# Properties

 Label 6840.2.a.bk.1.2 Level $6840$ Weight $2$ Character 6840.1 Self dual yes Analytic conductor $54.618$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$6840 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6840.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$54.6176749826$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.568.1 Defining polynomial: $$x^{3} - x^{2} - 6x - 2$$ x^3 - x^2 - 6*x - 2 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 2280) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-0.363328$$ of defining polynomial Character $$\chi$$ $$=$$ 6840.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{5} -2.77801 q^{7} +O(q^{10})$$ $$q+1.00000 q^{5} -2.77801 q^{7} +0.778008 q^{11} -5.50466 q^{13} +7.00933 q^{17} +1.00000 q^{19} +1.27334 q^{23} +1.00000 q^{25} -8.23132 q^{29} +5.00933 q^{31} -2.77801 q^{35} +8.51399 q^{37} -9.06068 q^{41} -10.5140 q^{43} +6.72666 q^{47} +0.717328 q^{49} -8.72666 q^{53} +0.778008 q^{55} +5.29200 q^{59} -7.73599 q^{61} -5.50466 q^{65} -1.45331 q^{67} +12.4626 q^{71} +14.5653 q^{73} -2.16131 q^{77} +2.28267 q^{79} -3.45331 q^{83} +7.00933 q^{85} +2.51399 q^{89} +15.2920 q^{91} +1.00000 q^{95} +8.51399 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{5} - 2 q^{7}+O(q^{10})$$ 3 * q + 3 * q^5 - 2 * q^7 $$3 q + 3 q^{5} - 2 q^{7} - 4 q^{11} - 6 q^{13} + 3 q^{19} + 8 q^{23} + 3 q^{25} - 10 q^{29} - 6 q^{31} - 2 q^{35} - 6 q^{37} - 4 q^{41} + 16 q^{47} + 19 q^{49} - 22 q^{53} - 4 q^{55} - 22 q^{59} + 2 q^{61} - 6 q^{65} + 4 q^{67} + 8 q^{71} + 10 q^{73} - 36 q^{77} - 10 q^{79} - 2 q^{83} - 24 q^{89} + 8 q^{91} + 3 q^{95} - 6 q^{97}+O(q^{100})$$ 3 * q + 3 * q^5 - 2 * q^7 - 4 * q^11 - 6 * q^13 + 3 * q^19 + 8 * q^23 + 3 * q^25 - 10 * q^29 - 6 * q^31 - 2 * q^35 - 6 * q^37 - 4 * q^41 + 16 * q^47 + 19 * q^49 - 22 * q^53 - 4 * q^55 - 22 * q^59 + 2 * q^61 - 6 * q^65 + 4 * q^67 + 8 * q^71 + 10 * q^73 - 36 * q^77 - 10 * q^79 - 2 * q^83 - 24 * q^89 + 8 * q^91 + 3 * q^95 - 6 * q^97

## 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
$$3$$ 0 0
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ −2.77801 −1.04999 −0.524994 0.851106i $$-0.675932\pi$$
−0.524994 + 0.851106i $$0.675932\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 0.778008 0.234578 0.117289 0.993098i $$-0.462580\pi$$
0.117289 + 0.993098i $$0.462580\pi$$
$$12$$ 0 0
$$13$$ −5.50466 −1.52672 −0.763360 0.645974i $$-0.776452\pi$$
−0.763360 + 0.645974i $$0.776452\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 7.00933 1.70001 0.850006 0.526773i $$-0.176598\pi$$
0.850006 + 0.526773i $$0.176598\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 1.27334 0.265510 0.132755 0.991149i $$-0.457618\pi$$
0.132755 + 0.991149i $$0.457618\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −8.23132 −1.52852 −0.764259 0.644909i $$-0.776895\pi$$
−0.764259 + 0.644909i $$0.776895\pi$$
$$30$$ 0 0
$$31$$ 5.00933 0.899702 0.449851 0.893104i $$-0.351477\pi$$
0.449851 + 0.893104i $$0.351477\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −2.77801 −0.469569
$$36$$ 0 0
$$37$$ 8.51399 1.39969 0.699846 0.714294i $$-0.253252\pi$$
0.699846 + 0.714294i $$0.253252\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −9.06068 −1.41504 −0.707520 0.706693i $$-0.750186\pi$$
−0.707520 + 0.706693i $$0.750186\pi$$
$$42$$ 0 0
$$43$$ −10.5140 −1.60337 −0.801684 0.597747i $$-0.796063\pi$$
−0.801684 + 0.597747i $$0.796063\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 6.72666 0.981184 0.490592 0.871389i $$-0.336781\pi$$
0.490592 + 0.871389i $$0.336781\pi$$
$$48$$ 0 0
$$49$$ 0.717328 0.102475
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −8.72666 −1.19870 −0.599349 0.800488i $$-0.704574\pi$$
−0.599349 + 0.800488i $$0.704574\pi$$
$$54$$ 0 0
$$55$$ 0.778008 0.104907
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 5.29200 0.688960 0.344480 0.938794i $$-0.388055\pi$$
0.344480 + 0.938794i $$0.388055\pi$$
$$60$$ 0 0
$$61$$ −7.73599 −0.990491 −0.495246 0.868753i $$-0.664922\pi$$
