# Properties

 Label 2352.2.b.l.1567.7 Level $2352$ Weight $2$ Character 2352.1567 Analytic conductor $18.781$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2352 = 2^{4} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2352.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$18.7808145554$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.339738624.1 Defining polynomial: $$x^{8} - 4 x^{6} + 14 x^{4} - 8 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1567.7 Root $$-1.60021 - 0.923880i$$ of defining polynomial Character $$\chi$$ $$=$$ 2352.1567 Dual form 2352.2.b.l.1567.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} +3.21486i q^{5} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} +3.21486i q^{5} +1.00000 q^{9} +1.36710i q^{11} -2.93015i q^{13} +3.21486i q^{15} +6.91037i q^{17} -7.35449 q^{19} +3.62774i q^{23} -5.33530 q^{25} +1.00000 q^{27} -1.11185 q^{29} -8.70319 q^{31} +1.36710i q^{33} +7.63792 q^{37} -2.93015i q^{39} -0.833147i q^{41} -4.82362i q^{43} +3.21486i q^{45} -2.95367 q^{47} +6.91037i q^{51} -4.57794 q^{53} -4.39502 q^{55} -7.35449 q^{57} +14.0985 q^{59} +11.0618i q^{61} +9.42002 q^{65} +12.1545i q^{67} +3.62774i q^{69} -12.6249i q^{71} -6.49011i q^{73} -5.33530 q^{75} -7.79367i q^{79} +1.00000 q^{81} -8.87502 q^{83} -22.2159 q^{85} -1.11185 q^{87} +12.6146i q^{89} -8.70319 q^{93} -23.6436i q^{95} +13.8811i q^{97} +1.36710i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 8q^{3} + 8q^{9} + O(q^{10})$$ $$8q + 8q^{3} + 8q^{9} - 24q^{25} + 8q^{27} + 16q^{29} - 32q^{31} - 16q^{47} + 16q^{53} - 64q^{55} + 48q^{59} - 16q^{65} - 24q^{75} + 8q^{81} - 64q^{85} + 16q^{87} - 32q^{93} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times$$.

 $$n$$ $$785$$ $$1471$$ $$1765$$ $$2257$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$-1$$

## 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
$$3$$ 1.00000 0.577350
$$4$$ 0 0
$$5$$ 3.21486i 1.43773i 0.695151 + 0.718864i $$0.255338\pi$$
−0.695151 + 0.718864i $$0.744662\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 1.36710i 0.412195i 0.978531 + 0.206098i $$0.0660765\pi$$
−0.978531 + 0.206098i $$0.933924\pi$$
$$12$$ 0 0
$$13$$ − 2.93015i − 0.812678i −0.913722 0.406339i $$-0.866805\pi$$
0.913722 0.406339i $$-0.133195\pi$$
$$14$$ 0 0
$$15$$ 3.21486i 0.830072i
$$16$$ 0 0
$$17$$ 6.91037i 1.67601i 0.545661 + 0.838006i $$0.316279\pi$$
−0.545661 + 0.838006i $$0.683721\pi$$
$$18$$ 0 0
$$19$$ −7.35449 −1.68724 −0.843618 0.536943i $$-0.819579\pi$$
−0.843618 + 0.536943i $$0.819579\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 3.62774i 0.756436i 0.925717 + 0.378218i $$0.123463\pi$$
−0.925717 + 0.378218i $$0.876537\pi$$
$$24$$ 0 0
$$25$$ −5.33530 −1.06706
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ −1.11185 −0.206466 −0.103233 0.994657i $$-0.532919\pi$$
−0.103233 + 0.994657i $$0.532919\pi$$
$$30$$ 0 0
$$31$$ −8.70319 −1.56314 −0.781569 0.623819i $$-0.785580\pi$$
−0.781569 + 0.623819i $$0.785580\pi$$
$$32$$ 0 0
$$33$$ 1.36710i 0.237981i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 7.63792 1.25567 0.627833 0.778348i $$-0.283942\pi$$
0.627833 + 0.778348i $$0.283942\pi$$
$$38$$ 0 0
$$39$$ − 2.93015i − 0.469200i
$$40$$ 0 0
$$41$$ − 0.833147i − 0.130116i −0.997881 0.0650579i $$-0.979277\pi$$
0.997881 0.0650579i $$-0.0207232\pi$$
$$42$$ 0 0
$$43$$ − 4.82362i − 0.735595i −0.929906 0.367797i $$-0.880112\pi$$
0.929906 0.367797i $$-0.119888\pi$$
$$44$$ 0 0
$$45$$ 3.21486i 0.479243i
$$46$$ 0 0
$$47$$ −2.95367 −0.430837 −0.215418 0.976522i $$-0.569112\pi$$
−0.215418 + 0.976522i $$0.569112\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 6.91037i 0.967646i
$$52$$ 0 0
$$53$$ −4.57794 −0.628829 −0.314414 0.949286i $$-0.601808\pi$$
−0.314414 + 0.949286i $$0.601808\pi$$
$$54$$ 0 0
$$55$$ −4.39502 −0.592625
$$56$$ 0 0
$$57$$ −7.35449 −0.974127
$$58$$ 0 0
$$59$$ 14.0985 1.83546 0.917732 0.397200i $$-0.130018\pi$$
0.917732 + 0.397200i $$0.130018\pi$$
$$60$$ 0 0
$$61$$ 11.0618i 1.41632i 0.706052 + 0.708160i $$0.250475\pi$$
−0.706052 + 0.708160i $$0.749525\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 9.42002 1.16841
$$66$$ 0 0
$$67$$ 12.1545i 1.48490i 0.669900 + 0.742452i $$0.266337\pi$$
−0.669900 + 0.742452i $$0.733663\pi$$
$$68$$ 0 0
$$69$$ 3.62774i 0.436728i
$$70$$ 0 0
