# Properties

 Label 4200.2.a.bq.1.1 Level $4200$ Weight $2$ Character 4200.1 Self dual yes Analytic conductor $33.537$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4200 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4200.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$33.5371688489$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.148.1 Defining polynomial: $$x^{3} - x^{2} - 3x + 1$$ x^3 - x^2 - 3*x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 840) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$0.311108$$ of defining polynomial Character $$\chi$$ $$=$$ 4200.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} +1.00000 q^{7} +1.00000 q^{9} -3.80642 q^{11} +0.622216 q^{13} -4.42864 q^{17} +0.622216 q^{19} +1.00000 q^{21} +2.62222 q^{23} +1.00000 q^{27} +9.61285 q^{29} -0.622216 q^{31} -3.80642 q^{33} -1.24443 q^{37} +0.622216 q^{39} +4.62222 q^{41} -4.85728 q^{43} +11.6128 q^{47} +1.00000 q^{49} -4.42864 q^{51} +13.4795 q^{53} +0.622216 q^{57} +11.6128 q^{59} -8.10171 q^{61} +1.00000 q^{63} +2.62222 q^{69} +2.56199 q^{71} +10.9906 q^{73} -3.80642 q^{77} -6.75557 q^{79} +1.00000 q^{81} +11.6128 q^{83} +9.61285 q^{87} -8.23506 q^{89} +0.622216 q^{91} -0.622216 q^{93} +4.23506 q^{97} -3.80642 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{3} + 3 q^{7} + 3 q^{9}+O(q^{10})$$ 3 * q + 3 * q^3 + 3 * q^7 + 3 * q^9 $$3 q + 3 q^{3} + 3 q^{7} + 3 q^{9} + 2 q^{11} + 2 q^{13} + 2 q^{19} + 3 q^{21} + 8 q^{23} + 3 q^{27} + 2 q^{29} - 2 q^{31} + 2 q^{33} - 4 q^{37} + 2 q^{39} + 14 q^{41} + 12 q^{43} + 8 q^{47} + 3 q^{49} + 14 q^{53} + 2 q^{57} + 8 q^{59} + 2 q^{61} + 3 q^{63} + 8 q^{69} - 6 q^{71} + 6 q^{73} + 2 q^{77} - 20 q^{79} + 3 q^{81} + 8 q^{83} + 2 q^{87} + 2 q^{89} + 2 q^{91} - 2 q^{93} - 14 q^{97} + 2 q^{99}+O(q^{100})$$ 3 * q + 3 * q^3 + 3 * q^7 + 3 * q^9 + 2 * q^11 + 2 * q^13 + 2 * q^19 + 3 * q^21 + 8 * q^23 + 3 * q^27 + 2 * q^29 - 2 * q^31 + 2 * q^33 - 4 * q^37 + 2 * q^39 + 14 * q^41 + 12 * q^43 + 8 * q^47 + 3 * q^49 + 14 * q^53 + 2 * q^57 + 8 * q^59 + 2 * q^61 + 3 * q^63 + 8 * q^69 - 6 * q^71 + 6 * q^73 + 2 * q^77 - 20 * q^79 + 3 * q^81 + 8 * q^83 + 2 * q^87 + 2 * q^89 + 2 * q^91 - 2 * q^93 - 14 * q^97 + 2 * 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
$$3$$ 1.00000 0.577350
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 1.00000 0.377964
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −3.80642 −1.14768 −0.573840 0.818967i $$-0.694547\pi$$
−0.573840 + 0.818967i $$0.694547\pi$$
$$12$$ 0 0
$$13$$ 0.622216 0.172572 0.0862858 0.996270i $$-0.472500\pi$$
0.0862858 + 0.996270i $$0.472500\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −4.42864 −1.07410 −0.537051 0.843550i $$-0.680462\pi$$
−0.537051 + 0.843550i $$0.680462\pi$$
$$18$$ 0 0
$$19$$ 0.622216 0.142746 0.0713730 0.997450i $$-0.477262\pi$$
0.0713730 + 0.997450i $$0.477262\pi$$
$$20$$ 0 0
$$21$$ 1.00000 0.218218
$$22$$ 0 0
$$23$$ 2.62222 0.546770 0.273385 0.961905i $$-0.411857\pi$$
0.273385 + 0.961905i $$0.411857\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ 9.61285 1.78506 0.892531 0.450987i $$-0.148928\pi$$
0.892531 + 0.450987i $$0.148928\pi$$
$$30$$ 0 0
$$31$$ −0.622216 −0.111753 −0.0558766 0.998438i $$-0.517795\pi$$
−0.0558766 + 0.998438i $$0.517795\pi$$
$$32$$ 0 0
$$33$$ −3.80642 −0.662613
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −1.24443 −0.204583 −0.102292 0.994754i $$-0.532617\pi$$
−0.102292 + 0.994754i $$0.532617\pi$$
$$38$$ 0 0
$$39$$ 0.622216 0.0996342
$$40$$ 0 0
$$41$$ 4.62222 0.721869 0.360934 0.932591i $$-0.382458\pi$$
0.360934 + 0.932591i $$0.382458\pi$$
$$42$$ 0 0
$$43$$ −4.85728 −0.740728 −0.370364 0.928887i $$-0.620767\pi$$
−0.370364 + 0.928887i $$0.620767\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 11.6128 1.69391 0.846954 0.531666i $$-0.178434\pi$$
0.846954 + 0.531666i $$0.178434\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ −4.42864 −0.620134
$$52$$ 0 0
$$53$$ 13.4795 1.85155 0.925775 0.378074i $$-0.123413\pi$$
0.925775 + 0.378074i $$0.123413\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0.622216 0.0824145
$$58$$ 0 0
$$59$$ 11.6128 1.51186 0.755932 0.654650i $$-0.227184\pi$$
0.755932 + 0.654650i $$0.227184\pi$$
