# Properties

 Label 1400.2.g.i.449.3 Level $1400$ Weight $2$ Character 1400.449 Analytic conductor $11.179$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1400 = 2^{3} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1400.g (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.1790562830$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{33})$$ Defining polynomial: $$x^{4} + 17 x^{2} + 64$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 280) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 449.3 Root $$2.37228i$$ of defining polynomial Character $$\chi$$ $$=$$ 1400.449 Dual form 1400.2.g.i.449.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.37228i q^{3} +1.00000i q^{7} -2.62772 q^{9} +O(q^{10})$$ $$q+2.37228i q^{3} +1.00000i q^{7} -2.62772 q^{9} +6.37228 q^{11} +4.37228i q^{13} +0.372281i q^{17} +4.74456 q^{19} -2.37228 q^{21} -4.74456i q^{23} +0.883156i q^{27} +4.37228 q^{29} -8.00000 q^{31} +15.1168i q^{33} +2.00000i q^{37} -10.3723 q^{39} +6.74456 q^{41} -8.74456i q^{43} +7.11684i q^{47} -1.00000 q^{49} -0.883156 q^{51} +10.7446i q^{53} +11.2554i q^{57} -8.00000 q^{59} -2.74456 q^{61} -2.62772i q^{63} +4.00000i q^{67} +11.2554 q^{69} +8.00000 q^{71} -6.00000i q^{73} +6.37228i q^{77} -15.1168 q^{79} -9.97825 q^{81} -9.48913i q^{83} +10.3723i q^{87} -14.7446 q^{89} -4.37228 q^{91} -18.9783i q^{93} +9.86141i q^{97} -16.7446 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 22q^{9} + O(q^{10})$$ $$4q - 22q^{9} + 14q^{11} - 4q^{19} + 2q^{21} + 6q^{29} - 32q^{31} - 30q^{39} + 4q^{41} - 4q^{49} - 38q^{51} - 32q^{59} + 12q^{61} + 68q^{69} + 32q^{71} - 26q^{79} + 52q^{81} - 36q^{89} - 6q^{91} - 44q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1400\mathbb{Z}\right)^\times$$.

 $$n$$ $$351$$ $$701$$ $$801$$ $$1177$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$

## 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
$$3$$ 2.37228i 1.36964i 0.728714 + 0.684819i $$0.240119\pi$$
−0.728714 + 0.684819i $$0.759881\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 1.00000i 0.377964i
$$8$$ 0 0
$$9$$ −2.62772 −0.875906
$$10$$ 0 0
$$11$$ 6.37228 1.92132 0.960658 0.277736i $$-0.0895839\pi$$
0.960658 + 0.277736i $$0.0895839\pi$$
$$12$$ 0 0
$$13$$ 4.37228i 1.21265i 0.795216 + 0.606326i $$0.207357\pi$$
−0.795216 + 0.606326i $$0.792643\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 0.372281i 0.0902915i 0.998980 + 0.0451457i $$0.0143752\pi$$
−0.998980 + 0.0451457i $$0.985625\pi$$
$$18$$ 0 0
$$19$$ 4.74456 1.08848 0.544239 0.838930i $$-0.316819\pi$$
0.544239 + 0.838930i $$0.316819\pi$$
$$20$$ 0 0
$$21$$ −2.37228 −0.517674
$$22$$ 0 0
$$23$$ − 4.74456i − 0.989310i −0.869090 0.494655i $$-0.835294\pi$$
0.869090 0.494655i $$-0.164706\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 0.883156i 0.169963i
$$28$$ 0 0
$$29$$ 4.37228 0.811912 0.405956 0.913893i $$-0.366939\pi$$
0.405956 + 0.913893i $$0.366939\pi$$
$$30$$ 0 0
$$31$$ −8.00000 −1.43684 −0.718421 0.695608i $$-0.755135\pi$$
−0.718421 + 0.695608i $$0.755135\pi$$
$$32$$ 0 0
$$33$$ 15.1168i 2.63150i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 2.00000i 0.328798i 0.986394 + 0.164399i $$0.0525685\pi$$
−0.986394 + 0.164399i $$0.947432\pi$$
$$38$$ 0 0
$$39$$ −10.3723 −1.66089
$$40$$ 0 0
$$41$$ 6.74456 1.05332 0.526662 0.850075i $$-0.323443\pi$$
0.526662 + 0.850075i $$0.323443\pi$$
$$42$$ 0 0
$$43$$ − 8.74456i − 1.33353i −0.745267 0.666767i $$-0.767678\pi$$
0.745267 0.666767i $$-0.232322\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 7.11684i 1.03810i 0.854744 + 0.519049i $$0.173714\pi$$
−0.854744 + 0.519049i $$0.826286\pi$$
$$48$$ 0 0
$$49$$ −1.00000 −0.142857
$$50$$ 0 0
$$51$$ −0.883156 −0.123667
$$52$$ 0 0
$$53$$ 10.7446i 1.47588i 0.674867 + 0.737940i $$0.264201\pi$$
−0.674867 + 0.737940i $$0.735799\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 11.2554i 1.49082i
$$58$$ 0 0
$$59$$ −8.00000 −1.04151 −0.520756 0.853706i $$-0.674350\pi$$
−0.520756 + 0.853706i $$0.674350\pi$$
$$60$$ 0 0
$$61$$ −2.74456 −0.351405 −0.175703 0.984443i $$-0.556220\pi$$
−0.175703 + 0.984443i $$0.556220\pi$$
$$62$$ 0 0
$$63$$ − 2.62772i − 0.331061i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 4.00000i 0.488678i 0.969690 + 0.244339i $$0.0785709\pi$$
−0.969690 + 0.244339i $$0.921429\pi$$
$$68$$ 0 0
