# Properties

 Label 9408.2.a.z.1.1 Level $9408$ Weight $2$ Character 9408.1 Self dual yes Analytic conductor $75.123$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$9408 = 2^{6} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9408.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$75.1232582216$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 42) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 9408.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} +1.00000 q^{5} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} +1.00000 q^{5} +1.00000 q^{9} -5.00000 q^{11} -1.00000 q^{15} +4.00000 q^{17} +8.00000 q^{19} -4.00000 q^{23} -4.00000 q^{25} -1.00000 q^{27} +5.00000 q^{29} -3.00000 q^{31} +5.00000 q^{33} +4.00000 q^{37} -2.00000 q^{43} +1.00000 q^{45} +6.00000 q^{47} -4.00000 q^{51} +9.00000 q^{53} -5.00000 q^{55} -8.00000 q^{57} -11.0000 q^{59} -6.00000 q^{61} +2.00000 q^{67} +4.00000 q^{69} +2.00000 q^{71} -10.0000 q^{73} +4.00000 q^{75} +3.00000 q^{79} +1.00000 q^{81} -7.00000 q^{83} +4.00000 q^{85} -5.00000 q^{87} +6.00000 q^{89} +3.00000 q^{93} +8.00000 q^{95} -7.00000 q^{97} -5.00000 q^{99} +O(q^{100})$$

## 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
$$3$$ −1.00000 −0.577350
$$4$$ 0 0
$$5$$ 1.00000 0.447214 0.223607 0.974679i $$-0.428217\pi$$
0.223607 + 0.974679i $$0.428217\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −5.00000 −1.50756 −0.753778 0.657129i $$-0.771771\pi$$
−0.753778 + 0.657129i $$0.771771\pi$$
$$12$$ 0 0
$$13$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$14$$ 0 0
$$15$$ −1.00000 −0.258199
$$16$$ 0 0
$$17$$ 4.00000 0.970143 0.485071 0.874475i $$-0.338794\pi$$
0.485071 + 0.874475i $$0.338794\pi$$
$$18$$ 0 0
$$19$$ 8.00000 1.83533 0.917663 0.397360i $$-0.130073\pi$$
0.917663 + 0.397360i $$0.130073\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −4.00000 −0.834058 −0.417029 0.908893i $$-0.636929\pi$$
−0.417029 + 0.908893i $$0.636929\pi$$
$$24$$ 0 0
$$25$$ −4.00000 −0.800000
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ 5.00000 0.928477 0.464238 0.885710i $$-0.346328\pi$$
0.464238 + 0.885710i $$0.346328\pi$$
$$30$$ 0 0
$$31$$ −3.00000 −0.538816 −0.269408 0.963026i $$-0.586828\pi$$
−0.269408 + 0.963026i $$0.586828\pi$$
$$32$$ 0 0
$$33$$ 5.00000 0.870388
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 4.00000 0.657596 0.328798 0.944400i $$-0.393356\pi$$
0.328798 + 0.944400i $$0.393356\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$42$$ 0 0
$$43$$ −2.00000 −0.304997 −0.152499 0.988304i $$-0.548732\pi$$
−0.152499 + 0.988304i $$0.548732\pi$$
$$44$$ 0 0
$$45$$ 1.00000 0.149071
$$46$$ 0 0
$$47$$ 6.00000 0.875190 0.437595 0.899172i $$-0.355830\pi$$
0.437595 + 0.899172i $$0.355830\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ −4.00000 −0.560112
$$52$$ 0 0
$$53$$ 9.00000 1.23625 0.618123 0.786082i $$-0.287894\pi$$
0.618123 + 0.786082i $$0.287894\pi$$
$$54$$ 0 0
$$55$$ −5.00000 −0.674200
$$56$$ 0 0
$$57$$ −8.00000 −1.05963
$$58$$ 0 0
$$59$$ −11.0000 −1.43208 −0.716039 0.698060i $$-0.754047\pi$$
−0.716039 + 0.698060i $$0.754047\pi$$
$$60$$ 0 0
$$61$$ −6.00000 −0.768221 −0.384111 0.923287i $$-0.625492\pi$$
−0.384111 + 0.923287i $$0.625492\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 2.00000 0.244339 0.122169 0.992509i $$-0.461015\pi$$
0.122169 + 0.992509i $$0.461015\pi$$
$$68$$ 0 0
$$69$$ 4.00000 0.481543
$$70$$ 0 0
$$71$$ 2.00000 0.237356 0.118678 0.992933i $$-0.462134\pi$$
0.118678 + 0.992933i $$0.462134\pi$$
$$72$$ 0 0
$$73$$ −10.0000 −1.17041 −0.585206 0.810885i $$-0.698986\pi$$
−0.585206 + 0.810885i $$0.698986\pi$$
$$74$$ 0 0
$$75$$ 4.00000 0.461880
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 3.00000 0.337526 0.168763 0.985657i $$-0.446023\pi$$
0.168763 + 0.985657i $$0.446023\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −7.00000 −0.768350 −0.384175 0.923260i $$-0.625514\pi$$
−0.384175 + 0.923260i $$0.625514\pi$$
$$84$$ 0 0
$$85$$ 4.00000 0.433861
$$86$$ 0 0
$$87$$ −5.00000 −0.536056
$$88$$ 0 0
$$89$$ 6.00000 0.635999 0.317999 0.948091i $$-0.396989\pi$$
0.317999 + 0.948091i $$0.396989\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 3.00000 0.311086
$$94$$ 0 0
$$95$$ 8.00000 0.820783
$$96$$ 0 0
