# Properties

 Label 9408.2.a.db.1.1 Level $9408$ Weight $2$ Character 9408.1 Self dual yes Analytic conductor $75.123$ Analytic rank $1$ 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: $$1$$ 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

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} +3.00000 q^{5} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} +3.00000 q^{5} +1.00000 q^{9} -3.00000 q^{11} -4.00000 q^{13} +3.00000 q^{15} -4.00000 q^{19} +4.00000 q^{25} +1.00000 q^{27} -9.00000 q^{29} +1.00000 q^{31} -3.00000 q^{33} -8.00000 q^{37} -4.00000 q^{39} +10.0000 q^{43} +3.00000 q^{45} +6.00000 q^{47} +3.00000 q^{53} -9.00000 q^{55} -4.00000 q^{57} +3.00000 q^{59} -10.0000 q^{61} -12.0000 q^{65} +10.0000 q^{67} -6.00000 q^{71} -2.00000 q^{73} +4.00000 q^{75} -1.00000 q^{79} +1.00000 q^{81} -9.00000 q^{83} -9.00000 q^{87} -6.00000 q^{89} +1.00000 q^{93} -12.0000 q^{95} +1.00000 q^{97} -3.00000 q^{99} +O(q^{100})$$

## 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
$$5$$ 3.00000 1.34164 0.670820 0.741620i $$-0.265942\pi$$
0.670820 + 0.741620i $$0.265942\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −3.00000 −0.904534 −0.452267 0.891883i $$-0.649385\pi$$
−0.452267 + 0.891883i $$0.649385\pi$$
$$12$$ 0 0
$$13$$ −4.00000 −1.10940 −0.554700 0.832050i $$-0.687167\pi$$
−0.554700 + 0.832050i $$0.687167\pi$$
$$14$$ 0 0
$$15$$ 3.00000 0.774597
$$16$$ 0 0
$$17$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$18$$ 0 0
$$19$$ −4.00000 −0.917663 −0.458831 0.888523i $$-0.651732\pi$$
−0.458831 + 0.888523i $$0.651732\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 0 0
$$25$$ 4.00000 0.800000
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ −9.00000 −1.67126 −0.835629 0.549294i $$-0.814897\pi$$
−0.835629 + 0.549294i $$0.814897\pi$$
$$30$$ 0 0
$$31$$ 1.00000 0.179605 0.0898027 0.995960i $$-0.471376\pi$$
0.0898027 + 0.995960i $$0.471376\pi$$
$$32$$ 0 0
$$33$$ −3.00000 −0.522233
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −8.00000 −1.31519 −0.657596 0.753371i $$-0.728427\pi$$
−0.657596 + 0.753371i $$0.728427\pi$$
$$38$$ 0 0
$$39$$ −4.00000 −0.640513
$$40$$ 0 0
$$41$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$42$$ 0 0
$$43$$ 10.0000 1.52499 0.762493 0.646997i $$-0.223975\pi$$
0.762493 + 0.646997i $$0.223975\pi$$
$$44$$ 0 0
$$45$$ 3.00000 0.447214
$$46$$ 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
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 3.00000 0.412082 0.206041 0.978543i $$-0.433942\pi$$
0.206041 + 0.978543i $$0.433942\pi$$
$$54$$ 0 0
$$55$$ −9.00000 −1.21356
$$56$$ 0 0
$$57$$ −4.00000 −0.529813
$$58$$ 0 0
$$59$$ 3.00000 0.390567 0.195283 0.980747i $$-0.437437\pi$$
0.195283 + 0.980747i $$0.437437\pi$$
$$60$$ 0 0
$$61$$ −10.0000 −1.28037 −0.640184 0.768221i $$-0.721142\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −12.0000 −1.48842
$$66$$ 0 0
$$67$$ 10.0000 1.22169 0.610847 0.791748i $$-0.290829\pi$$
0.610847 + 0.791748i $$0.290829\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −6.00000 −0.712069 −0.356034 0.934473i $$-0.615871\pi$$
−0.356034 + 0.934473i $$0.615871\pi$$
$$72$$ 0 0
$$73$$ −2.00000 −0.234082 −0.117041 0.993127i $$-0.537341\pi$$
−0.117041 + 0.993127i $$0.537341\pi$$
$$74$$ 0 0
$$75$$ 4.00000 0.461880
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −1.00000 −0.112509 −0.0562544 0.998416i $$-0.517916\pi$$
−0.0562544 + 0.998416i $$0.517916\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −9.00000 −0.987878 −0.493939 0.869496i $$-0.664443\pi$$
−0.493939 + 0.869496i $$0.664443\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ −9.00000 −0.964901
$$88$$ 0 0
$$89$$ −6.00000 −0.635999 −0.317999 0.948091i $$-0.603011\pi$$
−0.317999 + 0.948091i $$0.603011\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 1.00000 0.103695
$$94$$ 0 0
$$95$$ −12.0000 −1.23117
$$96$$ 0 0