$$69$$ 11.2554 1.35500
$$70$$ 0 0
$$71$$ 8.00000 0.949425 0.474713 0.880141i $$-0.342552\pi$$
0.474713 + 0.880141i $$0.342552\pi$$
$$72$$ 0 0
$$73$$ − 6.00000i − 0.702247i −0.936329 0.351123i $$-0.885800\pi$$
0.936329 0.351123i $$-0.114200\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 6.37228i 0.726189i
$$78$$ 0 0
$$79$$ −15.1168 −1.70078 −0.850389 0.526155i $$-0.823633\pi$$
−0.850389 + 0.526155i $$0.823633\pi$$
$$80$$ 0 0
$$81$$ −9.97825 −1.10869
$$82$$ 0 0
$$83$$ − 9.48913i − 1.04157i −0.853689 0.520783i $$-0.825640\pi$$
0.853689 0.520783i $$-0.174360\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 10.3723i 1.11203i
$$88$$ 0 0
$$89$$ −14.7446 −1.56292 −0.781460 0.623955i $$-0.785525\pi$$
−0.781460 + 0.623955i $$0.785525\pi$$
$$90$$ 0 0
$$91$$ −4.37228 −0.458340
$$92$$ 0 0
$$93$$ − 18.9783i − 1.96795i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 9.86141i 1.00127i 0.865657 + 0.500637i $$0.166901\pi$$
−0.865657 + 0.500637i $$0.833099\pi$$
$$98$$ 0 0
$$99$$ −16.7446 −1.68289
$$100$$ 0 0
$$101$$ −6.00000 −0.597022 −0.298511 0.954406i $$-0.596490\pi$$
−0.298511 + 0.954406i $$0.596490\pi$$
$$102$$ 0 0
$$103$$ − 5.62772i − 0.554516i −0.960796 0.277258i $$-0.910574\pi$$
0.960796 0.277258i $$-0.0894256\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ − 8.74456i − 0.845369i −0.906277 0.422684i $$-0.861088\pi$$
0.906277 0.422684i $$-0.138912\pi$$
$$108$$ 0 0
$$109$$ −0.372281 −0.0356581 −0.0178290 0.999841i $$-0.505675\pi$$
−0.0178290 + 0.999841i $$0.505675\pi$$
$$110$$ 0 0
$$111$$ −4.74456 −0.450334
$$112$$ 0 0
$$113$$ 2.00000i 0.188144i 0.995565 + 0.0940721i $$0.0299884\pi$$
−0.995565 + 0.0940721i $$0.970012\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ − 11.4891i − 1.06217i
$$118$$ 0 0
$$119$$ −0.372281 −0.0341270
$$120$$ 0 0
$$121$$ 29.6060 2.69145
$$122$$ 0 0
$$123$$ 16.0000i 1.44267i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ − 8.00000i − 0.709885i −0.934888 0.354943i $$-0.884500\pi$$
0.934888 0.354943i $$-0.115500\pi$$
$$128$$ 0 0
$$129$$ 20.7446 1.82646
$$130$$ 0 0
$$131$$ −4.74456 −0.414534 −0.207267 0.978284i $$-0.566457\pi$$
−0.207267 + 0.978284i $$0.566457\pi$$
$$132$$ 0 0
$$133$$ 4.74456i 0.411406i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 14.7446i − 1.25971i −0.776712 0.629857i $$-0.783114\pi$$
0.776712 0.629857i $$-0.216886\pi$$
$$138$$ 0 0
$$139$$ 4.74456 0.402429 0.201214 0.979547i $$-0.435511\pi$$
0.201214 + 0.979547i $$0.435511\pi$$
$$140$$ 0 0
$$141$$ −16.8832 −1.42182
$$142$$ 0 0
$$143$$ 27.8614i 2.32989i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ − 2.37228i − 0.195662i
$$148$$ 0 0
$$149$$ −15.4891 −1.26892 −0.634459 0.772956i $$-0.718777\pi$$
−0.634459 + 0.772956i $$0.718777\pi$$
$$150$$ 0 0
$$151$$ 15.1168 1.23019 0.615096 0.788452i $$-0.289117\pi$$
0.615096 + 0.788452i $$0.289117\pi$$
$$152$$ 0 0
$$153$$ − 0.978251i − 0.0790869i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 15.4891i 1.23617i 0.786113 + 0.618083i $$0.212091\pi$$
−0.786113 + 0.618083i $$0.787909\pi$$
$$158$$ 0 0
$$159$$ −25.4891 −2.02142
$$160$$ 0 0
$$161$$ 4.74456 0.373924
$$162$$ 0 0
$$163$$ − 16.7446i − 1.31154i −0.754963 0.655768i $$-0.772345\pi$$
0.754963 0.655768i $$-0.227655\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 5.62772i 0.435486i 0.976006 + 0.217743i $$0.0698695\pi$$
−0.976006 + 0.217743i $$0.930131\pi$$
$$168$$ 0 0
$$169$$ −6.11684 −0.470526
$$170$$ 0 0
$$171$$ −12.4674 −0.953404
$$172$$ 0 0
$$173$$ − 0.372281i − 0.0283040i −0.999900 0.0141520i $$-0.995495\pi$$
0.999900 0.0141520i $$-0.00450488\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ − 18.9783i − 1.42649i
$$178$$ 0 0
$$179$$ 22.9783 1.71748 0.858738 0.512416i $$-0.171249\pi$$
0.858738 + 0.512416i $$0.171249\pi$$
$$180$$ 0 0
$$181$$ 16.2337 1.20664 0.603320 0.797499i $$-0.293844\pi$$
0.603320 + 0.797499i $$0.293844\pi$$
$$182$$ 0 0
$$183$$ − 6.51087i − 0.481298i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 2.37228i 0.173478i
$$188$$ 0 0
$$189$$ −0.883156 −0.0642401
$$190$$ 0 0
$$191$$ 3.86141 0.279402 0.139701 0.990194i $$-0.455386\pi$$
0.139701 + 0.990194i $$0.455386\pi$$
$$192$$ 0 0
$$193$$ 6.74456i 0.485484i 0.970091 + 0.242742i $$0.0780469\pi$$
−0.970091 + 0.242742i $$0.921953\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 8.23369i 0.586626i 0.956016 + 0.293313i $$0.0947578\pi$$