$$97$$ −7.00000 −0.710742 −0.355371 0.934725i $$-0.615646\pi$$
−0.355371 + 0.934725i $$0.615646\pi$$
$$98$$ 0 0
$$99$$ −5.00000 −0.502519
$$100$$ 0 0
$$101$$ 10.0000 0.995037 0.497519 0.867453i $$-0.334245\pi$$
0.497519 + 0.867453i $$0.334245\pi$$
$$102$$ 0 0
$$103$$ −8.00000 −0.788263 −0.394132 0.919054i $$-0.628955\pi$$
−0.394132 + 0.919054i $$0.628955\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −3.00000 −0.290021 −0.145010 0.989430i $$-0.546322\pi$$
−0.145010 + 0.989430i $$0.546322\pi$$
$$108$$ 0 0
$$109$$ 2.00000 0.191565 0.0957826 0.995402i $$-0.469465\pi$$
0.0957826 + 0.995402i $$0.469465\pi$$
$$110$$ 0 0
$$111$$ −4.00000 −0.379663
$$112$$ 0 0
$$113$$ 16.0000 1.50515 0.752577 0.658505i $$-0.228811\pi$$
0.752577 + 0.658505i $$0.228811\pi$$
$$114$$ 0 0
$$115$$ −4.00000 −0.373002
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 14.0000 1.27273
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −9.00000 −0.804984
$$126$$ 0 0
$$127$$ 9.00000 0.798621 0.399310 0.916816i $$-0.369250\pi$$
0.399310 + 0.916816i $$0.369250\pi$$
$$128$$ 0 0
$$129$$ 2.00000 0.176090
$$130$$ 0 0
$$131$$ 1.00000 0.0873704 0.0436852 0.999045i $$-0.486090\pi$$
0.0436852 + 0.999045i $$0.486090\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ −1.00000 −0.0860663
$$136$$ 0 0
$$137$$ −2.00000 −0.170872 −0.0854358 0.996344i $$-0.527228\pi$$
−0.0854358 + 0.996344i $$0.527228\pi$$
$$138$$ 0 0
$$139$$ −14.0000 −1.18746 −0.593732 0.804663i $$-0.702346\pi$$
−0.593732 + 0.804663i $$0.702346\pi$$
$$140$$ 0 0
$$141$$ −6.00000 −0.505291
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 5.00000 0.415227
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 18.0000 1.47462 0.737309 0.675556i $$-0.236096\pi$$
0.737309 + 0.675556i $$0.236096\pi$$
$$150$$ 0 0
$$151$$ 19.0000 1.54620 0.773099 0.634285i $$-0.218706\pi$$
0.773099 + 0.634285i $$0.218706\pi$$
$$152$$ 0 0
$$153$$ 4.00000 0.323381
$$154$$ 0 0
$$155$$ −3.00000 −0.240966
$$156$$ 0 0
$$157$$ −4.00000 −0.319235 −0.159617 0.987179i $$-0.551026\pi$$
−0.159617 + 0.987179i $$0.551026\pi$$
$$158$$ 0 0
$$159$$ −9.00000 −0.713746
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 4.00000 0.313304 0.156652 0.987654i $$-0.449930\pi$$
0.156652 + 0.987654i $$0.449930\pi$$
$$164$$ 0 0
$$165$$ 5.00000 0.389249
$$166$$ 0 0
$$167$$ 14.0000 1.08335 0.541676 0.840587i $$-0.317790\pi$$
0.541676 + 0.840587i $$0.317790\pi$$
$$168$$ 0 0
$$169$$ −13.0000 −1.00000
$$170$$ 0 0
$$171$$ 8.00000 0.611775
$$172$$ 0 0
$$173$$ 22.0000 1.67263 0.836315 0.548250i $$-0.184706\pi$$
0.836315 + 0.548250i $$0.184706\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 11.0000 0.826811
$$178$$ 0 0
$$179$$ −12.0000 −0.896922 −0.448461 0.893802i $$-0.648028\pi$$
−0.448461 + 0.893802i $$0.648028\pi$$
$$180$$ 0 0
$$181$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$182$$ 0 0
$$183$$ 6.00000 0.443533
$$184$$ 0 0
$$185$$ 4.00000 0.294086
$$186$$ 0 0
$$187$$ −20.0000 −1.46254
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 24.0000 1.73658 0.868290 0.496058i $$-0.165220\pi$$
0.868290 + 0.496058i $$0.165220\pi$$
$$192$$ 0 0
$$193$$ 5.00000 0.359908 0.179954 0.983675i $$-0.442405\pi$$
0.179954 + 0.983675i $$0.442405\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −2.00000 −0.142494 −0.0712470 0.997459i $$-0.522698\pi$$
−0.0712470 + 0.997459i $$0.522698\pi$$
$$198$$ 0 0
$$199$$ 4.00000 0.283552 0.141776 0.989899i $$-0.454719\pi$$
0.141776 + 0.989899i $$0.454719\pi$$
$$200$$ 0 0
$$201$$ −2.00000 −0.141069
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ −4.00000 −0.278019
$$208$$ 0 0
$$209$$ −40.0000 −2.76686
$$210$$ 0 0
$$211$$ −2.00000 −0.137686 −0.0688428 0.997628i $$-0.521931\pi$$
−0.0688428 + 0.997628i $$0.521931\pi$$
$$212$$ 0 0
$$213$$ −2.00000 −0.137038
$$214$$ 0 0
$$215$$ −2.00000 −0.136399
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 10.0000 0.675737
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 7.00000 0.468755 0.234377 0.972146i $$-0.424695\pi$$
0.234377 + 0.972146i $$0.424695\pi$$
$$224$$ 0 0
$$225$$ −4.00000 −0.266667
$$226$$ 0 0
$$227$$ 3.00000 0.199117 0.0995585 0.995032i $$-0.468257\pi$$
0.0995585 + 0.995032i $$0.468257\pi$$
$$228$$ 0 0
$$229$$ −20.0000 −1.32164 −0.660819 0.750546i $$-0.729791\pi$$