$$97$$ 1.00000 0.101535 0.0507673 0.998711i $$-0.483833\pi$$
0.0507673 + 0.998711i $$0.483833\pi$$
$$98$$ 0 0
$$99$$ −3.00000 −0.301511
$$100$$ 0 0
$$101$$ −18.0000 −1.79107 −0.895533 0.444994i $$-0.853206\pi$$
−0.895533 + 0.444994i $$0.853206\pi$$
$$102$$ 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 $$-0.453678\pi$$
0.145010 + 0.989430i $$0.453678\pi$$
$$108$$ 0 0
$$109$$ −14.0000 −1.34096 −0.670478 0.741929i $$-0.733911\pi$$
−0.670478 + 0.741929i $$0.733911\pi$$
$$110$$ 0 0
$$111$$ −8.00000 −0.759326
$$112$$ 0 0
$$113$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −4.00000 −0.369800
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −2.00000 −0.181818
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −3.00000 −0.268328
$$126$$ 0 0
$$127$$ 5.00000 0.443678 0.221839 0.975083i $$-0.428794\pi$$
0.221839 + 0.975083i $$0.428794\pi$$
$$128$$ 0 0
$$129$$ 10.0000 0.880451
$$130$$ 0 0
$$131$$ −9.00000 −0.786334 −0.393167 0.919467i $$-0.628621\pi$$
−0.393167 + 0.919467i $$0.628621\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 3.00000 0.258199
$$136$$ 0 0
$$137$$ 18.0000 1.53784 0.768922 0.639343i $$-0.220793\pi$$
0.768922 + 0.639343i $$0.220793\pi$$
$$138$$ 0 0
$$139$$ 2.00000 0.169638 0.0848189 0.996396i $$-0.472969\pi$$
0.0848189 + 0.996396i $$0.472969\pi$$
$$140$$ 0 0
$$141$$ 6.00000 0.505291
$$142$$ 0 0
$$143$$ 12.0000 1.00349
$$144$$ 0 0
$$145$$ −27.0000 −2.24223
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −18.0000 −1.47462 −0.737309 0.675556i $$-0.763904\pi$$
−0.737309 + 0.675556i $$0.763904\pi$$
$$150$$ 0 0
$$151$$ −1.00000 −0.0813788 −0.0406894 0.999172i $$-0.512955\pi$$
−0.0406894 + 0.999172i $$0.512955\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 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
$$159$$ 3.00000 0.237915
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 16.0000 1.25322 0.626608 0.779334i $$-0.284443\pi$$
0.626608 + 0.779334i $$0.284443\pi$$
$$164$$ 0 0
$$165$$ −9.00000 −0.700649
$$166$$ 0 0
$$167$$ −6.00000 −0.464294 −0.232147 0.972681i $$-0.574575\pi$$
−0.232147 + 0.972681i $$0.574575\pi$$
$$168$$ 0 0
$$169$$ 3.00000 0.230769
$$170$$ 0 0
$$171$$ −4.00000 −0.305888
$$172$$ 0 0
$$173$$ 18.0000 1.36851 0.684257 0.729241i $$-0.260127\pi$$
0.684257 + 0.729241i $$0.260127\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 3.00000 0.225494
$$178$$ 0 0
$$179$$ −12.0000 −0.896922 −0.448461 0.893802i $$-0.648028\pi$$
−0.448461 + 0.893802i $$0.648028\pi$$
$$180$$ 0 0
$$181$$ 8.00000 0.594635 0.297318 0.954779i $$-0.403908\pi$$
0.297318 + 0.954779i $$0.403908\pi$$
$$182$$ 0 0
$$183$$ −10.0000 −0.739221
$$184$$ 0 0
$$185$$ −24.0000 −1.76452
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ 0 0
$$193$$ −19.0000 −1.36765 −0.683825 0.729646i $$-0.739685\pi$$
−0.683825 + 0.729646i $$0.739685\pi$$
$$194$$ 0 0
$$195$$ −12.0000 −0.859338
$$196$$ 0 0
$$197$$ −6.00000 −0.427482 −0.213741 0.976890i $$-0.568565\pi$$
−0.213741 + 0.976890i $$0.568565\pi$$
$$198$$ 0 0
$$199$$ −20.0000 −1.41776 −0.708881 0.705328i $$-0.750800\pi$$
−0.708881 + 0.705328i $$0.750800\pi$$
$$200$$ 0 0
$$201$$ 10.0000 0.705346
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 12.0000 0.830057
$$210$$ 0 0
$$211$$ −14.0000 −0.963800 −0.481900 0.876226i $$-0.660053\pi$$
−0.481900 + 0.876226i $$0.660053\pi$$
$$212$$ 0 0
$$213$$ −6.00000 −0.411113
$$214$$ 0 0
$$215$$ 30.0000 2.04598
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ −2.00000 −0.135147
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 19.0000 1.27233 0.636167 0.771551i $$-0.280519\pi$$
0.636167 + 0.771551i $$0.280519\pi$$
$$224$$ 0 0
$$225$$ 4.00000 0.266667
$$226$$ 0 0
$$227$$ −27.0000 −1.79205 −0.896026 0.444001i $$-0.853559\pi$$
−0.896026 + 0.444001i $$0.853559\pi$$
$$228$$ 0 0
$$229$$ −4.00000 −0.264327 −0.132164 0.991228i $$-0.542192\pi$$
