# Properties

 Label 9196.2.a.f.1.1 Level $9196$ Weight $2$ Character 9196.1 Self dual yes Analytic conductor $73.430$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$9196 = 2^{2} \cdot 11^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9196.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$73.4304296988$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 76) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 9196.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.00000 q^{3} -1.00000 q^{5} +3.00000 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q+2.00000 q^{3} -1.00000 q^{5} +3.00000 q^{7} +1.00000 q^{9} +4.00000 q^{13} -2.00000 q^{15} +3.00000 q^{17} +1.00000 q^{19} +6.00000 q^{21} +8.00000 q^{23} -4.00000 q^{25} -4.00000 q^{27} +2.00000 q^{29} +4.00000 q^{31} -3.00000 q^{35} +10.0000 q^{37} +8.00000 q^{39} -10.0000 q^{41} -1.00000 q^{43} -1.00000 q^{45} -1.00000 q^{47} +2.00000 q^{49} +6.00000 q^{51} -4.00000 q^{53} +2.00000 q^{57} +6.00000 q^{59} +13.0000 q^{61} +3.00000 q^{63} -4.00000 q^{65} -12.0000 q^{67} +16.0000 q^{69} +2.00000 q^{71} -9.00000 q^{73} -8.00000 q^{75} -8.00000 q^{79} -11.0000 q^{81} +12.0000 q^{83} -3.00000 q^{85} +4.00000 q^{87} +12.0000 q^{89} +12.0000 q^{91} +8.00000 q^{93} -1.00000 q^{95} -8.00000 q^{97} +O(q^{100})$$

## 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
$$3$$ 2.00000 1.15470 0.577350 0.816497i $$-0.304087\pi$$
0.577350 + 0.816497i $$0.304087\pi$$
$$4$$ 0 0
$$5$$ −1.00000 −0.447214 −0.223607 0.974679i $$-0.571783\pi$$
−0.223607 + 0.974679i $$0.571783\pi$$
$$6$$ 0 0
$$7$$ 3.00000 1.13389 0.566947 0.823754i $$-0.308125\pi$$
0.566947 + 0.823754i $$0.308125\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 0 0
$$12$$ 0 0
$$13$$ 4.00000 1.10940 0.554700 0.832050i $$-0.312833\pi$$
0.554700 + 0.832050i $$0.312833\pi$$
$$14$$ 0 0
$$15$$ −2.00000 −0.516398
$$16$$ 0 0
$$17$$ 3.00000 0.727607 0.363803 0.931476i $$-0.381478\pi$$
0.363803 + 0.931476i $$0.381478\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ 6.00000 1.30931
$$22$$ 0 0
$$23$$ 8.00000 1.66812 0.834058 0.551677i $$-0.186012\pi$$
0.834058 + 0.551677i $$0.186012\pi$$
$$24$$ 0 0
$$25$$ −4.00000 −0.800000
$$26$$ 0 0
$$27$$ −4.00000 −0.769800
$$28$$ 0 0
$$29$$ 2.00000 0.371391 0.185695 0.982607i $$-0.440546\pi$$
0.185695 + 0.982607i $$0.440546\pi$$
$$30$$ 0 0
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −3.00000 −0.507093
$$36$$ 0 0
$$37$$ 10.0000 1.64399 0.821995 0.569495i $$-0.192861\pi$$
0.821995 + 0.569495i $$0.192861\pi$$
$$38$$ 0 0
$$39$$ 8.00000 1.28103
$$40$$ 0 0
$$41$$ −10.0000 −1.56174 −0.780869 0.624695i $$-0.785223\pi$$
−0.780869 + 0.624695i $$0.785223\pi$$
$$42$$ 0 0
$$43$$ −1.00000 −0.152499 −0.0762493 0.997089i $$-0.524294\pi$$
−0.0762493 + 0.997089i $$0.524294\pi$$
$$44$$ 0 0
$$45$$ −1.00000 −0.149071
$$46$$ 0 0
$$47$$ −1.00000 −0.145865 −0.0729325 0.997337i $$-0.523236\pi$$
−0.0729325 + 0.997337i $$0.523236\pi$$
$$48$$ 0 0
$$49$$ 2.00000 0.285714
$$50$$ 0 0
$$51$$ 6.00000 0.840168
$$52$$ 0 0
$$53$$ −4.00000 −0.549442 −0.274721 0.961524i $$-0.588586\pi$$
−0.274721 + 0.961524i $$0.588586\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 2.00000 0.264906
$$58$$ 0 0
$$59$$ 6.00000 0.781133 0.390567 0.920575i $$-0.372279\pi$$
0.390567 + 0.920575i $$0.372279\pi$$
$$60$$ 0 0
$$61$$ 13.0000 1.66448 0.832240 0.554416i $$-0.187058\pi$$
0.832240 + 0.554416i $$0.187058\pi$$
$$62$$ 0 0
$$63$$ 3.00000 0.377964
$$64$$ 0 0
$$65$$ −4.00000 −0.496139
$$66$$ 0 0
$$67$$ −12.0000 −1.46603 −0.733017 0.680211i $$-0.761888\pi$$
−0.733017 + 0.680211i $$0.761888\pi$$
$$68$$ 0 0
$$69$$ 16.0000 1.92617
$$70$$ 0 0
$$71$$ 2.00000 0.237356 0.118678 0.992933i $$-0.462134\pi$$
0.118678 + 0.992933i $$0.462134\pi$$
$$72$$ 0 0
$$73$$ −9.00000 −1.05337 −0.526685 0.850060i $$-0.676565\pi$$
−0.526685 + 0.850060i $$0.676565\pi$$
$$74$$ 0 0
$$75$$ −8.00000 −0.923760
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −8.00000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ 0 0
$$81$$ −11.0000 −1.22222
$$82$$ 0 0
$$83$$ 12.0000 1.31717 0.658586 0.752506i $$-0.271155\pi$$
0.658586 + 0.752506i $$0.271155\pi$$
$$84$$ 0 0