−0.660819 + 0.750546i $$0.729791\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −4.00000 −0.262049 −0.131024 0.991379i $$-0.541827\pi$$
−0.131024 + 0.991379i $$0.541827\pi$$
$$234$$ 0 0
$$235$$ 6.00000 0.391397
$$236$$ 0 0
$$237$$ −3.00000 −0.194871
$$238$$ 0 0
$$239$$ −12.0000 −0.776215 −0.388108 0.921614i $$-0.626871\pi$$
−0.388108 + 0.921614i $$0.626871\pi$$
$$240$$ 0 0
$$241$$ 25.0000 1.61039 0.805196 0.593009i $$-0.202060\pi$$
0.805196 + 0.593009i $$0.202060\pi$$
$$242$$ 0 0
$$243$$ −1.00000 −0.0641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 7.00000 0.443607
$$250$$ 0 0
$$251$$ 21.0000 1.32551 0.662754 0.748837i $$-0.269387\pi$$
0.662754 + 0.748837i $$0.269387\pi$$
$$252$$ 0 0
$$253$$ 20.0000 1.25739
$$254$$ 0 0
$$255$$ −4.00000 −0.250490
$$256$$ 0 0
$$257$$ 6.00000 0.374270 0.187135 0.982334i $$-0.440080\pi$$
0.187135 + 0.982334i $$0.440080\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 5.00000 0.309492
$$262$$ 0 0
$$263$$ −30.0000 −1.84988 −0.924940 0.380114i $$-0.875885\pi$$
−0.924940 + 0.380114i $$0.875885\pi$$
$$264$$ 0 0
$$265$$ 9.00000 0.552866
$$266$$ 0 0
$$267$$ −6.00000 −0.367194
$$268$$ 0 0
$$269$$ 31.0000 1.89010 0.945052 0.326921i $$-0.106011\pi$$
0.945052 + 0.326921i $$0.106011\pi$$
$$270$$ 0 0
$$271$$ −15.0000 −0.911185 −0.455593 0.890188i $$-0.650573\pi$$
−0.455593 + 0.890188i $$0.650573\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 20.0000 1.20605
$$276$$ 0 0
$$277$$ 16.0000 0.961347 0.480673 0.876900i $$-0.340392\pi$$
0.480673 + 0.876900i $$0.340392\pi$$
$$278$$ 0 0
$$279$$ −3.00000 −0.179605
$$280$$ 0 0
$$281$$ 2.00000 0.119310 0.0596550 0.998219i $$-0.481000\pi$$
0.0596550 + 0.998219i $$0.481000\pi$$
$$282$$ 0 0
$$283$$ 10.0000 0.594438 0.297219 0.954809i $$-0.403941\pi$$
0.297219 + 0.954809i $$0.403941\pi$$
$$284$$ 0 0
$$285$$ −8.00000 −0.473879
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −1.00000 −0.0588235
$$290$$ 0 0
$$291$$ 7.00000 0.410347
$$292$$ 0 0
$$293$$ −21.0000 −1.22683 −0.613417 0.789760i $$-0.710205\pi$$
−0.613417 + 0.789760i $$0.710205\pi$$
$$294$$ 0 0
$$295$$ −11.0000 −0.640445
$$296$$ 0 0
$$297$$ 5.00000 0.290129
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ −10.0000 −0.574485
$$304$$ 0 0
$$305$$ −6.00000 −0.343559
$$306$$ 0 0
$$307$$ 28.0000 1.59804 0.799022 0.601302i $$-0.205351\pi$$
0.799022 + 0.601302i $$0.205351\pi$$
$$308$$ 0 0
$$309$$ 8.00000 0.455104
$$310$$ 0 0
$$311$$ 32.0000 1.81455 0.907277 0.420534i $$-0.138157\pi$$
0.907277 + 0.420534i $$0.138157\pi$$
$$312$$ 0 0
$$313$$ −1.00000 −0.0565233 −0.0282617 0.999601i $$-0.508997\pi$$
−0.0282617 + 0.999601i $$0.508997\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −3.00000 −0.168497 −0.0842484 0.996445i $$-0.526849\pi$$
−0.0842484 + 0.996445i $$0.526849\pi$$
$$318$$ 0 0
$$319$$ −25.0000 −1.39973
$$320$$ 0 0
$$321$$ 3.00000 0.167444
$$322$$ 0 0
$$323$$ 32.0000 1.78053
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −2.00000 −0.110600
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 4.00000 0.219860 0.109930 0.993939i $$-0.464937\pi$$
0.109930 + 0.993939i $$0.464937\pi$$
$$332$$ 0 0
$$333$$ 4.00000 0.219199
$$334$$ 0 0
$$335$$ 2.00000 0.109272
$$336$$ 0 0
$$337$$ 9.00000 0.490261 0.245131 0.969490i $$-0.421169\pi$$
0.245131 + 0.969490i $$0.421169\pi$$
$$338$$ 0 0
$$339$$ −16.0000 −0.869001
$$340$$ 0 0
$$341$$ 15.0000 0.812296
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 4.00000 0.215353
$$346$$ 0 0
$$347$$ −12.0000 −0.644194 −0.322097 0.946707i $$-0.604388\pi$$
−0.322097 + 0.946707i $$0.604388\pi$$
$$348$$ 0 0
$$349$$ −14.0000 −0.749403 −0.374701 0.927146i $$-0.622255\pi$$
−0.374701 + 0.927146i $$0.622255\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −24.0000 −1.27739 −0.638696 0.769460i $$-0.720526\pi$$
−0.638696 + 0.769460i $$0.720526\pi$$
$$354$$ 0 0
$$355$$ 2.00000 0.106149
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 10.0000 0.527780 0.263890 0.964553i $$-0.414994\pi$$
0.263890 + 0.964553i $$0.414994\pi$$
$$360$$ 0 0
$$361$$ 45.0000 2.36842
$$362$$ 0 0
$$363$$ −14.0000 −0.734809
$$364$$ 0 0
$$365$$ −10.0000 −0.523424
$$366$$ 0 0
$$367$$ −17.0000 −0.887393 −0.443696 0.896177i $$-0.646333\pi$$