−0.956016 + 0.293313i $$0.905242\pi$$
$$198$$ 0 0
$$199$$ 16.0000 1.13421 0.567105 0.823646i $$-0.308063\pi$$
0.567105 + 0.823646i $$0.308063\pi$$
$$200$$ 0 0
$$201$$ −9.48913 −0.669311
$$202$$ 0 0
$$203$$ 4.37228i 0.306874i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 12.4674i 0.866543i
$$208$$ 0 0
$$209$$ 30.2337 2.09131
$$210$$ 0 0
$$211$$ −14.3723 −0.989429 −0.494714 0.869056i $$-0.664727\pi$$
−0.494714 + 0.869056i $$0.664727\pi$$
$$212$$ 0 0
$$213$$ 18.9783i 1.30037i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ − 8.00000i − 0.543075i
$$218$$ 0 0
$$219$$ 14.2337 0.961823
$$220$$ 0 0
$$221$$ −1.62772 −0.109492
$$222$$ 0 0
$$223$$ − 5.62772i − 0.376860i −0.982087 0.188430i $$-0.939660\pi$$
0.982087 0.188430i $$-0.0603398\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ − 19.8614i − 1.31825i −0.752034 0.659124i $$-0.770927\pi$$
0.752034 0.659124i $$-0.229073\pi$$
$$228$$ 0 0
$$229$$ 12.2337 0.808425 0.404212 0.914665i $$-0.367546\pi$$
0.404212 + 0.914665i $$0.367546\pi$$
$$230$$ 0 0
$$231$$ −15.1168 −0.994615
$$232$$ 0 0
$$233$$ − 1.25544i − 0.0822464i −0.999154 0.0411232i $$-0.986906\pi$$
0.999154 0.0411232i $$-0.0130936\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ − 35.8614i − 2.32945i
$$238$$ 0 0
$$239$$ −13.6277 −0.881504 −0.440752 0.897629i $$-0.645288\pi$$
−0.440752 + 0.897629i $$0.645288\pi$$
$$240$$ 0 0
$$241$$ 26.0000 1.67481 0.837404 0.546585i $$-0.184072\pi$$
0.837404 + 0.546585i $$0.184072\pi$$
$$242$$ 0 0
$$243$$ − 21.0217i − 1.34855i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 20.7446i 1.31994i
$$248$$ 0 0
$$249$$ 22.5109 1.42657
$$250$$ 0 0
$$251$$ 4.74456 0.299474 0.149737 0.988726i $$-0.452157\pi$$
0.149737 + 0.988726i $$0.452157\pi$$
$$252$$ 0 0
$$253$$ − 30.2337i − 1.90078i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 23.4891i 1.46521i 0.680653 + 0.732606i $$0.261696\pi$$
−0.680653 + 0.732606i $$0.738304\pi$$
$$258$$ 0 0
$$259$$ −2.00000 −0.124274
$$260$$ 0 0
$$261$$ −11.4891 −0.711159
$$262$$ 0 0
$$263$$ 22.2337i 1.37099i 0.728078 + 0.685494i $$0.240414\pi$$
−0.728078 + 0.685494i $$0.759586\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ − 34.9783i − 2.14063i
$$268$$ 0 0
$$269$$ 6.00000 0.365826 0.182913 0.983129i $$-0.441447\pi$$
0.182913 + 0.983129i $$0.441447\pi$$
$$270$$ 0 0
$$271$$ 9.48913 0.576423 0.288212 0.957567i $$-0.406939\pi$$
0.288212 + 0.957567i $$0.406939\pi$$
$$272$$ 0 0
$$273$$ − 10.3723i − 0.627759i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 24.9783i − 1.50080i −0.660985 0.750399i $$-0.729861\pi$$
0.660985 0.750399i $$-0.270139\pi$$
$$278$$ 0 0
$$279$$ 21.0217 1.25854
$$280$$ 0 0
$$281$$ 18.6060 1.10994 0.554970 0.831871i $$-0.312730\pi$$
0.554970 + 0.831871i $$0.312730\pi$$
$$282$$ 0 0
$$283$$ − 8.88316i − 0.528049i −0.964516 0.264024i $$-0.914950\pi$$
0.964516 0.264024i $$-0.0850500\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 6.74456i 0.398119i
$$288$$ 0 0
$$289$$ 16.8614 0.991847
$$290$$ 0 0
$$291$$ −23.3940 −1.37138
$$292$$ 0 0
$$293$$ 25.1168i 1.46734i 0.679505 + 0.733671i $$0.262195\pi$$
−0.679505 + 0.733671i $$0.737805\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 5.62772i 0.326553i
$$298$$ 0 0
$$299$$ 20.7446 1.19969
$$300$$ 0 0
$$301$$ 8.74456 0.504028
$$302$$ 0 0
$$303$$ − 14.2337i − 0.817704i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 31.1168i − 1.77593i −0.459909 0.887966i $$-0.652118\pi$$
0.459909 0.887966i $$-0.347882\pi$$
$$308$$ 0 0
$$309$$ 13.3505 0.759485
$$310$$ 0 0
$$311$$ −12.7446 −0.722678 −0.361339 0.932435i $$-0.617680\pi$$
−0.361339 + 0.932435i $$0.617680\pi$$
$$312$$ 0 0
$$313$$ 2.88316i 0.162966i 0.996675 + 0.0814828i $$0.0259656\pi$$
−0.996675 + 0.0814828i $$0.974034\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 14.0000i − 0.786318i −0.919470 0.393159i $$-0.871382\pi$$
0.919470 0.393159i $$-0.128618\pi$$
$$318$$ 0 0
$$319$$ 27.8614 1.55994
$$320$$ 0 0
$$321$$ 20.7446 1.15785
$$322$$ 0 0
$$323$$ 1.76631i 0.0982802i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ − 0.883156i − 0.0488386i
$$328$$ 0 0
$$329$$ −7.11684 −0.392364
$$330$$ 0 0
$$331$$ 12.0000 0.659580 0.329790 0.944054i $$-0.393022\pi$$
0.329790 + 0.944054i $$0.393022\pi$$
$$332$$ 0 0