−0.132164 + 0.991228i $$0.542192\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −24.0000 −1.57229 −0.786146 0.618041i $$-0.787927\pi$$
−0.786146 + 0.618041i $$0.787927\pi$$
$$234$$ 0 0
$$235$$ 18.0000 1.17419
$$236$$ 0 0
$$237$$ −1.00000 −0.0649570
$$238$$ 0 0
$$239$$ −24.0000 −1.55243 −0.776215 0.630468i $$-0.782863\pi$$
−0.776215 + 0.630468i $$0.782863\pi$$
$$240$$ 0 0
$$241$$ 1.00000 0.0644157 0.0322078 0.999481i $$-0.489746\pi$$
0.0322078 + 0.999481i $$0.489746\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 16.0000 1.01806
$$248$$ 0 0
$$249$$ −9.00000 −0.570352
$$250$$ 0 0
$$251$$ 27.0000 1.70422 0.852112 0.523359i $$-0.175321\pi$$
0.852112 + 0.523359i $$0.175321\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −6.00000 −0.374270 −0.187135 0.982334i $$-0.559920\pi$$
−0.187135 + 0.982334i $$0.559920\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −9.00000 −0.557086
$$262$$ 0 0
$$263$$ −6.00000 −0.369976 −0.184988 0.982741i $$-0.559225\pi$$
−0.184988 + 0.982741i $$0.559225\pi$$
$$264$$ 0 0
$$265$$ 9.00000 0.552866
$$266$$ 0 0
$$267$$ −6.00000 −0.367194
$$268$$ 0 0
$$269$$ 21.0000 1.28039 0.640196 0.768211i $$-0.278853\pi$$
0.640196 + 0.768211i $$0.278853\pi$$
$$270$$ 0 0
$$271$$ −11.0000 −0.668202 −0.334101 0.942537i $$-0.608433\pi$$
−0.334101 + 0.942537i $$0.608433\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −12.0000 −0.723627
$$276$$ 0 0
$$277$$ −8.00000 −0.480673 −0.240337 0.970690i $$-0.577258\pi$$
−0.240337 + 0.970690i $$0.577258\pi$$
$$278$$ 0 0
$$279$$ 1.00000 0.0598684
$$280$$ 0 0
$$281$$ 6.00000 0.357930 0.178965 0.983855i $$-0.442725\pi$$
0.178965 + 0.983855i $$0.442725\pi$$
$$282$$ 0 0
$$283$$ 14.0000 0.832214 0.416107 0.909316i $$-0.363394\pi$$
0.416107 + 0.909316i $$0.363394\pi$$
$$284$$ 0 0
$$285$$ −12.0000 −0.710819
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −17.0000 −1.00000
$$290$$ 0 0
$$291$$ 1.00000 0.0586210
$$292$$ 0 0
$$293$$ 33.0000 1.92788 0.963940 0.266119i $$-0.0857413\pi$$
0.963940 + 0.266119i $$0.0857413\pi$$
$$294$$ 0 0
$$295$$ 9.00000 0.524000
$$296$$ 0 0
$$297$$ −3.00000 −0.174078
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ −18.0000 −1.03407
$$304$$ 0 0
$$305$$ −30.0000 −1.71780
$$306$$ 0 0
$$307$$ 8.00000 0.456584 0.228292 0.973593i $$-0.426686\pi$$
0.228292 + 0.973593i $$0.426686\pi$$
$$308$$ 0 0
$$309$$ −8.00000 −0.455104
$$310$$ 0 0
$$311$$ −24.0000 −1.36092 −0.680458 0.732787i $$-0.738219\pi$$
−0.680458 + 0.732787i $$0.738219\pi$$
$$312$$ 0 0
$$313$$ 31.0000 1.75222 0.876112 0.482108i $$-0.160129\pi$$
0.876112 + 0.482108i $$0.160129\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −9.00000 −0.505490 −0.252745 0.967533i $$-0.581333\pi$$
−0.252745 + 0.967533i $$0.581333\pi$$
$$318$$ 0 0
$$319$$ 27.0000 1.51171
$$320$$ 0 0
$$321$$ 3.00000 0.167444
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ −16.0000 −0.887520
$$326$$ 0 0
$$327$$ −14.0000 −0.774202
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −20.0000 −1.09930 −0.549650 0.835395i $$-0.685239\pi$$
−0.549650 + 0.835395i $$0.685239\pi$$
$$332$$ 0 0
$$333$$ −8.00000 −0.438397
$$334$$ 0 0
$$335$$ 30.0000 1.63908
$$336$$ 0 0
$$337$$ −7.00000 −0.381314 −0.190657 0.981657i $$-0.561062\pi$$
−0.190657 + 0.981657i $$0.561062\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −3.00000 −0.162459
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −12.0000 −0.644194 −0.322097 0.946707i $$-0.604388\pi$$
−0.322097 + 0.946707i $$0.604388\pi$$
$$348$$ 0 0
$$349$$ 26.0000 1.39175 0.695874 0.718164i $$-0.255017\pi$$
0.695874 + 0.718164i $$0.255017\pi$$
$$350$$ 0 0
$$351$$ −4.00000 −0.213504
$$352$$ 0 0
$$353$$ −24.0000 −1.27739 −0.638696 0.769460i $$-0.720526\pi$$
−0.638696 + 0.769460i $$0.720526\pi$$
$$354$$ 0 0
$$355$$ −18.0000 −0.955341
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 30.0000 1.58334 0.791670 0.610949i $$-0.209212\pi$$
0.791670 + 0.610949i $$0.209212\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ 0 0