−0.443696 + 0.896177i $$0.646333\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 32.0000 1.65690 0.828449 0.560065i $$-0.189224\pi$$
0.828449 + 0.560065i $$0.189224\pi$$
$$374$$ 0 0
$$375$$ 9.00000 0.464758
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ −16.0000 −0.821865 −0.410932 0.911666i $$-0.634797\pi$$
−0.410932 + 0.911666i $$0.634797\pi$$
$$380$$ 0 0
$$381$$ −9.00000 −0.461084
$$382$$ 0 0
$$383$$ 34.0000 1.73732 0.868659 0.495410i $$-0.164982\pi$$
0.868659 + 0.495410i $$0.164982\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −2.00000 −0.101666
$$388$$ 0 0
$$389$$ 2.00000 0.101404 0.0507020 0.998714i $$-0.483854\pi$$
0.0507020 + 0.998714i $$0.483854\pi$$
$$390$$ 0 0
$$391$$ −16.0000 −0.809155
$$392$$ 0 0
$$393$$ −1.00000 −0.0504433
$$394$$ 0 0
$$395$$ 3.00000 0.150946
$$396$$ 0 0
$$397$$ 36.0000 1.80679 0.903394 0.428811i $$-0.141067\pi$$
0.903394 + 0.428811i $$0.141067\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 24.0000 1.19850 0.599251 0.800561i $$-0.295465\pi$$
0.599251 + 0.800561i $$0.295465\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 1.00000 0.0496904
$$406$$ 0 0
$$407$$ −20.0000 −0.991363
$$408$$ 0 0
$$409$$ 25.0000 1.23617 0.618085 0.786111i $$-0.287909\pi$$
0.618085 + 0.786111i $$0.287909\pi$$
$$410$$ 0 0
$$411$$ 2.00000 0.0986527
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −7.00000 −0.343616
$$416$$ 0 0
$$417$$ 14.0000 0.685583
$$418$$ 0 0
$$419$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$420$$ 0 0
$$421$$ −30.0000 −1.46211 −0.731055 0.682318i $$-0.760972\pi$$
−0.731055 + 0.682318i $$0.760972\pi$$
$$422$$ 0 0
$$423$$ 6.00000 0.291730
$$424$$ 0 0
$$425$$ −16.0000 −0.776114
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 12.0000 0.578020 0.289010 0.957326i $$-0.406674\pi$$
0.289010 + 0.957326i $$0.406674\pi$$
$$432$$ 0 0
$$433$$ −14.0000 −0.672797 −0.336399 0.941720i $$-0.609209\pi$$
−0.336399 + 0.941720i $$0.609209\pi$$
$$434$$ 0 0
$$435$$ −5.00000 −0.239732
$$436$$ 0 0
$$437$$ −32.0000 −1.53077
$$438$$ 0 0
$$439$$ −15.0000 −0.715911 −0.357955 0.933739i $$-0.616526\pi$$
−0.357955 + 0.933739i $$0.616526\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −17.0000 −0.807694 −0.403847 0.914826i $$-0.632327\pi$$
−0.403847 + 0.914826i $$0.632327\pi$$
$$444$$ 0 0
$$445$$ 6.00000 0.284427
$$446$$ 0 0
$$447$$ −18.0000 −0.851371
$$448$$ 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
$$453$$ −19.0000 −0.892698
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 31.0000 1.45012 0.725059 0.688686i $$-0.241812\pi$$
0.725059 + 0.688686i $$0.241812\pi$$
$$458$$ 0 0
$$459$$ −4.00000 −0.186704
$$460$$ 0 0
$$461$$ −14.0000 −0.652045 −0.326023 0.945362i $$-0.605709\pi$$
−0.326023 + 0.945362i $$0.605709\pi$$
$$462$$ 0 0
$$463$$ 16.0000 0.743583 0.371792 0.928316i $$-0.378744\pi$$
0.371792 + 0.928316i $$0.378744\pi$$
$$464$$ 0 0
$$465$$ 3.00000 0.139122
$$466$$ 0 0
$$467$$ −20.0000 −0.925490 −0.462745 0.886492i $$-0.653135\pi$$
−0.462745 + 0.886492i $$0.653135\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 4.00000 0.184310
$$472$$ 0 0
$$473$$ 10.0000 0.459800
$$474$$ 0 0
$$475$$ −32.0000 −1.46826
$$476$$ 0 0
$$477$$ 9.00000 0.412082
$$478$$ 0 0
$$479$$ −38.0000 −1.73626 −0.868132 0.496333i $$-0.834679\pi$$
−0.868132 + 0.496333i $$0.834679\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −7.00000 −0.317854
$$486$$ 0 0
$$487$$ 5.00000 0.226572 0.113286 0.993562i $$-0.463862\pi$$
0.113286 + 0.993562i $$0.463862\pi$$
$$488$$ 0 0
$$489$$ −4.00000 −0.180886
$$490$$ 0 0
$$491$$ −9.00000 −0.406164 −0.203082 0.979162i $$-0.565096\pi$$
−0.203082 + 0.979162i $$0.565096\pi$$
$$492$$ 0 0
$$493$$ 20.0000 0.900755
$$494$$ 0 0
$$495$$ −5.00000 −0.224733
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −10.0000 −0.447661 −0.223831 0.974628i $$-0.571856\pi$$
−0.223831 + 0.974628i $$0.571856\pi$$
$$500$$ 0 0
$$501$$ −14.0000 −0.625474
$$502$$ 0 0
$$503$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$504$$ 0 0
$$505$$ 10.0000 0.444994
$$506$$ 0 0
$$507$$ 13.0000 0.577350
$$508$$ 0 0
$$509$$ 15.0000 0.664863 0.332432 0.943127i $$-0.392131\pi$$
0.332432 + 0.943127i $$0.392131\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ −8.00000 −0.353209
$$514$$ 0 0