$$85$$ −3.00000 −0.325396
$$86$$ 0 0
$$87$$ 4.00000 0.428845
$$88$$ 0 0
$$89$$ 12.0000 1.27200 0.635999 0.771690i $$-0.280588\pi$$
0.635999 + 0.771690i $$0.280588\pi$$
$$90$$ 0 0
$$91$$ 12.0000 1.25794
$$92$$ 0 0
$$93$$ 8.00000 0.829561
$$94$$ 0 0
$$95$$ −1.00000 −0.102598
$$96$$ 0 0
$$97$$ −8.00000 −0.812277 −0.406138 0.913812i $$-0.633125\pi$$
−0.406138 + 0.913812i $$0.633125\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 10.0000 0.995037 0.497519 0.867453i $$-0.334245\pi$$
0.497519 + 0.867453i $$0.334245\pi$$
$$102$$ 0 0
$$103$$ −6.00000 −0.591198 −0.295599 0.955312i $$-0.595519\pi$$
−0.295599 + 0.955312i $$0.595519\pi$$
$$104$$ 0 0
$$105$$ −6.00000 −0.585540
$$106$$ 0 0
$$107$$ −2.00000 −0.193347 −0.0966736 0.995316i $$-0.530820\pi$$
−0.0966736 + 0.995316i $$0.530820\pi$$
$$108$$ 0 0
$$109$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$110$$ 0 0
$$111$$ 20.0000 1.89832
$$112$$ 0 0
$$113$$ −10.0000 −0.940721 −0.470360 0.882474i $$-0.655876\pi$$
−0.470360 + 0.882474i $$0.655876\pi$$
$$114$$ 0 0
$$115$$ −8.00000 −0.746004
$$116$$ 0 0
$$117$$ 4.00000 0.369800
$$118$$ 0 0
$$119$$ 9.00000 0.825029
$$120$$ 0 0
$$121$$ 0 0
$$122$$ 0 0
$$123$$ −20.0000 −1.80334
$$124$$ 0 0
$$125$$ 9.00000 0.804984
$$126$$ 0 0
$$127$$ −6.00000 −0.532414 −0.266207 0.963916i $$-0.585770\pi$$
−0.266207 + 0.963916i $$0.585770\pi$$
$$128$$ 0 0
$$129$$ −2.00000 −0.176090
$$130$$ 0 0
$$131$$ 9.00000 0.786334 0.393167 0.919467i $$-0.371379\pi$$
0.393167 + 0.919467i $$0.371379\pi$$
$$132$$ 0 0
$$133$$ 3.00000 0.260133
$$134$$ 0 0
$$135$$ 4.00000 0.344265
$$136$$ 0 0
$$137$$ −11.0000 −0.939793 −0.469897 0.882721i $$-0.655709\pi$$
−0.469897 + 0.882721i $$0.655709\pi$$
$$138$$ 0 0
$$139$$ 3.00000 0.254457 0.127228 0.991873i $$-0.459392\pi$$
0.127228 + 0.991873i $$0.459392\pi$$
$$140$$ 0 0
$$141$$ −2.00000 −0.168430
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −2.00000 −0.166091
$$146$$ 0 0
$$147$$ 4.00000 0.329914
$$148$$ 0 0
$$149$$ 15.0000 1.22885 0.614424 0.788976i $$-0.289388\pi$$
0.614424 + 0.788976i $$0.289388\pi$$
$$150$$ 0 0
$$151$$ −2.00000 −0.162758 −0.0813788 0.996683i $$-0.525932\pi$$
−0.0813788 + 0.996683i $$0.525932\pi$$
$$152$$ 0 0
$$153$$ 3.00000 0.242536
$$154$$ 0 0
$$155$$ −4.00000 −0.321288
$$156$$ 0 0
$$157$$ −2.00000 −0.159617 −0.0798087 0.996810i $$-0.525431\pi$$
−0.0798087 + 0.996810i $$0.525431\pi$$
$$158$$ 0 0
$$159$$ −8.00000 −0.634441
$$160$$ 0 0
$$161$$ 24.0000 1.89146
$$162$$ 0 0
$$163$$ −4.00000 −0.313304 −0.156652 0.987654i $$-0.550070\pi$$
−0.156652 + 0.987654i $$0.550070\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 6.00000 0.464294 0.232147 0.972681i $$-0.425425\pi$$
0.232147 + 0.972681i $$0.425425\pi$$
$$168$$ 0 0
$$169$$ 3.00000 0.230769
$$170$$ 0 0
$$171$$ 1.00000 0.0764719
$$172$$ 0 0
$$173$$ −6.00000 −0.456172 −0.228086 0.973641i $$-0.573247\pi$$
−0.228086 + 0.973641i $$0.573247\pi$$
$$174$$ 0 0
$$175$$ −12.0000 −0.907115
$$176$$ 0 0
$$177$$ 12.0000 0.901975
$$178$$ 0 0
$$179$$ 18.0000 1.34538 0.672692 0.739923i $$-0.265138\pi$$
0.672692 + 0.739923i $$0.265138\pi$$
$$180$$ 0 0
$$181$$ 10.0000 0.743294 0.371647 0.928374i $$-0.378793\pi$$
0.371647 + 0.928374i $$0.378793\pi$$
$$182$$ 0 0
$$183$$ 26.0000 1.92198
$$184$$ 0 0
$$185$$ −10.0000 −0.735215
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ −12.0000 −0.872872
$$190$$ 0 0
$$191$$ 25.0000 1.80894 0.904468 0.426541i $$-0.140268\pi$$
0.904468 + 0.426541i $$0.140268\pi$$
$$192$$ 0 0
$$193$$ −12.0000 −0.863779 −0.431889 0.901927i $$-0.642153\pi$$
−0.431889 + 0.901927i $$0.642153\pi$$
$$194$$ 0 0
$$195$$ −8.00000 −0.572892
$$196$$ 0 0
$$197$$ −2.00000 −0.142494 −0.0712470 0.997459i $$-0.522698\pi$$
−0.0712470 + 0.997459i $$0.522698\pi$$
$$198$$ 0 0
$$199$$ −7.00000 −0.496217 −0.248108 0.968732i $$-0.579809\pi$$
−0.248108 + 0.968732i $$0.579809\pi$$
$$200$$ 0 0
$$201$$ −24.0000 −1.69283
$$202$$ 0 0
$$203$$ 6.00000 0.421117
$$204$$ 0 0
$$205$$ 10.0000 0.698430
$$206$$ 0 0
$$207$$ 8.00000 0.556038
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −18.0000 −1.23917 −0.619586 0.784929i $$-0.712699\pi$$
−0.619586 + 0.784929i $$0.712699\pi$$
$$212$$ 0 0