$$71$$ − 12.6249i − 1.49830i −0.662402 0.749148i $$-0.730463\pi$$
0.662402 0.749148i $$-0.269537\pi$$
$$72$$ 0 0
$$73$$ − 6.49011i − 0.759610i −0.925067 0.379805i $$-0.875991\pi$$
0.925067 0.379805i $$-0.124009\pi$$
$$74$$ 0 0
$$75$$ −5.33530 −0.616068
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ − 7.79367i − 0.876856i −0.898766 0.438428i $$-0.855535\pi$$
0.898766 0.438428i $$-0.144465\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −8.87502 −0.974159 −0.487080 0.873358i $$-0.661938\pi$$
−0.487080 + 0.873358i $$0.661938\pi$$
$$84$$ 0 0
$$85$$ −22.2159 −2.40965
$$86$$ 0 0
$$87$$ −1.11185 −0.119203
$$88$$ 0 0
$$89$$ 12.6146i 1.33715i 0.743646 + 0.668574i $$0.233095\pi$$
−0.743646 + 0.668574i $$0.766905\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ −8.70319 −0.902478
$$94$$ 0 0
$$95$$ − 23.6436i − 2.42579i
$$96$$ 0 0
$$97$$ 13.8811i 1.40942i 0.709497 + 0.704708i $$0.248922\pi$$
−0.709497 + 0.704708i $$0.751078\pi$$
$$98$$ 0 0
$$99$$ 1.36710i 0.137398i
$$100$$ 0 0
$$101$$ − 1.65891i − 0.165068i −0.996588 0.0825338i $$-0.973699\pi$$
0.996588 0.0825338i $$-0.0263013\pi$$
$$102$$ 0 0
$$103$$ −1.30791 −0.128872 −0.0644359 0.997922i $$-0.520525\pi$$
−0.0644359 + 0.997922i $$0.520525\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 7.15204i 0.691414i 0.938343 + 0.345707i $$0.112361\pi$$
−0.938343 + 0.345707i $$0.887639\pi$$
$$108$$ 0 0
$$109$$ −15.0443 −1.44098 −0.720491 0.693464i $$-0.756083\pi$$
−0.720491 + 0.693464i $$0.756083\pi$$
$$110$$ 0 0
$$111$$ 7.63792 0.724959
$$112$$ 0 0
$$113$$ −15.4530 −1.45369 −0.726846 0.686800i $$-0.759015\pi$$
−0.726846 + 0.686800i $$0.759015\pi$$
$$114$$ 0 0
$$115$$ −11.6627 −1.08755
$$116$$ 0 0
$$117$$ − 2.93015i − 0.270893i
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 9.13104 0.830095
$$122$$ 0 0
$$123$$ − 0.833147i − 0.0751224i
$$124$$ 0 0
$$125$$ − 1.07795i − 0.0964149i
$$126$$ 0 0
$$127$$ − 2.16478i − 0.192094i −0.995377 0.0960468i $$-0.969380\pi$$
0.995377 0.0960468i $$-0.0306198\pi$$
$$128$$ 0 0
$$129$$ − 4.82362i − 0.424696i
$$130$$ 0 0
$$131$$ 4.97877 0.434997 0.217499 0.976061i $$-0.430210\pi$$
0.217499 + 0.976061i $$0.430210\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 3.21486i 0.276691i
$$136$$ 0 0
$$137$$ −3.68980 −0.315241 −0.157620 0.987500i $$-0.550382\pi$$
−0.157620 + 0.987500i $$0.550382\pi$$
$$138$$ 0 0
$$139$$ 12.0577 1.02272 0.511360 0.859367i $$-0.329142\pi$$
0.511360 + 0.859367i $$0.329142\pi$$
$$140$$ 0 0
$$141$$ −2.95367 −0.248744
$$142$$ 0 0
$$143$$ 4.00580 0.334982
$$144$$ 0 0
$$145$$ − 3.57445i − 0.296842i
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 3.30262 0.270561 0.135280 0.990807i $$-0.456806\pi$$
0.135280 + 0.990807i $$0.456806\pi$$
$$150$$ 0 0
$$151$$ − 18.9511i − 1.54222i −0.636701 0.771111i $$-0.719701\pi$$
0.636701 0.771111i $$-0.280299\pi$$
$$152$$ 0 0
$$153$$ 6.91037i 0.558671i
$$154$$ 0 0
$$155$$ − 27.9795i − 2.24737i
$$156$$ 0 0
$$157$$ 0.886436i 0.0707453i 0.999374 + 0.0353726i $$0.0112618\pi$$
−0.999374 + 0.0353726i $$0.988738\pi$$
$$158$$ 0 0
$$159$$ −4.57794 −0.363055
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 19.3538i 1.51591i 0.652310 + 0.757953i $$0.273800\pi$$
−0.652310 + 0.757953i $$0.726200\pi$$
$$164$$ 0 0
$$165$$ −4.39502 −0.342152
$$166$$ 0 0
$$167$$ −25.4855 −1.97213 −0.986065 0.166360i $$-0.946799\pi$$
−0.986065 + 0.166360i $$0.946799\pi$$
$$168$$ 0 0
$$169$$ 4.41421 0.339555
$$170$$ 0 0
$$171$$ −7.35449 −0.562412
$$172$$ 0 0
$$173$$ 21.6924i 1.64925i 0.565683 + 0.824623i $$0.308613\pi$$
−0.565683 + 0.824623i $$0.691387\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 14.0985 1.05971
$$178$$ 0 0
$$179$$ 12.2824i 0.918032i 0.888428 + 0.459016i $$0.151798\pi$$
−0.888428 + 0.459016i $$0.848202\pi$$
$$180$$ 0 0
$$181$$ 2.74444i 0.203993i 0.994785 + 0.101996i $$0.0325230\pi$$
−0.994785 + 0.101996i $$0.967477\pi$$
$$182$$ 0 0
$$183$$ 11.0618i 0.817713i
$$184$$ 0 0
$$185$$ 24.5548i 1.80531i
$$186$$ 0 0
$$187$$ −9.44716 −0.690844
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ − 22.8857i − 1.65595i −0.560766 0.827974i $$-0.689493\pi$$
0.560766 0.827974i $$-0.310507\pi$$
$$192$$ 0 0
$$193$$ 14.6163 1.05211 0.526053 0.850452i $$-0.323671\pi$$
0.526053 + 0.850452i $$0.323671\pi$$
$$194$$ 0 0
$$195$$ 9.42002 0.674581
$$196$$ 0 0
$$197$$ 24.5430 1.74861 0.874307 0.485374i $$-0.161316\pi$$
