# Properties

 Label 1850.2.a.l.1.1 Level $1850$ Weight $2$ Character 1850.1 Self dual yes Analytic conductor $14.772$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1850 = 2 \cdot 5^{2} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1850.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$14.7723243739$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 370) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 1850.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{7} +1.00000 q^{8} -3.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{7} +1.00000 q^{8} -3.00000 q^{9} -3.00000 q^{11} -4.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} -3.00000 q^{17} -3.00000 q^{18} -3.00000 q^{22} -8.00000 q^{23} -4.00000 q^{26} +1.00000 q^{28} -3.00000 q^{29} -7.00000 q^{31} +1.00000 q^{32} -3.00000 q^{34} -3.00000 q^{36} -1.00000 q^{37} +11.0000 q^{41} +11.0000 q^{43} -3.00000 q^{44} -8.00000 q^{46} +4.00000 q^{47} -6.00000 q^{49} -4.00000 q^{52} +11.0000 q^{53} +1.00000 q^{56} -3.00000 q^{58} -12.0000 q^{59} -15.0000 q^{61} -7.00000 q^{62} -3.00000 q^{63} +1.00000 q^{64} -4.00000 q^{67} -3.00000 q^{68} +6.00000 q^{71} -3.00000 q^{72} +2.00000 q^{73} -1.00000 q^{74} -3.00000 q^{77} -8.00000 q^{79} +9.00000 q^{81} +11.0000 q^{82} +12.0000 q^{83} +11.0000 q^{86} -3.00000 q^{88} -4.00000 q^{91} -8.00000 q^{92} +4.00000 q^{94} -1.00000 q^{97} -6.00000 q^{98} +9.00000 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 1.00000 0.377964 0.188982 0.981981i $$-0.439481\pi$$
0.188982 + 0.981981i $$0.439481\pi$$
$$8$$ 1.00000 0.353553
$$9$$ −3.00000 −1.00000
$$10$$ 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 $$0.687167\pi$$
$$14$$ 1.00000 0.267261
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −3.00000 −0.727607 −0.363803 0.931476i $$-0.618522\pi$$
−0.363803 + 0.931476i $$0.618522\pi$$
$$18$$ −3.00000 −0.707107
$$19$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −3.00000 −0.639602
$$23$$ −8.00000 −1.66812 −0.834058 0.551677i $$-0.813988\pi$$
−0.834058 + 0.551677i $$0.813988\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −4.00000 −0.784465
$$27$$ 0 0
$$28$$ 1.00000 0.188982
$$29$$ −3.00000 −0.557086 −0.278543 0.960424i $$-0.589851\pi$$
−0.278543 + 0.960424i $$0.589851\pi$$
$$30$$ 0 0
$$31$$ −7.00000 −1.25724 −0.628619 0.777714i $$-0.716379\pi$$
−0.628619 + 0.777714i $$0.716379\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 0 0
$$34$$ −3.00000 −0.514496
$$35$$ 0 0
$$36$$ −3.00000 −0.500000
$$37$$ −1.00000 −0.164399
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 11.0000 1.71791 0.858956 0.512050i $$-0.171114\pi$$
0.858956 + 0.512050i $$0.171114\pi$$
$$42$$ 0 0
$$43$$ 11.0000 1.67748 0.838742 0.544529i $$-0.183292\pi$$
0.838742 + 0.544529i $$0.183292\pi$$
$$44$$ −3.00000 −0.452267
$$45$$ 0 0
$$46$$ −8.00000 −1.17954
$$47$$ 4.00000 0.583460 0.291730 0.956501i $$-0.405769\pi$$
0.291730 + 0.956501i $$0.405769\pi$$
$$48$$ 0 0
$$49$$ −6.00000 −0.857143
$$50$$ 0 0
$$51$$ 0 0
$$52$$ −4.00000 −0.554700
$$53$$ 11.0000 1.51097 0.755483 0.655168i $$-0.227402\pi$$