$$515$$ −8.00000 −0.352522
$$516$$ 0 0
$$517$$ −30.0000 −1.31940
$$518$$ 0 0
$$519$$ −22.0000 −0.965693
$$520$$ 0 0
$$521$$ 18.0000 0.788594 0.394297 0.918983i $$-0.370988\pi$$
0.394297 + 0.918983i $$0.370988\pi$$
$$522$$ 0 0
$$523$$ 8.00000 0.349816 0.174908 0.984585i $$-0.444037\pi$$
0.174908 + 0.984585i $$0.444037\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −12.0000 −0.522728
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ 0 0
$$531$$ −11.0000 −0.477359
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ −3.00000 −0.129701
$$536$$ 0 0
$$537$$ 12.0000 0.517838
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 18.0000 0.773880 0.386940 0.922105i $$-0.373532\pi$$
0.386940 + 0.922105i $$0.373532\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 2.00000 0.0856706
$$546$$ 0 0
$$547$$ 12.0000 0.513083 0.256541 0.966533i $$-0.417417\pi$$
0.256541 + 0.966533i $$0.417417\pi$$
$$548$$ 0 0
$$549$$ −6.00000 −0.256074
$$550$$ 0 0
$$551$$ 40.0000 1.70406
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ −4.00000 −0.169791
$$556$$ 0 0
$$557$$ 23.0000 0.974541 0.487271 0.873251i $$-0.337993\pi$$
0.487271 + 0.873251i $$0.337993\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 20.0000 0.844401
$$562$$ 0 0
$$563$$ 17.0000 0.716465 0.358232 0.933632i $$-0.383380\pi$$
0.358232 + 0.933632i $$0.383380\pi$$
$$564$$ 0 0
$$565$$ 16.0000 0.673125
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 24.0000 1.00613 0.503066 0.864248i $$-0.332205\pi$$
0.503066 + 0.864248i $$0.332205\pi$$
$$570$$ 0 0
$$571$$ 30.0000 1.25546 0.627730 0.778431i $$-0.283984\pi$$
0.627730 + 0.778431i $$0.283984\pi$$
$$572$$ 0 0
$$573$$ −24.0000 −1.00261
$$574$$ 0 0
$$575$$ 16.0000 0.667246
$$576$$ 0 0
$$577$$ −31.0000 −1.29055 −0.645273 0.763952i $$-0.723257\pi$$
−0.645273 + 0.763952i $$0.723257\pi$$
$$578$$ 0 0
$$579$$ −5.00000 −0.207793
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −45.0000 −1.86371
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 35.0000 1.44460 0.722302 0.691577i $$-0.243084\pi$$
0.722302 + 0.691577i $$0.243084\pi$$
$$588$$ 0 0
$$589$$ −24.0000 −0.988903
$$590$$ 0 0
$$591$$ 2.00000 0.0822690
$$592$$ 0 0
$$593$$ −36.0000 −1.47834 −0.739171 0.673517i $$-0.764783\pi$$
−0.739171 + 0.673517i $$0.764783\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −4.00000 −0.163709
$$598$$ 0 0
$$599$$ −30.0000 −1.22577 −0.612883 0.790173i $$-0.709990\pi$$
−0.612883 + 0.790173i $$0.709990\pi$$
$$600$$ 0 0
$$601$$ −35.0000 −1.42768 −0.713840 0.700309i $$-0.753046\pi$$
−0.713840 + 0.700309i $$0.753046\pi$$
$$602$$ 0 0
$$603$$ 2.00000 0.0814463
$$604$$ 0 0
$$605$$ 14.0000 0.569181
$$606$$ 0 0
$$607$$ 27.0000 1.09590 0.547948 0.836512i $$-0.315409\pi$$
0.547948 + 0.836512i $$0.315409\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ −12.0000 −0.484675 −0.242338 0.970192i $$-0.577914\pi$$
−0.242338 + 0.970192i $$0.577914\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 2.00000 0.0805170 0.0402585 0.999189i $$-0.487182\pi$$
0.0402585 + 0.999189i $$0.487182\pi$$
$$618$$ 0 0
$$619$$ 10.0000 0.401934 0.200967 0.979598i $$-0.435592\pi$$
0.200967 + 0.979598i $$0.435592\pi$$
$$620$$ 0 0
$$621$$ 4.00000 0.160514
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 11.0000 0.440000
$$626$$ 0 0
$$627$$ 40.0000 1.59745
$$628$$ 0 0
$$629$$ 16.0000 0.637962
$$630$$ 0 0
$$631$$ −19.0000 −0.756378 −0.378189 0.925728i $$-0.623453\pi$$
−0.378189 + 0.925728i $$0.623453\pi$$
$$632$$ 0 0
$$633$$ 2.00000 0.0794929
$$634$$ 0 0
$$635$$ 9.00000 0.357154
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 2.00000 0.0791188
$$640$$ 0 0
$$641$$ 26.0000 1.02694 0.513469 0.858108i $$-0.328360\pi$$
0.513469 + 0.858108i $$0.328360\pi$$
$$642$$ 0 0
$$643$$ 14.0000 0.552106 0.276053 0.961142i $$-0.410973\pi$$
0.276053 + 0.961142i $$0.410973\pi$$
$$644$$ 0 0
$$645$$ 2.00000 0.0787499
$$646$$ 0 0
$$647$$ 18.0000 0.707653 0.353827 0.935311i $$-0.384880\pi$$
0.353827 + 0.935311i $$0.384880\pi$$
$$648$$ 0 0
$$649$$ 55.0000 2.15894
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 39.0000 1.52619 0.763094 0.646288i $$-0.223679\pi$$
0.763094 + 0.646288i $$0.223679\pi$$
$$654$$ 0 0
$$655$$ 1.00000 0.0390732
$$656$$ 0 0
$$657$$ −10.0000 −0.390137
$$658$$ 0 0