0.874307 + 0.485374i $$0.161316\pi$$
$$198$$ 0 0
$$199$$ −6.04659 −0.428631 −0.214316 0.976764i $$-0.568752\pi$$
−0.214316 + 0.976764i $$0.568752\pi$$
$$200$$ 0 0
$$201$$ 12.1545i 0.857309i
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 2.67845 0.187071
$$206$$ 0 0
$$207$$ 3.62774i 0.252145i
$$208$$ 0 0
$$209$$ − 10.0543i − 0.695471i
$$210$$ 0 0
$$211$$ 9.33513i 0.642657i 0.946968 + 0.321328i $$0.104129\pi$$
−0.946968 + 0.321328i $$0.895871\pi$$
$$212$$ 0 0
$$213$$ − 12.6249i − 0.865042i
$$214$$ 0 0
$$215$$ 15.5072 1.05758
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ − 6.49011i − 0.438561i
$$220$$ 0 0
$$221$$ 20.2484 1.36206
$$222$$ 0 0
$$223$$ 5.48794 0.367500 0.183750 0.982973i $$-0.441176\pi$$
0.183750 + 0.982973i $$0.441176\pi$$
$$224$$ 0 0
$$225$$ −5.33530 −0.355687
$$226$$ 0 0
$$227$$ −1.03813 −0.0689028 −0.0344514 0.999406i $$-0.510968\pi$$
−0.0344514 + 0.999406i $$0.510968\pi$$
$$228$$ 0 0
$$229$$ 15.6039i 1.03113i 0.856850 + 0.515566i $$0.172418\pi$$
−0.856850 + 0.515566i $$0.827582\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 9.21561 0.603734 0.301867 0.953350i $$-0.402390\pi$$
0.301867 + 0.953350i $$0.402390\pi$$
$$234$$ 0 0
$$235$$ − 9.49562i − 0.619426i
$$236$$ 0 0
$$237$$ − 7.79367i − 0.506253i
$$238$$ 0 0
$$239$$ 1.54809i 0.100137i 0.998746 + 0.0500687i $$0.0159440\pi$$
−0.998746 + 0.0500687i $$0.984056\pi$$
$$240$$ 0 0
$$241$$ 3.68527i 0.237389i 0.992931 + 0.118695i $$0.0378709\pi$$
−0.992931 + 0.118695i $$0.962129\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 21.5498i 1.37118i
$$248$$ 0 0
$$249$$ −8.87502 −0.562431
$$250$$ 0 0
$$251$$ 26.1940 1.65335 0.826676 0.562679i $$-0.190229\pi$$
0.826676 + 0.562679i $$0.190229\pi$$
$$252$$ 0 0
$$253$$ −4.95947 −0.311799
$$254$$ 0 0
$$255$$ −22.2159 −1.39121
$$256$$ 0 0
$$257$$ 0.833147i 0.0519703i 0.999662 + 0.0259851i $$0.00827226\pi$$
−0.999662 + 0.0259851i $$0.991728\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −1.11185 −0.0688220
$$262$$ 0 0
$$263$$ 5.04212i 0.310911i 0.987843 + 0.155455i $$0.0496845\pi$$
−0.987843 + 0.155455i $$0.950316\pi$$
$$264$$ 0 0
$$265$$ − 14.7174i − 0.904085i
$$266$$ 0 0
$$267$$ 12.6146i 0.772002i
$$268$$ 0 0
$$269$$ 4.85594i 0.296072i 0.988982 + 0.148036i $$0.0472951\pi$$
−0.988982 + 0.148036i $$0.952705\pi$$
$$270$$ 0 0
$$271$$ −22.2488 −1.35152 −0.675759 0.737122i $$-0.736184\pi$$
−0.675759 + 0.737122i $$0.736184\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ − 7.29388i − 0.439837i
$$276$$ 0 0
$$277$$ 0.670606 0.0402928 0.0201464 0.999797i $$-0.493587\pi$$
0.0201464 + 0.999797i $$0.493587\pi$$
$$278$$ 0 0
$$279$$ −8.70319 −0.521046
$$280$$ 0 0
$$281$$ 25.5971 1.52700 0.763499 0.645810i $$-0.223480\pi$$
0.763499 + 0.645810i $$0.223480\pi$$
$$282$$ 0 0
$$283$$ 11.1115 0.660510 0.330255 0.943892i $$-0.392865\pi$$
0.330255 + 0.943892i $$0.392865\pi$$
$$284$$ 0 0
$$285$$ − 23.6436i − 1.40053i
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −30.7533 −1.80902
$$290$$ 0 0
$$291$$ 13.8811i 0.813727i
$$292$$ 0 0
$$293$$ − 19.2818i − 1.12645i −0.826303 0.563226i $$-0.809560\pi$$
0.826303 0.563226i $$-0.190440\pi$$
$$294$$ 0 0
$$295$$ 45.3245i 2.63890i
$$296$$ 0 0
$$297$$ 1.36710i 0.0793270i
$$298$$ 0 0
$$299$$ 10.6298 0.614738
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ − 1.65891i − 0.0953019i
$$304$$ 0 0
$$305$$ −35.5622 −2.03628
$$306$$ 0 0
$$307$$ 26.0058 1.48423 0.742115 0.670273i $$-0.233823\pi$$
0.742115 + 0.670273i $$0.233823\pi$$
$$308$$ 0 0
$$309$$ −1.30791 −0.0744042
$$310$$ 0 0
$$311$$ 19.0772 1.08177 0.540885 0.841096i $$-0.318089\pi$$
0.540885 + 0.841096i $$0.318089\pi$$
$$312$$ 0 0
$$313$$ − 21.3583i − 1.20724i −0.797272 0.603620i $$-0.793724\pi$$
0.797272 0.603620i $$-0.206276\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 7.74131 0.434795 0.217398 0.976083i $$-0.430243\pi$$
0.217398 + 0.976083i $$0.430243\pi$$
$$318$$ 0 0
$$319$$ − 1.52001i − 0.0851043i
$$320$$ 0 0
$$321$$ 7.15204i 0.399188i
$$322$$ 0 0
$$323$$ − 50.8223i − 2.82783i
$$324$$ 0 0
$$325$$ 15.6332i 0.867176i
$$326$$ 0 0
$$327$$ −15.0443 −0.831951
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ − 0.675988i − 0.0371557i −0.999827 0.0185778i $$-0.994086\pi$$
0.999827 0.0185778i $$-0.00591385\pi$$
$$332$$ 0 0
$$333$$ 7.63792 0.418555
$$334$$ 0 0
$$335$$ −39.0748 −2.13489