0.755483 + 0.655168i $$0.227402\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 1.00000 0.133631
$$57$$ 0 0
$$58$$ −3.00000 −0.393919
$$59$$ −12.0000 −1.56227 −0.781133 0.624364i $$-0.785358\pi$$
−0.781133 + 0.624364i $$0.785358\pi$$
$$60$$ 0 0
$$61$$ −15.0000 −1.92055 −0.960277 0.279050i $$-0.909981\pi$$
−0.960277 + 0.279050i $$0.909981\pi$$
$$62$$ −7.00000 −0.889001
$$63$$ −3.00000 −0.377964
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −4.00000 −0.488678 −0.244339 0.969690i $$-0.578571\pi$$
−0.244339 + 0.969690i $$0.578571\pi$$
$$68$$ −3.00000 −0.363803
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 6.00000 0.712069 0.356034 0.934473i $$-0.384129\pi$$
0.356034 + 0.934473i $$0.384129\pi$$
$$72$$ −3.00000 −0.353553
$$73$$ 2.00000 0.234082 0.117041 0.993127i $$-0.462659\pi$$
0.117041 + 0.993127i $$0.462659\pi$$
$$74$$ −1.00000 −0.116248
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −3.00000 −0.341882
$$78$$ 0 0
$$79$$ −8.00000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ 0 0
$$81$$ 9.00000 1.00000
$$82$$ 11.0000 1.21475
$$83$$ 12.0000 1.31717 0.658586 0.752506i $$-0.271155\pi$$
0.658586 + 0.752506i $$0.271155\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 11.0000 1.18616
$$87$$ 0 0
$$88$$ −3.00000 −0.319801
$$89$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$90$$ 0 0
$$91$$ −4.00000 −0.419314
$$92$$ −8.00000 −0.834058
$$93$$ 0 0
$$94$$ 4.00000 0.412568
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −1.00000 −0.101535 −0.0507673 0.998711i $$-0.516167\pi$$
−0.0507673 + 0.998711i $$0.516167\pi$$
$$98$$ −6.00000 −0.606092
$$99$$ 9.00000 0.904534
$$100$$ 0 0
$$101$$ 14.0000 1.39305 0.696526 0.717532i $$-0.254728\pi$$
0.696526 + 0.717532i $$0.254728\pi$$
$$102$$ 0 0
$$103$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$104$$ −4.00000 −0.392232
$$105$$ 0 0
$$106$$ 11.0000 1.06841
$$107$$ −10.0000 −0.966736 −0.483368 0.875417i $$-0.660587\pi$$
−0.483368 + 0.875417i $$0.660587\pi$$
$$108$$ 0 0
$$109$$ −11.0000 −1.05361 −0.526804 0.849987i $$-0.676610\pi$$
−0.526804 + 0.849987i $$0.676610\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 1.00000 0.0944911
$$113$$ −1.00000 −0.0940721 −0.0470360 0.998893i $$-0.514978\pi$$
−0.0470360 + 0.998893i $$0.514978\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −3.00000 −0.278543
$$117$$ 12.0000 1.10940
$$118$$ −12.0000 −1.10469
$$119$$ −3.00000 −0.275010
$$120$$ 0 0
$$121$$ −2.00000 −0.181818
$$122$$ −15.0000 −1.35804
$$123$$ 0 0
$$124$$ −7.00000 −0.628619
$$125$$ 0 0
$$126$$ −3.00000 −0.267261
$$127$$ 16.0000 1.41977 0.709885 0.704317i $$-0.248747\pi$$
0.709885 + 0.704317i $$0.248747\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −10.0000 −0.873704 −0.436852 0.899533i $$-0.643907\pi$$
−0.436852 + 0.899533i $$0.643907\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ −4.00000 −0.345547
$$135$$ 0 0
$$136$$ −3.00000 −0.257248
$$137$$ 6.00000 0.512615 0.256307 0.966595i $$-0.417494\pi$$
0.256307 + 0.966595i $$0.417494\pi$$
$$138$$ 0 0
$$139$$ 1.00000 0.0848189 0.0424094 0.999100i $$-0.486497\pi$$