$$60$$ 0 0
$$61$$ −8.10171 −1.03732 −0.518659 0.854981i $$-0.673569\pi$$
−0.518659 + 0.854981i $$0.673569\pi$$
$$62$$ 0 0
$$63$$ 1.00000 0.125988
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$68$$ 0 0
$$69$$ 2.62222 0.315678
$$70$$ 0 0
$$71$$ 2.56199 0.304053 0.152026 0.988376i $$-0.451420\pi$$
0.152026 + 0.988376i $$0.451420\pi$$
$$72$$ 0 0
$$73$$ 10.9906 1.28636 0.643178 0.765717i $$-0.277615\pi$$
0.643178 + 0.765717i $$0.277615\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −3.80642 −0.433782
$$78$$ 0 0
$$79$$ −6.75557 −0.760061 −0.380030 0.924974i $$-0.624086\pi$$
−0.380030 + 0.924974i $$0.624086\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 11.6128 1.27468 0.637338 0.770585i $$-0.280036\pi$$
0.637338 + 0.770585i $$0.280036\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 9.61285 1.03061
$$88$$ 0 0
$$89$$ −8.23506 −0.872915 −0.436457 0.899725i $$-0.643767\pi$$
−0.436457 + 0.899725i $$0.643767\pi$$
$$90$$ 0 0
$$91$$ 0.622216 0.0652259
$$92$$ 0 0
$$93$$ −0.622216 −0.0645208
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 4.23506 0.430006 0.215003 0.976613i $$-0.431024\pi$$
0.215003 + 0.976613i $$0.431024\pi$$
$$98$$ 0 0
$$99$$ −3.80642 −0.382560
$$100$$ 0 0
$$101$$ 18.7239 1.86310 0.931550 0.363613i $$-0.118457\pi$$
0.931550 + 0.363613i $$0.118457\pi$$
$$102$$ 0 0
$$103$$ −0.857279 −0.0844702 −0.0422351 0.999108i $$-0.513448\pi$$
−0.0422351 + 0.999108i $$0.513448\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −11.0923 −1.07234 −0.536169 0.844111i $$-0.680129\pi$$
−0.536169 + 0.844111i $$0.680129\pi$$
$$108$$ 0 0
$$109$$ 5.61285 0.537613 0.268807 0.963194i $$-0.413371\pi$$
0.268807 + 0.963194i $$0.413371\pi$$
$$110$$ 0 0
$$111$$ −1.24443 −0.118116
$$112$$ 0 0
$$113$$ 16.2351 1.52727 0.763633 0.645650i $$-0.223414\pi$$
0.763633 + 0.645650i $$0.223414\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 0.622216 0.0575239
$$118$$ 0 0
$$119$$ −4.42864 −0.405973
$$120$$ 0 0
$$121$$ 3.48886 0.317169
$$122$$ 0 0
$$123$$ 4.62222 0.416771
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 15.3461 1.36175 0.680875 0.732400i $$-0.261600\pi$$
0.680875 + 0.732400i $$0.261600\pi$$
$$128$$ 0 0
$$129$$ −4.85728 −0.427660
$$130$$ 0 0
$$131$$ −4.00000 −0.349482 −0.174741 0.984614i $$-0.555909\pi$$
−0.174741 + 0.984614i $$0.555909\pi$$
$$132$$ 0 0
$$133$$ 0.622216 0.0539529
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −17.0923 −1.46030 −0.730149 0.683288i $$-0.760549\pi$$
−0.730149 + 0.683288i $$0.760549\pi$$
$$138$$ 0 0
$$139$$ −13.4795 −1.14332 −0.571658 0.820492i $$-0.693700\pi$$
−0.571658 + 0.820492i $$0.693700\pi$$
$$140$$ 0 0
$$141$$ 11.6128 0.977978
$$142$$ 0 0
$$143$$ −2.36842 −0.198057
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 1.00000 0.0824786
$$148$$ 0 0
$$149$$ −9.34614 −0.765666 −0.382833 0.923818i $$-0.625051\pi$$
−0.382833 + 0.923818i $$0.625051\pi$$
$$150$$ 0 0
$$151$$ −7.14272 −0.581266 −0.290633 0.956835i $$-0.593866\pi$$
−0.290633 + 0.956835i $$0.593866\pi$$
$$152$$ 0 0
$$153$$ −4.42864 −0.358034
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −6.99063 −0.557913 −0.278957 0.960304i $$-0.589989\pi$$
−0.278957 + 0.960304i $$0.589989\pi$$
$$158$$ 0 0
$$159$$ 13.4795 1.06899
$$160$$ 0 0
$$161$$ 2.62222 0.206660
$$162$$ 0 0
$$163$$ 15.6128 1.22289 0.611446 0.791286i $$-0.290588\pi$$
0.611446 + 0.791286i $$0.290588\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −1.51114 −0.116935 −0.0584677 0.998289i $$-0.518621\pi$$
−0.0584677 + 0.998289i $$0.518621\pi$$
$$168$$ 0 0
$$169$$ −12.6128 −0.970219
$$170$$ 0 0
$$171$$ 0.622216 0.0475820
$$172$$ 0 0
$$173$$ −6.53035 −0.496493 −0.248247 0.968697i $$-0.579854\pi$$
−0.248247 + 0.968697i $$0.579854\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 11.6128 0.872875
$$178$$ 0 0
$$179$$ −6.29529 −0.470532 −0.235266 0.971931i $$-0.575596\pi$$
−0.235266 + 0.971931i $$0.575596\pi$$
$$180$$ 0 0
$$181$$ −6.85728 −0.509698 −0.254849 0.966981i $$-0.582026\pi$$
−0.254849 + 0.966981i $$0.582026\pi$$
$$182$$ 0 0
$$183$$ −8.10171 −0.598896
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 16.8573 1.23273
$$188$$ 0 0
$$189$$ 1.00000 0.0727393
$$190$$ 0 0