$$336$$ 0 0
$$337$$ −0.176755 −0.00962847 −0.00481423 0.999988i $$-0.501532\pi$$
−0.00481423 + 0.999988i $$0.501532\pi$$
$$338$$ 0 0
$$339$$ −15.4530 −0.839290
$$340$$ 0 0
$$341$$ − 11.8981i − 0.644318i
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ −11.6627 −0.627896
$$346$$ 0 0
$$347$$ − 6.22637i − 0.334249i −0.985936 0.167125i $$-0.946552\pi$$
0.985936 0.167125i $$-0.0534482\pi$$
$$348$$ 0 0
$$349$$ 17.4414i 0.933616i 0.884359 + 0.466808i $$0.154596\pi$$
−0.884359 + 0.466808i $$0.845404\pi$$
$$350$$ 0 0
$$351$$ − 2.93015i − 0.156400i
$$352$$ 0 0
$$353$$ − 8.70218i − 0.463170i −0.972815 0.231585i $$-0.925609\pi$$
0.972815 0.231585i $$-0.0743912\pi$$
$$354$$ 0 0
$$355$$ 40.5871 2.15414
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 0.764263i 0.0403362i 0.999797 + 0.0201681i $$0.00642015\pi$$
−0.999797 + 0.0201681i $$0.993580\pi$$
$$360$$ 0 0
$$361$$ 35.0886 1.84677
$$362$$ 0 0
$$363$$ 9.13104 0.479256
$$364$$ 0 0
$$365$$ 20.8648 1.09211
$$366$$ 0 0
$$367$$ 26.5162 1.38413 0.692067 0.721833i $$-0.256700\pi$$
0.692067 + 0.721833i $$0.256700\pi$$
$$368$$ 0 0
$$369$$ − 0.833147i − 0.0433719i
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −20.1310 −1.04235 −0.521173 0.853451i $$-0.674505\pi$$
−0.521173 + 0.853451i $$0.674505\pi$$
$$374$$ 0 0
$$375$$ − 1.07795i − 0.0556652i
$$376$$ 0 0
$$377$$ 3.25790i 0.167790i
$$378$$ 0 0
$$379$$ 10.2608i 0.527061i 0.964651 + 0.263531i $$0.0848870\pi$$
−0.964651 + 0.263531i $$0.915113\pi$$
$$380$$ 0 0
$$381$$ − 2.16478i − 0.110905i
$$382$$ 0 0
$$383$$ −2.34315 −0.119729 −0.0598646 0.998207i $$-0.519067\pi$$
−0.0598646 + 0.998207i $$0.519067\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ − 4.82362i − 0.245198i
$$388$$ 0 0
$$389$$ 24.0440 1.21908 0.609540 0.792755i $$-0.291354\pi$$
0.609540 + 0.792755i $$0.291354\pi$$
$$390$$ 0 0
$$391$$ −25.0690 −1.26780
$$392$$ 0 0
$$393$$ 4.97877 0.251146
$$394$$ 0 0
$$395$$ 25.0555 1.26068
$$396$$ 0 0
$$397$$ − 18.3526i − 0.921088i −0.887637 0.460544i $$-0.847654\pi$$
0.887637 0.460544i $$-0.152346\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −2.96972 −0.148301 −0.0741503 0.997247i $$-0.523624\pi$$
−0.0741503 + 0.997247i $$0.523624\pi$$
$$402$$ 0 0
$$403$$ 25.5016i 1.27033i
$$404$$ 0 0
$$405$$ 3.21486i 0.159748i
$$406$$ 0 0
$$407$$ 10.4418i 0.517580i
$$408$$ 0 0
$$409$$ − 23.9502i − 1.18426i −0.805842 0.592131i $$-0.798287\pi$$
0.805842 0.592131i $$-0.201713\pi$$
$$410$$ 0 0
$$411$$ −3.68980 −0.182004
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ − 28.5319i − 1.40058i
$$416$$ 0 0
$$417$$ 12.0577 0.590467
$$418$$ 0 0
$$419$$ −36.3065 −1.77369 −0.886844 0.462069i $$-0.847107\pi$$
−0.886844 + 0.462069i $$0.847107\pi$$
$$420$$ 0 0
$$421$$ −22.6274 −1.10279 −0.551396 0.834243i $$-0.685905\pi$$
−0.551396 + 0.834243i $$0.685905\pi$$
$$422$$ 0 0
$$423$$ −2.95367 −0.143612
$$424$$ 0 0
$$425$$ − 36.8689i − 1.78841i
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 4.00580 0.193402
$$430$$ 0 0
$$431$$ 30.6040i 1.47414i 0.675816 + 0.737071i $$0.263792\pi$$
−0.675816 + 0.737071i $$0.736208\pi$$
$$432$$ 0 0
$$433$$ 14.4650i 0.695146i 0.937653 + 0.347573i $$0.112994\pi$$
−0.937653 + 0.347573i $$0.887006\pi$$
$$434$$ 0 0
$$435$$ − 3.57445i − 0.171382i
$$436$$ 0 0
$$437$$ − 26.6802i − 1.27629i
$$438$$ 0 0
$$439$$ −9.10981 −0.434788 −0.217394 0.976084i $$-0.569756\pi$$
−0.217394 + 0.976084i $$0.569756\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ − 2.83319i − 0.134609i −0.997732 0.0673045i $$-0.978560\pi$$
0.997732 0.0673045i $$-0.0214399\pi$$
$$444$$ 0 0
$$445$$ −40.5542 −1.92245
$$446$$ 0 0
$$447$$ 3.30262 0.156208
$$448$$ 0 0
$$449$$ 10.3630 0.489058 0.244529 0.969642i $$-0.421367\pi$$
0.244529 + 0.969642i $$0.421367\pi$$
$$450$$ 0 0
$$451$$ 1.13899 0.0536331
$$452$$ 0 0
$$453$$ − 18.9511i − 0.890402i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −6.65951 −0.311519 −0.155759 0.987795i $$-0.549782\pi$$
−0.155759 + 0.987795i $$0.549782\pi$$
$$458$$ 0 0
$$459$$ 6.91037i 0.322549i
$$460$$ 0 0
$$461$$ 1.71311i 0.0797873i 0.999204 + 0.0398936i $$0.0127019\pi$$
−0.999204 + 0.0398936i $$0.987298\pi$$
$$462$$ 0 0
$$463$$ 24.1733i 1.12343i 0.827331 + 0.561715i $$0.189858\pi$$
−0.827331 + 0.561715i $$0.810142\pi$$
$$464$$ 0 0
$$465$$ − 27.9795i − 1.29752i
$$466$$ 0 0
$$467$$ 37.2654 1.72444 0.862220 0.506535i $$-0.169074\pi$$
