0 0
$$993$$ −43.3137 −1.37452
$$994$$ 0 0
$$995$$ 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 0
$$999$$ −20.6863 −0.654485
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 4400.2.a.bn.1.2 2
4.3 odd 2 275.2.a.c.1.2 2
5.2 odd 4 4400.2.b.q.4049.1 4
5.3 odd 4 4400.2.b.q.4049.4 4
5.4 even 2 880.2.a.m.1.1 2
12.11 even 2 2475.2.a.x.1.1 2
15.14 odd 2 7920.2.a.ch.1.1 2
20.3 even 4 275.2.b.d.199.2 4
20.7 even 4 275.2.b.d.199.3 4
20.19 odd 2 55.2.a.b.1.1 2
40.19 odd 2 3520.2.a.bn.1.1 2
40.29 even 2 3520.2.a.bo.1.2 2
44.43 even 2 3025.2.a.o.1.1 2
55.54 odd 2 9680.2.a.bn.1.1 2
60.23 odd 4 2475.2.c.l.199.3 4
60.47 odd 4 2475.2.c.l.199.2 4
60.59 even 2 495.2.a.b.1.2 2
140.139 even 2 2695.2.a.f.1.1 2
220.19 even 10 605.2.g.l.251.2 8
220.39 even 10 605.2.g.l.366.1 8
220.59 odd 10 605.2.g.f.511.1 8
220.79 even 10 605.2.g.l.81.1 8
220.119 odd 10 605.2.g.f.81.2 8
220.139 even 10 605.2.g.l.511.2 8
220.159 odd 10 605.2.g.f.366.2 8
220.179 odd 10 605.2.g.f.251.1 8
220.219 even 2 605.2.a.d.1.2 2
260.259 odd 2 9295.2.a.g.1.2 2
660.659 odd 2 5445.2.a.y.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
55.2.a.b.1.1 2 20.19 odd 2
275.2.a.c.1.2 2 4.3 odd 2
275.2.b.d.199.2 4 20.3 even 4
275.2.b.d.199.3 4 20.7 even 4
495.2.a.b.1.2 2 60.59 even 2
605.2.a.d.1.2 2 220.219 even 2
605.2.g.f.81.2 8 220.119 odd 10
605.2.g.f.251.1 8 220.179 odd 10
605.2.g.f.366.2 8 220.159 odd 10
605.2.g.f.511.1 8 220.59 odd 10
605.2.g.l.81.1 8 220.79 even 10
605.2.g.l.251.2 8 220.19 even 10
605.2.g.l.366.1 8 220.39 even 10
605.2.g.l.511.2 8 220.139 even 10
880.2.a.m.1.1 2 5.4 even 2
2475.2.a.x.1.1 2 12.11 even 2
2475.2.c.l.199.2 4 60.47 odd 4
2475.2.c.l.199.3 4 60.23 odd 4
2695.2.a.f.1.1 2 140.139 even 2
3025.2.a.o.1.1 2 44.43 even 2
3520.2.a.bn.1.1 2 40.19 odd 2
3520.2.a.bo.1.2 2 40.29 even 2
4400.2.a.bn.1.2 2 1.1 even 1 trivial
4400.2.b.q.4049.1 4 5.2 odd 4
4400.2.b.q.4049.4 4 5.3 odd 4
5445.2.a.y.1.1 2 660.659 odd 2
7920.2.a.ch.1.1 2 15.14 odd 2
9295.2.a.g.1.2 2 260.259 odd 2
9680.2.a.bn.1.1 2 55.54 odd 2