−0.495246 + 0.868753i $$0.664922\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −5.50466 −0.682770
$$66$$ 0 0
$$67$$ −1.45331 −0.177550 −0.0887752 0.996052i $$-0.528295\pi$$
−0.0887752 + 0.996052i $$0.528295\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 12.4626 1.47904 0.739522 0.673133i $$-0.235052\pi$$
0.739522 + 0.673133i $$0.235052\pi$$
$$72$$ 0 0
$$73$$ 14.5653 1.70474 0.852372 0.522935i $$-0.175163\pi$$
0.852372 + 0.522935i $$0.175163\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −2.16131 −0.246304
$$78$$ 0 0
$$79$$ 2.28267 0.256821 0.128410 0.991721i $$-0.459013\pi$$
0.128410 + 0.991721i $$0.459013\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ −3.45331 −0.379050 −0.189525 0.981876i $$-0.560695\pi$$
−0.189525 + 0.981876i $$0.560695\pi$$
$$84$$ 0 0
$$85$$ 7.00933 0.760268
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 2.51399 0.266483 0.133241 0.991084i $$-0.457461\pi$$
0.133241 + 0.991084i $$0.457461\pi$$
$$90$$ 0 0
$$91$$ 15.2920 1.60304
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 1.00000 0.102598
$$96$$ 0 0
$$97$$ 8.51399 0.864465 0.432233 0.901762i $$-0.357726\pi$$
0.432233 + 0.901762i $$0.357726\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −15.5560 −1.54788 −0.773941 0.633258i $$-0.781717\pi$$
−0.773941 + 0.633258i $$0.781717\pi$$
$$102$$ 0 0
$$103$$ −1.45331 −0.143199 −0.0715996 0.997433i $$-0.522810\pi$$
−0.0715996 + 0.997433i $$0.522810\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −14.0187 −1.35523 −0.677617 0.735415i $$-0.736987\pi$$
−0.677617 + 0.735415i $$0.736987\pi$$
$$108$$ 0 0
$$109$$ −7.29200 −0.698447 −0.349224 0.937039i $$-0.613555\pi$$
−0.349224 + 0.937039i $$0.613555\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −19.7360 −1.85661 −0.928303 0.371825i $$-0.878732\pi$$
−0.928303 + 0.371825i $$0.878732\pi$$
$$114$$ 0 0
$$115$$ 1.27334 0.118740
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −19.4720 −1.78499
$$120$$ 0 0
$$121$$ −10.3947 −0.944973
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ 1.45331 0.128961 0.0644803 0.997919i $$-0.479461\pi$$
0.0644803 + 0.997919i $$0.479461\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −15.7873 −1.37935 −0.689673 0.724121i $$-0.742246\pi$$
−0.689673 + 0.724121i $$0.742246\pi$$
$$132$$ 0 0
$$133$$ −2.77801 −0.240884
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 0.282672 0.0241503 0.0120752 0.999927i $$-0.496156\pi$$
0.0120752 + 0.999927i $$0.496156\pi$$
$$138$$ 0 0
$$139$$ 16.4626 1.39634 0.698172 0.715931i $$-0.253997\pi$$
0.698172 + 0.715931i $$0.253997\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −4.28267 −0.358135
$$144$$ 0 0
$$145$$ −8.23132 −0.683574
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −14.0000 −1.14692 −0.573462 0.819232i $$-0.694400\pi$$
−0.573462 + 0.819232i $$0.694400\pi$$
$$150$$ 0 0
$$151$$ −21.5747 −1.75572 −0.877861 0.478916i $$-0.841030\pi$$
−0.877861 + 0.478916i $$0.841030\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 5.00933 0.402359
$$156$$ 0 0
$$157$$ −13.8387 −1.10445 −0.552224 0.833696i $$-0.686221\pi$$
−0.552224 + 0.833696i $$0.686221\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −3.53736 −0.278783
$$162$$ 0 0
$$163$$ 18.6167 1.45817 0.729086 0.684422i $$-0.239945\pi$$
0.729086 + 0.684422i $$0.239945\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 9.73599 0.753393 0.376697 0.926337i $$-0.377060\pi$$
0.376697 + 0.926337i $$0.377060\pi$$
$$168$$ 0 0
$$169$$ 17.3013 1.33087
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −4.82936 −0.367169 −0.183585 0.983004i $$-0.558770\pi$$
−0.183585 + 0.983004i $$0.558770\pi$$
$$174$$ 0 0
$$175$$ −2.77801 −0.209998
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −23.8387 −1.78179 −0.890894 0.454212i $$-0.849921\pi$$
−0.890894 + 0.454212i $$0.849921\pi$$
$$180$$ 0 0
$$181$$ −14.7267 −1.09462 −0.547312 0.836929i $$-0.684349\pi$$
−0.547312 + 0.836929i $$0.684349\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 8.51399 0.625961
$$186$$ 0 0
$$187$$ 5.45331 0.398786