$$363$$ −2.00000 −0.104973
$$364$$ 0 0
$$365$$ −6.00000 −0.314054
$$366$$ 0 0
$$367$$ 19.0000 0.991792 0.495896 0.868382i $$-0.334840\pi$$
0.495896 + 0.868382i $$0.334840\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −8.00000 −0.414224 −0.207112 0.978317i $$-0.566407\pi$$
−0.207112 + 0.978317i $$0.566407\pi$$
$$374$$ 0 0
$$375$$ −3.00000 −0.154919
$$376$$ 0 0
$$377$$ 36.0000 1.85409
$$378$$ 0 0
$$379$$ −8.00000 −0.410932 −0.205466 0.978664i $$-0.565871\pi$$
−0.205466 + 0.978664i $$0.565871\pi$$
$$380$$ 0 0
$$381$$ 5.00000 0.256158
$$382$$ 0 0
$$383$$ −18.0000 −0.919757 −0.459879 0.887982i $$-0.652107\pi$$
−0.459879 + 0.887982i $$0.652107\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 10.0000 0.508329
$$388$$ 0 0
$$389$$ 6.00000 0.304212 0.152106 0.988364i $$-0.451394\pi$$
0.152106 + 0.988364i $$0.451394\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ −9.00000 −0.453990
$$394$$ 0 0
$$395$$ −3.00000 −0.150946
$$396$$ 0 0
$$397$$ −4.00000 −0.200754 −0.100377 0.994949i $$-0.532005\pi$$
−0.100377 + 0.994949i $$0.532005\pi$$
$$398$$ 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
$$403$$ −4.00000 −0.199254
$$404$$ 0 0
$$405$$ 3.00000 0.149071
$$406$$ 0 0
$$407$$ 24.0000 1.18964
$$408$$ 0 0
$$409$$ 25.0000 1.23617 0.618085 0.786111i $$-0.287909\pi$$
0.618085 + 0.786111i $$0.287909\pi$$
$$410$$ 0 0
$$411$$ 18.0000 0.887875
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −27.0000 −1.32538
$$416$$ 0 0
$$417$$ 2.00000 0.0979404
$$418$$ 0 0
$$419$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$420$$ 0 0
$$421$$ 22.0000 1.07221 0.536107 0.844150i $$-0.319894\pi$$
0.536107 + 0.844150i $$0.319894\pi$$
$$422$$ 0 0
$$423$$ 6.00000 0.291730
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 12.0000 0.579365
$$430$$ 0 0
$$431$$ 12.0000 0.578020 0.289010 0.957326i $$-0.406674\pi$$
0.289010 + 0.957326i $$0.406674\pi$$
$$432$$ 0 0
$$433$$ 34.0000 1.63394 0.816968 0.576683i $$-0.195653\pi$$
0.816968 + 0.576683i $$0.195653\pi$$
$$434$$ 0 0
$$435$$ −27.0000 −1.29455
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ −35.0000 −1.67046 −0.835229 0.549902i $$-0.814665\pi$$
−0.835229 + 0.549902i $$0.814665\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 33.0000 1.56788 0.783939 0.620838i $$-0.213208\pi$$
0.783939 + 0.620838i $$0.213208\pi$$
$$444$$ 0 0
$$445$$ −18.0000 −0.853282
$$446$$ 0 0
$$447$$ −18.0000 −0.851371
$$448$$ 0 0
$$449$$ 12.0000 0.566315 0.283158 0.959073i $$-0.408618\pi$$
0.283158 + 0.959073i $$0.408618\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ −1.00000 −0.0469841
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −1.00000 −0.0467780 −0.0233890 0.999726i $$-0.507446\pi$$
−0.0233890 + 0.999726i $$0.507446\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 30.0000 1.39724 0.698620 0.715493i $$-0.253798\pi$$
0.698620 + 0.715493i $$0.253798\pi$$
$$462$$ 0 0
$$463$$ 8.00000 0.371792 0.185896 0.982569i $$-0.440481\pi$$
0.185896 + 0.982569i $$0.440481\pi$$
$$464$$ 0 0
$$465$$ 3.00000 0.139122
$$466$$ 0 0
$$467$$ 36.0000 1.66588 0.832941 0.553362i $$-0.186655\pi$$
0.832941 + 0.553362i $$0.186655\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −4.00000 −0.184310
$$472$$ 0 0
$$473$$ −30.0000 −1.37940
$$474$$ 0 0
$$475$$ −16.0000 −0.734130
$$476$$ 0 0
$$477$$ 3.00000 0.137361
$$478$$ 0 0
$$479$$ 18.0000 0.822441 0.411220 0.911536i $$-0.365103\pi$$
0.411220 + 0.911536i $$0.365103\pi$$
$$480$$ 0 0
$$481$$ 32.0000 1.45907
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 3.00000 0.136223
$$486$$ 0 0
$$487$$ 41.0000 1.85789 0.928944 0.370221i $$-0.120718\pi$$
0.928944 + 0.370221i $$0.120718\pi$$
$$488$$ 0 0
$$489$$ 16.0000 0.723545
$$490$$ 0 0
$$491$$ 33.0000 1.48927 0.744635 0.667472i $$-0.232624\pi$$
0.744635 + 0.667472i $$0.232624\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ −9.00000 −0.404520