0.0424094 + 0.999100i $$0.486497\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 6.00000 0.503509
$$143$$ 12.0000 1.00349
$$144$$ −3.00000 −0.250000
$$145$$ 0 0
$$146$$ 2.00000 0.165521
$$147$$ 0 0
$$148$$ −1.00000 −0.0821995
$$149$$ −4.00000 −0.327693 −0.163846 0.986486i $$-0.552390\pi$$
−0.163846 + 0.986486i $$0.552390\pi$$
$$150$$ 0 0
$$151$$ −14.0000 −1.13930 −0.569652 0.821886i $$-0.692922\pi$$
−0.569652 + 0.821886i $$0.692922\pi$$
$$152$$ 0 0
$$153$$ 9.00000 0.727607
$$154$$ −3.00000 −0.241747
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −5.00000 −0.399043 −0.199522 0.979893i $$-0.563939\pi$$
−0.199522 + 0.979893i $$0.563939\pi$$
$$158$$ −8.00000 −0.636446
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −8.00000 −0.630488
$$162$$ 9.00000 0.707107
$$163$$ 3.00000 0.234978 0.117489 0.993074i $$-0.462515\pi$$
0.117489 + 0.993074i $$0.462515\pi$$
$$164$$ 11.0000 0.858956
$$165$$ 0 0
$$166$$ 12.0000 0.931381
$$167$$ −14.0000 −1.08335 −0.541676 0.840587i $$-0.682210\pi$$
−0.541676 + 0.840587i $$0.682210\pi$$
$$168$$ 0 0
$$169$$ 3.00000 0.230769
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 11.0000 0.838742
$$173$$ −13.0000 −0.988372 −0.494186 0.869356i $$-0.664534\pi$$
−0.494186 + 0.869356i $$0.664534\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −3.00000 −0.226134
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 10.0000 0.747435 0.373718 0.927543i $$-0.378083\pi$$
0.373718 + 0.927543i $$0.378083\pi$$
$$180$$ 0 0
$$181$$ −16.0000 −1.18927 −0.594635 0.803996i $$-0.702704\pi$$
−0.594635 + 0.803996i $$0.702704\pi$$
$$182$$ −4.00000 −0.296500
$$183$$ 0 0
$$184$$ −8.00000 −0.589768
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 9.00000 0.658145
$$188$$ 4.00000 0.291730
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −17.0000 −1.23008 −0.615038 0.788497i $$-0.710860\pi$$
−0.615038 + 0.788497i $$0.710860\pi$$
$$192$$ 0 0
$$193$$ −14.0000 −1.00774 −0.503871 0.863779i $$-0.668091\pi$$
−0.503871 + 0.863779i $$0.668091\pi$$
$$194$$ −1.00000 −0.0717958
$$195$$ 0 0
$$196$$ −6.00000 −0.428571
$$197$$ 2.00000 0.142494 0.0712470 0.997459i $$-0.477302\pi$$
0.0712470 + 0.997459i $$0.477302\pi$$
$$198$$ 9.00000 0.639602
$$199$$ 16.0000 1.13421 0.567105 0.823646i $$-0.308063\pi$$
0.567105 + 0.823646i $$0.308063\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 14.0000 0.985037
$$203$$ −3.00000 −0.210559
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 24.0000 1.66812
$$208$$ −4.00000 −0.277350
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −1.00000 −0.0688428 −0.0344214 0.999407i $$-0.510959\pi$$
−0.0344214 + 0.999407i $$0.510959\pi$$
$$212$$ 11.0000 0.755483
$$213$$ 0 0
$$214$$ −10.0000 −0.683586
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −7.00000 −0.475191
$$218$$ −11.0000 −0.745014
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 12.0000 0.807207
$$222$$ 0 0
$$223$$ 11.0000 0.736614 0.368307 0.929704i $$-0.379937\pi$$
0.368307 + 0.929704i $$0.379937\pi$$
$$224$$ 1.00000 0.0668153
$$225$$ 0 0
$$226$$ −1.00000 −0.0665190
$$227$$ 17.0000 1.12833 0.564165 0.825662i $$-0.309198\pi$$