$$191$$ −10.5620 −0.764239 −0.382119 0.924113i $$-0.624806\pi$$
−0.382119 + 0.924113i $$0.624806\pi$$
$$192$$ 0 0
$$193$$ 5.24443 0.377502 0.188751 0.982025i $$-0.439556\pi$$
0.188751 + 0.982025i $$0.439556\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 17.7462 1.26436 0.632182 0.774820i $$-0.282159\pi$$
0.632182 + 0.774820i $$0.282159\pi$$
$$198$$ 0 0
$$199$$ −20.2351 −1.43443 −0.717213 0.696854i $$-0.754582\pi$$
−0.717213 + 0.696854i $$0.754582\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 9.61285 0.674690
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 2.62222 0.182257
$$208$$ 0 0
$$209$$ −2.36842 −0.163827
$$210$$ 0 0
$$211$$ 21.3274 1.46824 0.734120 0.679020i $$-0.237595\pi$$
0.734120 + 0.679020i $$0.237595\pi$$
$$212$$ 0 0
$$213$$ 2.56199 0.175545
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −0.622216 −0.0422387
$$218$$ 0 0
$$219$$ 10.9906 0.742678
$$220$$ 0 0
$$221$$ −2.75557 −0.185360
$$222$$ 0 0
$$223$$ −9.71456 −0.650535 −0.325267 0.945622i $$-0.605454\pi$$
−0.325267 + 0.945622i $$0.605454\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −11.3461 −0.753070 −0.376535 0.926402i $$-0.622885\pi$$
−0.376535 + 0.926402i $$0.622885\pi$$
$$228$$ 0 0
$$229$$ 1.34614 0.0889555 0.0444778 0.999010i $$-0.485838\pi$$
0.0444778 + 0.999010i $$0.485838\pi$$
$$230$$ 0 0
$$231$$ −3.80642 −0.250444
$$232$$ 0 0
$$233$$ 15.3778 1.00743 0.503716 0.863869i $$-0.331966\pi$$
0.503716 + 0.863869i $$0.331966\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −6.75557 −0.438821
$$238$$ 0 0
$$239$$ 7.53972 0.487704 0.243852 0.969812i $$-0.421589\pi$$
0.243852 + 0.969812i $$0.421589\pi$$
$$240$$ 0 0
$$241$$ 23.9813 1.54477 0.772385 0.635155i $$-0.219064\pi$$
0.772385 + 0.635155i $$0.219064\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0.387152 0.0246339
$$248$$ 0 0
$$249$$ 11.6128 0.735934
$$250$$ 0 0
$$251$$ 14.1017 0.890092 0.445046 0.895508i $$-0.353187\pi$$
0.445046 + 0.895508i $$0.353187\pi$$
$$252$$ 0 0
$$253$$ −9.98126 −0.627517
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 17.0192 1.06163 0.530815 0.847488i $$-0.321886\pi$$
0.530815 + 0.847488i $$0.321886\pi$$
$$258$$ 0 0
$$259$$ −1.24443 −0.0773252
$$260$$ 0 0
$$261$$ 9.61285 0.595020
$$262$$ 0 0
$$263$$ 12.6035 0.777164 0.388582 0.921414i $$-0.372965\pi$$
0.388582 + 0.921414i $$0.372965\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −8.23506 −0.503978
$$268$$ 0 0
$$269$$ 3.76494 0.229552 0.114776 0.993391i $$-0.463385\pi$$
0.114776 + 0.993391i $$0.463385\pi$$
$$270$$ 0 0
$$271$$ −17.8666 −1.08532 −0.542661 0.839952i $$-0.682583\pi$$
−0.542661 + 0.839952i $$0.682583\pi$$
$$272$$ 0 0
$$273$$ 0.622216 0.0376582
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −1.24443 −0.0747706 −0.0373853 0.999301i $$-0.511903\pi$$
−0.0373853 + 0.999301i $$0.511903\pi$$
$$278$$ 0 0
$$279$$ −0.622216 −0.0372511
$$280$$ 0 0
$$281$$ −8.95899 −0.534448 −0.267224 0.963634i $$-0.586106\pi$$
−0.267224 + 0.963634i $$0.586106\pi$$
$$282$$ 0 0
$$283$$ 30.5718 1.81731 0.908654 0.417551i $$-0.137111\pi$$
0.908654 + 0.417551i $$0.137111\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 4.62222 0.272841
$$288$$ 0 0
$$289$$ 2.61285 0.153697
$$290$$ 0 0
$$291$$ 4.23506 0.248264
$$292$$ 0 0
$$293$$ −5.67307 −0.331424 −0.165712 0.986174i $$-0.552992\pi$$
−0.165712 + 0.986174i $$0.552992\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −3.80642 −0.220871
$$298$$ 0 0
$$299$$ 1.63158 0.0943569
$$300$$ 0 0
$$301$$ −4.85728 −0.279969
$$302$$ 0 0
$$303$$ 18.7239 1.07566
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −4.85728 −0.277220 −0.138610 0.990347i $$-0.544263\pi$$
−0.138610 + 0.990347i $$0.544263\pi$$
$$308$$ 0 0
$$309$$ −0.857279 −0.0487689
$$310$$ 0 0
$$311$$ −34.5718 −1.96039 −0.980195 0.198037i $$-0.936543\pi$$
−0.980195 + 0.198037i $$0.936543\pi$$
$$312$$ 0 0
$$313$$ −6.33677 −0.358176 −0.179088 0.983833i $$-0.557315\pi$$
−0.179088 + 0.983833i $$0.557315\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 15.9684 0.896872 0.448436 0.893815i $$-0.351981\pi$$
0.448436 + 0.893815i $$0.351981\pi$$
$$318$$ 0 0
$$319$$ −36.5906 −2.04868
$$320$$ 0 0
$$321$$ −11.0923 −0.619114
$$322$$ 0 0