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −2.00000 −0.0895323 −0.0447661 0.998997i $$-0.514254\pi$$
−0.0447661 + 0.998997i $$0.514254\pi$$
$$500$$ 0 0
$$501$$ −6.00000 −0.268060
$$502$$ 0 0
$$503$$ 12.0000 0.535054 0.267527 0.963550i $$-0.413794\pi$$
0.267527 + 0.963550i $$0.413794\pi$$
$$504$$ 0 0
$$505$$ −54.0000 −2.40297
$$506$$ 0 0
$$507$$ 3.00000 0.133235
$$508$$ 0 0
$$509$$ −3.00000 −0.132973 −0.0664863 0.997787i $$-0.521179\pi$$
−0.0664863 + 0.997787i $$0.521179\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ −4.00000 −0.176604
$$514$$ 0 0
$$515$$ −24.0000 −1.05757
$$516$$ 0 0
$$517$$ −18.0000 −0.791639
$$518$$ 0 0
$$519$$ 18.0000 0.790112
$$520$$ 0 0
$$521$$ 18.0000 0.788594 0.394297 0.918983i $$-0.370988\pi$$
0.394297 + 0.918983i $$0.370988\pi$$
$$522$$ 0 0
$$523$$ −4.00000 −0.174908 −0.0874539 0.996169i $$-0.527873\pi$$
−0.0874539 + 0.996169i $$0.527873\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −23.0000 −1.00000
$$530$$ 0 0
$$531$$ 3.00000 0.130189
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 9.00000 0.389104
$$536$$ 0 0
$$537$$ −12.0000 −0.517838
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −26.0000 −1.11783 −0.558914 0.829226i $$-0.688782\pi$$
−0.558914 + 0.829226i $$0.688782\pi$$
$$542$$ 0 0
$$543$$ 8.00000 0.343313
$$544$$ 0 0
$$545$$ −42.0000 −1.79908
$$546$$ 0 0
$$547$$ −8.00000 −0.342055 −0.171028 0.985266i $$-0.554709\pi$$
−0.171028 + 0.985266i $$0.554709\pi$$
$$548$$ 0 0
$$549$$ −10.0000 −0.426790
$$550$$ 0 0
$$551$$ 36.0000 1.53365
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ −24.0000 −1.01874
$$556$$ 0 0
$$557$$ −3.00000 −0.127114 −0.0635570 0.997978i $$-0.520244\pi$$
−0.0635570 + 0.997978i $$0.520244\pi$$
$$558$$ 0 0
$$559$$ −40.0000 −1.69182
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 39.0000 1.64365 0.821827 0.569737i $$-0.192955\pi$$
0.821827 + 0.569737i $$0.192955\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −36.0000 −1.50920 −0.754599 0.656186i $$-0.772169\pi$$
−0.754599 + 0.656186i $$0.772169\pi$$
$$570$$ 0 0
$$571$$ 34.0000 1.42286 0.711428 0.702759i $$-0.248049\pi$$
0.711428 + 0.702759i $$0.248049\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −23.0000 −0.957503 −0.478751 0.877951i $$-0.658910\pi$$
−0.478751 + 0.877951i $$0.658910\pi$$
$$578$$ 0 0
$$579$$ −19.0000 −0.789613
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −9.00000 −0.372742
$$584$$ 0 0
$$585$$ −12.0000 −0.496139
$$586$$ 0 0
$$587$$ 21.0000 0.866763 0.433381 0.901211i $$-0.357320\pi$$
0.433381 + 0.901211i $$0.357320\pi$$
$$588$$ 0 0
$$589$$ −4.00000 −0.164817
$$590$$ 0 0
$$591$$ −6.00000 −0.246807
$$592$$ 0 0
$$593$$ −24.0000 −0.985562 −0.492781 0.870153i $$-0.664020\pi$$
−0.492781 + 0.870153i $$0.664020\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −20.0000 −0.818546
$$598$$ 0 0
$$599$$ −18.0000 −0.735460 −0.367730 0.929933i $$-0.619865\pi$$
−0.367730 + 0.929933i $$0.619865\pi$$
$$600$$ 0 0
$$601$$ −11.0000 −0.448699 −0.224350 0.974509i $$-0.572026\pi$$
−0.224350 + 0.974509i $$0.572026\pi$$
$$602$$ 0 0
$$603$$ 10.0000 0.407231
$$604$$ 0 0
$$605$$ −6.00000 −0.243935
$$606$$ 0 0
$$607$$ 7.00000 0.284121 0.142061 0.989858i $$-0.454627\pi$$
0.142061 + 0.989858i $$0.454627\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −24.0000 −0.970936
$$612$$ 0 0
$$613$$ 16.0000 0.646234 0.323117 0.946359i $$-0.395269\pi$$
0.323117 + 0.946359i $$0.395269\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −6.00000 −0.241551 −0.120775 0.992680i $$-0.538538\pi$$
−0.120775 + 0.992680i $$0.538538\pi$$
$$618$$ 0 0
$$619$$ −34.0000 −1.36658 −0.683288 0.730149i $$-0.739451\pi$$
−0.683288 + 0.730149i $$0.739451\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −29.0000 −1.16000
$$626$$ 0 0
$$627$$ 12.0000 0.479234
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ −7.00000 −0.278666 −0.139333 0.990246i $$-0.544496\pi$$