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 3.22199 0.233135 0.116568 0.993183i $$-0.462811\pi$$
0.116568 + 0.993183i $$0.462811\pi$$
$$192$$ 0 0
$$193$$ 5.50466 0.396234 0.198117 0.980178i $$-0.436517\pi$$
0.198117 + 0.980178i $$0.436517\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −6.17997 −0.440305 −0.220152 0.975466i $$-0.570655\pi$$
−0.220152 + 0.975466i $$0.570655\pi$$
$$198$$ 0 0
$$199$$ −17.0280 −1.20708 −0.603541 0.797332i $$-0.706244\pi$$
−0.603541 + 0.797332i $$0.706244\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 22.8667 1.60493
$$204$$ 0 0
$$205$$ −9.06068 −0.632825
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 0.778008 0.0538159
$$210$$ 0 0
$$211$$ 8.56534 0.589663 0.294831 0.955549i $$-0.404737\pi$$
0.294831 + 0.955549i $$0.404737\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −10.5140 −0.717048
$$216$$ 0 0
$$217$$ −13.9160 −0.944677
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −38.5840 −2.59544
$$222$$ 0 0
$$223$$ −1.98134 −0.132681 −0.0663403 0.997797i $$-0.521132\pi$$
−0.0663403 + 0.997797i $$0.521132\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −0.282672 −0.0187616 −0.00938081 0.999956i $$-0.502986\pi$$
−0.00938081 + 0.999956i $$0.502986\pi$$
$$228$$ 0 0
$$229$$ 4.26401 0.281774 0.140887 0.990026i $$-0.455005\pi$$
0.140887 + 0.990026i $$0.455005\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 6.62395 0.433950 0.216975 0.976177i $$-0.430381\pi$$
0.216975 + 0.976177i $$0.430381\pi$$
$$234$$ 0 0
$$235$$ 6.72666 0.438799
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −4.21266 −0.272495 −0.136247 0.990675i $$-0.543504\pi$$
−0.136247 + 0.990675i $$0.543504\pi$$
$$240$$ 0 0
$$241$$ −25.1120 −1.61761 −0.808804 0.588078i $$-0.799885\pi$$
−0.808804 + 0.588078i $$0.799885\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 0.717328 0.0458284
$$246$$ 0 0
$$247$$ −5.50466 −0.350253
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 22.3340 1.40971 0.704856 0.709351i $$-0.251012\pi$$
0.704856 + 0.709351i $$0.251012\pi$$
$$252$$ 0 0
$$253$$ 0.990671 0.0622830
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 0.829359 0.0517340 0.0258670 0.999665i $$-0.491765\pi$$
0.0258670 + 0.999665i $$0.491765\pi$$
$$258$$ 0 0
$$259$$ −23.6519 −1.46966
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −5.73599 −0.353696 −0.176848 0.984238i $$-0.556590\pi$$
−0.176848 + 0.984238i $$0.556590\pi$$
$$264$$ 0 0
$$265$$ −8.72666 −0.536074
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 15.1379 0.922977 0.461488 0.887146i $$-0.347316\pi$$
0.461488 + 0.887146i $$0.347316\pi$$
$$270$$ 0 0
$$271$$ −24.6680 −1.49848 −0.749239 0.662300i $$-0.769580\pi$$
−0.749239 + 0.662300i $$0.769580\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0.778008 0.0469156
$$276$$ 0 0
$$277$$ −4.28267 −0.257321 −0.128660 0.991689i $$-0.541068\pi$$
−0.128660 + 0.991689i $$0.541068\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −30.5140 −1.82031 −0.910156 0.414265i $$-0.864039\pi$$
−0.910156 + 0.414265i $$0.864039\pi$$
$$282$$ 0 0
$$283$$ 17.5233 1.04165 0.520827 0.853662i $$-0.325624\pi$$
0.520827 + 0.853662i $$0.325624\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 25.1706 1.48578
$$288$$ 0 0
$$289$$ 32.1307 1.89004
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −5.71733 −0.334010 −0.167005 0.985956i $$-0.553410\pi$$
−0.167005 + 0.985956i $$0.553410\pi$$
$$294$$ 0 0
$$295$$ 5.29200 0.308112
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −7.00933 −0.405360
$$300$$ 0 0
$$301$$ 29.2080 1.68352
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −7.73599 −0.442961
$$306$$ 0 0
$$307$$ −8.46264 −0.482988 −0.241494 0.970402i $$-0.577637\pi$$
−0.241494 + 0.970402i $$0.577637\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 10.2313 0.580165 0.290082 0.957002i $$-0.406317\pi$$
0.290082 + 0.957002i $$0.406317\pi$$
$$312$$ 0 0
$$313$$ 30.1400 1.70361 0.851807 0.523855i $$-0.175507\pi$$
0.851807 + 0.523855i $$0.175507\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −7.83869 −0.440265 −0.220132 0.975470i $$-0.570649\pi$$