$$323$$ −2.75557 −0.153324
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 5.61285 0.310391
$$328$$ 0 0
$$329$$ 11.6128 0.640237
$$330$$ 0 0
$$331$$ −27.6128 −1.51774 −0.758870 0.651243i $$-0.774248\pi$$
−0.758870 + 0.651243i $$0.774248\pi$$
$$332$$ 0 0
$$333$$ −1.24443 −0.0681944
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 16.0000 0.871576 0.435788 0.900049i $$-0.356470\pi$$
0.435788 + 0.900049i $$0.356470\pi$$
$$338$$ 0 0
$$339$$ 16.2351 0.881768
$$340$$ 0 0
$$341$$ 2.36842 0.128257
$$342$$ 0 0
$$343$$ 1.00000 0.0539949
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −11.8666 −0.637035 −0.318517 0.947917i $$-0.603185\pi$$
−0.318517 + 0.947917i $$0.603185\pi$$
$$348$$ 0 0
$$349$$ 21.8163 1.16780 0.583899 0.811826i $$-0.301526\pi$$
0.583899 + 0.811826i $$0.301526\pi$$
$$350$$ 0 0
$$351$$ 0.622216 0.0332114
$$352$$ 0 0
$$353$$ −2.79706 −0.148872 −0.0744361 0.997226i $$-0.523716\pi$$
−0.0744361 + 0.997226i $$0.523716\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ −4.42864 −0.234388
$$358$$ 0 0
$$359$$ −13.0509 −0.688798 −0.344399 0.938823i $$-0.611917\pi$$
−0.344399 + 0.938823i $$0.611917\pi$$
$$360$$ 0 0
$$361$$ −18.6128 −0.979624
$$362$$ 0 0
$$363$$ 3.48886 0.183118
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −10.4889 −0.547514 −0.273757 0.961799i $$-0.588266\pi$$
−0.273757 + 0.961799i $$0.588266\pi$$
$$368$$ 0 0
$$369$$ 4.62222 0.240623
$$370$$ 0 0
$$371$$ 13.4795 0.699820
$$372$$ 0 0
$$373$$ 30.1847 1.56290 0.781452 0.623966i $$-0.214479\pi$$
0.781452 + 0.623966i $$0.214479\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 5.98126 0.308051
$$378$$ 0 0
$$379$$ −12.8573 −0.660434 −0.330217 0.943905i $$-0.607122\pi$$
−0.330217 + 0.943905i $$0.607122\pi$$
$$380$$ 0 0
$$381$$ 15.3461 0.786207
$$382$$ 0 0
$$383$$ −24.4701 −1.25037 −0.625183 0.780479i $$-0.714975\pi$$
−0.625183 + 0.780479i $$0.714975\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −4.85728 −0.246909
$$388$$ 0 0
$$389$$ −1.61285 −0.0817746 −0.0408873 0.999164i $$-0.513018\pi$$
−0.0408873 + 0.999164i $$0.513018\pi$$
$$390$$ 0 0
$$391$$ −11.6128 −0.587287
$$392$$ 0 0
$$393$$ −4.00000 −0.201773
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 22.2163 1.11501 0.557503 0.830175i $$-0.311760\pi$$
0.557503 + 0.830175i $$0.311760\pi$$
$$398$$ 0 0
$$399$$ 0.622216 0.0311497
$$400$$ 0 0
$$401$$ 19.9813 0.997817 0.498908 0.866655i $$-0.333734\pi$$
0.498908 + 0.866655i $$0.333734\pi$$
$$402$$ 0 0
$$403$$ −0.387152 −0.0192854
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 4.73683 0.234796
$$408$$ 0 0
$$409$$ −5.73329 −0.283493 −0.141747 0.989903i $$-0.545272\pi$$
−0.141747 + 0.989903i $$0.545272\pi$$
$$410$$ 0 0
$$411$$ −17.0923 −0.843103
$$412$$ 0 0
$$413$$ 11.6128 0.571431
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −13.4795 −0.660094
$$418$$ 0 0
$$419$$ 26.3684 1.28818 0.644091 0.764949i $$-0.277236\pi$$
0.644091 + 0.764949i $$0.277236\pi$$
$$420$$ 0 0
$$421$$ −19.3274 −0.941960 −0.470980 0.882144i $$-0.656100\pi$$
−0.470980 + 0.882144i $$0.656100\pi$$
$$422$$ 0 0
$$423$$ 11.6128 0.564636
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −8.10171 −0.392069
$$428$$ 0 0
$$429$$ −2.36842 −0.114348
$$430$$ 0 0
$$431$$ −26.9491 −1.29809 −0.649047 0.760748i $$-0.724832\pi$$
−0.649047 + 0.760748i $$0.724832\pi$$
$$432$$ 0 0
$$433$$ −2.13335 −0.102522 −0.0512612 0.998685i $$-0.516324\pi$$
−0.0512612 + 0.998685i $$0.516324\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 1.63158 0.0780492
$$438$$ 0 0
$$439$$ −10.5205 −0.502116 −0.251058 0.967972i $$-0.580779\pi$$
−0.251058 + 0.967972i $$0.580779\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ 0 0
$$443$$ −6.88892 −0.327303 −0.163651 0.986518i $$-0.552327\pi$$
−0.163651 + 0.986518i $$0.552327\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ −9.34614 −0.442057
$$448$$ 0 0
$$449$$ 39.9180 1.88385 0.941923 0.335829i $$-0.109016\pi$$
0.941923 + 0.335829i $$0.109016\pi$$
$$450$$ 0 0
$$451$$ −17.5941 −0.828474
$$452$$ 0 0
$$453$$ −7.14272 −0.335594
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −8.47013 −0.396216 −0.198108 0.980180i $$-0.563480\pi$$
−0.198108 + 0.980180i $$0.563480\pi$$
$$458$$ 0 0
$$459$$ −4.42864 −0.206711