0.564165 + 0.825662i $$0.309198\pi$$
$$228$$ 0 0
$$229$$ 20.0000 1.32164 0.660819 0.750546i $$-0.270209\pi$$
0.660819 + 0.750546i $$0.270209\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −3.00000 −0.196960
$$233$$ 6.00000 0.393073 0.196537 0.980497i $$-0.437031\pi$$
0.196537 + 0.980497i $$0.437031\pi$$
$$234$$ 12.0000 0.784465
$$235$$ 0 0
$$236$$ −12.0000 −0.781133
$$237$$ 0 0
$$238$$ −3.00000 −0.194461
$$239$$ −17.0000 −1.09964 −0.549819 0.835284i $$-0.685303\pi$$
−0.549819 + 0.835284i $$0.685303\pi$$
$$240$$ 0 0
$$241$$ −14.0000 −0.901819 −0.450910 0.892570i $$-0.648900\pi$$
−0.450910 + 0.892570i $$0.648900\pi$$
$$242$$ −2.00000 −0.128565
$$243$$ 0 0
$$244$$ −15.0000 −0.960277
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0 0
$$248$$ −7.00000 −0.444500
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 20.0000 1.26239 0.631194 0.775625i $$-0.282565\pi$$
0.631194 + 0.775625i $$0.282565\pi$$
$$252$$ −3.00000 −0.188982
$$253$$ 24.0000 1.50887
$$254$$ 16.0000 1.00393
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 6.00000 0.374270 0.187135 0.982334i $$-0.440080\pi$$
0.187135 + 0.982334i $$0.440080\pi$$
$$258$$ 0 0
$$259$$ −1.00000 −0.0621370
$$260$$ 0 0
$$261$$ 9.00000 0.557086
$$262$$ −10.0000 −0.617802
$$263$$ 31.0000 1.91154 0.955771 0.294112i $$-0.0950239\pi$$
0.955771 + 0.294112i $$0.0950239\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ −4.00000 −0.244339
$$269$$ −16.0000 −0.975537 −0.487769 0.872973i $$-0.662189\pi$$
−0.487769 + 0.872973i $$0.662189\pi$$
$$270$$ 0 0
$$271$$ 22.0000 1.33640 0.668202 0.743980i $$-0.267064\pi$$
0.668202 + 0.743980i $$0.267064\pi$$
$$272$$ −3.00000 −0.181902
$$273$$ 0 0
$$274$$ 6.00000 0.362473
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$278$$ 1.00000 0.0599760
$$279$$ 21.0000 1.25724
$$280$$ 0 0
$$281$$ 6.00000 0.357930 0.178965 0.983855i $$-0.442725\pi$$
0.178965 + 0.983855i $$0.442725\pi$$
$$282$$ 0 0
$$283$$ −8.00000 −0.475551 −0.237775 0.971320i $$-0.576418\pi$$
−0.237775 + 0.971320i $$0.576418\pi$$
$$284$$ 6.00000 0.356034
$$285$$ 0 0
$$286$$ 12.0000 0.709575
$$287$$ 11.0000 0.649309
$$288$$ −3.00000 −0.176777
$$289$$ −8.00000 −0.470588
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 2.00000 0.117041
$$293$$ −15.0000 −0.876309 −0.438155 0.898900i $$-0.644368\pi$$
−0.438155 + 0.898900i $$0.644368\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −1.00000 −0.0581238
$$297$$ 0 0
$$298$$ −4.00000 −0.231714
$$299$$ 32.0000 1.85061
$$300$$ 0 0
$$301$$ 11.0000 0.634029
$$302$$ −14.0000 −0.805609
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 9.00000 0.514496
$$307$$ −8.00000 −0.456584 −0.228292 0.973593i $$-0.573314\pi$$
−0.228292 + 0.973593i $$0.573314\pi$$
$$308$$ −3.00000 −0.170941
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 3.00000 0.170114 0.0850572 0.996376i $$-0.472893\pi$$
0.0850572 + 0.996376i $$0.472893\pi$$
$$312$$ 0 0
$$313$$ −14.0000 −0.791327 −0.395663 0.918396i $$-0.629485\pi$$
−0.395663 + 0.918396i $$0.629485\pi$$
$$314$$ −5.00000 −0.282166
$$315$$ 0 0