$$213$$ 4.00000 0.274075
$$214$$ 0 0
$$215$$ 1.00000 0.0681994
$$216$$ 0 0
$$217$$ 12.0000 0.814613
$$218$$ 0 0
$$219$$ −18.0000 −1.21633
$$220$$ 0 0
$$221$$ 12.0000 0.807207
$$222$$ 0 0
$$223$$ 2.00000 0.133930 0.0669650 0.997755i $$-0.478668\pi$$
0.0669650 + 0.997755i $$0.478668\pi$$
$$224$$ 0 0
$$225$$ −4.00000 −0.266667
$$226$$ 0 0
$$227$$ −4.00000 −0.265489 −0.132745 0.991150i $$-0.542379\pi$$
−0.132745 + 0.991150i $$0.542379\pi$$
$$228$$ 0 0
$$229$$ 17.0000 1.12339 0.561696 0.827344i $$-0.310149\pi$$
0.561696 + 0.827344i $$0.310149\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −3.00000 −0.196537 −0.0982683 0.995160i $$-0.531330\pi$$
−0.0982683 + 0.995160i $$0.531330\pi$$
$$234$$ 0 0
$$235$$ 1.00000 0.0652328
$$236$$ 0 0
$$237$$ −16.0000 −1.03931
$$238$$ 0 0
$$239$$ −21.0000 −1.35838 −0.679189 0.733964i $$-0.737668\pi$$
−0.679189 + 0.733964i $$0.737668\pi$$
$$240$$ 0 0
$$241$$ 26.0000 1.67481 0.837404 0.546585i $$-0.184072\pi$$
0.837404 + 0.546585i $$0.184072\pi$$
$$242$$ 0 0
$$243$$ −10.0000 −0.641500
$$244$$ 0 0
$$245$$ −2.00000 −0.127775
$$246$$ 0 0
$$247$$ 4.00000 0.254514
$$248$$ 0 0
$$249$$ 24.0000 1.52094
$$250$$ 0 0
$$251$$ 11.0000 0.694314 0.347157 0.937807i $$-0.387147\pi$$
0.347157 + 0.937807i $$0.387147\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ −6.00000 −0.375735
$$256$$ 0 0
$$257$$ 32.0000 1.99611 0.998053 0.0623783i $$-0.0198685\pi$$
0.998053 + 0.0623783i $$0.0198685\pi$$
$$258$$ 0 0
$$259$$ 30.0000 1.86411
$$260$$ 0 0
$$261$$ 2.00000 0.123797
$$262$$ 0 0
$$263$$ 21.0000 1.29492 0.647458 0.762101i $$-0.275832\pi$$
0.647458 + 0.762101i $$0.275832\pi$$
$$264$$ 0 0
$$265$$ 4.00000 0.245718
$$266$$ 0 0
$$267$$ 24.0000 1.46878
$$268$$ 0 0
$$269$$ −24.0000 −1.46331 −0.731653 0.681677i $$-0.761251\pi$$
−0.731653 + 0.681677i $$0.761251\pi$$
$$270$$ 0 0
$$271$$ 8.00000 0.485965 0.242983 0.970031i $$-0.421874\pi$$
0.242983 + 0.970031i $$0.421874\pi$$
$$272$$ 0 0
$$273$$ 24.0000 1.45255
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −1.00000 −0.0600842 −0.0300421 0.999549i $$-0.509564\pi$$
−0.0300421 + 0.999549i $$0.509564\pi$$
$$278$$ 0 0
$$279$$ 4.00000 0.239474
$$280$$ 0 0
$$281$$ −22.0000 −1.31241 −0.656205 0.754583i $$-0.727839\pi$$
−0.656205 + 0.754583i $$0.727839\pi$$
$$282$$ 0 0
$$283$$ 3.00000 0.178331 0.0891657 0.996017i $$-0.471580\pi$$
0.0891657 + 0.996017i $$0.471580\pi$$
$$284$$ 0 0
$$285$$ −2.00000 −0.118470
$$286$$ 0 0
$$287$$ −30.0000 −1.77084
$$288$$ 0 0
$$289$$ −8.00000 −0.470588
$$290$$ 0 0
$$291$$ −16.0000 −0.937937
$$292$$ 0 0
$$293$$ 12.0000 0.701047 0.350524 0.936554i $$-0.386004\pi$$
0.350524 + 0.936554i $$0.386004\pi$$
$$294$$ 0 0
$$295$$ −6.00000 −0.349334
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 32.0000 1.85061
$$300$$ 0 0
$$301$$ −3.00000 −0.172917
$$302$$ 0 0
$$303$$ 20.0000 1.14897
$$304$$ 0 0
$$305$$ −13.0000 −0.744378
$$306$$ 0 0
$$307$$ −12.0000 −0.684876 −0.342438 0.939540i $$-0.611253\pi$$
−0.342438 + 0.939540i $$0.611253\pi$$
$$308$$ 0 0
$$309$$ −12.0000 −0.682656
$$310$$ 0 0
$$311$$ 7.00000 0.396934 0.198467 0.980108i $$-0.436404\pi$$
0.198467 + 0.980108i $$0.436404\pi$$
$$312$$ 0 0
$$313$$ −10.0000 −0.565233 −0.282617 0.959233i $$-0.591202\pi$$
−0.282617 + 0.959233i $$0.591202\pi$$
$$314$$ 0 0
$$315$$ −3.00000 −0.169031
$$316$$ 0 0
$$317$$ 30.0000 1.68497 0.842484 0.538721i $$-0.181092\pi$$
0.842484 + 0.538721i $$0.181092\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ −4.00000 −0.223258
$$322$$ 0 0
$$323$$ 3.00000 0.166924
$$324$$ 0 0
$$325$$ −16.0000 −0.887520
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −3.00000 −0.165395
$$330$$ 0 0
$$331$$ −4.00000 −0.219860 −0.109930 0.993939i $$-0.535063\pi$$
−0.109930 + 0.993939i $$0.535063\pi$$
$$332$$ 0 0
$$333$$ 10.0000 0.547997
$$334$$ 0 0
$$335$$ 12.0000 0.655630
$$336$$ 0 0
$$337$$ 32.0000 1.74315 0.871576 0.490261i $$-0.163099\pi$$
0.871576 + 0.490261i $$0.163099\pi$$
$$338$$ 0 0
$$339$$ −20.0000 −1.08625
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ −15.0000 −0.809924
$$344$$ 0 0
$$345$$ −16.0000 −0.861411
$$346$$ 0 0
$$347$$ −19.0000 −1.01997 −0.509987 0.860182i $$-0.670350\pi$$