$$460$$ 0 0
$$461$$ 40.1146 1.86832 0.934162 0.356849i $$-0.116149\pi$$
0.934162 + 0.356849i $$0.116149\pi$$
$$462$$ 0 0
$$463$$ 33.5941 1.56125 0.780625 0.624999i $$-0.214901\pi$$
0.780625 + 0.624999i $$0.214901\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 11.3461 0.525037 0.262518 0.964927i $$-0.415447\pi$$
0.262518 + 0.964927i $$0.415447\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −6.99063 −0.322111
$$472$$ 0 0
$$473$$ 18.4889 0.850119
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 13.4795 0.617184
$$478$$ 0 0
$$479$$ 36.2864 1.65797 0.828984 0.559273i $$-0.188919\pi$$
0.828984 + 0.559273i $$0.188919\pi$$
$$480$$ 0 0
$$481$$ −0.774305 −0.0353053
$$482$$ 0 0
$$483$$ 2.62222 0.119315
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −38.8385 −1.75994 −0.879971 0.475027i $$-0.842438\pi$$
−0.879971 + 0.475027i $$0.842438\pi$$
$$488$$ 0 0
$$489$$ 15.6128 0.706037
$$490$$ 0 0
$$491$$ −28.7467 −1.29732 −0.648660 0.761079i $$-0.724670\pi$$
−0.648660 + 0.761079i $$0.724670\pi$$
$$492$$ 0 0
$$493$$ −42.5718 −1.91734
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 2.56199 0.114921
$$498$$ 0 0
$$499$$ −5.63158 −0.252104 −0.126052 0.992024i $$-0.540231\pi$$
−0.126052 + 0.992024i $$0.540231\pi$$
$$500$$ 0 0
$$501$$ −1.51114 −0.0675126
$$502$$ 0 0
$$503$$ −34.9590 −1.55874 −0.779372 0.626561i $$-0.784462\pi$$
−0.779372 + 0.626561i $$0.784462\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −12.6128 −0.560156
$$508$$ 0 0
$$509$$ −10.9906 −0.487151 −0.243576 0.969882i $$-0.578320\pi$$
−0.243576 + 0.969882i $$0.578320\pi$$
$$510$$ 0 0
$$511$$ 10.9906 0.486197
$$512$$ 0 0
$$513$$ 0.622216 0.0274715
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −44.2034 −1.94406
$$518$$ 0 0
$$519$$ −6.53035 −0.286651
$$520$$ 0 0
$$521$$ 6.90766 0.302630 0.151315 0.988486i $$-0.451649\pi$$
0.151315 + 0.988486i $$0.451649\pi$$
$$522$$ 0 0
$$523$$ 37.7146 1.64914 0.824571 0.565758i $$-0.191416\pi$$
0.824571 + 0.565758i $$0.191416\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 2.75557 0.120034
$$528$$ 0 0
$$529$$ −16.1240 −0.701043
$$530$$ 0 0
$$531$$ 11.6128 0.503955
$$532$$ 0 0
$$533$$ 2.87601 0.124574
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −6.29529 −0.271662
$$538$$ 0 0
$$539$$ −3.80642 −0.163954
$$540$$ 0 0
$$541$$ −3.12399 −0.134311 −0.0671553 0.997743i $$-0.521392\pi$$
−0.0671553 + 0.997743i $$0.521392\pi$$
$$542$$ 0 0
$$543$$ −6.85728 −0.294274
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −5.51114 −0.235639 −0.117820 0.993035i $$-0.537590\pi$$
−0.117820 + 0.993035i $$0.537590\pi$$
$$548$$ 0 0
$$549$$ −8.10171 −0.345773
$$550$$ 0 0
$$551$$ 5.98126 0.254810
$$552$$ 0 0
$$553$$ −6.75557 −0.287276
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 36.7052 1.55525 0.777624 0.628729i $$-0.216425\pi$$
0.777624 + 0.628729i $$0.216425\pi$$
$$558$$ 0 0
$$559$$ −3.02227 −0.127829
$$560$$ 0 0
$$561$$ 16.8573 0.711715
$$562$$ 0 0
$$563$$ −27.4924 −1.15867 −0.579333 0.815091i $$-0.696687\pi$$
−0.579333 + 0.815091i $$0.696687\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 1.00000 0.0419961
$$568$$ 0 0
$$569$$ 23.2444 0.974457 0.487229 0.873274i $$-0.338008\pi$$
0.487229 + 0.873274i $$0.338008\pi$$
$$570$$ 0 0
$$571$$ −25.5111 −1.06761 −0.533804 0.845608i $$-0.679238\pi$$
−0.533804 + 0.845608i $$0.679238\pi$$
$$572$$ 0 0
$$573$$ −10.5620 −0.441234
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 26.0701 1.08531 0.542656 0.839955i $$-0.317419\pi$$
0.542656 + 0.839955i $$0.317419\pi$$
$$578$$ 0 0
$$579$$ 5.24443 0.217951
$$580$$ 0 0
$$581$$ 11.6128 0.481782
$$582$$ 0 0
$$583$$ −51.3087 −2.12499
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −12.2667 −0.506301 −0.253151 0.967427i $$-0.581467\pi$$
−0.253151 + 0.967427i $$0.581467\pi$$
$$588$$ 0 0
$$589$$ −0.387152 −0.0159523
$$590$$ 0 0
$$591$$ 17.7462 0.729981
$$592$$ 0 0
$$593$$ −14.9175 −0.612588 −0.306294 0.951937i $$-0.599089\pi$$
−0.306294 + 0.951937i $$0.599089\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −20.2351 −0.828166
$$598$$ 0 0
$$599$$ −26.5620 −1.08529 −0.542647 0.839961i $$-0.682578\pi$$
−0.542647 + 0.839961i $$0.682578\pi$$
$$600$$ 0 0