0.862220 + 0.506535i $$0.169074\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 0.886436i 0.0408448i
$$472$$ 0 0
$$473$$ 6.59436 0.303209
$$474$$ 0 0
$$475$$ 39.2385 1.80038
$$476$$ 0 0
$$477$$ −4.57794 −0.209610
$$478$$ 0 0
$$479$$ 21.8596 0.998790 0.499395 0.866374i $$-0.333556\pi$$
0.499395 + 0.866374i $$0.333556\pi$$
$$480$$ 0 0
$$481$$ − 22.3803i − 1.02045i
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −44.6259 −2.02636
$$486$$ 0 0
$$487$$ 3.37269i 0.152831i 0.997076 + 0.0764155i $$0.0243475\pi$$
−0.997076 + 0.0764155i $$0.975652\pi$$
$$488$$ 0 0
$$489$$ 19.3538i 0.875208i
$$490$$ 0 0
$$491$$ 9.68824i 0.437224i 0.975812 + 0.218612i $$0.0701529\pi$$
−0.975812 + 0.218612i $$0.929847\pi$$
$$492$$ 0 0
$$493$$ − 7.68332i − 0.346039i
$$494$$ 0 0
$$495$$ −4.39502 −0.197542
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 13.7401i 0.615089i 0.951534 + 0.307545i $$0.0995074\pi$$
−0.951534 + 0.307545i $$0.900493\pi$$
$$500$$ 0 0
$$501$$ −25.4855 −1.13861
$$502$$ 0 0
$$503$$ −1.92715 −0.0859273 −0.0429637 0.999077i $$-0.513680\pi$$
−0.0429637 + 0.999077i $$0.513680\pi$$
$$504$$ 0 0
$$505$$ 5.33316 0.237322
$$506$$ 0 0
$$507$$ 4.41421 0.196042
$$508$$ 0 0
$$509$$ − 14.1621i − 0.627723i −0.949469 0.313862i $$-0.898377\pi$$
0.949469 0.313862i $$-0.101623\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ −7.35449 −0.324709
$$514$$ 0 0
$$515$$ − 4.20473i − 0.185283i
$$516$$ 0 0
$$517$$ − 4.03795i − 0.177589i
$$518$$ 0 0
$$519$$ 21.6924i 0.952193i
$$520$$ 0 0
$$521$$ 21.6026i 0.946427i 0.880948 + 0.473213i $$0.156906\pi$$
−0.880948 + 0.473213i $$0.843094\pi$$
$$522$$ 0 0
$$523$$ 13.9650 0.610648 0.305324 0.952249i $$-0.401235\pi$$
0.305324 + 0.952249i $$0.401235\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ − 60.1423i − 2.61984i
$$528$$ 0 0
$$529$$ 9.83952 0.427805
$$530$$ 0 0
$$531$$ 14.0985 0.611821
$$532$$ 0 0
$$533$$ −2.44125 −0.105742
$$534$$ 0 0
$$535$$ −22.9928 −0.994064
$$536$$ 0 0
$$537$$ 12.2824i 0.530026i
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 3.88005 0.166816 0.0834081 0.996515i $$-0.473419\pi$$
0.0834081 + 0.996515i $$0.473419\pi$$
$$542$$ 0 0
$$543$$ 2.74444i 0.117775i
$$544$$ 0 0
$$545$$ − 48.3652i − 2.07174i
$$546$$ 0 0
$$547$$ − 35.1640i − 1.50351i −0.659445 0.751753i $$-0.729209\pi$$
0.659445 0.751753i $$-0.270791\pi$$
$$548$$ 0 0
$$549$$ 11.0618i 0.472107i
$$550$$ 0 0
$$551$$ 8.17712 0.348357
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 24.5548i 1.04229i
$$556$$ 0 0
$$557$$ 42.4346 1.79801 0.899006 0.437936i $$-0.144290\pi$$
0.899006 + 0.437936i $$0.144290\pi$$
$$558$$ 0 0
$$559$$ −14.1339 −0.597802
$$560$$ 0 0
$$561$$ −9.44716 −0.398859
$$562$$ 0 0
$$563$$ −16.5488 −0.697447 −0.348724 0.937226i $$-0.613385\pi$$
−0.348724 + 0.937226i $$0.613385\pi$$
$$564$$ 0 0
$$565$$ − 49.6790i − 2.09001i
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −24.2566 −1.01689 −0.508446 0.861094i $$-0.669780\pi$$
−0.508446 + 0.861094i $$0.669780\pi$$
$$570$$ 0 0
$$571$$ 4.62563i 0.193576i 0.995305 + 0.0967882i $$0.0308569\pi$$
−0.995305 + 0.0967882i $$0.969143\pi$$
$$572$$ 0 0
$$573$$ − 22.8857i − 0.956062i
$$574$$ 0 0
$$575$$ − 19.3551i − 0.807163i
$$576$$ 0 0
$$577$$ − 26.7907i − 1.11531i −0.830072 0.557656i $$-0.811701\pi$$
0.830072 0.557656i $$-0.188299\pi$$
$$578$$ 0 0
$$579$$ 14.6163 0.607434
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ − 6.25850i − 0.259200i
$$584$$ 0 0
$$585$$ 9.42002 0.389470
$$586$$ 0 0
$$587$$ 27.8591 1.14987 0.574933 0.818200i $$-0.305028\pi$$
0.574933 + 0.818200i $$0.305028\pi$$
$$588$$ 0 0
$$589$$ 64.0075 2.63738
$$590$$ 0 0
$$591$$ 24.5430 1.00956
$$592$$ 0 0
$$593$$ 1.56906i 0.0644334i 0.999481 + 0.0322167i $$0.0102567\pi$$
−0.999481 + 0.0322167i $$0.989743\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −6.04659 −0.247470
$$598$$ 0 0
$$599$$ − 40.1156i − 1.63908i −0.573021 0.819540i $$-0.694229\pi$$
0.573021 0.819540i $$-0.305771\pi$$
$$600$$ 0 0
$$601$$ − 24.7827i − 1.01091i −0.862854 0.505453i $$-0.831325\pi$$
0.862854 0.505453i $$-0.168675\pi$$
$$602$$ 0 0
$$603$$ 12.1545i 0.494968i
$$604$$ 0 0
$$605$$ 29.3550i 1.19345i
$$606$$ 0 0
$$607$$ −4.66240 −0.189241 −0.0946205 0.995513i $$-0.530164\pi$$
−0.0946205 + 0.995513i $$0.530164\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 8.65470i 0.350132i