−0.139333 + 0.990246i $$0.544496\pi$$
$$632$$ 0 0
$$633$$ −14.0000 −0.556450
$$634$$ 0 0
$$635$$ 15.0000 0.595257
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ −6.00000 −0.237356
$$640$$ 0 0
$$641$$ −30.0000 −1.18493 −0.592464 0.805597i $$-0.701845\pi$$
−0.592464 + 0.805597i $$0.701845\pi$$
$$642$$ 0 0
$$643$$ −34.0000 −1.34083 −0.670415 0.741987i $$-0.733884\pi$$
−0.670415 + 0.741987i $$0.733884\pi$$
$$644$$ 0 0
$$645$$ 30.0000 1.18125
$$646$$ 0 0
$$647$$ 18.0000 0.707653 0.353827 0.935311i $$-0.384880\pi$$
0.353827 + 0.935311i $$0.384880\pi$$
$$648$$ 0 0
$$649$$ −9.00000 −0.353281
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −3.00000 −0.117399 −0.0586995 0.998276i $$-0.518695\pi$$
−0.0586995 + 0.998276i $$0.518695\pi$$
$$654$$ 0 0
$$655$$ −27.0000 −1.05498
$$656$$ 0 0
$$657$$ −2.00000 −0.0780274
$$658$$ 0 0
$$659$$ 24.0000 0.934907 0.467454 0.884018i $$-0.345171\pi$$
0.467454 + 0.884018i $$0.345171\pi$$
$$660$$ 0 0
$$661$$ 14.0000 0.544537 0.272268 0.962221i $$-0.412226\pi$$
0.272268 + 0.962221i $$0.412226\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 19.0000 0.734582
$$670$$ 0 0
$$671$$ 30.0000 1.15814
$$672$$ 0 0
$$673$$ 29.0000 1.11787 0.558934 0.829212i $$-0.311211\pi$$
0.558934 + 0.829212i $$0.311211\pi$$
$$674$$ 0 0
$$675$$ 4.00000 0.153960
$$676$$ 0 0
$$677$$ −33.0000 −1.26829 −0.634147 0.773213i $$-0.718648\pi$$
−0.634147 + 0.773213i $$0.718648\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −27.0000 −1.03464
$$682$$ 0 0
$$683$$ −33.0000 −1.26271 −0.631355 0.775494i $$-0.717501\pi$$
−0.631355 + 0.775494i $$0.717501\pi$$
$$684$$ 0 0
$$685$$ 54.0000 2.06323
$$686$$ 0 0
$$687$$ −4.00000 −0.152610
$$688$$ 0 0
$$689$$ −12.0000 −0.457164
$$690$$ 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
$$695$$ 6.00000 0.227593
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ −24.0000 −0.907763
$$700$$ 0 0
$$701$$ 15.0000 0.566542 0.283271 0.959040i $$-0.408580\pi$$
0.283271 + 0.959040i $$0.408580\pi$$
$$702$$ 0 0
$$703$$ 32.0000 1.20690
$$704$$ 0 0
$$705$$ 18.0000 0.677919
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 10.0000 0.375558 0.187779 0.982211i $$-0.439871\pi$$
0.187779 + 0.982211i $$0.439871\pi$$
$$710$$ 0 0
$$711$$ −1.00000 −0.0375029
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 36.0000 1.34632
$$716$$ 0 0
$$717$$ −24.0000 −0.896296
$$718$$ 0 0
$$719$$ 18.0000 0.671287 0.335643 0.941989i $$-0.391046\pi$$
0.335643 + 0.941989i $$0.391046\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 1.00000 0.0371904
$$724$$ 0 0
$$725$$ −36.0000 −1.33701
$$726$$ 0 0
$$727$$ 13.0000 0.482143 0.241072 0.970507i $$-0.422501\pi$$
0.241072 + 0.970507i $$0.422501\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ −10.0000 −0.369358 −0.184679 0.982799i $$-0.559125\pi$$
−0.184679 + 0.982799i $$0.559125\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −30.0000 −1.10506
$$738$$ 0 0
$$739$$ −50.0000 −1.83928 −0.919640 0.392763i $$-0.871519\pi$$
−0.919640 + 0.392763i $$0.871519\pi$$
$$740$$ 0 0
$$741$$ 16.0000 0.587775
$$742$$ 0 0
$$743$$ 42.0000 1.54083 0.770415 0.637542i $$-0.220049\pi$$
0.770415 + 0.637542i $$0.220049\pi$$
$$744$$ 0 0
$$745$$ −54.0000 −1.97841
$$746$$ 0 0
$$747$$ −9.00000 −0.329293
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −7.00000 −0.255434 −0.127717 0.991811i $$-0.540765\pi$$
−0.127717 + 0.991811i $$0.540765\pi$$
$$752$$ 0 0
$$753$$ 27.0000 0.983935
$$754$$ 0 0
$$755$$ −3.00000 −0.109181
$$756$$ 0 0
$$757$$ −38.0000 −1.38113 −0.690567 0.723269i $$-0.742639\pi$$
−0.690567 + 0.723269i $$0.742639\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −12.0000 −0.435000 −0.217500 0.976060i $$-0.569790\pi$$
−0.217500 + 0.976060i $$0.569790\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −12.0000 −0.433295
$$768$$ 0 0
$$769$$ 19.0000 0.685158 0.342579 0.939489i $$-0.388700\pi$$
0.342579 + 0.939489i $$0.388700\pi$$