$$333$$ − 5.25544i − 0.287996i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 7.48913i 0.407959i 0.978975 + 0.203979i $$0.0653875\pi$$
−0.978975 + 0.203979i $$0.934612\pi$$
$$338$$ 0 0
$$339$$ −4.74456 −0.257689
$$340$$ 0 0
$$341$$ −50.9783 −2.76063
$$342$$ 0 0
$$343$$ − 1.00000i − 0.0539949i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 24.7446i 1.32836i 0.747574 + 0.664179i $$0.231219\pi$$
−0.747574 + 0.664179i $$0.768781\pi$$
$$348$$ 0 0
$$349$$ −19.4891 −1.04323 −0.521614 0.853181i $$-0.674670\pi$$
−0.521614 + 0.853181i $$0.674670\pi$$
$$350$$ 0 0
$$351$$ −3.86141 −0.206107
$$352$$ 0 0
$$353$$ − 1.86141i − 0.0990727i −0.998772 0.0495363i $$-0.984226\pi$$
0.998772 0.0495363i $$-0.0157744\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ − 0.883156i − 0.0467416i
$$358$$ 0 0
$$359$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$360$$ 0 0
$$361$$ 3.51087 0.184783
$$362$$ 0 0
$$363$$ 70.2337i 3.68631i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 19.8614i − 1.03676i −0.855151 0.518378i $$-0.826536\pi$$
0.855151 0.518378i $$-0.173464\pi$$
$$368$$ 0 0
$$369$$ −17.7228 −0.922613
$$370$$ 0 0
$$371$$ −10.7446 −0.557830
$$372$$ 0 0
$$373$$ − 30.7446i − 1.59189i −0.605367 0.795947i $$-0.706974\pi$$
0.605367 0.795947i $$-0.293026\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 19.1168i 0.984568i
$$378$$ 0 0
$$379$$ −12.0000 −0.616399 −0.308199 0.951322i $$-0.599726\pi$$
−0.308199 + 0.951322i $$0.599726\pi$$
$$380$$ 0 0
$$381$$ 18.9783 0.972285
$$382$$ 0 0
$$383$$ − 17.4891i − 0.893653i −0.894621 0.446826i $$-0.852554\pi$$
0.894621 0.446826i $$-0.147446\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 22.9783i 1.16805i
$$388$$ 0 0
$$389$$ −17.8614 −0.905609 −0.452805 0.891610i $$-0.649576\pi$$
−0.452805 + 0.891610i $$0.649576\pi$$
$$390$$ 0 0
$$391$$ 1.76631 0.0893262
$$392$$ 0 0
$$393$$ − 11.2554i − 0.567762i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 31.6277i − 1.58735i −0.608342 0.793675i $$-0.708165\pi$$
0.608342 0.793675i $$-0.291835\pi$$
$$398$$ 0 0
$$399$$ −11.2554 −0.563477
$$400$$ 0 0
$$401$$ −38.6060 −1.92789 −0.963945 0.266101i $$-0.914264\pi$$
−0.963945 + 0.266101i $$0.914264\pi$$
$$402$$ 0 0
$$403$$ − 34.9783i − 1.74239i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 12.7446i 0.631725i
$$408$$ 0 0
$$409$$ −11.4891 −0.568101 −0.284050 0.958809i $$-0.591678\pi$$
−0.284050 + 0.958809i $$0.591678\pi$$
$$410$$ 0 0
$$411$$ 34.9783 1.72535
$$412$$ 0 0
$$413$$ − 8.00000i − 0.393654i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 11.2554i 0.551181i
$$418$$ 0 0
$$419$$ 14.5109 0.708903 0.354451 0.935074i $$-0.384668\pi$$
0.354451 + 0.935074i $$0.384668\pi$$
$$420$$ 0 0
$$421$$ −18.6060 −0.906799 −0.453400 0.891307i $$-0.649789\pi$$
−0.453400 + 0.891307i $$0.649789\pi$$
$$422$$ 0 0
$$423$$ − 18.7011i − 0.909277i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ − 2.74456i − 0.132819i
$$428$$ 0 0
$$429$$ −66.0951 −3.19110
$$430$$ 0 0
$$431$$ 18.3723 0.884962 0.442481 0.896778i $$-0.354098\pi$$
0.442481 + 0.896778i $$0.354098\pi$$
$$432$$ 0 0
$$433$$ 28.9783i 1.39261i 0.717748 + 0.696303i $$0.245173\pi$$
−0.717748 + 0.696303i $$0.754827\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ − 22.5109i − 1.07684i
$$438$$ 0 0
$$439$$ 6.23369 0.297518 0.148759 0.988874i $$-0.452472\pi$$
0.148759 + 0.988874i $$0.452472\pi$$
$$440$$ 0 0
$$441$$ 2.62772 0.125129
$$442$$ 0 0
$$443$$ − 16.7446i − 0.795558i −0.917481 0.397779i $$-0.869781\pi$$
0.917481 0.397779i $$-0.130219\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ − 36.7446i − 1.73796i
$$448$$ 0 0
$$449$$ −17.1168 −0.807794 −0.403897 0.914805i $$-0.632345\pi$$
−0.403897 + 0.914805i $$0.632345\pi$$
$$450$$ 0 0
$$451$$ 42.9783 2.02377
$$452$$ 0 0
$$453$$ 35.8614i 1.68492i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 5.25544i − 0.245839i −0.992417 0.122919i $$-0.960774\pi$$
0.992417 0.122919i $$-0.0392257\pi$$
$$458$$ 0 0
$$459$$ −0.328782 −0.0153463
$$460$$ 0 0
$$461$$ −1.25544 −0.0584715 −0.0292358 0.999573i $$-0.509307\pi$$
−0.0292358 + 0.999573i $$0.509307\pi$$
$$462$$ 0 0
$$463$$ − 6.51087i − 0.302586i −0.988489 0.151293i $$-0.951656\pi$$
0.988489 0.151293i $$-0.0483437\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ − 8.60597i − 0.398237i −0.979975 0.199118i $$-0.936192\pi$$