$$612$$ 0 0
$$613$$ 28.5817 1.15440 0.577202 0.816601i $$-0.304144\pi$$
0.577202 + 0.816601i $$0.304144\pi$$
$$614$$ 0 0
$$615$$ 2.67845 0.108005
$$616$$ 0 0
$$617$$ 10.1524 0.408720 0.204360 0.978896i $$-0.434489\pi$$
0.204360 + 0.978896i $$0.434489\pi$$
$$618$$ 0 0
$$619$$ −8.64022 −0.347280 −0.173640 0.984809i $$-0.555553\pi$$
−0.173640 + 0.984809i $$0.555553\pi$$
$$620$$ 0 0
$$621$$ 3.62774i 0.145576i
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −23.2111 −0.928442
$$626$$ 0 0
$$627$$ − 10.0543i − 0.401530i
$$628$$ 0 0
$$629$$ 52.7809i 2.10451i
$$630$$ 0 0
$$631$$ 5.46211i 0.217443i 0.994072 + 0.108722i $$0.0346757\pi$$
−0.994072 + 0.108722i $$0.965324\pi$$
$$632$$ 0 0
$$633$$ 9.33513i 0.371038i
$$634$$ 0 0
$$635$$ 6.95947 0.276178
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ − 12.6249i − 0.499432i
$$640$$ 0 0
$$641$$ 1.42822 0.0564114 0.0282057 0.999602i $$-0.491021\pi$$
0.0282057 + 0.999602i $$0.491021\pi$$
$$642$$ 0 0
$$643$$ −24.6036 −0.970270 −0.485135 0.874439i $$-0.661229\pi$$
−0.485135 + 0.874439i $$0.661229\pi$$
$$644$$ 0 0
$$645$$ 15.5072 0.610597
$$646$$ 0 0
$$647$$ 20.2785 0.797229 0.398614 0.917119i $$-0.369491\pi$$
0.398614 + 0.917119i $$0.369491\pi$$
$$648$$ 0 0
$$649$$ 19.2740i 0.756570i
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 42.8841 1.67818 0.839092 0.543990i $$-0.183087\pi$$
0.839092 + 0.543990i $$0.183087\pi$$
$$654$$ 0 0
$$655$$ 16.0060i 0.625407i
$$656$$ 0 0
$$657$$ − 6.49011i − 0.253203i
$$658$$ 0 0
$$659$$ 31.4717i 1.22596i 0.790098 + 0.612981i $$0.210030\pi$$
−0.790098 + 0.612981i $$0.789970\pi$$
$$660$$ 0 0
$$661$$ − 12.3154i − 0.479015i −0.970895 0.239507i $$-0.923014\pi$$
0.970895 0.239507i $$-0.0769859\pi$$
$$662$$ 0 0
$$663$$ 20.2484 0.786384
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ − 4.03351i − 0.156178i
$$668$$ 0 0
$$669$$ 5.48794 0.212176
$$670$$ 0 0
$$671$$ −15.1226 −0.583801
$$672$$ 0 0
$$673$$ 37.1864 1.43343 0.716716 0.697365i $$-0.245645\pi$$
0.716716 + 0.697365i $$0.245645\pi$$
$$674$$ 0 0
$$675$$ −5.33530 −0.205356
$$676$$ 0 0
$$677$$ 35.6046i 1.36840i 0.729296 + 0.684198i $$0.239848\pi$$
−0.729296 + 0.684198i $$0.760152\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −1.03813 −0.0397810
$$682$$ 0 0
$$683$$ 32.2056i 1.23231i 0.787623 + 0.616157i $$0.211311\pi$$
−0.787623 + 0.616157i $$0.788689\pi$$
$$684$$ 0 0
$$685$$ − 11.8622i − 0.453230i
$$686$$ 0 0
$$687$$ 15.6039i 0.595325i
$$688$$ 0 0
$$689$$ 13.4141i 0.511035i
$$690$$ 0 0
$$691$$ −38.6168 −1.46905 −0.734527 0.678580i $$-0.762596\pi$$
−0.734527 + 0.678580i $$0.762596\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 38.7637i 1.47039i
$$696$$ 0 0
$$697$$ 5.75736 0.218076
$$698$$ 0 0
$$699$$ 9.21561 0.348566
$$700$$ 0 0
$$701$$ 30.8725 1.16604 0.583018 0.812459i $$-0.301872\pi$$
0.583018 + 0.812459i $$0.301872\pi$$
$$702$$ 0 0
$$703$$ −56.1730 −2.11861
$$704$$ 0 0
$$705$$ − 9.49562i − 0.357626i
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 39.8452 1.49642 0.748209 0.663463i $$-0.230914\pi$$
0.748209 + 0.663463i $$0.230914\pi$$
$$710$$ 0 0
$$711$$ − 7.79367i − 0.292285i
$$712$$ 0 0
$$713$$ − 31.5729i − 1.18241i
$$714$$ 0 0
$$715$$ 12.8781i 0.481613i
$$716$$ 0 0
$$717$$ 1.54809i 0.0578144i
$$718$$ 0 0
$$719$$ 2.09266 0.0780431 0.0390216 0.999238i $$-0.487576\pi$$
0.0390216 + 0.999238i $$0.487576\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 3.68527i 0.137057i
$$724$$ 0 0
$$725$$ 5.93207 0.220312
$$726$$ 0 0
$$727$$ 21.5164 0.798001 0.399000 0.916951i $$-0.369357\pi$$
0.399000 + 0.916951i $$0.369357\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 33.3330 1.23287
$$732$$ 0 0
$$733$$ 7.72856i 0.285461i 0.989762 + 0.142730i $$0.0455882\pi$$
−0.989762 + 0.142730i $$0.954412\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −16.6163 −0.612070
$$738$$ 0 0
$$739$$ 16.1063i 0.592481i 0.955113 + 0.296240i $$0.0957329\pi$$
−0.955113 + 0.296240i $$0.904267\pi$$
$$740$$ 0 0
$$741$$ 21.5498i 0.791651i
$$742$$ 0 0
$$743$$ 29.3296i 1.07600i 0.842945 + 0.537999i $$0.180820\pi$$
−0.842945 + 0.537999i $$0.819180\pi$$
$$744$$ 0 0
$$745$$ 10.6174i 0.388993i
$$746$$ 0 0
$$747$$ −8.87502 −0.324720
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ − 13.5991i − 0.496238i −0.968730 0.248119i $$-0.920188\pi$$
0.968730 0.248119i $$-0.0798125\pi$$