−0.509987 + 0.860182i $$0.670350\pi$$
$$348$$ 0 0
$$349$$ 11.0000 0.588817 0.294408 0.955680i $$-0.404877\pi$$
0.294408 + 0.955680i $$0.404877\pi$$
$$350$$ 0 0
$$351$$ −16.0000 −0.854017
$$352$$ 0 0
$$353$$ −6.00000 −0.319348 −0.159674 0.987170i $$-0.551044\pi$$
−0.159674 + 0.987170i $$0.551044\pi$$
$$354$$ 0 0
$$355$$ −2.00000 −0.106149
$$356$$ 0 0
$$357$$ 18.0000 0.952661
$$358$$ 0 0
$$359$$ −21.0000 −1.10834 −0.554169 0.832404i $$-0.686964\pi$$
−0.554169 + 0.832404i $$0.686964\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 9.00000 0.471082
$$366$$ 0 0
$$367$$ 16.0000 0.835193 0.417597 0.908633i $$-0.362873\pi$$
0.417597 + 0.908633i $$0.362873\pi$$
$$368$$ 0 0
$$369$$ −10.0000 −0.520579
$$370$$ 0 0
$$371$$ −12.0000 −0.623009
$$372$$ 0 0
$$373$$ 4.00000 0.207112 0.103556 0.994624i $$-0.466978\pi$$
0.103556 + 0.994624i $$0.466978\pi$$
$$374$$ 0 0
$$375$$ 18.0000 0.929516
$$376$$ 0 0
$$377$$ 8.00000 0.412021
$$378$$ 0 0
$$379$$ −30.0000 −1.54100 −0.770498 0.637442i $$-0.779993\pi$$
−0.770498 + 0.637442i $$0.779993\pi$$
$$380$$ 0 0
$$381$$ −12.0000 −0.614779
$$382$$ 0 0
$$383$$ 4.00000 0.204390 0.102195 0.994764i $$-0.467413\pi$$
0.102195 + 0.994764i $$0.467413\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −1.00000 −0.0508329
$$388$$ 0 0
$$389$$ −21.0000 −1.06474 −0.532371 0.846511i $$-0.678699\pi$$
−0.532371 + 0.846511i $$0.678699\pi$$
$$390$$ 0 0
$$391$$ 24.0000 1.21373
$$392$$ 0 0
$$393$$ 18.0000 0.907980
$$394$$ 0 0
$$395$$ 8.00000 0.402524
$$396$$ 0 0
$$397$$ 5.00000 0.250943 0.125471 0.992097i $$-0.459956\pi$$
0.125471 + 0.992097i $$0.459956\pi$$
$$398$$ 0 0
$$399$$ 6.00000 0.300376
$$400$$ 0 0
$$401$$ 28.0000 1.39825 0.699127 0.714998i $$-0.253572\pi$$
0.699127 + 0.714998i $$0.253572\pi$$
$$402$$ 0 0
$$403$$ 16.0000 0.797017
$$404$$ 0 0
$$405$$ 11.0000 0.546594
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 20.0000 0.988936 0.494468 0.869196i $$-0.335363\pi$$
0.494468 + 0.869196i $$0.335363\pi$$
$$410$$ 0 0
$$411$$ −22.0000 −1.08518
$$412$$ 0 0
$$413$$ 18.0000 0.885722
$$414$$ 0 0
$$415$$ −12.0000 −0.589057
$$416$$ 0 0
$$417$$ 6.00000 0.293821
$$418$$ 0 0
$$419$$ −36.0000 −1.75872 −0.879358 0.476162i $$-0.842028\pi$$
−0.879358 + 0.476162i $$0.842028\pi$$
$$420$$ 0 0
$$421$$ −40.0000 −1.94948 −0.974740 0.223341i $$-0.928304\pi$$
−0.974740 + 0.223341i $$0.928304\pi$$
$$422$$ 0 0
$$423$$ −1.00000 −0.0486217
$$424$$ 0 0
$$425$$ −12.0000 −0.582086
$$426$$ 0 0
$$427$$ 39.0000 1.88734
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −24.0000 −1.15604 −0.578020 0.816023i $$-0.696174\pi$$
−0.578020 + 0.816023i $$0.696174\pi$$
$$432$$ 0 0
$$433$$ 2.00000 0.0961139 0.0480569 0.998845i $$-0.484697\pi$$
0.0480569 + 0.998845i $$0.484697\pi$$
$$434$$ 0 0
$$435$$ −4.00000 −0.191785
$$436$$ 0 0
$$437$$ 8.00000 0.382692
$$438$$ 0 0
$$439$$ −2.00000 −0.0954548 −0.0477274 0.998860i $$-0.515198\pi$$
−0.0477274 + 0.998860i $$0.515198\pi$$
$$440$$ 0 0
$$441$$ 2.00000 0.0952381
$$442$$ 0 0
$$443$$ −5.00000 −0.237557 −0.118779 0.992921i $$-0.537898\pi$$
−0.118779 + 0.992921i $$0.537898\pi$$
$$444$$ 0 0
$$445$$ −12.0000 −0.568855
$$446$$ 0 0
$$447$$ 30.0000 1.41895
$$448$$ 0 0
$$449$$ 16.0000 0.755087 0.377543 0.925992i $$-0.376769\pi$$
0.377543 + 0.925992i $$0.376769\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ −4.00000 −0.187936
$$454$$ 0 0
$$455$$ −12.0000 −0.562569
$$456$$ 0 0
$$457$$ 13.0000 0.608114 0.304057 0.952654i $$-0.401659\pi$$
0.304057 + 0.952654i $$0.401659\pi$$
$$458$$ 0 0
$$459$$ −12.0000 −0.560112
$$460$$ 0 0
$$461$$ 19.0000 0.884918 0.442459 0.896789i $$-0.354106\pi$$
0.442459 + 0.896789i $$0.354106\pi$$
$$462$$ 0 0
$$463$$ 19.0000 0.883005 0.441502 0.897260i $$-0.354446\pi$$
0.441502 + 0.897260i $$0.354446\pi$$
$$464$$ 0 0
$$465$$ −8.00000 −0.370991
$$466$$ 0 0
$$467$$ −5.00000 −0.231372 −0.115686 0.993286i $$-0.536907\pi$$
−0.115686 + 0.993286i $$0.536907\pi$$
$$468$$ 0 0
$$469$$ −36.0000 −1.66233
$$470$$ 0 0
$$471$$ −4.00000 −0.184310
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ −4.00000 −0.183533
$$476$$ 0 0
$$477$$ −4.00000 −0.183147
$$478$$ 0 0
$$479$$ −4.00000 −0.182765 −0.0913823 0.995816i $$-0.529129\pi$$