$$770$$ 0 0
$$771$$ −6.00000 −0.216085
$$772$$ 0 0
$$773$$ 6.00000 0.215805 0.107903 0.994161i $$-0.465587\pi$$
0.107903 + 0.994161i $$0.465587\pi$$
$$774$$ 0 0
$$775$$ 4.00000 0.143684
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 18.0000 0.644091
$$782$$ 0 0
$$783$$ −9.00000 −0.321634
$$784$$ 0 0
$$785$$ −12.0000 −0.428298
$$786$$ 0 0
$$787$$ 50.0000 1.78231 0.891154 0.453701i $$-0.149897\pi$$
0.891154 + 0.453701i $$0.149897\pi$$
$$788$$ 0 0
$$789$$ −6.00000 −0.213606
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 40.0000 1.42044
$$794$$ 0 0
$$795$$ 9.00000 0.319197
$$796$$ 0 0
$$797$$ −33.0000 −1.16892 −0.584460 0.811423i $$-0.698694\pi$$
−0.584460 + 0.811423i $$0.698694\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ −6.00000 −0.212000
$$802$$ 0 0
$$803$$ 6.00000 0.211735
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 21.0000 0.739235
$$808$$ 0 0
$$809$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$810$$ 0 0
$$811$$ 2.00000 0.0702295 0.0351147 0.999383i $$-0.488820\pi$$
0.0351147 + 0.999383i $$0.488820\pi$$
$$812$$ 0 0
$$813$$ −11.0000 −0.385787
$$814$$ 0 0
$$815$$ 48.0000 1.68137
$$816$$ 0 0
$$817$$ −40.0000 −1.39942
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 3.00000 0.104701 0.0523504 0.998629i $$-0.483329\pi$$
0.0523504 + 0.998629i $$0.483329\pi$$
$$822$$ 0 0
$$823$$ −40.0000 −1.39431 −0.697156 0.716919i $$-0.745552\pi$$
−0.697156 + 0.716919i $$0.745552\pi$$
$$824$$ 0 0
$$825$$ −12.0000 −0.417786
$$826$$ 0 0
$$827$$ −15.0000 −0.521601 −0.260801 0.965393i $$-0.583986\pi$$
−0.260801 + 0.965393i $$0.583986\pi$$
$$828$$ 0 0
$$829$$ −4.00000 −0.138926 −0.0694629 0.997585i $$-0.522129\pi$$
−0.0694629 + 0.997585i $$0.522129\pi$$
$$830$$ 0 0
$$831$$ −8.00000 −0.277517
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −18.0000 −0.622916
$$836$$ 0 0
$$837$$ 1.00000 0.0345651
$$838$$ 0 0
$$839$$ 24.0000 0.828572 0.414286 0.910147i $$-0.364031\pi$$
0.414286 + 0.910147i $$0.364031\pi$$
$$840$$ 0 0
$$841$$ 52.0000 1.79310
$$842$$ 0 0
$$843$$ 6.00000 0.206651
$$844$$ 0 0
$$845$$ 9.00000 0.309609
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 14.0000 0.480479
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ −10.0000 −0.342393 −0.171197 0.985237i $$-0.554763\pi$$
−0.171197 + 0.985237i $$0.554763\pi$$
$$854$$ 0 0
$$855$$ −12.0000 −0.410391
$$856$$ 0 0
$$857$$ 42.0000 1.43469 0.717346 0.696717i $$-0.245357\pi$$
0.717346 + 0.696717i $$0.245357\pi$$
$$858$$ 0 0
$$859$$ 50.0000 1.70598 0.852989 0.521929i $$-0.174787\pi$$
0.852989 + 0.521929i $$0.174787\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 6.00000 0.204242 0.102121 0.994772i $$-0.467437\pi$$
0.102121 + 0.994772i $$0.467437\pi$$
$$864$$ 0 0
$$865$$ 54.0000 1.83606
$$866$$ 0 0
$$867$$ −17.0000 −0.577350
$$868$$ 0 0
$$869$$ 3.00000 0.101768
$$870$$ 0 0
$$871$$ −40.0000 −1.35535
$$872$$ 0 0
$$873$$ 1.00000 0.0338449
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −32.0000 −1.08056 −0.540282 0.841484i $$-0.681682\pi$$
−0.540282 + 0.841484i $$0.681682\pi$$
$$878$$ 0 0
$$879$$ 33.0000 1.11306
$$880$$ 0 0
$$881$$ 6.00000 0.202145 0.101073 0.994879i $$-0.467773\pi$$
0.101073 + 0.994879i $$0.467773\pi$$
$$882$$ 0 0
$$883$$ −32.0000 −1.07689 −0.538443 0.842662i $$-0.680987\pi$$
−0.538443 + 0.842662i $$0.680987\pi$$
$$884$$ 0 0
$$885$$ 9.00000 0.302532
$$886$$ 0 0
$$887$$ 24.0000 0.805841 0.402921 0.915235i $$-0.367995\pi$$
0.402921 + 0.915235i $$0.367995\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −3.00000 −0.100504
$$892$$ 0 0
$$893$$ −24.0000 −0.803129
$$894$$ 0 0
$$895$$ −36.0000 −1.20335
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −9.00000 −0.300167
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 24.0000 0.797787
$$906$$ 0 0
$$907$$ −8.00000 −0.265636 −0.132818 0.991140i $$-0.542403\pi$$
−0.132818 + 0.991140i $$0.542403\pi$$
$$908$$ 0 0