−0.220132 + 0.975470i $$0.570649\pi$$
$$318$$ 0 0
$$319$$ −6.40403 −0.358557
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 7.00933 0.390009
$$324$$ 0 0
$$325$$ −5.50466 −0.305344
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −18.6867 −1.03023
$$330$$ 0 0
$$331$$ −7.29200 −0.400805 −0.200402 0.979714i $$-0.564225\pi$$
−0.200402 + 0.979714i $$0.564225\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −1.45331 −0.0794030
$$336$$ 0 0
$$337$$ −10.4953 −0.571717 −0.285859 0.958272i $$-0.592279\pi$$
−0.285859 + 0.958272i $$0.592279\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 3.89730 0.211050
$$342$$ 0 0
$$343$$ 17.4533 0.942390
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 5.43466 0.291748 0.145874 0.989303i $$-0.453401\pi$$
0.145874 + 0.989303i $$0.453401\pi$$
$$348$$ 0 0
$$349$$ −1.43466 −0.0767954 −0.0383977 0.999263i $$-0.512225\pi$$
−0.0383977 + 0.999263i $$0.512225\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 5.73599 0.305296 0.152648 0.988281i $$-0.451220\pi$$
0.152648 + 0.988281i $$0.451220\pi$$
$$354$$ 0 0
$$355$$ 12.4626 0.661448
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −28.7780 −1.51885 −0.759423 0.650598i $$-0.774518\pi$$
−0.759423 + 0.650598i $$0.774518\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 14.5653 0.762385
$$366$$ 0 0
$$367$$ −31.9087 −1.66562 −0.832810 0.553559i $$-0.813269\pi$$
−0.832810 + 0.553559i $$0.813269\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 24.2427 1.25862
$$372$$ 0 0
$$373$$ −23.5233 −1.21799 −0.608996 0.793174i $$-0.708427\pi$$
−0.608996 + 0.793174i $$0.708427\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 45.3107 2.33362
$$378$$ 0 0
$$379$$ 27.2920 1.40190 0.700948 0.713212i $$-0.252761\pi$$
0.700948 + 0.713212i $$0.252761\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −5.65872 −0.289147 −0.144574 0.989494i $$-0.546181\pi$$
−0.144574 + 0.989494i $$0.546181\pi$$
$$384$$ 0 0
$$385$$ −2.16131 −0.110151
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −30.9253 −1.56797 −0.783987 0.620777i $$-0.786817\pi$$
−0.783987 + 0.620777i $$0.786817\pi$$
$$390$$ 0 0
$$391$$ 8.92528 0.451371
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 2.28267 0.114854
$$396$$ 0 0
$$397$$ −11.1893 −0.561575 −0.280787 0.959770i $$-0.590596\pi$$
−0.280787 + 0.959770i $$0.590596\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 7.60737 0.379894 0.189947 0.981794i $$-0.439168\pi$$
0.189947 + 0.981794i $$0.439168\pi$$
$$402$$ 0 0
$$403$$ −27.5747 −1.37359
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 6.62395 0.328337
$$408$$ 0 0
$$409$$ −18.6680 −0.923076 −0.461538 0.887121i $$-0.652702\pi$$
−0.461538 + 0.887121i $$0.652702\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ −14.7012 −0.723400
$$414$$ 0 0
$$415$$ −3.45331 −0.169516
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −30.8994 −1.50953 −0.754766 0.655994i $$-0.772250\pi$$
−0.754766 + 0.655994i $$0.772250\pi$$
$$420$$ 0 0
$$421$$ −21.7360 −1.05935 −0.529674 0.848202i $$-0.677686\pi$$
−0.529674 + 0.848202i $$0.677686\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 7.00933 0.340002
$$426$$ 0 0
$$427$$ 21.4906 1.04000
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −24.1400 −1.16278 −0.581392 0.813624i $$-0.697492\pi$$
−0.581392 + 0.813624i $$0.697492\pi$$
$$432$$ 0 0
$$433$$ −6.53265 −0.313939 −0.156970 0.987603i $$-0.550172\pi$$
−0.156970 + 0.987603i $$0.550172\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 1.27334 0.0609123
$$438$$ 0 0
$$439$$ 27.2080 1.29856 0.649282 0.760547i $$-0.275069\pi$$
0.649282 + 0.760547i $$0.275069\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 36.1587 1.71795 0.858975 0.512017i $$-0.171102\pi$$
0.858975 + 0.512017i $$0.171102\pi$$
$$444$$ 0 0
$$445$$ 2.51399 0.119175
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 38.6540 1.82420 0.912098 0.409973i $$-0.134462\pi$$
0.912098 + 0.409973i $$0.134462\pi$$
$$450$$ 0 0
$$451$$ −7.04928 −0.331938