$$601$$ 39.7146 1.61999 0.809995 0.586436i $$-0.199470\pi$$
0.809995 + 0.586436i $$0.199470\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 6.28544 0.255118 0.127559 0.991831i $$-0.459286\pi$$
0.127559 + 0.991831i $$0.459286\pi$$
$$608$$ 0 0
$$609$$ 9.61285 0.389532
$$610$$ 0 0
$$611$$ 7.22570 0.292320
$$612$$ 0 0
$$613$$ −45.7146 −1.84639 −0.923197 0.384328i $$-0.874433\pi$$
−0.923197 + 0.384328i $$0.874433\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 4.88892 0.196821 0.0984103 0.995146i $$-0.468624\pi$$
0.0984103 + 0.995146i $$0.468624\pi$$
$$618$$ 0 0
$$619$$ 12.2351 0.491769 0.245884 0.969299i $$-0.420922\pi$$
0.245884 + 0.969299i $$0.420922\pi$$
$$620$$ 0 0
$$621$$ 2.62222 0.105226
$$622$$ 0 0
$$623$$ −8.23506 −0.329931
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ −2.36842 −0.0945854
$$628$$ 0 0
$$629$$ 5.51114 0.219743
$$630$$ 0 0
$$631$$ 1.24443 0.0495400 0.0247700 0.999693i $$-0.492115\pi$$
0.0247700 + 0.999693i $$0.492115\pi$$
$$632$$ 0 0
$$633$$ 21.3274 0.847688
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 0.622216 0.0246531
$$638$$ 0 0
$$639$$ 2.56199 0.101351
$$640$$ 0 0
$$641$$ −48.1847 −1.90318 −0.951590 0.307369i $$-0.900551\pi$$
−0.951590 + 0.307369i $$0.900551\pi$$
$$642$$ 0 0
$$643$$ 4.85728 0.191552 0.0957762 0.995403i $$-0.469467\pi$$
0.0957762 + 0.995403i $$0.469467\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 0.203420 0.00799728 0.00399864 0.999992i $$-0.498727\pi$$
0.00399864 + 0.999992i $$0.498727\pi$$
$$648$$ 0 0
$$649$$ −44.2034 −1.73514
$$650$$ 0 0
$$651$$ −0.622216 −0.0243866
$$652$$ 0 0
$$653$$ −27.3145 −1.06890 −0.534449 0.845200i $$-0.679481\pi$$
−0.534449 + 0.845200i $$0.679481\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 10.9906 0.428785
$$658$$ 0 0
$$659$$ 33.3176 1.29787 0.648934 0.760845i $$-0.275215\pi$$
0.648934 + 0.760845i $$0.275215\pi$$
$$660$$ 0 0
$$661$$ 14.5906 0.567508 0.283754 0.958897i $$-0.408420\pi$$
0.283754 + 0.958897i $$0.408420\pi$$
$$662$$ 0 0
$$663$$ −2.75557 −0.107017
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 25.2070 0.976017
$$668$$ 0 0
$$669$$ −9.71456 −0.375587
$$670$$ 0 0
$$671$$ 30.8385 1.19051
$$672$$ 0 0
$$673$$ 4.53341 0.174750 0.0873751 0.996175i $$-0.472152\pi$$
0.0873751 + 0.996175i $$0.472152\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −27.2672 −1.04796 −0.523981 0.851730i $$-0.675554\pi$$
−0.523981 + 0.851730i $$0.675554\pi$$
$$678$$ 0 0
$$679$$ 4.23506 0.162527
$$680$$ 0 0
$$681$$ −11.3461 −0.434785
$$682$$ 0 0
$$683$$ −29.5812 −1.13189 −0.565947 0.824442i $$-0.691489\pi$$
−0.565947 + 0.824442i $$0.691489\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 1.34614 0.0513585
$$688$$ 0 0
$$689$$ 8.38715 0.319525
$$690$$ 0 0
$$691$$ 2.99063 0.113769 0.0568845 0.998381i $$-0.481883\pi$$
0.0568845 + 0.998381i $$0.481883\pi$$
$$692$$ 0 0
$$693$$ −3.80642 −0.144594
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −20.4701 −0.775361
$$698$$ 0 0
$$699$$ 15.3778 0.581641
$$700$$ 0 0
$$701$$ −41.0420 −1.55013 −0.775067 0.631879i $$-0.782284\pi$$
−0.775067 + 0.631879i $$0.782284\pi$$
$$702$$ 0 0
$$703$$ −0.774305 −0.0292035
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 18.7239 0.704186
$$708$$ 0 0
$$709$$ −41.4291 −1.55590 −0.777952 0.628324i $$-0.783741\pi$$
−0.777952 + 0.628324i $$0.783741\pi$$
$$710$$ 0 0
$$711$$ −6.75557 −0.253354
$$712$$ 0 0
$$713$$ −1.63158 −0.0611033
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 7.53972 0.281576
$$718$$ 0 0
$$719$$ −18.9590 −0.707051 −0.353525 0.935425i $$-0.615017\pi$$
−0.353525 + 0.935425i $$0.615017\pi$$
$$720$$ 0 0
$$721$$ −0.857279 −0.0319267
$$722$$ 0 0
$$723$$ 23.9813 0.891873
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 41.7975 1.55018 0.775092 0.631848i $$-0.217703\pi$$
0.775092 + 0.631848i $$0.217703\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 21.5111 0.795618
$$732$$ 0 0
$$733$$ 15.3145 0.565654 0.282827 0.959171i $$-0.408728\pi$$
0.282827 + 0.959171i $$0.408728\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ −32.7368 −1.20424 −0.602122 0.798404i $$-0.705678\pi$$
−0.602122 + 0.798404i $$0.705678\pi$$
$$740$$ 0 0
$$741$$ 0.387152 0.0142224
$$742$$ 0 0