0.979975 0.199118i $$-0.0638078\pi$$
$$468$$ 0 0
$$469$$ −4.00000 −0.184703
$$470$$ 0 0
$$471$$ −36.7446 −1.69310
$$472$$ 0 0
$$473$$ − 55.7228i − 2.56214i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ − 28.2337i − 1.29273i
$$478$$ 0 0
$$479$$ −22.2337 −1.01588 −0.507942 0.861392i $$-0.669593\pi$$
−0.507942 + 0.861392i $$0.669593\pi$$
$$480$$ 0 0
$$481$$ −8.74456 −0.398718
$$482$$ 0 0
$$483$$ 11.2554i 0.512140i
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 30.2337i − 1.37002i −0.728534 0.685010i $$-0.759798\pi$$
0.728534 0.685010i $$-0.240202\pi$$
$$488$$ 0 0
$$489$$ 39.7228 1.79633
$$490$$ 0 0
$$491$$ 35.1168 1.58480 0.792400 0.610001i $$-0.208831\pi$$
0.792400 + 0.610001i $$0.208831\pi$$
$$492$$ 0 0
$$493$$ 1.62772i 0.0733088i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 8.00000i 0.358849i
$$498$$ 0 0
$$499$$ −31.8614 −1.42631 −0.713156 0.701005i $$-0.752735\pi$$
−0.713156 + 0.701005i $$0.752735\pi$$
$$500$$ 0 0
$$501$$ −13.3505 −0.596458
$$502$$ 0 0
$$503$$ − 18.3723i − 0.819180i −0.912270 0.409590i $$-0.865672\pi$$
0.912270 0.409590i $$-0.134328\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ − 14.5109i − 0.644451i
$$508$$ 0 0
$$509$$ 40.9783 1.81633 0.908165 0.418613i $$-0.137484\pi$$
0.908165 + 0.418613i $$0.137484\pi$$
$$510$$ 0 0
$$511$$ 6.00000 0.265424
$$512$$ 0 0
$$513$$ 4.19019i 0.185001i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 45.3505i 1.99451i
$$518$$ 0 0
$$519$$ 0.883156 0.0387662
$$520$$ 0 0
$$521$$ −30.0000 −1.31432 −0.657162 0.753749i $$-0.728243\pi$$
−0.657162 + 0.753749i $$0.728243\pi$$
$$522$$ 0 0
$$523$$ − 36.4674i − 1.59461i −0.603579 0.797304i $$-0.706259\pi$$
0.603579 0.797304i $$-0.293741\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ − 2.97825i − 0.129735i
$$528$$ 0 0
$$529$$ 0.489125 0.0212663
$$530$$ 0 0
$$531$$ 21.0217 0.912266
$$532$$ 0 0
$$533$$ 29.4891i 1.27732i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 54.5109i 2.35232i
$$538$$ 0 0
$$539$$ −6.37228 −0.274474
$$540$$ 0 0
$$541$$ −31.3505 −1.34786 −0.673932 0.738793i $$-0.735396\pi$$
−0.673932 + 0.738793i $$0.735396\pi$$
$$542$$ 0 0
$$543$$ 38.5109i 1.65266i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 30.9783i 1.32453i 0.749268 + 0.662267i $$0.230406\pi$$
−0.749268 + 0.662267i $$0.769594\pi$$
$$548$$ 0 0
$$549$$ 7.21194 0.307798
$$550$$ 0 0
$$551$$ 20.7446 0.883748
$$552$$ 0 0
$$553$$ − 15.1168i − 0.642834i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 3.76631i 0.159584i 0.996812 + 0.0797919i $$0.0254256\pi$$
−0.996812 + 0.0797919i $$0.974574\pi$$
$$558$$ 0 0
$$559$$ 38.2337 1.61711
$$560$$ 0 0
$$561$$ −5.62772 −0.237602
$$562$$ 0 0
$$563$$ 17.4891i 0.737079i 0.929612 + 0.368539i $$0.120142\pi$$
−0.929612 + 0.368539i $$0.879858\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ − 9.97825i − 0.419047i
$$568$$ 0 0
$$569$$ 24.9783 1.04714 0.523571 0.851982i $$-0.324599\pi$$
0.523571 + 0.851982i $$0.324599\pi$$
$$570$$ 0 0
$$571$$ −20.0000 −0.836974 −0.418487 0.908223i $$-0.637439\pi$$
−0.418487 + 0.908223i $$0.637439\pi$$
$$572$$ 0 0
$$573$$ 9.16034i 0.382679i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 22.6060i 0.941099i 0.882374 + 0.470549i $$0.155944\pi$$
−0.882374 + 0.470549i $$0.844056\pi$$
$$578$$ 0 0
$$579$$ −16.0000 −0.664937
$$580$$ 0 0
$$581$$ 9.48913 0.393675
$$582$$ 0 0
$$583$$ 68.4674i 2.83563i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 34.9783i 1.44371i 0.692046 + 0.721853i $$0.256710\pi$$
−0.692046 + 0.721853i $$0.743290\pi$$
$$588$$ 0 0
$$589$$ −37.9565 −1.56397
$$590$$ 0 0
$$591$$ −19.5326 −0.803465
$$592$$ 0 0
$$593$$ − 19.6277i − 0.806014i −0.915197 0.403007i $$-0.867965\pi$$
0.915197 0.403007i $$-0.132035\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 37.9565i 1.55346i
$$598$$ 0 0
$$599$$ 32.6060 1.33224 0.666122 0.745843i $$-0.267953\pi$$
0.666122 + 0.745843i $$0.267953\pi$$
$$600$$ 0 0
$$601$$ 16.5109 0.673493 0.336746 0.941595i $$-0.390674\pi$$
0.336746 + 0.941595i $$0.390674\pi$$
$$602$$ 0 0
$$603$$ − 10.5109i − 0.428036i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ − 24.6060i − 0.998725i −0.866393 0.499363i $$-0.833568\pi$$
0.866393 0.499363i $$-0.166432\pi$$
$$608$$ 0 0
$$609$$ −10.3723 −0.420306
$$610$$ 0 0
$$611$$ −31.1168 −1.25885
$$612$$ 0 0