−0.0913823 + 0.995816i $$0.529129\pi$$
$$480$$ 0 0
$$481$$ 40.0000 1.82384
$$482$$ 0 0
$$483$$ 48.0000 2.18408
$$484$$ 0 0
$$485$$ 8.00000 0.363261
$$486$$ 0 0
$$487$$ −26.0000 −1.17817 −0.589086 0.808070i $$-0.700512\pi$$
−0.589086 + 0.808070i $$0.700512\pi$$
$$488$$ 0 0
$$489$$ −8.00000 −0.361773
$$490$$ 0 0
$$491$$ 28.0000 1.26362 0.631811 0.775122i $$-0.282312\pi$$
0.631811 + 0.775122i $$0.282312\pi$$
$$492$$ 0 0
$$493$$ 6.00000 0.270226
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 6.00000 0.269137
$$498$$ 0 0
$$499$$ 19.0000 0.850557 0.425278 0.905063i $$-0.360176\pi$$
0.425278 + 0.905063i $$0.360176\pi$$
$$500$$ 0 0
$$501$$ 12.0000 0.536120
$$502$$ 0 0
$$503$$ 36.0000 1.60516 0.802580 0.596544i $$-0.203460\pi$$
0.802580 + 0.596544i $$0.203460\pi$$
$$504$$ 0 0
$$505$$ −10.0000 −0.444994
$$506$$ 0 0
$$507$$ 6.00000 0.266469
$$508$$ 0 0
$$509$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$510$$ 0 0
$$511$$ −27.0000 −1.19441
$$512$$ 0 0
$$513$$ −4.00000 −0.176604
$$514$$ 0 0
$$515$$ 6.00000 0.264392
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ −12.0000 −0.526742
$$520$$ 0 0
$$521$$ 32.0000 1.40195 0.700973 0.713188i $$-0.252749\pi$$
0.700973 + 0.713188i $$0.252749\pi$$
$$522$$ 0 0
$$523$$ −26.0000 −1.13690 −0.568450 0.822718i $$-0.692457\pi$$
−0.568450 + 0.822718i $$0.692457\pi$$
$$524$$ 0 0
$$525$$ −24.0000 −1.04745
$$526$$ 0 0
$$527$$ 12.0000 0.522728
$$528$$ 0 0
$$529$$ 41.0000 1.78261
$$530$$ 0 0
$$531$$ 6.00000 0.260378
$$532$$ 0 0
$$533$$ −40.0000 −1.73259
$$534$$ 0 0
$$535$$ 2.00000 0.0864675
$$536$$ 0 0
$$537$$ 36.0000 1.55351
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −11.0000 −0.472927 −0.236463 0.971640i $$-0.575988\pi$$
−0.236463 + 0.971640i $$0.575988\pi$$
$$542$$ 0 0
$$543$$ 20.0000 0.858282
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −28.0000 −1.19719 −0.598597 0.801050i $$-0.704275\pi$$
−0.598597 + 0.801050i $$0.704275\pi$$
$$548$$ 0 0
$$549$$ 13.0000 0.554826
$$550$$ 0 0
$$551$$ 2.00000 0.0852029
$$552$$ 0 0
$$553$$ −24.0000 −1.02058
$$554$$ 0 0
$$555$$ −20.0000 −0.848953
$$556$$ 0 0
$$557$$ −1.00000 −0.0423714 −0.0211857 0.999776i $$-0.506744\pi$$
−0.0211857 + 0.999776i $$0.506744\pi$$
$$558$$ 0 0
$$559$$ −4.00000 −0.169182
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −42.0000 −1.77009 −0.885044 0.465506i $$-0.845872\pi$$
−0.885044 + 0.465506i $$0.845872\pi$$
$$564$$ 0 0
$$565$$ 10.0000 0.420703
$$566$$ 0 0
$$567$$ −33.0000 −1.38587
$$568$$ 0 0
$$569$$ 8.00000 0.335377 0.167689 0.985840i $$-0.446370\pi$$
0.167689 + 0.985840i $$0.446370\pi$$
$$570$$ 0 0
$$571$$ −20.0000 −0.836974 −0.418487 0.908223i $$-0.637439\pi$$
−0.418487 + 0.908223i $$0.637439\pi$$
$$572$$ 0 0
$$573$$ 50.0000 2.08878
$$574$$ 0 0
$$575$$ −32.0000 −1.33449
$$576$$ 0 0
$$577$$ −13.0000 −0.541197 −0.270599 0.962692i $$-0.587222\pi$$
−0.270599 + 0.962692i $$0.587222\pi$$
$$578$$ 0 0
$$579$$ −24.0000 −0.997406
$$580$$ 0 0
$$581$$ 36.0000 1.49353
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ −4.00000 −0.165380
$$586$$ 0 0
$$587$$ 3.00000 0.123823 0.0619116 0.998082i $$-0.480280\pi$$
0.0619116 + 0.998082i $$0.480280\pi$$
$$588$$ 0 0
$$589$$ 4.00000 0.164817
$$590$$ 0 0
$$591$$ −4.00000 −0.164538
$$592$$ 0 0
$$593$$ −22.0000 −0.903432 −0.451716 0.892162i $$-0.649188\pi$$
−0.451716 + 0.892162i $$0.649188\pi$$
$$594$$ 0 0
$$595$$ −9.00000 −0.368964
$$596$$ 0 0
$$597$$ −14.0000 −0.572982
$$598$$ 0 0
$$599$$ −12.0000 −0.490307 −0.245153 0.969484i $$-0.578838\pi$$
−0.245153 + 0.969484i $$0.578838\pi$$
$$600$$ 0 0
$$601$$ −10.0000 −0.407909 −0.203954 0.978980i $$-0.565379\pi$$
−0.203954 + 0.978980i $$0.565379\pi$$
$$602$$ 0 0
$$603$$ −12.0000 −0.488678
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −32.0000 −1.29884 −0.649420 0.760430i $$-0.724988\pi$$
−0.649420 + 0.760430i $$0.724988\pi$$
$$608$$ 0 0
$$609$$ 12.0000 0.486265
$$610$$ 0 0
$$611$$ −4.00000 −0.161823
$$612$$ 0 0
$$613$$ 47.0000 1.89831 0.949156 0.314806i $$-0.101939\pi$$
0.949156 + 0.314806i $$0.101939\pi$$
$$614$$ 0 0
$$615$$ 20.0000 0.806478
$$616$$ 0 0
$$617$$ 9.00000 0.362326 0.181163 0.983453i $$-0.442014\pi$$