$$316$$ −8.00000 −0.450035
$$317$$ −21.0000 −1.17948 −0.589739 0.807594i $$-0.700769\pi$$
−0.589739 + 0.807594i $$0.700769\pi$$
$$318$$ 0 0
$$319$$ 9.00000 0.503903
$$320$$ 0 0
$$321$$ 0 0
$$322$$ −8.00000 −0.445823
$$323$$ 0 0
$$324$$ 9.00000 0.500000
$$325$$ 0 0
$$326$$ 3.00000 0.166155
$$327$$ 0 0
$$328$$ 11.0000 0.607373
$$329$$ 4.00000 0.220527
$$330$$ 0 0
$$331$$ 2.00000 0.109930 0.0549650 0.998488i $$-0.482495\pi$$
0.0549650 + 0.998488i $$0.482495\pi$$
$$332$$ 12.0000 0.658586
$$333$$ 3.00000 0.164399
$$334$$ −14.0000 −0.766046
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −28.0000 −1.52526 −0.762629 0.646837i $$-0.776092\pi$$
−0.762629 + 0.646837i $$0.776092\pi$$
$$338$$ 3.00000 0.163178
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 21.0000 1.13721
$$342$$ 0 0
$$343$$ −13.0000 −0.701934
$$344$$ 11.0000 0.593080
$$345$$ 0 0
$$346$$ −13.0000 −0.698884
$$347$$ 12.0000 0.644194 0.322097 0.946707i $$-0.395612\pi$$
0.322097 + 0.946707i $$0.395612\pi$$
$$348$$ 0 0
$$349$$ −6.00000 −0.321173 −0.160586 0.987022i $$-0.551338\pi$$
−0.160586 + 0.987022i $$0.551338\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −3.00000 −0.159901
$$353$$ −3.00000 −0.159674 −0.0798369 0.996808i $$-0.525440\pi$$
−0.0798369 + 0.996808i $$0.525440\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 10.0000 0.528516
$$359$$ −14.0000 −0.738892 −0.369446 0.929252i $$-0.620452\pi$$
−0.369446 + 0.929252i $$0.620452\pi$$
$$360$$ 0 0
$$361$$ −19.0000 −1.00000
$$362$$ −16.0000 −0.840941
$$363$$ 0 0
$$364$$ −4.00000 −0.209657
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 13.0000 0.678594 0.339297 0.940679i $$-0.389811\pi$$
0.339297 + 0.940679i $$0.389811\pi$$
$$368$$ −8.00000 −0.417029
$$369$$ −33.0000 −1.71791
$$370$$ 0 0
$$371$$ 11.0000 0.571092
$$372$$ 0 0
$$373$$ 38.0000 1.96757 0.983783 0.179364i $$-0.0574041\pi$$
0.983783 + 0.179364i $$0.0574041\pi$$
$$374$$ 9.00000 0.465379
$$375$$ 0 0
$$376$$ 4.00000 0.206284
$$377$$ 12.0000 0.618031
$$378$$ 0 0
$$379$$ 12.0000 0.616399 0.308199 0.951322i $$-0.400274\pi$$
0.308199 + 0.951322i $$0.400274\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ −17.0000 −0.869796
$$383$$ −16.0000 −0.817562 −0.408781 0.912633i $$-0.634046\pi$$
−0.408781 + 0.912633i $$0.634046\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −14.0000 −0.712581
$$387$$ −33.0000 −1.67748
$$388$$ −1.00000 −0.0507673
$$389$$ 23.0000 1.16615 0.583073 0.812420i $$-0.301850\pi$$
0.583073 + 0.812420i $$0.301850\pi$$
$$390$$ 0 0
$$391$$ 24.0000 1.21373
$$392$$ −6.00000 −0.303046
$$393$$ 0 0
$$394$$ 2.00000 0.100759
$$395$$ 0 0
$$396$$ 9.00000 0.452267
$$397$$ 14.0000 0.702640 0.351320 0.936255i $$-0.385733\pi$$
0.351320 + 0.936255i $$0.385733\pi$$
$$398$$ 16.0000 0.802008
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 10.0000 0.499376 0.249688 0.968326i $$-0.419672\pi$$
0.249688 + 0.968326i $$0.419672\pi$$
$$402$$ 0 0
$$403$$ 28.0000 1.39478
$$404$$ 14.0000 0.696526
$$405$$ 0 0
$$406$$ −3.00000 −0.148888
$$407$$ 3.00000 0.148704
$$408$$ 0 0
$$409$$ −4.00000 −0.197787 −0.0988936 0.995098i $$-0.531530\pi$$