$$613$$ − 8.51087i − 0.343751i −0.985119 0.171875i $$-0.945017\pi$$
0.985119 0.171875i $$-0.0549827\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 6.00000i 0.241551i 0.992680 + 0.120775i $$0.0385381\pi$$
−0.992680 + 0.120775i $$0.961462\pi$$
$$618$$ 0 0
$$619$$ 12.7446 0.512247 0.256124 0.966644i $$-0.417555\pi$$
0.256124 + 0.966644i $$0.417555\pi$$
$$620$$ 0 0
$$621$$ 4.19019 0.168146
$$622$$ 0 0
$$623$$ − 14.7446i − 0.590728i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 71.7228i 2.86433i
$$628$$ 0 0
$$629$$ −0.744563 −0.0296877
$$630$$ 0 0
$$631$$ 2.37228 0.0944390 0.0472195 0.998885i $$-0.484964\pi$$
0.0472195 + 0.998885i $$0.484964\pi$$
$$632$$ 0 0
$$633$$ − 34.0951i − 1.35516i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 4.37228i − 0.173236i
$$638$$ 0 0
$$639$$ −21.0217 −0.831608
$$640$$ 0 0
$$641$$ 2.00000 0.0789953 0.0394976 0.999220i $$-0.487424\pi$$
0.0394976 + 0.999220i $$0.487424\pi$$
$$642$$ 0 0
$$643$$ 18.3723i 0.724532i 0.932075 + 0.362266i $$0.117997\pi$$
−0.932075 + 0.362266i $$0.882003\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 16.0000i 0.629025i 0.949253 + 0.314512i $$0.101841\pi$$
−0.949253 + 0.314512i $$0.898159\pi$$
$$648$$ 0 0
$$649$$ −50.9783 −2.00107
$$650$$ 0 0
$$651$$ 18.9783 0.743816
$$652$$ 0 0
$$653$$ 6.00000i 0.234798i 0.993085 + 0.117399i $$0.0374557\pi$$
−0.993085 + 0.117399i $$0.962544\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 15.7663i 0.615102i
$$658$$ 0 0
$$659$$ 11.1168 0.433051 0.216525 0.976277i $$-0.430528\pi$$
0.216525 + 0.976277i $$0.430528\pi$$
$$660$$ 0 0
$$661$$ 14.7446 0.573497 0.286749 0.958006i $$-0.407426\pi$$
0.286749 + 0.958006i $$0.407426\pi$$
$$662$$ 0 0
$$663$$ − 3.86141i − 0.149965i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ − 20.7446i − 0.803233i
$$668$$ 0 0
$$669$$ 13.3505 0.516161
$$670$$ 0 0
$$671$$ −17.4891 −0.675160
$$672$$ 0 0
$$673$$ 25.7228i 0.991542i 0.868453 + 0.495771i $$0.165114\pi$$
−0.868453 + 0.495771i $$0.834886\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ − 15.3505i − 0.589969i −0.955502 0.294984i $$-0.904686\pi$$
0.955502 0.294984i $$-0.0953145\pi$$
$$678$$ 0 0
$$679$$ −9.86141 −0.378446
$$680$$ 0 0
$$681$$ 47.1168 1.80552
$$682$$ 0 0
$$683$$ 36.0000i 1.37750i 0.724998 + 0.688751i $$0.241841\pi$$
−0.724998 + 0.688751i $$0.758159\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 29.0217i 1.10725i
$$688$$ 0 0
$$689$$ −46.9783 −1.78973
$$690$$ 0 0
$$691$$ −9.48913 −0.360983 −0.180492 0.983577i $$-0.557769\pi$$
−0.180492 + 0.983577i $$0.557769\pi$$
$$692$$ 0 0
$$693$$ − 16.7446i − 0.636073i
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 2.51087i 0.0951062i
$$698$$ 0 0
$$699$$ 2.97825 0.112648
$$700$$ 0 0
$$701$$ 2.13859 0.0807736 0.0403868 0.999184i $$-0.487141\pi$$
0.0403868 + 0.999184i $$0.487141\pi$$
$$702$$ 0 0
$$703$$ 9.48913i 0.357889i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ − 6.00000i − 0.225653i
$$708$$ 0 0
$$709$$ −6.60597 −0.248092 −0.124046 0.992276i $$-0.539587\pi$$
−0.124046 + 0.992276i $$0.539587\pi$$
$$710$$ 0 0
$$711$$ 39.7228 1.48972
$$712$$ 0 0
$$713$$ 37.9565i 1.42148i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ − 32.3288i − 1.20734i
$$718$$ 0 0
$$719$$ −3.25544 −0.121407 −0.0607037 0.998156i $$-0.519334\pi$$
−0.0607037 + 0.998156i $$0.519334\pi$$
$$720$$ 0 0
$$721$$ 5.62772 0.209587
$$722$$ 0 0
$$723$$ 61.6793i 2.29388i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 32.0000i − 1.18681i −0.804902 0.593407i $$-0.797782\pi$$
0.804902 0.593407i $$-0.202218\pi$$
$$728$$ 0 0
$$729$$ 19.9348 0.738324
$$730$$ 0 0
$$731$$ 3.25544 0.120407
$$732$$ 0 0
$$733$$ − 10.1386i − 0.374477i −0.982314 0.187239i $$-0.940046\pi$$
0.982314 0.187239i $$-0.0599538\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 25.4891i 0.938904i
$$738$$ 0 0
$$739$$ 20.6060 0.758003 0.379001 0.925396i $$-0.376268\pi$$
0.379001 + 0.925396i $$0.376268\pi$$
$$740$$ 0 0
$$741$$ −49.2119 −1.80785
$$742$$ 0 0
$$743$$ − 6.51087i − 0.238861i −0.992843 0.119430i $$-0.961893\pi$$
0.992843 0.119430i $$-0.0381068\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 24.9348i 0.912315i
$$748$$ 0 0
$$749$$ 8.74456 0.319519
$$750$$ 0 0
$$751$$ 20.1386 0.734868 0.367434 0.930050i $$-0.380236\pi$$
0.367434 + 0.930050i $$0.380236\pi$$
$$752$$ 0 0