$$659$$ 40.0000 1.55818 0.779089 0.626913i $$-0.215682\pi$$
0.779089 + 0.626913i $$0.215682\pi$$
$$660$$ 0 0
$$661$$ 10.0000 0.388955 0.194477 0.980907i $$-0.437699\pi$$
0.194477 + 0.980907i $$0.437699\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −20.0000 −0.774403
$$668$$ 0 0
$$669$$ −7.00000 −0.270636
$$670$$ 0 0
$$671$$ 30.0000 1.15814
$$672$$ 0 0
$$673$$ −19.0000 −0.732396 −0.366198 0.930537i $$-0.619341\pi$$
−0.366198 + 0.930537i $$0.619341\pi$$
$$674$$ 0 0
$$675$$ 4.00000 0.153960
$$676$$ 0 0
$$677$$ −27.0000 −1.03769 −0.518847 0.854867i $$-0.673639\pi$$
−0.518847 + 0.854867i $$0.673639\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −3.00000 −0.114960
$$682$$ 0 0
$$683$$ 9.00000 0.344375 0.172188 0.985064i $$-0.444916\pi$$
0.172188 + 0.985064i $$0.444916\pi$$
$$684$$ 0 0
$$685$$ −2.00000 −0.0764161
$$686$$ 0 0
$$687$$ 20.0000 0.763048
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 8.00000 0.304334 0.152167 0.988355i $$-0.451375\pi$$
0.152167 + 0.988355i $$0.451375\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −14.0000 −0.531050
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 4.00000 0.151294
$$700$$ 0 0
$$701$$ 5.00000 0.188847 0.0944237 0.995532i $$-0.469899\pi$$
0.0944237 + 0.995532i $$0.469899\pi$$
$$702$$ 0 0
$$703$$ 32.0000 1.20690
$$704$$ 0 0
$$705$$ −6.00000 −0.225973
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −38.0000 −1.42712 −0.713560 0.700594i $$-0.752918\pi$$
−0.713560 + 0.700594i $$0.752918\pi$$
$$710$$ 0 0
$$711$$ 3.00000 0.112509
$$712$$ 0 0
$$713$$ 12.0000 0.449404
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 12.0000 0.448148
$$718$$ 0 0
$$719$$ 6.00000 0.223762 0.111881 0.993722i $$-0.464312\pi$$
0.111881 + 0.993722i $$0.464312\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ −25.0000 −0.929760
$$724$$ 0 0
$$725$$ −20.0000 −0.742781
$$726$$ 0 0
$$727$$ −7.00000 −0.259616 −0.129808 0.991539i $$-0.541436\pi$$
−0.129808 + 0.991539i $$0.541436\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −8.00000 −0.295891
$$732$$ 0 0
$$733$$ −6.00000 −0.221615 −0.110808 0.993842i $$-0.535344\pi$$
−0.110808 + 0.993842i $$0.535344\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −10.0000 −0.368355
$$738$$ 0 0
$$739$$ 30.0000 1.10357 0.551784 0.833987i $$-0.313947\pi$$
0.551784 + 0.833987i $$0.313947\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 30.0000 1.10059 0.550297 0.834969i $$-0.314515\pi$$
0.550297 + 0.834969i $$0.314515\pi$$
$$744$$ 0 0
$$745$$ 18.0000 0.659469
$$746$$ 0 0
$$747$$ −7.00000 −0.256117
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 45.0000 1.64207 0.821037 0.570875i $$-0.193396\pi$$
0.821037 + 0.570875i $$0.193396\pi$$
$$752$$ 0 0
$$753$$ −21.0000 −0.765283
$$754$$ 0 0
$$755$$ 19.0000 0.691481
$$756$$ 0 0
$$757$$ 54.0000 1.96266 0.981332 0.192323i $$-0.0616021\pi$$
0.981332 + 0.192323i $$0.0616021\pi$$
$$758$$ 0 0
$$759$$ −20.0000 −0.725954
$$760$$ 0 0
$$761$$ −8.00000 −0.290000 −0.145000 0.989432i $$-0.546318\pi$$
−0.145000 + 0.989432i $$0.546318\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 4.00000 0.144620
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 35.0000 1.26213 0.631066 0.775729i $$-0.282618\pi$$
0.631066 + 0.775729i $$0.282618\pi$$
$$770$$ 0 0
$$771$$ −6.00000 −0.216085
$$772$$ 0 0
$$773$$ 10.0000 0.359675 0.179838 0.983696i $$-0.442443\pi$$
0.179838 + 0.983696i $$0.442443\pi$$
$$774$$ 0 0
$$775$$ 12.0000 0.431053
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ −10.0000 −0.357828
$$782$$ 0 0
$$783$$ −5.00000 −0.178685
$$784$$ 0 0
$$785$$ −4.00000 −0.142766
$$786$$ 0 0
$$787$$ −18.0000 −0.641631 −0.320815 0.947142i $$-0.603957\pi$$
−0.320815 + 0.947142i $$0.603957\pi$$
$$788$$ 0 0
$$789$$ 30.0000 1.06803
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ −9.00000 −0.319197
$$796$$ 0 0
$$797$$ 21.0000 0.743858 0.371929 0.928261i $$-0.378696\pi$$
0.371929 + 0.928261i $$0.378696\pi$$
$$798$$ 0 0
$$799$$ 24.0000 0.849059
$$800$$ 0 0
$$801$$ 6.00000 0.212000
$$802$$ 0 0
$$803$$ 50.0000 1.76446
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −31.0000 −1.09125
$$808$$ 0 0
$$809$$ 40.0000 1.40633 0.703163 0.711029i $$-0.251771\pi$$
0.703163 + 0.711029i $$0.251771\pi$$
$$810$$ 0 0