$$743$$ −37.3778 −1.37126 −0.685629 0.727951i $$-0.740473\pi$$
−0.685629 + 0.727951i $$0.740473\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 11.6128 0.424892
$$748$$ 0 0
$$749$$ −11.0923 −0.405305
$$750$$ 0 0
$$751$$ 20.3497 0.742570 0.371285 0.928519i $$-0.378917\pi$$
0.371285 + 0.928519i $$0.378917\pi$$
$$752$$ 0 0
$$753$$ 14.1017 0.513895
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −6.95899 −0.252929 −0.126464 0.991971i $$-0.540363\pi$$
−0.126464 + 0.991971i $$0.540363\pi$$
$$758$$ 0 0
$$759$$ −9.98126 −0.362297
$$760$$ 0 0
$$761$$ 48.6419 1.76327 0.881634 0.471934i $$-0.156444\pi$$
0.881634 + 0.471934i $$0.156444\pi$$
$$762$$ 0 0
$$763$$ 5.61285 0.203199
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 7.22570 0.260905
$$768$$ 0 0
$$769$$ −24.6923 −0.890426 −0.445213 0.895425i $$-0.646872\pi$$
−0.445213 + 0.895425i $$0.646872\pi$$
$$770$$ 0 0
$$771$$ 17.0192 0.612932
$$772$$ 0 0
$$773$$ 36.0415 1.29632 0.648161 0.761503i $$-0.275538\pi$$
0.648161 + 0.761503i $$0.275538\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ −1.24443 −0.0446437
$$778$$ 0 0
$$779$$ 2.87601 0.103044
$$780$$ 0 0
$$781$$ −9.75203 −0.348955
$$782$$ 0 0
$$783$$ 9.61285 0.343535
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −32.2034 −1.14793 −0.573964 0.818881i $$-0.694595\pi$$
−0.573964 + 0.818881i $$0.694595\pi$$
$$788$$ 0 0
$$789$$ 12.6035 0.448696
$$790$$ 0 0
$$791$$ 16.2351 0.577252
$$792$$ 0 0
$$793$$ −5.04101 −0.179012
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −17.5526 −0.621746 −0.310873 0.950451i $$-0.600621\pi$$
−0.310873 + 0.950451i $$0.600621\pi$$
$$798$$ 0 0
$$799$$ −51.4291 −1.81943
$$800$$ 0 0
$$801$$ −8.23506 −0.290972
$$802$$ 0 0
$$803$$ −41.8350 −1.47633
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 3.76494 0.132532
$$808$$ 0 0
$$809$$ 46.1659 1.62311 0.811554 0.584277i $$-0.198622\pi$$
0.811554 + 0.584277i $$0.198622\pi$$
$$810$$ 0 0
$$811$$ 18.5205 0.650343 0.325171 0.945655i $$-0.394578\pi$$
0.325171 + 0.945655i $$0.394578\pi$$
$$812$$ 0 0
$$813$$ −17.8666 −0.626611
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −3.02227 −0.105736
$$818$$ 0 0
$$819$$ 0.622216 0.0217420
$$820$$ 0 0
$$821$$ 43.0607 1.50283 0.751414 0.659831i $$-0.229372\pi$$
0.751414 + 0.659831i $$0.229372\pi$$
$$822$$ 0 0
$$823$$ 14.5718 0.507942 0.253971 0.967212i $$-0.418263\pi$$
0.253971 + 0.967212i $$0.418263\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −55.8292 −1.94137 −0.970685 0.240354i $$-0.922736\pi$$
−0.970685 + 0.240354i $$0.922736\pi$$
$$828$$ 0 0
$$829$$ −37.3087 −1.29578 −0.647892 0.761732i $$-0.724349\pi$$
−0.647892 + 0.761732i $$0.724349\pi$$
$$830$$ 0 0
$$831$$ −1.24443 −0.0431688
$$832$$ 0 0
$$833$$ −4.42864 −0.153443
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −0.622216 −0.0215069
$$838$$ 0 0
$$839$$ −51.0420 −1.76216 −0.881082 0.472963i $$-0.843184\pi$$
−0.881082 + 0.472963i $$0.843184\pi$$
$$840$$ 0 0
$$841$$ 63.4068 2.18644
$$842$$ 0 0
$$843$$ −8.95899 −0.308564
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 3.48886 0.119879
$$848$$ 0 0
$$849$$ 30.5718 1.04922
$$850$$ 0 0
$$851$$ −3.26317 −0.111860
$$852$$ 0 0
$$853$$ 26.4197 0.904595 0.452297 0.891867i $$-0.350605\pi$$
0.452297 + 0.891867i $$0.350605\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −0.161933 −0.00553154 −0.00276577 0.999996i $$-0.500880\pi$$
−0.00276577 + 0.999996i $$0.500880\pi$$
$$858$$ 0 0
$$859$$ 51.3403 1.75171 0.875854 0.482575i $$-0.160299\pi$$
0.875854 + 0.482575i $$0.160299\pi$$
$$860$$ 0 0
$$861$$ 4.62222 0.157525
$$862$$ 0 0
$$863$$ 12.8702 0.438106 0.219053 0.975713i $$-0.429703\pi$$
0.219053 + 0.975713i $$0.429703\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 2.61285 0.0887370
$$868$$ 0 0
$$869$$ 25.7146 0.872307
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ 4.23506 0.143335
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −43.2257 −1.45963 −0.729814 0.683646i $$-0.760393\pi$$
−0.729814 + 0.683646i $$0.760393\pi$$
$$878$$ 0 0
$$879$$ −5.67307 −0.191348
$$880$$ 0 0
$$881$$ −16.5018 −0.555959 −0.277979 0.960587i $$-0.589665\pi$$
−0.277979 + 0.960587i $$0.589665\pi$$
$$882$$ 0 0
$$883$$ −46.1847 −1.55424 −0.777119 0.629353i $$-0.783320\pi$$