$$753$$ 11.2554i 0.410171i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 3.76631i 0.136889i 0.997655 + 0.0684445i $$0.0218036\pi$$
−0.997655 + 0.0684445i $$0.978196\pi$$
$$758$$ 0 0
$$759$$ 71.7228 2.60337
$$760$$ 0 0
$$761$$ 4.97825 0.180461 0.0902307 0.995921i $$-0.471240\pi$$
0.0902307 + 0.995921i $$0.471240\pi$$
$$762$$ 0 0
$$763$$ − 0.372281i − 0.0134775i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ − 34.9783i − 1.26299i
$$768$$ 0 0
$$769$$ −3.48913 −0.125821 −0.0629105 0.998019i $$-0.520038\pi$$
−0.0629105 + 0.998019i $$0.520038\pi$$
$$770$$ 0 0
$$771$$ −55.7228 −2.00681
$$772$$ 0 0
$$773$$ 4.37228i 0.157260i 0.996904 + 0.0786300i $$0.0250546\pi$$
−0.996904 + 0.0786300i $$0.974945\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ − 4.74456i − 0.170210i
$$778$$ 0 0
$$779$$ 32.0000 1.14652
$$780$$ 0 0
$$781$$ 50.9783 1.82415
$$782$$ 0 0
$$783$$ 3.86141i 0.137995i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 31.1168i 1.10920i 0.832119 + 0.554598i $$0.187128\pi$$
−0.832119 + 0.554598i $$0.812872\pi$$
$$788$$ 0 0
$$789$$ −52.7446 −1.87776
$$790$$ 0 0
$$791$$ −2.00000 −0.0711118
$$792$$ 0 0
$$793$$ − 12.0000i − 0.426132i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ − 15.6277i − 0.553562i −0.960933 0.276781i $$-0.910732\pi$$
0.960933 0.276781i $$-0.0892677\pi$$
$$798$$ 0 0
$$799$$ −2.64947 −0.0937314
$$800$$ 0 0
$$801$$ 38.7446 1.36897
$$802$$ 0 0
$$803$$ − 38.2337i − 1.34924i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 14.2337i 0.501050i
$$808$$ 0 0
$$809$$ −20.3723 −0.716251 −0.358126 0.933673i $$-0.616584\pi$$
−0.358126 + 0.933673i $$0.616584\pi$$
$$810$$ 0 0
$$811$$ 12.7446 0.447522 0.223761 0.974644i $$-0.428166\pi$$
0.223761 + 0.974644i $$0.428166\pi$$
$$812$$ 0 0
$$813$$ 22.5109i 0.789491i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ − 41.4891i − 1.45152i
$$818$$ 0 0
$$819$$ 11.4891 0.401463
$$820$$ 0 0
$$821$$ −25.1168 −0.876584 −0.438292 0.898833i $$-0.644416\pi$$
−0.438292 + 0.898833i $$0.644416\pi$$
$$822$$ 0 0
$$823$$ − 8.00000i − 0.278862i −0.990232 0.139431i $$-0.955473\pi$$
0.990232 0.139431i $$-0.0445274\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 24.7446i − 0.860453i −0.902721 0.430226i $$-0.858434\pi$$
0.902721 0.430226i $$-0.141566\pi$$
$$828$$ 0 0
$$829$$ 12.2337 0.424894 0.212447 0.977173i $$-0.431857\pi$$
0.212447 + 0.977173i $$0.431857\pi$$
$$830$$ 0 0
$$831$$ 59.2554 2.05555
$$832$$ 0 0
$$833$$ − 0.372281i − 0.0128988i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ − 7.06525i − 0.244211i
$$838$$ 0 0
$$839$$ 11.2554 0.388581 0.194290 0.980944i $$-0.437760\pi$$
0.194290 + 0.980944i $$0.437760\pi$$
$$840$$ 0 0
$$841$$ −9.88316 −0.340798
$$842$$ 0 0
$$843$$ 44.1386i 1.52021i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 29.6060i 1.01727i
$$848$$ 0 0
$$849$$ 21.0733 0.723235
$$850$$ 0 0
$$851$$ 9.48913 0.325283
$$852$$ 0 0
$$853$$ − 39.4891i − 1.35208i −0.736864 0.676041i $$-0.763694\pi$$
0.736864 0.676041i $$-0.236306\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ − 18.0000i − 0.614868i −0.951569 0.307434i $$-0.900530\pi$$
0.951569 0.307434i $$-0.0994704\pi$$
$$858$$ 0 0
$$859$$ −44.4674 −1.51721 −0.758604 0.651552i $$-0.774118\pi$$
−0.758604 + 0.651552i $$0.774118\pi$$
$$860$$ 0 0
$$861$$ −16.0000 −0.545279
$$862$$ 0 0
$$863$$ 25.4891i 0.867660i 0.900995 + 0.433830i $$0.142838\pi$$
−0.900995 + 0.433830i $$0.857162\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 40.0000i 1.35847i
$$868$$ 0 0
$$869$$ −96.3288 −3.26773
$$870$$ 0 0
$$871$$ −17.4891 −0.592596
$$872$$ 0 0
$$873$$ − 25.9130i − 0.877022i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 7.02175i 0.237108i 0.992948 + 0.118554i $$0.0378258\pi$$
−0.992948 + 0.118554i $$0.962174\pi$$
$$878$$ 0 0
$$879$$ −59.5842 −2.00973
$$880$$ 0 0
$$881$$ −25.2554 −0.850877 −0.425439 0.904987i $$-0.639880\pi$$
−0.425439 + 0.904987i $$0.639880\pi$$
$$882$$ 0 0
$$883$$ − 10.5109i − 0.353719i −0.984236 0.176860i $$-0.943406\pi$$
0.984236 0.176860i $$-0.0565938\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 13.0217i 0.437228i 0.975811 + 0.218614i $$0.0701535\pi$$
−0.975811 + 0.218614i $$0.929847\pi$$
$$888$$ 0 0
$$889$$ 8.00000 0.268311
$$890$$ 0 0
$$891$$ −63.5842 −2.13015
$$892$$ 0 0
$$893$$ 33.7663i 1.12995i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 49.2119i 1.64314i