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 15.2920 0.716900
$$456$$ 0 0
$$457$$ 40.1587 1.87854 0.939272 0.343174i $$-0.111502\pi$$
0.939272 + 0.343174i $$0.111502\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 22.9253 1.06774 0.533868 0.845568i $$-0.320738\pi$$
0.533868 + 0.845568i $$0.320738\pi$$
$$462$$ 0 0
$$463$$ −5.11929 −0.237914 −0.118957 0.992899i $$-0.537955\pi$$
−0.118957 + 0.992899i $$0.537955\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −35.0280 −1.62090 −0.810451 0.585807i $$-0.800778\pi$$
−0.810451 + 0.585807i $$0.800778\pi$$
$$468$$ 0 0
$$469$$ 4.03731 0.186426
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −8.17997 −0.376115
$$474$$ 0 0
$$475$$ 1.00000 0.0458831
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 40.8153 1.86490 0.932450 0.361299i $$-0.117667\pi$$
0.932450 + 0.361299i $$0.117667\pi$$
$$480$$ 0 0
$$481$$ −46.8667 −2.13694
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 8.51399 0.386601
$$486$$ 0 0
$$487$$ −21.9160 −0.993107 −0.496553 0.868006i $$-0.665401\pi$$
−0.496553 + 0.868006i $$0.665401\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 15.9927 0.721742 0.360871 0.932616i $$-0.382479\pi$$
0.360871 + 0.932616i $$0.382479\pi$$
$$492$$ 0 0
$$493$$ −57.6960 −2.59850
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −34.6213 −1.55298
$$498$$ 0 0
$$499$$ −0.668047 −0.0299059 −0.0149530 0.999888i $$-0.504760\pi$$
−0.0149530 + 0.999888i $$0.504760\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −12.4881 −0.556816 −0.278408 0.960463i $$-0.589807\pi$$
−0.278408 + 0.960463i $$0.589807\pi$$
$$504$$ 0 0
$$505$$ −15.5560 −0.692234
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 1.32469 0.0587161 0.0293580 0.999569i $$-0.490654\pi$$
0.0293580 + 0.999569i $$0.490654\pi$$
$$510$$ 0 0
$$511$$ −40.4626 −1.78996
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −1.45331 −0.0640406
$$516$$ 0 0
$$517$$ 5.23339 0.230164
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −32.3272 −1.41628 −0.708141 0.706071i $$-0.750466\pi$$
−0.708141 + 0.706071i $$0.750466\pi$$
$$522$$ 0 0
$$523$$ 33.2334 1.45319 0.726597 0.687063i $$-0.241101\pi$$
0.726597 + 0.687063i $$0.241101\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 35.1120 1.52950
$$528$$ 0 0
$$529$$ −21.3786 −0.929504
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 49.8760 2.16037
$$534$$ 0 0
$$535$$ −14.0187 −0.606079
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 0.558087 0.0240385
$$540$$ 0 0
$$541$$ 11.0934 0.476941 0.238471 0.971150i $$-0.423354\pi$$
0.238471 + 0.971150i $$0.423354\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −7.29200 −0.312355
$$546$$ 0 0
$$547$$ −5.55602 −0.237558 −0.118779 0.992921i $$-0.537898\pi$$
−0.118779 + 0.992921i $$0.537898\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −8.23132 −0.350666
$$552$$ 0 0
$$553$$ −6.34128 −0.269659
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −30.2427 −1.28143 −0.640713 0.767781i $$-0.721361\pi$$
−0.640713 + 0.767781i $$0.721361\pi$$
$$558$$ 0 0
$$559$$ 57.8760 2.44789
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −32.8853 −1.38595 −0.692976 0.720961i $$-0.743701\pi$$
−0.692976 + 0.720961i $$0.743701\pi$$
$$564$$ 0 0
$$565$$ −19.7360 −0.830299
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 9.48601 0.397674 0.198837 0.980033i $$-0.436284\pi$$
0.198837 + 0.980033i $$0.436284\pi$$
$$570$$ 0 0
$$571$$ −45.0653 −1.88592 −0.942962 0.332900i $$-0.891973\pi$$
−0.942962 + 0.332900i $$0.891973\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 1.27334 0.0531021
$$576$$ 0 0
$$577$$ −34.6680 −1.44325 −0.721625 0.692284i $$-0.756604\pi$$
−0.721625 + 0.692284i $$0.756604\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 9.59333 0.397998
$$582$$ 0 0
$$583$$ −6.78941 −0.281189
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 35.3879 1.46062 0.730308 0.683118i $$-0.239377\pi$$
0.730308 + 0.683118i $$0.239377\pi$$
$$588$$ 0 0
$$589$$ 5.00933 0.206406
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −25.2080 −1.03517 −0.517583 0.855633i $$-0.673168\pi$$