−0.777119 + 0.629353i $$0.783320\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −37.3274 −1.25333 −0.626666 0.779288i $$-0.715581\pi$$
−0.626666 + 0.779288i $$0.715581\pi$$
$$888$$ 0 0
$$889$$ 15.3461 0.514693
$$890$$ 0 0
$$891$$ −3.80642 −0.127520
$$892$$ 0 0
$$893$$ 7.22570 0.241799
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 1.63158 0.0544770
$$898$$ 0 0
$$899$$ −5.98126 −0.199486
$$900$$ 0 0
$$901$$ −59.6958 −1.98876
$$902$$ 0 0
$$903$$ −4.85728 −0.161640
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 24.6735 0.819272 0.409636 0.912249i $$-0.365656\pi$$
0.409636 + 0.912249i $$0.365656\pi$$
$$908$$ 0 0
$$909$$ 18.7239 0.621033
$$910$$ 0 0
$$911$$ 29.4380 0.975325 0.487662 0.873032i $$-0.337850\pi$$
0.487662 + 0.873032i $$0.337850\pi$$
$$912$$ 0 0
$$913$$ −44.2034 −1.46292
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −4.00000 −0.132092
$$918$$ 0 0
$$919$$ 40.0197 1.32013 0.660064 0.751209i $$-0.270529\pi$$
0.660064 + 0.751209i $$0.270529\pi$$
$$920$$ 0 0
$$921$$ −4.85728 −0.160053
$$922$$ 0 0
$$923$$ 1.59411 0.0524708
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ −0.857279 −0.0281567
$$928$$ 0 0
$$929$$ −22.4572 −0.736797 −0.368399 0.929668i $$-0.620094\pi$$
−0.368399 + 0.929668i $$0.620094\pi$$
$$930$$ 0 0
$$931$$ 0.622216 0.0203923
$$932$$ 0 0
$$933$$ −34.5718 −1.13183
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −33.8292 −1.10515 −0.552575 0.833463i $$-0.686355\pi$$
−0.552575 + 0.833463i $$0.686355\pi$$
$$938$$ 0 0
$$939$$ −6.33677 −0.206793
$$940$$ 0 0
$$941$$ 47.8479 1.55980 0.779899 0.625906i $$-0.215271\pi$$
0.779899 + 0.625906i $$0.215271\pi$$
$$942$$ 0 0
$$943$$ 12.1204 0.394696
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 44.8069 1.45603 0.728014 0.685562i $$-0.240443\pi$$
0.728014 + 0.685562i $$0.240443\pi$$
$$948$$ 0 0
$$949$$ 6.83854 0.221989
$$950$$ 0 0
$$951$$ 15.9684 0.517809
$$952$$ 0 0
$$953$$ 15.7017 0.508626 0.254313 0.967122i $$-0.418151\pi$$
0.254313 + 0.967122i $$0.418151\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ −36.5906 −1.18281
$$958$$ 0 0
$$959$$ −17.0923 −0.551941
$$960$$ 0 0
$$961$$ −30.6128 −0.987511
$$962$$ 0 0
$$963$$ −11.0923 −0.357446
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −46.9590 −1.51010 −0.755050 0.655668i $$-0.772387\pi$$
−0.755050 + 0.655668i $$0.772387\pi$$
$$968$$ 0 0
$$969$$ −2.75557 −0.0885216
$$970$$ 0 0
$$971$$ −18.7556 −0.601895 −0.300947 0.953641i $$-0.597303\pi$$
−0.300947 + 0.953641i $$0.597303\pi$$
$$972$$ 0 0
$$973$$ −13.4795 −0.432133
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 37.8292 1.21026 0.605131 0.796126i $$-0.293121\pi$$
0.605131 + 0.796126i $$0.293121\pi$$
$$978$$ 0 0
$$979$$ 31.3461 1.00183
$$980$$ 0 0
$$981$$ 5.61285 0.179204
$$982$$ 0 0
$$983$$ −17.6316 −0.562360 −0.281180 0.959655i $$-0.590726\pi$$
−0.281180 + 0.959655i $$0.590726\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 11.6128 0.369641
$$988$$ 0 0
$$989$$ −12.7368 −0.405008
$$990$$ 0 0
$$991$$ −37.1240 −1.17928 −0.589641 0.807665i $$-0.700731\pi$$
−0.589641 + 0.807665i $$0.700731\pi$$
$$992$$ 0 0
$$993$$ −27.6128 −0.876267
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 42.4197 1.34345 0.671723 0.740802i $$-0.265554\pi$$
0.671723 + 0.740802i $$0.265554\pi$$
$$998$$ 0 0
$$999$$ −1.24443 −0.0393721
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 4200.2.a.bq.1.1 3
4.3 odd 2 8400.2.a.dh.1.3 3
5.2 odd 4 840.2.t.e.169.2 6
5.3 odd 4 840.2.t.e.169.5 yes 6
5.4 even 2 4200.2.a.bo.1.1 3
15.2 even 4 2520.2.t.j.1009.4 6
15.8 even 4 2520.2.t.j.1009.3 6
20.3 even 4 1680.2.t.i.1009.2 6
20.7 even 4 1680.2.t.i.1009.5 6
20.19 odd 2 8400.2.a.dk.1.3 3
60.23 odd 4 5040.2.t.ba.1009.3 6
60.47 odd 4 5040.2.t.ba.1009.4 6

By twisted newform
Twist Min Dim Char Parity Ord Type
840.2.t.e.169.2 6 5.2 odd 4
840.2.t.e.169.5 yes 6 5.3 odd 4
1680.2.t.i.1009.2 6 20.3 even 4
1680.2.t.i.1009.5 6 20.7 even 4
2520.2.t.j.1009.3 6 15.8 even 4
2520.2.t.j.1009.4 6 15.2 even 4
4200.2.a.bo.1.1 3 5.4 even 2
4200.2.a.bq.1.1 3 1.1 even 1 trivial
5040.2.t.ba.1009.3 6 60.23 odd 4
5040.2.t.ba.1009.4 6 60.47 odd 4
8400.2.a.dh.1.3 3 4.3 odd 2
8400.2.a.dk.1.3 3 20.19 odd 2