$$898$$ 0 0
$$899$$ −34.9783 −1.16659
$$900$$ 0 0
$$901$$ −4.00000 −0.133259
$$902$$ 0 0
$$903$$ 20.7446i 0.690336i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ − 11.7228i − 0.389250i −0.980878 0.194625i $$-0.937651\pi$$
0.980878 0.194625i $$-0.0623489\pi$$
$$908$$ 0 0
$$909$$ 15.7663 0.522936
$$910$$ 0 0
$$911$$ −45.9565 −1.52261 −0.761303 0.648396i $$-0.775440\pi$$
−0.761303 + 0.648396i $$0.775440\pi$$
$$912$$ 0 0
$$913$$ − 60.4674i − 2.00118i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ − 4.74456i − 0.156679i
$$918$$ 0 0
$$919$$ −13.6277 −0.449537 −0.224768 0.974412i $$-0.572163\pi$$
−0.224768 + 0.974412i $$0.572163\pi$$
$$920$$ 0 0
$$921$$ 73.8179 2.43238
$$922$$ 0 0
$$923$$ 34.9783i 1.15132i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 14.7881i 0.485704i
$$928$$ 0 0
$$929$$ 7.76631 0.254804 0.127402 0.991851i $$-0.459336\pi$$
0.127402 + 0.991851i $$0.459336\pi$$
$$930$$ 0 0
$$931$$ −4.74456 −0.155497
$$932$$ 0 0
$$933$$ − 30.2337i − 0.989807i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 28.0951i − 0.917827i −0.888481 0.458913i $$-0.848239\pi$$
0.888481 0.458913i $$-0.151761\pi$$
$$938$$ 0 0
$$939$$ −6.83966 −0.223204
$$940$$ 0 0
$$941$$ 32.2337 1.05079 0.525394 0.850859i $$-0.323918\pi$$
0.525394 + 0.850859i $$0.323918\pi$$
$$942$$ 0 0
$$943$$ − 32.0000i − 1.04206i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 28.0000i 0.909878i 0.890523 + 0.454939i $$0.150339\pi$$
−0.890523 + 0.454939i $$0.849661\pi$$
$$948$$ 0 0
$$949$$ 26.2337 0.851582
$$950$$ 0 0
$$951$$ 33.2119 1.07697
$$952$$ 0 0
$$953$$ 37.2554i 1.20682i 0.797430 + 0.603411i $$0.206192\pi$$
−0.797430 + 0.603411i $$0.793808\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 66.0951i 2.13655i
$$958$$ 0 0
$$959$$ 14.7446 0.476127
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ 0 0
$$963$$ 22.9783i 0.740464i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 1.76631i − 0.0568008i −0.999597 0.0284004i $$-0.990959\pi$$
0.999597 0.0284004i $$-0.00904134\pi$$
$$968$$ 0 0
$$969$$ −4.19019 −0.134608
$$970$$ 0 0
$$971$$ 33.4891 1.07472 0.537359 0.843354i $$-0.319422\pi$$
0.537359 + 0.843354i $$0.319422\pi$$
$$972$$ 0 0
$$973$$ 4.74456i 0.152104i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ − 52.9783i − 1.69492i −0.530856 0.847462i $$-0.678129\pi$$
0.530856 0.847462i $$-0.321871\pi$$
$$978$$ 0 0
$$979$$ −93.9565 −3.00286
$$980$$ 0 0
$$981$$ 0.978251 0.0312331
$$982$$ 0 0
$$983$$ − 10.3723i − 0.330824i −0.986225 0.165412i $$-0.947105\pi$$
0.986225 0.165412i $$-0.0528954\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ − 16.8832i − 0.537397i
$$988$$ 0 0
$$989$$ −41.4891 −1.31928
$$990$$ 0 0
$$991$$ −37.9565 −1.20573 −0.602864 0.797844i $$-0.705974\pi$$
−0.602864 + 0.797844i $$0.705974\pi$$
$$992$$ 0 0
$$993$$ 28.4674i 0.903385i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 6.88316i 0.217992i 0.994042 + 0.108996i $$0.0347635\pi$$
−0.994042 + 0.108996i $$0.965236\pi$$
$$998$$ 0 0
$$999$$ −1.76631 −0.0558836
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 1400.2.g.i.449.3 4
4.3 odd 2 2800.2.g.r.449.2 4
5.2 odd 4 280.2.a.c.1.2 2
5.3 odd 4 1400.2.a.r.1.1 2
5.4 even 2 inner 1400.2.g.i.449.2 4
15.2 even 4 2520.2.a.x.1.1 2
20.3 even 4 2800.2.a.bk.1.2 2
20.7 even 4 560.2.a.h.1.1 2
20.19 odd 2 2800.2.g.r.449.3 4
35.2 odd 12 1960.2.q.t.361.1 4
35.12 even 12 1960.2.q.r.361.2 4
35.13 even 4 9800.2.a.bu.1.2 2
35.17 even 12 1960.2.q.r.961.2 4
35.27 even 4 1960.2.a.s.1.1 2
35.32 odd 12 1960.2.q.t.961.1 4
40.27 even 4 2240.2.a.bg.1.2 2
40.37 odd 4 2240.2.a.bk.1.1 2
60.47 odd 4 5040.2.a.by.1.2 2
140.27 odd 4 3920.2.a.bt.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
280.2.a.c.1.2 2 5.2 odd 4
560.2.a.h.1.1 2 20.7 even 4
1400.2.a.r.1.1 2 5.3 odd 4
1400.2.g.i.449.2 4 5.4 even 2 inner
1400.2.g.i.449.3 4 1.1 even 1 trivial
1960.2.a.s.1.1 2 35.27 even 4
1960.2.q.r.361.2 4 35.12 even 12
1960.2.q.r.961.2 4 35.17 even 12
1960.2.q.t.361.1 4 35.2 odd 12
1960.2.q.t.961.1 4 35.32 odd 12
2240.2.a.bg.1.2 2 40.27 even 4
2240.2.a.bk.1.1 2 40.37 odd 4
2520.2.a.x.1.1 2 15.2 even 4
2800.2.a.bk.1.2 2 20.3 even 4
2800.2.g.r.449.2 4 4.3 odd 2
2800.2.g.r.449.3 4 20.19 odd 2
3920.2.a.bt.1.2 2 140.27 odd 4
5040.2.a.by.1.2 2 60.47 odd 4
9800.2.a.bu.1.2 2 35.13 even 4