$$811$$ −14.0000 −0.491606 −0.245803 0.969320i $$-0.579052\pi$$
−0.245803 + 0.969320i $$0.579052\pi$$
$$812$$ 0 0
$$813$$ 15.0000 0.526073
$$814$$ 0 0
$$815$$ 4.00000 0.140114
$$816$$ 0 0
$$817$$ −16.0000 −0.559769
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 25.0000 0.872506 0.436253 0.899824i $$-0.356305\pi$$
0.436253 + 0.899824i $$0.356305\pi$$
$$822$$ 0 0
$$823$$ 40.0000 1.39431 0.697156 0.716919i $$-0.254448\pi$$
0.697156 + 0.716919i $$0.254448\pi$$
$$824$$ 0 0
$$825$$ −20.0000 −0.696311
$$826$$ 0 0
$$827$$ −9.00000 −0.312961 −0.156480 0.987681i $$-0.550015\pi$$
−0.156480 + 0.987681i $$0.550015\pi$$
$$828$$ 0 0
$$829$$ −32.0000 −1.11141 −0.555703 0.831381i $$-0.687551\pi$$
−0.555703 + 0.831381i $$0.687551\pi$$
$$830$$ 0 0
$$831$$ −16.0000 −0.555034
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 14.0000 0.484490
$$836$$ 0 0
$$837$$ 3.00000 0.103695
$$838$$ 0 0
$$839$$ 28.0000 0.966667 0.483334 0.875436i $$-0.339426\pi$$
0.483334 + 0.875436i $$0.339426\pi$$
$$840$$ 0 0
$$841$$ −4.00000 −0.137931
$$842$$ 0 0
$$843$$ −2.00000 −0.0688837
$$844$$ 0 0
$$845$$ −13.0000 −0.447214
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ −10.0000 −0.343199
$$850$$ 0 0
$$851$$ −16.0000 −0.548473
$$852$$ 0 0
$$853$$ 14.0000 0.479351 0.239675 0.970853i $$-0.422959\pi$$
0.239675 + 0.970853i $$0.422959\pi$$
$$854$$ 0 0
$$855$$ 8.00000 0.273594
$$856$$ 0 0
$$857$$ 18.0000 0.614868 0.307434 0.951569i $$-0.400530\pi$$
0.307434 + 0.951569i $$0.400530\pi$$
$$858$$ 0 0
$$859$$ −34.0000 −1.16007 −0.580033 0.814593i $$-0.696960\pi$$
−0.580033 + 0.814593i $$0.696960\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 10.0000 0.340404 0.170202 0.985409i $$-0.445558\pi$$
0.170202 + 0.985409i $$0.445558\pi$$
$$864$$ 0 0
$$865$$ 22.0000 0.748022
$$866$$ 0 0
$$867$$ 1.00000 0.0339618
$$868$$ 0 0
$$869$$ −15.0000 −0.508840
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ −7.00000 −0.236914
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 32.0000 1.08056 0.540282 0.841484i $$-0.318318\pi$$
0.540282 + 0.841484i $$0.318318\pi$$
$$878$$ 0 0
$$879$$ 21.0000 0.708312
$$880$$ 0 0
$$881$$ 42.0000 1.41502 0.707508 0.706705i $$-0.249819\pi$$
0.707508 + 0.706705i $$0.249819\pi$$
$$882$$ 0 0
$$883$$ 40.0000 1.34611 0.673054 0.739594i $$-0.264982\pi$$
0.673054 + 0.739594i $$0.264982\pi$$
$$884$$ 0 0
$$885$$ 11.0000 0.369761
$$886$$ 0 0
$$887$$ −36.0000 −1.20876 −0.604381 0.796696i $$-0.706579\pi$$
−0.604381 + 0.796696i $$0.706579\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −5.00000 −0.167506
$$892$$ 0 0
$$893$$ 48.0000 1.60626
$$894$$ 0 0
$$895$$ −12.0000 −0.401116
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −15.0000 −0.500278
$$900$$ 0 0
$$901$$ 36.0000 1.19933
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −12.0000 −0.398453 −0.199227 0.979953i $$-0.563843\pi$$
−0.199227 + 0.979953i $$0.563843\pi$$
$$908$$ 0 0
$$909$$ 10.0000 0.331679
$$910$$ 0 0
$$911$$ 30.0000 0.993944 0.496972 0.867766i $$-0.334445\pi$$
0.496972 + 0.867766i $$0.334445\pi$$
$$912$$ 0 0
$$913$$ 35.0000 1.15833
$$914$$ 0 0
$$915$$ 6.00000 0.198354
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −32.0000 −1.05558 −0.527791 0.849374i $$-0.676980\pi$$
−0.527791 + 0.849374i $$0.676980\pi$$
$$920$$ 0 0
$$921$$ −28.0000 −0.922631
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ −16.0000 −0.526077
$$926$$ 0 0
$$927$$ −8.00000 −0.262754
$$928$$ 0 0
$$929$$ 6.00000 0.196854 0.0984268 0.995144i $$-0.468619\pi$$
0.0984268 + 0.995144i $$0.468619\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ −32.0000 −1.04763
$$934$$ 0 0
$$935$$ −20.0000 −0.654070
$$936$$ 0 0
$$937$$ −35.0000 −1.14340 −0.571700 0.820463i $$-0.693716\pi$$
−0.571700 + 0.820463i $$0.693716\pi$$
$$938$$ 0 0
$$939$$ 1.00000 0.0326338
$$940$$ 0 0
$$941$$ −11.0000 −0.358590 −0.179295 0.983795i $$-0.557382\pi$$
−0.179295 + 0.983795i $$0.557382\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 32.0000 1.03986 0.519930 0.854209i $$-0.325958\pi$$
0.519930 + 0.854209i $$0.325958\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ 3.00000 0.0972817
$$952$$ 0 0
$$953$$ 2.00000 0.0647864 0.0323932 0.999475i $$-0.489687\pi$$
0.0323932 + 0.999475i $$0.489687\pi$$
$$954$$ 0 0