$$752$$ 0 0
$$753$$ 26.1940 0.954563
$$754$$ 0 0
$$755$$ 60.9252 2.21730
$$756$$ 0 0
$$757$$ −2.59688 −0.0943851 −0.0471926 0.998886i $$-0.515027\pi$$
−0.0471926 + 0.998886i $$0.515027\pi$$
$$758$$ 0 0
$$759$$ −4.95947 −0.180017
$$760$$ 0 0
$$761$$ − 17.2920i − 0.626833i −0.949616 0.313416i $$-0.898526\pi$$
0.949616 0.313416i $$-0.101474\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ −22.2159 −0.803216
$$766$$ 0 0
$$767$$ − 41.3106i − 1.49164i
$$768$$ 0 0
$$769$$ 41.4025i 1.49301i 0.665378 + 0.746507i $$0.268270\pi$$
−0.665378 + 0.746507i $$0.731730\pi$$
$$770$$ 0 0
$$771$$ 0.833147i 0.0300051i
$$772$$ 0 0
$$773$$ − 29.4669i − 1.05985i −0.848044 0.529925i $$-0.822220\pi$$
0.848044 0.529925i $$-0.177780\pi$$
$$774$$ 0 0
$$775$$ 46.4341 1.66796
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 6.12738i 0.219536i
$$780$$ 0 0
$$781$$ 17.2594 0.617591
$$782$$ 0 0
$$783$$ −1.11185 −0.0397344
$$784$$ 0 0
$$785$$ −2.84976 −0.101712
$$786$$ 0 0
$$787$$ −54.1841 −1.93146 −0.965728 0.259557i $$-0.916423\pi$$
−0.965728 + 0.259557i $$0.916423\pi$$
$$788$$ 0 0
$$789$$ 5.04212i 0.179504i
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 32.4128 1.15101
$$794$$ 0 0
$$795$$ − 14.7174i − 0.521974i
$$796$$ 0 0
$$797$$ − 0.512549i − 0.0181554i −0.999959 0.00907771i $$-0.997110\pi$$
0.999959 0.00907771i $$-0.00288956\pi$$
$$798$$ 0 0
$$799$$ − 20.4110i − 0.722088i
$$800$$ 0 0
$$801$$ 12.6146i 0.445716i
$$802$$ 0 0
$$803$$ 8.87261 0.313108
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 4.85594i 0.170937i
$$808$$ 0 0
$$809$$ −53.1407 −1.86833 −0.934164 0.356844i $$-0.883853\pi$$
−0.934164 + 0.356844i $$0.883853\pi$$
$$810$$ 0 0
$$811$$ 10.8127 0.379687 0.189843 0.981814i $$-0.439202\pi$$
0.189843 + 0.981814i $$0.439202\pi$$
$$812$$ 0 0
$$813$$ −22.2488 −0.780300
$$814$$ 0 0
$$815$$ −62.2196 −2.17946
$$816$$ 0 0
$$817$$ 35.4753i 1.24112i
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −2.70078 −0.0942579 −0.0471290 0.998889i $$-0.515007\pi$$
−0.0471290 + 0.998889i $$0.515007\pi$$
$$822$$ 0 0
$$823$$ − 12.6132i − 0.439670i −0.975537 0.219835i $$-0.929448\pi$$
0.975537 0.219835i $$-0.0705519\pi$$
$$824$$ 0 0
$$825$$ − 7.29388i − 0.253940i
$$826$$ 0 0
$$827$$ − 2.95803i − 0.102861i −0.998677 0.0514304i $$-0.983622\pi$$
0.998677 0.0514304i $$-0.0163780\pi$$
$$828$$ 0 0
$$829$$ 15.8101i 0.549106i 0.961572 + 0.274553i $$0.0885300\pi$$
−0.961572 + 0.274553i $$0.911470\pi$$
$$830$$ 0 0
$$831$$ 0.670606 0.0232631
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ − 81.9324i − 2.83539i
$$836$$ 0 0
$$837$$ −8.70319 −0.300826
$$838$$ 0 0
$$839$$ 5.13130 0.177152 0.0885761 0.996069i $$-0.471768\pi$$
0.0885761 + 0.996069i $$0.471768\pi$$
$$840$$ 0 0
$$841$$ −27.7638 −0.957372
$$842$$ 0 0
$$843$$ 25.5971 0.881612
$$844$$ 0 0
$$845$$ 14.1911i 0.488187i
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 11.1115 0.381345
$$850$$ 0 0
$$851$$ 27.7084i 0.949831i
$$852$$ 0 0
$$853$$ − 45.1627i − 1.54634i −0.634198 0.773170i $$-0.718670\pi$$
0.634198 0.773170i $$-0.281330\pi$$
$$854$$ 0 0
$$855$$ − 23.6436i − 0.808596i
$$856$$ 0 0
$$857$$ 1.47725i 0.0504618i 0.999682 + 0.0252309i $$0.00803210\pi$$
−0.999682 + 0.0252309i $$0.991968\pi$$
$$858$$ 0 0
$$859$$ 3.08183 0.105151 0.0525753 0.998617i $$-0.483257\pi$$
0.0525753 + 0.998617i $$0.483257\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 46.4000i 1.57948i 0.613445 + 0.789738i $$0.289783\pi$$
−0.613445 + 0.789738i $$0.710217\pi$$
$$864$$ 0 0
$$865$$ −69.7381 −2.37117
$$866$$ 0 0
$$867$$ −30.7533 −1.04444
$$868$$ 0 0
$$869$$ 10.6547 0.361436
$$870$$ 0 0
$$871$$ 35.6144 1.20675
$$872$$ 0 0
$$873$$ 13.8811i 0.469806i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −50.4774 −1.70450 −0.852251 0.523133i $$-0.824763\pi$$
−0.852251 + 0.523133i $$0.824763\pi$$
$$878$$ 0 0
$$879$$ − 19.2818i − 0.650357i
$$880$$ 0 0
$$881$$ 3.66920i 0.123619i 0.998088 + 0.0618093i $$0.0196870\pi$$
−0.998088 + 0.0618093i $$0.980313\pi$$
$$882$$ 0 0
$$883$$ − 18.0393i − 0.607071i −0.952820 0.303535i $$-0.901833\pi$$
0.952820 0.303535i $$-0.0981671\pi$$
$$884$$ 0 0
$$885$$ 45.3245i 1.52357i
$$886$$ 0 0
$$887$$ −12.8996 −0.433126 −0.216563 0.976269i $$-0.569485\pi$$
−0.216563 + 0.976269i $$0.569485\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 1.36710i 0.0457995i
$$892$$ 0 0