−0.0988936 + 0.995098i $$0.531530\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ −12.0000 −0.590481
$$414$$ 24.0000 1.17954
$$415$$ 0 0
$$416$$ −4.00000 −0.196116
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 20.0000 0.977064 0.488532 0.872546i $$-0.337533\pi$$
0.488532 + 0.872546i $$0.337533\pi$$
$$420$$ 0 0
$$421$$ −30.0000 −1.46211 −0.731055 0.682318i $$-0.760972\pi$$
−0.731055 + 0.682318i $$0.760972\pi$$
$$422$$ −1.00000 −0.0486792
$$423$$ −12.0000 −0.583460
$$424$$ 11.0000 0.534207
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −15.0000 −0.725901
$$428$$ −10.0000 −0.483368
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 23.0000 1.10787 0.553936 0.832560i $$-0.313125\pi$$
0.553936 + 0.832560i $$0.313125\pi$$
$$432$$ 0 0
$$433$$ −16.0000 −0.768911 −0.384455 0.923144i $$-0.625611\pi$$
−0.384455 + 0.923144i $$0.625611\pi$$
$$434$$ −7.00000 −0.336011
$$435$$ 0 0
$$436$$ −11.0000 −0.526804
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 7.00000 0.334092 0.167046 0.985949i $$-0.446577\pi$$
0.167046 + 0.985949i $$0.446577\pi$$
$$440$$ 0 0
$$441$$ 18.0000 0.857143
$$442$$ 12.0000 0.570782
$$443$$ 34.0000 1.61539 0.807694 0.589601i $$-0.200715\pi$$
0.807694 + 0.589601i $$0.200715\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 11.0000 0.520865
$$447$$ 0 0
$$448$$ 1.00000 0.0472456
$$449$$ 36.0000 1.69895 0.849473 0.527633i $$-0.176920\pi$$
0.849473 + 0.527633i $$0.176920\pi$$
$$450$$ 0 0
$$451$$ −33.0000 −1.55391
$$452$$ −1.00000 −0.0470360
$$453$$ 0 0
$$454$$ 17.0000 0.797850
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −37.0000 −1.73079 −0.865393 0.501093i $$-0.832931\pi$$
−0.865393 + 0.501093i $$0.832931\pi$$
$$458$$ 20.0000 0.934539
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 3.00000 0.139724 0.0698620 0.997557i $$-0.477744\pi$$
0.0698620 + 0.997557i $$0.477744\pi$$
$$462$$ 0 0
$$463$$ −4.00000 −0.185896 −0.0929479 0.995671i $$-0.529629\pi$$
−0.0929479 + 0.995671i $$0.529629\pi$$
$$464$$ −3.00000 −0.139272
$$465$$ 0 0
$$466$$ 6.00000 0.277945
$$467$$ 3.00000 0.138823 0.0694117 0.997588i $$-0.477888\pi$$
0.0694117 + 0.997588i $$0.477888\pi$$
$$468$$ 12.0000 0.554700
$$469$$ −4.00000 −0.184703
$$470$$ 0 0
$$471$$ 0 0
$$472$$ −12.0000 −0.552345
$$473$$ −33.0000 −1.51734
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −3.00000 −0.137505
$$477$$ −33.0000 −1.51097
$$478$$ −17.0000 −0.777562
$$479$$ −36.0000 −1.64488 −0.822441 0.568850i $$-0.807388\pi$$
−0.822441 + 0.568850i $$0.807388\pi$$
$$480$$ 0 0
$$481$$ 4.00000 0.182384
$$482$$ −14.0000 −0.637683
$$483$$ 0 0
$$484$$ −2.00000 −0.0909091
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 36.0000 1.63132 0.815658 0.578535i $$-0.196375\pi$$
0.815658 + 0.578535i $$0.196375\pi$$
$$488$$ −15.0000 −0.679018
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −36.0000 −1.62466 −0.812329 0.583200i $$-0.801800\pi$$
−0.812329 + 0.583200i $$0.801800\pi$$
$$492$$ 0 0
$$493$$ 9.00000 0.405340
$$494$$ 0 0
$$495$$ 0 0
$$496$$ −7.00000 −0.314309
$$497$$ 6.00000 0.269137
$$498$$ 0 0