$$955$$ 24.0000 0.776622
$$956$$ 0 0
$$957$$ 25.0000 0.808135
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −22.0000 −0.709677
$$962$$ 0 0
$$963$$ −3.00000 −0.0966736
$$964$$ 0 0
$$965$$ 5.00000 0.160956
$$966$$ 0 0
$$967$$ −61.0000 −1.96163 −0.980814 0.194946i $$-0.937547\pi$$
−0.980814 + 0.194946i $$0.937547\pi$$
$$968$$ 0 0
$$969$$ −32.0000 −1.02799
$$970$$ 0 0
$$971$$ 15.0000 0.481373 0.240686 0.970603i $$-0.422627\pi$$
0.240686 + 0.970603i $$0.422627\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −30.0000 −0.959785 −0.479893 0.877327i $$-0.659324\pi$$
−0.479893 + 0.877327i $$0.659324\pi$$
$$978$$ 0 0
$$979$$ −30.0000 −0.958804
$$980$$ 0 0
$$981$$ 2.00000 0.0638551
$$982$$ 0 0
$$983$$ 60.0000 1.91370 0.956851 0.290578i $$-0.0938475\pi$$
0.956851 + 0.290578i $$0.0938475\pi$$
$$984$$ 0 0
$$985$$ −2.00000 −0.0637253
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 8.00000 0.254385
$$990$$ 0 0
$$991$$ 47.0000 1.49300 0.746502 0.665383i $$-0.231732\pi$$
0.746502 + 0.665383i $$0.231732\pi$$
$$992$$ 0 0
$$993$$ −4.00000 −0.126936
$$994$$ 0 0
$$995$$ 4.00000 0.126809
$$996$$ 0 0
$$997$$ 38.0000 1.20347 0.601736 0.798695i $$-0.294476\pi$$
0.601736 + 0.798695i $$0.294476\pi$$
$$998$$ 0 0
$$999$$ −4.00000 −0.126554
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 9408.2.a.z.1.1 1
4.3 odd 2 9408.2.a.cr.1.1 1
7.3 odd 6 1344.2.q.g.961.1 2
7.5 odd 6 1344.2.q.g.193.1 2
7.6 odd 2 9408.2.a.ce.1.1 1
8.3 odd 2 2352.2.a.f.1.1 1
8.5 even 2 294.2.a.f.1.1 1
24.5 odd 2 882.2.a.d.1.1 1
24.11 even 2 7056.2.a.bl.1.1 1
28.3 even 6 1344.2.q.s.961.1 2
28.19 even 6 1344.2.q.s.193.1 2
28.27 even 2 9408.2.a.q.1.1 1
40.29 even 2 7350.2.a.q.1.1 1
56.3 even 6 336.2.q.b.289.1 2
56.5 odd 6 42.2.e.a.25.1 2
56.11 odd 6 2352.2.q.u.961.1 2
56.13 odd 2 294.2.a.e.1.1 1
56.19 even 6 336.2.q.b.193.1 2
56.27 even 2 2352.2.a.t.1.1 1
56.37 even 6 294.2.e.b.67.1 2
56.45 odd 6 42.2.e.a.37.1 yes 2
56.51 odd 6 2352.2.q.u.1537.1 2
56.53 even 6 294.2.e.b.79.1 2
168.5 even 6 126.2.g.c.109.1 2
168.53 odd 6 882.2.g.i.667.1 2
168.59 odd 6 1008.2.s.k.289.1 2
168.83 odd 2 7056.2.a.w.1.1 1
168.101 even 6 126.2.g.c.37.1 2
168.125 even 2 882.2.a.c.1.1 1
168.131 odd 6 1008.2.s.k.865.1 2
168.149 odd 6 882.2.g.i.361.1 2
280.69 odd 2 7350.2.a.bl.1.1 1
280.117 even 12 1050.2.o.a.949.2 4
280.157 even 12 1050.2.o.a.499.1 4
280.173 even 12 1050.2.o.a.949.1 4
280.213 even 12 1050.2.o.a.499.2 4
280.229 odd 6 1050.2.i.l.151.1 2
280.269 odd 6 1050.2.i.l.751.1 2
504.5 even 6 1134.2.e.e.865.1 2
504.61 odd 6 1134.2.h.e.109.1 2
504.101 even 6 1134.2.e.e.919.1 2
504.157 odd 6 1134.2.h.e.541.1 2
504.173 even 6 1134.2.h.l.109.1 2
504.229 odd 6 1134.2.e.l.865.1 2
504.437 even 6 1134.2.h.l.541.1 2
504.493 odd 6 1134.2.e.l.919.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
42.2.e.a.25.1 2 56.5 odd 6
42.2.e.a.37.1 yes 2 56.45 odd 6
126.2.g.c.37.1 2 168.101 even 6
126.2.g.c.109.1 2 168.5 even 6
294.2.a.e.1.1 1 56.13 odd 2
294.2.a.f.1.1 1 8.5 even 2
294.2.e.b.67.1 2 56.37 even 6
294.2.e.b.79.1 2 56.53 even 6
336.2.q.b.193.1 2 56.19 even 6
336.2.q.b.289.1 2 56.3 even 6
882.2.a.c.1.1 1 168.125 even 2
882.2.a.d.1.1 1 24.5 odd 2
882.2.g.i.361.1 2 168.149 odd 6
882.2.g.i.667.1 2 168.53 odd 6
1008.2.s.k.289.1 2 168.59 odd 6
1008.2.s.k.865.1 2 168.131 odd 6
1050.2.i.l.151.1 2 280.229 odd 6
1050.2.i.l.751.1 2 280.269 odd 6
1050.2.o.a.499.1 4 280.157 even 12
1050.2.o.a.499.2 4 280.213 even 12
1050.2.o.a.949.1 4 280.173 even 12
1050.2.o.a.949.2 4 280.117 even 12
1134.2.e.e.865.1 2 504.5 even 6
1134.2.e.e.919.1 2 504.101 even 6
1134.2.e.l.865.1 2 504.229 odd 6
1134.2.e.l.919.1 2 504.493 odd 6
1134.2.h.e.109.1 2 504.61 odd 6
1134.2.h.e.541.1 2 504.157 odd 6
1134.2.h.l.109.1 2 504.173 even 6
1134.2.h.l.541.1 2 504.437 even 6
1344.2.q.g.193.1 2 7.5 odd 6
1344.2.q.g.961.1 2 7.3 odd 6
1344.2.q.s.193.1 2 28.19 even 6
1344.2.q.s.961.1 2 28.3 even 6
2352.2.a.f.1.1 1 8.3 odd 2
2352.2.a.t.1.1 1 56.27 even 2
2352.2.q.u.961.1 2 56.11 odd 6
2352.2.q.u.1537.1 2 56.51 odd 6
7056.2.a.w.1.1 1 168.83 odd 2
7056.2.a.bl.1.1 1 24.11 even 2
7350.2.a.q.1.1 1 40.29 even 2
7350.2.a.bl.1.1 1 280.69 odd 2
9408.2.a.q.1.1 1 28.27 even 2
9408.2.a.z.1.1 1 1.1 even 1 trivial
9408.2.a.ce.1.1 1 7.6 odd 2
9408.2.a.cr.1.1 1 4.3 odd 2