−0.517583 + 0.855633i $$0.673168\pi$$
$$594$$ 0 0
$$595$$ −19.4720 −0.798273
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −32.6027 −1.33211 −0.666054 0.745903i $$-0.732018\pi$$
−0.666054 + 0.745903i $$0.732018\pi$$
$$600$$ 0 0
$$601$$ 5.89730 0.240556 0.120278 0.992740i $$-0.461621\pi$$
0.120278 + 0.992740i $$0.461621\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −10.3947 −0.422605
$$606$$ 0 0
$$607$$ 8.46264 0.343488 0.171744 0.985142i $$-0.445060\pi$$
0.171744 + 0.985142i $$0.445060\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −37.0280 −1.49799
$$612$$ 0 0
$$613$$ 46.4413 1.87575 0.937874 0.346976i $$-0.112791\pi$$
0.937874 + 0.346976i $$0.112791\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −4.42533 −0.178157 −0.0890785 0.996025i $$-0.528392\pi$$
−0.0890785 + 0.996025i $$0.528392\pi$$
$$618$$ 0 0
$$619$$ 28.9907 1.16523 0.582617 0.812747i $$-0.302029\pi$$
0.582617 + 0.812747i $$0.302029\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −6.98389 −0.279804
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 59.6774 2.37949
$$630$$ 0 0
$$631$$ −1.98134 −0.0788760 −0.0394380 0.999222i $$-0.512557\pi$$
−0.0394380 + 0.999222i $$0.512557\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 1.45331 0.0576730
$$636$$ 0 0
$$637$$ −3.94865 −0.156451
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 4.17271 0.164812 0.0824061 0.996599i $$-0.473740\pi$$
0.0824061 + 0.996599i $$0.473740\pi$$
$$642$$ 0 0
$$643$$ −14.6167 −0.576426 −0.288213 0.957566i $$-0.593061\pi$$
−0.288213 + 0.957566i $$0.593061\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −1.17064 −0.0460226 −0.0230113 0.999735i $$-0.507325\pi$$
−0.0230113 + 0.999735i $$0.507325\pi$$
$$648$$ 0 0
$$649$$ 4.11722 0.161615
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 17.6519 0.690774 0.345387 0.938460i $$-0.387748\pi$$
0.345387 + 0.938460i $$0.387748\pi$$
$$654$$ 0 0
$$655$$ −15.7873 −0.616862
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −28.5067 −1.11046 −0.555232 0.831695i $$-0.687371\pi$$
−0.555232 + 0.831695i $$0.687371\pi$$
$$660$$ 0 0
$$661$$ 29.5161 1.14804 0.574021 0.818841i $$-0.305383\pi$$
0.574021 + 0.818841i $$0.305383\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −2.77801 −0.107727
$$666$$ 0 0
$$667$$ −10.4813 −0.405838
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −6.01866 −0.232348
$$672$$ 0 0
$$673$$ 1.50466 0.0580005 0.0290003 0.999579i $$-0.490768\pi$$
0.0290003 + 0.999579i $$0.490768\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 9.85735 0.378849 0.189424 0.981895i $$-0.439338\pi$$
0.189424 + 0.981895i $$0.439338\pi$$
$$678$$ 0 0
$$679$$ −23.6519 −0.907678
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −1.65872 −0.0634691 −0.0317346 0.999496i $$-0.510103\pi$$
−0.0317346 + 0.999496i $$0.510103\pi$$
$$684$$ 0 0
$$685$$ 0.282672 0.0108004
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 48.0373 1.83008
$$690$$ 0 0
$$691$$ 19.2147 0.730963 0.365481 0.930819i $$-0.380904\pi$$
0.365481 + 0.930819i $$0.380904\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 16.4626 0.624464
$$696$$ 0 0
$$697$$ −63.5093 −2.40559
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −34.5653 −1.30552 −0.652758 0.757567i $$-0.726388\pi$$
−0.652758 + 0.757567i $$0.726388\pi$$
$$702$$ 0 0
$$703$$ 8.51399 0.321111
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 43.2147 1.62526
$$708$$ 0 0
$$709$$ 2.84802 0.106960 0.0534798 0.998569i $$-0.482969\pi$$
0.0534798 + 0.998569i $$0.482969\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 6.37860 0.238880
$$714$$ 0 0
$$715$$ −4.28267 −0.160163
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 6.12862 0.228559 0.114279 0.993449i $$-0.463544\pi$$
0.114279 + 0.993449i $$0.463544\pi$$
$$720$$ 0 0
$$721$$ 4.03731 0.150357
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −8.23132 −0.305704
$$726$$ 0 0
$$727$$ −10.6753 −0.395925 −0.197963 0.980210i $$-0.563432\pi$$