0.181163 + 0.983453i $$0.442014\pi$$
$$618$$ 0 0
$$619$$ 4.00000 0.160774 0.0803868 0.996764i $$-0.474384\pi$$
0.0803868 + 0.996764i $$0.474384\pi$$
$$620$$ 0 0
$$621$$ −32.0000 −1.28412
$$622$$ 0 0
$$623$$ 36.0000 1.44231
$$624$$ 0 0
$$625$$ 11.0000 0.440000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 30.0000 1.19618
$$630$$ 0 0
$$631$$ −7.00000 −0.278666 −0.139333 0.990246i $$-0.544496\pi$$
−0.139333 + 0.990246i $$0.544496\pi$$
$$632$$ 0 0
$$633$$ −36.0000 −1.43087
$$634$$ 0 0
$$635$$ 6.00000 0.238103
$$636$$ 0 0
$$637$$ 8.00000 0.316972
$$638$$ 0 0
$$639$$ 2.00000 0.0791188
$$640$$ 0 0
$$641$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$642$$ 0 0
$$643$$ −11.0000 −0.433798 −0.216899 0.976194i $$-0.569594\pi$$
−0.216899 + 0.976194i $$0.569594\pi$$
$$644$$ 0 0
$$645$$ 2.00000 0.0787499
$$646$$ 0 0
$$647$$ −7.00000 −0.275198 −0.137599 0.990488i $$-0.543939\pi$$
−0.137599 + 0.990488i $$0.543939\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 24.0000 0.940634
$$652$$ 0 0
$$653$$ −27.0000 −1.05659 −0.528296 0.849060i $$-0.677169\pi$$
−0.528296 + 0.849060i $$0.677169\pi$$
$$654$$ 0 0
$$655$$ −9.00000 −0.351659
$$656$$ 0 0
$$657$$ −9.00000 −0.351123
$$658$$ 0 0
$$659$$ 34.0000 1.32445 0.662226 0.749304i $$-0.269612\pi$$
0.662226 + 0.749304i $$0.269612\pi$$
$$660$$ 0 0
$$661$$ 16.0000 0.622328 0.311164 0.950356i $$-0.399281\pi$$
0.311164 + 0.950356i $$0.399281\pi$$
$$662$$ 0 0
$$663$$ 24.0000 0.932083
$$664$$ 0 0
$$665$$ −3.00000 −0.116335
$$666$$ 0 0
$$667$$ 16.0000 0.619522
$$668$$ 0 0
$$669$$ 4.00000 0.154649
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 10.0000 0.385472 0.192736 0.981251i $$-0.438264\pi$$
0.192736 + 0.981251i $$0.438264\pi$$
$$674$$ 0 0
$$675$$ 16.0000 0.615840
$$676$$ 0 0
$$677$$ 2.00000 0.0768662 0.0384331 0.999261i $$-0.487763\pi$$
0.0384331 + 0.999261i $$0.487763\pi$$
$$678$$ 0 0
$$679$$ −24.0000 −0.921035
$$680$$ 0 0
$$681$$ −8.00000 −0.306561
$$682$$ 0 0
$$683$$ −20.0000 −0.765279 −0.382639 0.923898i $$-0.624985\pi$$
−0.382639 + 0.923898i $$0.624985\pi$$
$$684$$ 0 0
$$685$$ 11.0000 0.420288
$$686$$ 0 0
$$687$$ 34.0000 1.29718
$$688$$ 0 0
$$689$$ −16.0000 −0.609551
$$690$$ 0 0
$$691$$ −33.0000 −1.25538 −0.627690 0.778464i $$-0.715999\pi$$
−0.627690 + 0.778464i $$0.715999\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −3.00000 −0.113796
$$696$$ 0 0
$$697$$ −30.0000 −1.13633
$$698$$ 0 0
$$699$$ −6.00000 −0.226941
$$700$$ 0 0
$$701$$ 10.0000 0.377695 0.188847 0.982006i $$-0.439525\pi$$
0.188847 + 0.982006i $$0.439525\pi$$
$$702$$ 0 0
$$703$$ 10.0000 0.377157
$$704$$ 0 0
$$705$$ 2.00000 0.0753244
$$706$$ 0 0
$$707$$ 30.0000 1.12827
$$708$$ 0 0
$$709$$ −6.00000 −0.225335 −0.112667 0.993633i $$-0.535939\pi$$
−0.112667 + 0.993633i $$0.535939\pi$$
$$710$$ 0 0
$$711$$ −8.00000 −0.300023
$$712$$ 0 0
$$713$$ 32.0000 1.19841
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −42.0000 −1.56852
$$718$$ 0 0
$$719$$ 13.0000 0.484818 0.242409 0.970174i $$-0.422062\pi$$
0.242409 + 0.970174i $$0.422062\pi$$
$$720$$ 0 0
$$721$$ −18.0000 −0.670355
$$722$$ 0 0
$$723$$ 52.0000 1.93390
$$724$$ 0 0
$$725$$ −8.00000 −0.297113
$$726$$ 0 0
$$727$$ 7.00000 0.259616 0.129808 0.991539i $$-0.458564\pi$$
0.129808 + 0.991539i $$0.458564\pi$$
$$728$$ 0 0
$$729$$ 13.0000 0.481481
$$730$$ 0 0
$$731$$ −3.00000 −0.110959
$$732$$ 0 0
$$733$$ 6.00000 0.221615 0.110808 0.993842i $$-0.464656\pi$$
0.110808 + 0.993842i $$0.464656\pi$$
$$734$$ 0 0
$$735$$ −4.00000 −0.147542
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 43.0000 1.58178 0.790890 0.611958i $$-0.209618\pi$$
0.790890 + 0.611958i $$0.209618\pi$$
$$740$$ 0 0
$$741$$ 8.00000 0.293887
$$742$$ 0 0
$$743$$ 40.0000 1.46746 0.733729 0.679442i $$-0.237778\pi$$
0.733729 + 0.679442i $$0.237778\pi$$
$$744$$ 0 0
$$745$$ −15.0000 −0.549557
$$746$$ 0 0
$$747$$ 12.0000 0.439057
$$748$$ 0 0
$$749$$ −6.00000 −0.219235
$$750$$ 0 0
$$751$$ −16.0000 −0.583848 −0.291924 0.956441i $$-0.594295\pi$$
−0.291924 + 0.956441i $$0.594295\pi$$
$$752$$ 0 0
$$753$$ 22.0000 0.801725
$$754$$ 0 0
$$755$$ 2.00000 0.0727875
$$756$$ 0 0
$$757$$ −5.00000 −0.181728 −0.0908640 0.995863i $$-0.528963\pi$$