$$893$$ 21.7227 0.726924
$$894$$ 0 0
$$895$$ −39.4863 −1.31988
$$896$$ 0 0
$$897$$ 10.6298 0.354919
$$898$$ 0 0
$$899$$ 9.67667 0.322735
$$900$$ 0 0
$$901$$ − 31.6353i − 1.05392i
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −8.82299 −0.293286
$$906$$ 0 0
$$907$$ − 0.765170i − 0.0254071i −0.999919 0.0127035i $$-0.995956\pi$$
0.999919 0.0127035i $$-0.00404377\pi$$
$$908$$ 0 0
$$909$$ − 1.65891i − 0.0550226i
$$910$$ 0 0
$$911$$ − 18.7415i − 0.620934i −0.950584 0.310467i $$-0.899515\pi$$
0.950584 0.310467i $$-0.100485\pi$$
$$912$$ 0 0
$$913$$ − 12.1330i − 0.401544i
$$914$$ 0 0
$$915$$ −35.5622 −1.17565
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ − 21.0460i − 0.694242i −0.937820 0.347121i $$-0.887159\pi$$
0.937820 0.347121i $$-0.112841\pi$$
$$920$$ 0 0
$$921$$ 26.0058 0.856920
$$922$$ 0 0
$$923$$ −36.9928 −1.21763
$$924$$ 0 0
$$925$$ −40.7506 −1.33987
$$926$$ 0 0
$$927$$ −1.30791 −0.0429573
$$928$$ 0 0
$$929$$ − 32.7286i − 1.07379i −0.843649 0.536895i $$-0.819597\pi$$
0.843649 0.536895i $$-0.180403\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ 19.0772 0.624561
$$934$$ 0 0
$$935$$ − 30.3713i − 0.993246i
$$936$$ 0 0
$$937$$ − 2.53824i − 0.0829207i −0.999140 0.0414603i $$-0.986799\pi$$
0.999140 0.0414603i $$-0.0132010\pi$$
$$938$$ 0 0
$$939$$ − 21.3583i − 0.697001i
$$940$$ 0 0
$$941$$ − 46.1741i − 1.50523i −0.658459 0.752617i $$-0.728791\pi$$
0.658459 0.752617i $$-0.271209\pi$$
$$942$$ 0 0
$$943$$ 3.02244 0.0984242
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 1.51338i 0.0491782i 0.999698 + 0.0245891i $$0.00782775\pi$$
−0.999698 + 0.0245891i $$0.992172\pi$$
$$948$$ 0 0
$$949$$ −19.0170 −0.617318
$$950$$ 0 0
$$951$$ 7.74131 0.251029
$$952$$ 0 0
$$953$$ −43.1132 −1.39657 −0.698287 0.715818i $$-0.746054\pi$$
−0.698287 + 0.715818i $$0.746054\pi$$
$$954$$ 0 0
$$955$$ 73.5741 2.38080
$$956$$ 0 0
$$957$$ − 1.52001i − 0.0491350i
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 44.7454 1.44340
$$962$$ 0 0
$$963$$ 7.15204i 0.230471i
$$964$$ 0 0
$$965$$ 46.9894i 1.51264i
$$966$$ 0 0
$$967$$ − 22.5218i − 0.724253i −0.932129 0.362127i $$-0.882051\pi$$
0.932129 0.362127i $$-0.117949\pi$$
$$968$$ 0 0
$$969$$ − 50.8223i − 1.63265i
$$970$$ 0 0
$$971$$ −24.4894 −0.785902 −0.392951 0.919559i $$-0.628546\pi$$
−0.392951 + 0.919559i $$0.628546\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 15.6332i 0.500665i
$$976$$ 0 0
$$977$$ 16.6382 0.532302 0.266151 0.963931i $$-0.414248\pi$$
0.266151 + 0.963931i $$0.414248\pi$$
$$978$$ 0 0
$$979$$ −17.2454 −0.551166
$$980$$ 0 0
$$981$$ −15.0443 −0.480327
$$982$$ 0 0
$$983$$ 5.35617 0.170835 0.0854177 0.996345i $$-0.472778\pi$$
0.0854177 + 0.996345i $$0.472778\pi$$
$$984$$ 0 0
$$985$$ 78.9021i 2.51403i
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 17.4988 0.556430
$$990$$ 0 0
$$991$$ − 4.67605i − 0.148540i −0.997238 0.0742698i $$-0.976337\pi$$
0.997238 0.0742698i $$-0.0236626\pi$$
$$992$$ 0 0
$$993$$ − 0.675988i − 0.0214518i
$$994$$ 0 0
$$995$$ − 19.4389i − 0.616255i
$$996$$ 0 0
$$997$$ − 0.906930i − 0.0287228i −0.999897 0.0143614i $$-0.995428\pi$$
0.999897 0.0143614i $$-0.00457153\pi$$
$$998$$ 0 0
$$999$$ 7.63792 0.241653
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 2352.2.b.l.1567.7 yes 8
3.2 odd 2 7056.2.b.w.1567.2 8
4.3 odd 2 2352.2.b.k.1567.7 yes 8
7.2 even 3 2352.2.bl.q.31.4 8
7.3 odd 6 2352.2.bl.r.607.4 8
7.4 even 3 2352.2.bl.o.607.1 8
7.5 odd 6 2352.2.bl.t.31.1 8
7.6 odd 2 2352.2.b.k.1567.2 8
12.11 even 2 7056.2.b.x.1567.2 8
21.20 even 2 7056.2.b.x.1567.7 8
28.3 even 6 2352.2.bl.q.607.4 8
28.11 odd 6 2352.2.bl.t.607.1 8
28.19 even 6 2352.2.bl.o.31.1 8
28.23 odd 6 2352.2.bl.r.31.4 8
28.27 even 2 inner 2352.2.b.l.1567.2 yes 8
84.83 odd 2 7056.2.b.w.1567.7 8

By twisted newform
Twist Min Dim Char Parity Ord Type
2352.2.b.k.1567.2 8 7.6 odd 2
2352.2.b.k.1567.7 yes 8 4.3 odd 2
2352.2.b.l.1567.2 yes 8 28.27 even 2 inner
2352.2.b.l.1567.7 yes 8 1.1 even 1 trivial
2352.2.bl.o.31.1 8 28.19 even 6
2352.2.bl.o.607.1 8 7.4 even 3
2352.2.bl.q.31.4 8 7.2 even 3
2352.2.bl.q.607.4 8 28.3 even 6
2352.2.bl.r.31.4 8 28.23 odd 6
2352.2.bl.r.607.4 8 7.3 odd 6
2352.2.bl.t.31.1 8 7.5 odd 6
2352.2.bl.t.607.1 8 28.11 odd 6
7056.2.b.w.1567.2 8 3.2 odd 2
7056.2.b.w.1567.7 8 84.83 odd 2
7056.2.b.x.1567.2 8 12.11 even 2
7056.2.b.x.1567.7 8 21.20 even 2