−0.197963 + 0.980210i $$0.563432\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −73.6960 −2.72575
$$732$$ 0 0
$$733$$ 40.6426 1.50117 0.750585 0.660774i $$-0.229772\pi$$
0.750585 + 0.660774i $$0.229772\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −1.13069 −0.0416495
$$738$$ 0 0
$$739$$ 26.2241 0.964668 0.482334 0.875987i $$-0.339789\pi$$
0.482334 + 0.875987i $$0.339789\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 8.56534 0.314232 0.157116 0.987580i $$-0.449780\pi$$
0.157116 + 0.987580i $$0.449780\pi$$
$$744$$ 0 0
$$745$$ −14.0000 −0.512920
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 38.9439 1.42298
$$750$$ 0 0
$$751$$ −41.3693 −1.50959 −0.754793 0.655963i $$-0.772263\pi$$
−0.754793 + 0.655963i $$0.772263\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −21.5747 −0.785183
$$756$$ 0 0
$$757$$ 34.1986 1.24297 0.621485 0.783426i $$-0.286530\pi$$
0.621485 + 0.783426i $$0.286530\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −6.00000 −0.217500 −0.108750 0.994069i $$-0.534685\pi$$
−0.108750 + 0.994069i $$0.534685\pi$$
$$762$$ 0 0
$$763$$ 20.2572 0.733361
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −29.1307 −1.05185
$$768$$ 0 0
$$769$$ −24.5840 −0.886522 −0.443261 0.896393i $$-0.646178\pi$$
−0.443261 + 0.896393i $$0.646178\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −42.2173 −1.51845 −0.759225 0.650828i $$-0.774422\pi$$
−0.759225 + 0.650828i $$0.774422\pi$$
$$774$$ 0 0
$$775$$ 5.00933 0.179940
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −9.06068 −0.324633
$$780$$ 0 0
$$781$$ 9.69603 0.346951
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −13.8387 −0.493924
$$786$$ 0 0
$$787$$ 24.8226 0.884829 0.442415 0.896811i $$-0.354122\pi$$
0.442415 + 0.896811i $$0.354122\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 54.8267 1.94941
$$792$$ 0 0
$$793$$ 42.5840 1.51220
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −11.1706 −0.395684 −0.197842 0.980234i $$-0.563393\pi$$
−0.197842 + 0.980234i $$0.563393\pi$$
$$798$$ 0 0
$$799$$ 47.1493 1.66802
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 11.3320 0.399896
$$804$$ 0 0
$$805$$ −3.53736 −0.124675
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −24.9066 −0.875670 −0.437835 0.899055i $$-0.644255\pi$$
−0.437835 + 0.899055i $$0.644255\pi$$
$$810$$ 0 0
$$811$$ 43.6774 1.53372 0.766860 0.641814i $$-0.221818\pi$$
0.766860 + 0.641814i $$0.221818\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 18.6167 0.652114
$$816$$ 0 0
$$817$$ −10.5140 −0.367838
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 16.8039 0.586461 0.293230 0.956042i $$-0.405270\pi$$
0.293230 + 0.956042i $$0.405270\pi$$
$$822$$ 0 0
$$823$$ 8.33402 0.290506 0.145253 0.989395i $$-0.453600\pi$$
0.145253 + 0.989395i $$0.453600\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 11.7173 0.407451 0.203726 0.979028i $$-0.434695\pi$$
0.203726 + 0.979028i $$0.434695\pi$$
$$828$$ 0 0
$$829$$ 16.2173 0.563250 0.281625 0.959525i $$-0.409127\pi$$
0.281625 + 0.959525i $$0.409127\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 5.02799 0.174209
$$834$$ 0 0
$$835$$ 9.73599 0.336928
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 35.6120 1.22946 0.614731 0.788737i $$-0.289264\pi$$
0.614731 + 0.788737i $$0.289264\pi$$
$$840$$ 0 0
$$841$$ 38.7546 1.33637
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 17.3013 0.595184
$$846$$ 0 0
$$847$$ 28.8766 0.992211
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 10.8412 0.371633
$$852$$ 0 0
$$853$$ −24.3854 −0.834939 −0.417470 0.908691i $$-0.637083\pi$$
−0.417470 + 0.908691i $$0.637083\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 39.2733 1.34155 0.670776 0.741660i $$-0.265961\pi$$
0.670776 + 0.741660i $$0.265961\pi$$
$$858$$ 0 0
$$859$$ 15.4720 0.527897 0.263948 0.964537i $$-0.414975\pi$$
0.263948 + 0.964537i $$0.414975\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −5.65872 −0.192625 −0.0963125 0.995351i $$-0.530705\pi$$
−0.0963125 + 0.995351i $$0.530705\pi$$
$$864$$ 0 0
$$865$$ −4.82936 −0.164203