−0.0908640 + 0.995863i $$0.528963\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −9.00000 −0.326250 −0.163125 0.986605i $$-0.552157\pi$$
−0.163125 + 0.986605i $$0.552157\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ −3.00000 −0.108465
$$766$$ 0 0
$$767$$ 24.0000 0.866590
$$768$$ 0 0
$$769$$ −31.0000 −1.11789 −0.558944 0.829205i $$-0.688793\pi$$
−0.558944 + 0.829205i $$0.688793\pi$$
$$770$$ 0 0
$$771$$ 64.0000 2.30490
$$772$$ 0 0
$$773$$ 18.0000 0.647415 0.323708 0.946157i $$-0.395071\pi$$
0.323708 + 0.946157i $$0.395071\pi$$
$$774$$ 0 0
$$775$$ −16.0000 −0.574737
$$776$$ 0 0
$$777$$ 60.0000 2.15249
$$778$$ 0 0
$$779$$ −10.0000 −0.358287
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ −8.00000 −0.285897
$$784$$ 0 0
$$785$$ 2.00000 0.0713831
$$786$$ 0 0
$$787$$ −4.00000 −0.142585 −0.0712923 0.997455i $$-0.522712\pi$$
−0.0712923 + 0.997455i $$0.522712\pi$$
$$788$$ 0 0
$$789$$ 42.0000 1.49524
$$790$$ 0 0
$$791$$ −30.0000 −1.06668
$$792$$ 0 0
$$793$$ 52.0000 1.84657
$$794$$ 0 0
$$795$$ 8.00000 0.283731
$$796$$ 0 0
$$797$$ −12.0000 −0.425062 −0.212531 0.977154i $$-0.568171\pi$$
−0.212531 + 0.977154i $$0.568171\pi$$
$$798$$ 0 0
$$799$$ −3.00000 −0.106132
$$800$$ 0 0
$$801$$ 12.0000 0.423999
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ −24.0000 −0.845889
$$806$$ 0 0
$$807$$ −48.0000 −1.68968
$$808$$ 0 0
$$809$$ −39.0000 −1.37117 −0.685583 0.727994i $$-0.740453\pi$$
−0.685583 + 0.727994i $$0.740453\pi$$
$$810$$ 0 0
$$811$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$812$$ 0 0
$$813$$ 16.0000 0.561144
$$814$$ 0 0
$$815$$ 4.00000 0.140114
$$816$$ 0 0
$$817$$ −1.00000 −0.0349856
$$818$$ 0 0
$$819$$ 12.0000 0.419314
$$820$$ 0 0
$$821$$ −45.0000 −1.57051 −0.785255 0.619172i $$-0.787468\pi$$
−0.785255 + 0.619172i $$0.787468\pi$$
$$822$$ 0 0
$$823$$ 53.0000 1.84746 0.923732 0.383040i $$-0.125123\pi$$
0.923732 + 0.383040i $$0.125123\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 28.0000 0.973655 0.486828 0.873498i $$-0.338154\pi$$
0.486828 + 0.873498i $$0.338154\pi$$
$$828$$ 0 0
$$829$$ −48.0000 −1.66711 −0.833554 0.552437i $$-0.813698\pi$$
−0.833554 + 0.552437i $$0.813698\pi$$
$$830$$ 0 0
$$831$$ −2.00000 −0.0693792
$$832$$ 0 0
$$833$$ 6.00000 0.207888
$$834$$ 0 0
$$835$$ −6.00000 −0.207639
$$836$$ 0 0
$$837$$ −16.0000 −0.553041
$$838$$ 0 0
$$839$$ −10.0000 −0.345238 −0.172619 0.984989i $$-0.555223\pi$$
−0.172619 + 0.984989i $$0.555223\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ 0 0
$$843$$ −44.0000 −1.51544
$$844$$ 0 0
$$845$$ −3.00000 −0.103203
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 6.00000 0.205919
$$850$$ 0 0
$$851$$ 80.0000 2.74236
$$852$$ 0 0
$$853$$ −42.0000 −1.43805 −0.719026 0.694983i $$-0.755412\pi$$
−0.719026 + 0.694983i $$0.755412\pi$$
$$854$$ 0 0
$$855$$ −1.00000 −0.0341993
$$856$$ 0 0
$$857$$ −18.0000 −0.614868 −0.307434 0.951569i $$-0.599470\pi$$
−0.307434 + 0.951569i $$0.599470\pi$$
$$858$$ 0 0
$$859$$ 1.00000 0.0341196 0.0170598 0.999854i $$-0.494569\pi$$
0.0170598 + 0.999854i $$0.494569\pi$$
$$860$$ 0 0
$$861$$ −60.0000 −2.04479
$$862$$ 0 0
$$863$$ 6.00000 0.204242 0.102121 0.994772i $$-0.467437\pi$$
0.102121 + 0.994772i $$0.467437\pi$$
$$864$$ 0 0
$$865$$ 6.00000 0.204006
$$866$$ 0 0
$$867$$ −16.0000 −0.543388
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −48.0000 −1.62642
$$872$$ 0 0
$$873$$ −8.00000 −0.270759
$$874$$ 0 0
$$875$$ 27.0000 0.912767
$$876$$ 0 0
$$877$$ −34.0000 −1.14810 −0.574049 0.818821i $$-0.694628\pi$$
−0.574049 + 0.818821i $$0.694628\pi$$
$$878$$ 0 0
$$879$$ 24.0000 0.809500
$$880$$ 0 0
$$881$$ −51.0000 −1.71823 −0.859117 0.511780i $$-0.828986\pi$$
−0.859117 + 0.511780i $$0.828986\pi$$
$$882$$ 0 0
$$883$$ 1.00000 0.0336527 0.0168263 0.999858i $$-0.494644\pi$$
0.0168263 + 0.999858i $$0.494644\pi$$
$$884$$ 0 0
$$885$$ −12.0000 −0.403376
$$886$$ 0 0
$$887$$ −22.0000 −0.738688 −0.369344 0.929293i $$-0.620418\pi$$
−0.369344 + 0.929293i $$0.620418\pi$$
$$888$$ 0 0
$$889$$ −18.0000 −0.603701
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −1.00000 −0.0334637
$$894$$ 0 0
$$895$$ −18.0000 −0.601674
$$896$$ 0 0
$$897$$ 64.0000 2.13690
$$898$$ 0 0