$$909$$ −18.0000 −0.597022
$$910$$ 0 0
$$911$$ 6.00000 0.198789 0.0993944 0.995048i $$-0.468309\pi$$
0.0993944 + 0.995048i $$0.468309\pi$$
$$912$$ 0 0
$$913$$ 27.0000 0.893570
$$914$$ 0 0
$$915$$ −30.0000 −0.991769
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 8.00000 0.263896 0.131948 0.991257i $$-0.457877\pi$$
0.131948 + 0.991257i $$0.457877\pi$$
$$920$$ 0 0
$$921$$ 8.00000 0.263609
$$922$$ 0 0
$$923$$ 24.0000 0.789970
$$924$$ 0 0
$$925$$ −32.0000 −1.05215
$$926$$ 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
$$933$$ −24.0000 −0.785725
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −35.0000 −1.14340 −0.571700 0.820463i $$-0.693716\pi$$
−0.571700 + 0.820463i $$0.693716\pi$$
$$938$$ 0 0
$$939$$ 31.0000 1.01165
$$940$$ 0 0
$$941$$ −9.00000 −0.293392 −0.146696 0.989182i $$-0.546864\pi$$
−0.146696 + 0.989182i $$0.546864\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$948$$ 0 0
$$949$$ 8.00000 0.259691
$$950$$ 0 0
$$951$$ −9.00000 −0.291845
$$952$$ 0 0
$$953$$ −6.00000 −0.194359 −0.0971795 0.995267i $$-0.530982\pi$$
−0.0971795 + 0.995267i $$0.530982\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 27.0000 0.872786
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −30.0000 −0.967742
$$962$$ 0 0
$$963$$ 3.00000 0.0966736
$$964$$ 0 0
$$965$$ −57.0000 −1.83489
$$966$$ 0 0
$$967$$ −1.00000 −0.0321578 −0.0160789 0.999871i $$-0.505118\pi$$
−0.0160789 + 0.999871i $$0.505118\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −39.0000 −1.25157 −0.625785 0.779996i $$-0.715221\pi$$
−0.625785 + 0.779996i $$0.715221\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ −16.0000 −0.512410
$$976$$ 0 0
$$977$$ 42.0000 1.34370 0.671850 0.740688i $$-0.265500\pi$$
0.671850 + 0.740688i $$0.265500\pi$$
$$978$$ 0 0
$$979$$ 18.0000 0.575282
$$980$$ 0 0
$$981$$ −14.0000 −0.446986
$$982$$ 0 0
$$983$$ −36.0000 −1.14822 −0.574111 0.818778i $$-0.694652\pi$$
−0.574111 + 0.818778i $$0.694652\pi$$
$$984$$ 0 0
$$985$$ −18.0000 −0.573528
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ −13.0000 −0.412959 −0.206479 0.978451i $$-0.566201\pi$$
−0.206479 + 0.978451i $$0.566201\pi$$
$$992$$ 0 0
$$993$$ −20.0000 −0.634681
$$994$$ 0 0
$$995$$ −60.0000 −1.90213
$$996$$ 0 0
$$997$$ 14.0000 0.443384 0.221692 0.975117i $$-0.428842\pi$$
0.221692 + 0.975117i $$0.428842\pi$$
$$998$$ 0 0
$$999$$ −8.00000 −0.253109
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.db.1.1 1
4.3 odd 2 9408.2.a.bm.1.1 1
7.3 odd 6 1344.2.q.v.961.1 2
7.5 odd 6 1344.2.q.v.193.1 2
7.6 odd 2 9408.2.a.d.1.1 1
8.3 odd 2 2352.2.a.n.1.1 1
8.5 even 2 294.2.a.a.1.1 1
24.5 odd 2 882.2.a.k.1.1 1
24.11 even 2 7056.2.a.bz.1.1 1
28.3 even 6 1344.2.q.j.961.1 2
28.19 even 6 1344.2.q.j.193.1 2
28.27 even 2 9408.2.a.bu.1.1 1
40.29 even 2 7350.2.a.cw.1.1 1
56.3 even 6 336.2.q.d.289.1 2
56.5 odd 6 42.2.e.b.25.1 2
56.11 odd 6 2352.2.q.m.961.1 2
56.13 odd 2 294.2.a.d.1.1 1
56.19 even 6 336.2.q.d.193.1 2
56.27 even 2 2352.2.a.m.1.1 1
56.37 even 6 294.2.e.f.67.1 2
56.45 odd 6 42.2.e.b.37.1 yes 2
56.51 odd 6 2352.2.q.m.1537.1 2
56.53 even 6 294.2.e.f.79.1 2
168.5 even 6 126.2.g.b.109.1 2
168.53 odd 6 882.2.g.b.667.1 2
168.59 odd 6 1008.2.s.n.289.1 2
168.83 odd 2 7056.2.a.g.1.1 1
168.101 even 6 126.2.g.b.37.1 2
168.125 even 2 882.2.a.g.1.1 1
168.131 odd 6 1008.2.s.n.865.1 2
168.149 odd 6 882.2.g.b.361.1 2
280.69 odd 2 7350.2.a.ce.1.1 1
280.117 even 12 1050.2.o.b.949.1 4
280.157 even 12 1050.2.o.b.499.2 4
280.173 even 12 1050.2.o.b.949.2 4
280.213 even 12 1050.2.o.b.499.1 4
280.229 odd 6 1050.2.i.e.151.1 2
280.269 odd 6 1050.2.i.e.751.1 2
504.5 even 6 1134.2.e.p.865.1 2
504.61 odd 6 1134.2.h.p.109.1 2
504.101 even 6 1134.2.e.p.919.1 2
504.157 odd 6 1134.2.h.p.541.1 2
504.173 even 6 1134.2.h.a.109.1 2
504.229 odd 6 1134.2.e.a.865.1 2
504.437 even 6 1134.2.h.a.541.1 2
504.493 odd 6 1134.2.e.a.919.1 2

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