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 1.77594 0.0602445
$$870$$ 0 0
$$871$$ 8.00000 0.271070
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ −2.77801 −0.0939138
$$876$$ 0 0
$$877$$ −3.06068 −0.103352 −0.0516759 0.998664i $$-0.516456\pi$$
−0.0516759 + 0.998664i $$0.516456\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −12.6867 −0.427426 −0.213713 0.976896i $$-0.568556\pi$$
−0.213713 + 0.976896i $$0.568556\pi$$
$$882$$ 0 0
$$883$$ −27.0793 −0.911292 −0.455646 0.890161i $$-0.650592\pi$$
−0.455646 + 0.890161i $$0.650592\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 12.8880 0.432736 0.216368 0.976312i $$-0.430579\pi$$
0.216368 + 0.976312i $$0.430579\pi$$
$$888$$ 0 0
$$889$$ −4.03731 −0.135407
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 6.72666 0.225099
$$894$$ 0 0
$$895$$ −23.8387 −0.796839
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −41.2334 −1.37521
$$900$$ 0 0
$$901$$ −61.1680 −2.03780
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −14.7267 −0.489531
$$906$$ 0 0
$$907$$ −13.5560 −0.450120 −0.225060 0.974345i $$-0.572258\pi$$
−0.225060 + 0.974345i $$0.572258\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 17.0934 0.566329 0.283164 0.959071i $$-0.408616\pi$$
0.283164 + 0.959071i $$0.408616\pi$$
$$912$$ 0 0
$$913$$ −2.68670 −0.0889169
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 43.8573 1.44830
$$918$$ 0 0
$$919$$ 24.7708 0.817112 0.408556 0.912733i $$-0.366033\pi$$
0.408556 + 0.912733i $$0.366033\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ −68.6027 −2.25808
$$924$$ 0 0
$$925$$ 8.51399 0.279938
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −1.57467 −0.0516634 −0.0258317 0.999666i $$-0.508223\pi$$
−0.0258317 + 0.999666i $$0.508223\pi$$
$$930$$ 0 0
$$931$$ 0.717328 0.0235095
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 5.45331 0.178342
$$936$$ 0 0
$$937$$ −27.5933 −0.901435 −0.450717 0.892667i $$-0.648832\pi$$
−0.450717 + 0.892667i $$0.648832\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −52.3713 −1.70726 −0.853628 0.520882i $$-0.825603\pi$$
−0.853628 + 0.520882i $$0.825603\pi$$
$$942$$ 0 0
$$943$$ −11.5374 −0.375708
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 46.2427 1.50269 0.751343 0.659912i $$-0.229406\pi$$
0.751343 + 0.659912i $$0.229406\pi$$
$$948$$ 0 0
$$949$$ −80.1773 −2.60267
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −37.1893 −1.20468 −0.602340 0.798240i $$-0.705765\pi$$
−0.602340 + 0.798240i $$0.705765\pi$$
$$954$$ 0 0
$$955$$ 3.22199 0.104261
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −0.785266 −0.0253576
$$960$$ 0 0
$$961$$ −5.90663 −0.190536
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 5.50466 0.177201
$$966$$ 0 0
$$967$$ 42.2500 1.35867 0.679334 0.733829i $$-0.262269\pi$$
0.679334 + 0.733829i $$0.262269\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 7.73599 0.248260 0.124130 0.992266i $$-0.460386\pi$$
0.124130 + 0.992266i $$0.460386\pi$$
$$972$$ 0 0
$$973$$ −45.7333 −1.46614
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 38.9880 1.24734 0.623669 0.781689i $$-0.285641\pi$$
0.623669 + 0.781689i $$0.285641\pi$$
$$978$$ 0 0
$$979$$ 1.95591 0.0625110
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 7.39470 0.235854 0.117927 0.993022i $$-0.462375\pi$$
0.117927 + 0.993022i $$0.462375\pi$$
$$984$$ 0 0
$$985$$ −6.17997 −0.196910
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −13.3879 −0.425711
$$990$$ 0 0
$$991$$ 44.6613 1.41871 0.709356 0.704850i $$-0.248986\pi$$
0.709356 + 0.704850i $$0.248986\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −17.0280 −0.539823
$$996$$ 0 0
$$997$$ −8.74531 −0.276967 −0.138483 0.990365i $$-0.544223\pi$$
−0.138483 + 0.990365i $$0.544223\pi$$
$$998$$ 0 0
$$999$$ 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 6840.2.a.bk.1.2 3
3.2 odd 2 2280.2.a.r.1.2 3
12.11 even 2 4560.2.a.bt.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
2280.2.a.r.1.2 3 3.2 odd 2
4560.2.a.bt.1.2 3 12.11 even 2
6840.2.a.bk.1.2 3 1.1 even 1 trivial