$$899$$ 8.00000 0.266815
$$900$$ 0 0
$$901$$ −12.0000 −0.399778
$$902$$ 0 0
$$903$$ −6.00000 −0.199667
$$904$$ 0 0
$$905$$ −10.0000 −0.332411
$$906$$ 0 0
$$907$$ −40.0000 −1.32818 −0.664089 0.747653i $$-0.731180\pi$$
−0.664089 + 0.747653i $$0.731180\pi$$
$$908$$ 0 0
$$909$$ 10.0000 0.331679
$$910$$ 0 0
$$911$$ −18.0000 −0.596367 −0.298183 0.954509i $$-0.596381\pi$$
−0.298183 + 0.954509i $$0.596381\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ −26.0000 −0.859533
$$916$$ 0 0
$$917$$ 27.0000 0.891619
$$918$$ 0 0
$$919$$ 28.0000 0.923635 0.461817 0.886975i $$-0.347198\pi$$
0.461817 + 0.886975i $$0.347198\pi$$
$$920$$ 0 0
$$921$$ −24.0000 −0.790827
$$922$$ 0 0
$$923$$ 8.00000 0.263323
$$924$$ 0 0
$$925$$ −40.0000 −1.31519
$$926$$ 0 0
$$927$$ −6.00000 −0.197066
$$928$$ 0 0
$$929$$ 14.0000 0.459325 0.229663 0.973270i $$-0.426238\pi$$
0.229663 + 0.973270i $$0.426238\pi$$
$$930$$ 0 0
$$931$$ 2.00000 0.0655474
$$932$$ 0 0
$$933$$ 14.0000 0.458339
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 31.0000 1.01273 0.506363 0.862320i $$-0.330990\pi$$
0.506363 + 0.862320i $$0.330990\pi$$
$$938$$ 0 0
$$939$$ −20.0000 −0.652675
$$940$$ 0 0
$$941$$ 26.0000 0.847576 0.423788 0.905761i $$-0.360700\pi$$
0.423788 + 0.905761i $$0.360700\pi$$
$$942$$ 0 0
$$943$$ −80.0000 −2.60516
$$944$$ 0 0
$$945$$ 12.0000 0.390360
$$946$$ 0 0
$$947$$ −12.0000 −0.389948 −0.194974 0.980808i $$-0.562462\pi$$
−0.194974 + 0.980808i $$0.562462\pi$$
$$948$$ 0 0
$$949$$ −36.0000 −1.16861
$$950$$ 0 0
$$951$$ 60.0000 1.94563
$$952$$ 0 0
$$953$$ 16.0000 0.518291 0.259145 0.965838i $$-0.416559\pi$$
0.259145 + 0.965838i $$0.416559\pi$$
$$954$$ 0 0
$$955$$ −25.0000 −0.808981
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −33.0000 −1.06563
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 0 0
$$963$$ −2.00000 −0.0644491
$$964$$ 0 0
$$965$$ 12.0000 0.386294
$$966$$ 0 0
$$967$$ −48.0000 −1.54358 −0.771788 0.635880i $$-0.780637\pi$$
−0.771788 + 0.635880i $$0.780637\pi$$
$$968$$ 0 0
$$969$$ 6.00000 0.192748
$$970$$ 0 0
$$971$$ 20.0000 0.641831 0.320915 0.947108i $$-0.396010\pi$$
0.320915 + 0.947108i $$0.396010\pi$$
$$972$$ 0 0
$$973$$ 9.00000 0.288527
$$974$$ 0 0
$$975$$ −32.0000 −1.02482
$$976$$ 0 0
$$977$$ 56.0000 1.79160 0.895799 0.444459i $$-0.146604\pi$$
0.895799 + 0.444459i $$0.146604\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ −12.0000 −0.382741 −0.191370 0.981518i $$-0.561293\pi$$
−0.191370 + 0.981518i $$0.561293\pi$$
$$984$$ 0 0
$$985$$ 2.00000 0.0637253
$$986$$ 0 0
$$987$$ −6.00000 −0.190982
$$988$$ 0 0
$$989$$ −8.00000 −0.254385
$$990$$ 0 0
$$991$$ −38.0000 −1.20711 −0.603555 0.797321i $$-0.706250\pi$$
−0.603555 + 0.797321i $$0.706250\pi$$
$$992$$ 0 0
$$993$$ −8.00000 −0.253872
$$994$$ 0 0
$$995$$ 7.00000 0.221915
$$996$$ 0 0
$$997$$ −37.0000 −1.17180 −0.585901 0.810383i $$-0.699259\pi$$
−0.585901 + 0.810383i $$0.699259\pi$$
$$998$$ 0 0
$$999$$ −40.0000 −1.26554
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 9196.2.a.f.1.1 1
11.10 odd 2 76.2.a.a.1.1 1
33.32 even 2 684.2.a.b.1.1 1
44.43 even 2 304.2.a.a.1.1 1
55.32 even 4 1900.2.c.b.1749.1 2
55.43 even 4 1900.2.c.b.1749.2 2
55.54 odd 2 1900.2.a.b.1.1 1
77.76 even 2 3724.2.a.a.1.1 1
88.21 odd 2 1216.2.a.c.1.1 1
88.43 even 2 1216.2.a.q.1.1 1
132.131 odd 2 2736.2.a.q.1.1 1
209.65 even 6 1444.2.e.c.653.1 2
209.87 odd 6 1444.2.e.a.653.1 2
209.164 even 6 1444.2.e.c.429.1 2
209.197 odd 6 1444.2.e.a.429.1 2
209.208 even 2 1444.2.a.a.1.1 1
220.219 even 2 7600.2.a.p.1.1 1
836.835 odd 2 5776.2.a.p.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
76.2.a.a.1.1 1 11.10 odd 2
304.2.a.a.1.1 1 44.43 even 2
684.2.a.b.1.1 1 33.32 even 2
1216.2.a.c.1.1 1 88.21 odd 2
1216.2.a.q.1.1 1 88.43 even 2
1444.2.a.a.1.1 1 209.208 even 2
1444.2.e.a.429.1 2 209.197 odd 6
1444.2.e.a.653.1 2 209.87 odd 6
1444.2.e.c.429.1 2 209.164 even 6
1444.2.e.c.653.1 2 209.65 even 6
1900.2.a.b.1.1 1 55.54 odd 2
1900.2.c.b.1749.1 2 55.32 even 4
1900.2.c.b.1749.2 2 55.43 even 4
2736.2.a.q.1.1 1 132.131 odd 2
3724.2.a.a.1.1 1 77.76 even 2
5776.2.a.p.1.1 1 836.835 odd 2
7600.2.a.p.1.1 1 220.219 even 2
9196.2.a.f.1.1 1 1.1 even 1 trivial