$$499$$ −18.0000 −0.805791 −0.402895 0.915246i $$-0.631996\pi$$
−0.402895 + 0.915246i $$0.631996\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 20.0000 0.892644
$$503$$ −26.0000 −1.15928 −0.579641 0.814872i $$-0.696807\pi$$
−0.579641 + 0.814872i $$0.696807\pi$$
$$504$$ −3.00000 −0.133631
$$505$$ 0 0
$$506$$ 24.0000 1.06693
$$507$$ 0 0
$$508$$ 16.0000 0.709885
$$509$$ 30.0000 1.32973 0.664863 0.746965i $$-0.268490\pi$$
0.664863 + 0.746965i $$0.268490\pi$$
$$510$$ 0 0
$$511$$ 2.00000 0.0884748
$$512$$ 1.00000 0.0441942
$$513$$ 0 0
$$514$$ 6.00000 0.264649
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −12.0000 −0.527759
$$518$$ −1.00000 −0.0439375
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −7.00000 −0.306676 −0.153338 0.988174i $$-0.549002\pi$$
−0.153338 + 0.988174i $$0.549002\pi$$
$$522$$ 9.00000 0.393919
$$523$$ 4.00000 0.174908 0.0874539 0.996169i $$-0.472127\pi$$
0.0874539 + 0.996169i $$0.472127\pi$$
$$524$$ −10.0000 −0.436852
$$525$$ 0 0
$$526$$ 31.0000 1.35166
$$527$$ 21.0000 0.914774
$$528$$ 0 0
$$529$$ 41.0000 1.78261
$$530$$ 0 0
$$531$$ 36.0000 1.56227
$$532$$ 0 0
$$533$$ −44.0000 −1.90585
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −4.00000 −0.172774
$$537$$ 0 0
$$538$$ −16.0000 −0.689809
$$539$$ 18.0000 0.775315
$$540$$ 0 0
$$541$$ 26.0000 1.11783 0.558914 0.829226i $$-0.311218\pi$$
0.558914 + 0.829226i $$0.311218\pi$$
$$542$$ 22.0000 0.944981
$$543$$ 0 0
$$544$$ −3.00000 −0.128624
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 35.0000 1.49649 0.748246 0.663421i $$-0.230896\pi$$
0.748246 + 0.663421i $$0.230896\pi$$
$$548$$ 6.00000 0.256307
$$549$$ 45.0000 1.92055
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ −8.00000 −0.340195
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 1.00000 0.0424094
$$557$$ −6.00000 −0.254228 −0.127114 0.991888i $$-0.540571\pi$$
−0.127114 + 0.991888i $$0.540571\pi$$
$$558$$ 21.0000 0.889001
$$559$$ −44.0000 −1.86100
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 6.00000 0.253095
$$563$$ −19.0000 −0.800755 −0.400377 0.916350i $$-0.631121\pi$$
−0.400377 + 0.916350i $$0.631121\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −8.00000 −0.336265
$$567$$ 9.00000 0.377964
$$568$$ 6.00000 0.251754
$$569$$ 14.0000 0.586911 0.293455 0.955973i $$-0.405195\pi$$
0.293455 + 0.955973i $$0.405195\pi$$
$$570$$ 0 0
$$571$$ 7.00000 0.292941 0.146470 0.989215i $$-0.453209\pi$$
0.146470 + 0.989215i $$0.453209\pi$$
$$572$$ 12.0000 0.501745
$$573$$ 0 0
$$574$$ 11.0000 0.459131
$$575$$ 0 0
$$576$$ −3.00000 −0.125000
$$577$$ −2.00000 −0.0832611 −0.0416305 0.999133i $$-0.513255\pi$$
−0.0416305 + 0.999133i $$0.513255\pi$$
$$578$$ −8.00000 −0.332756
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 12.0000 0.497844
$$582$$ 0 0
$$583$$ −33.0000 −1.36672
$$584$$ 2.00000 0.0827606
$$585$$ 0 0
$$586$$ −15.0000 −0.619644
$$587$$ 15.0000 0.619116 0.309558 0.950881i $$-0.399819\pi$$
0.309558 + 0.950881i $$0.399819\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 0 0
$$592$$ −1.00000 −0.0410997
$$593$$ −36.0000 −1.47834 −0.739171 0.673517i $$-0.764783\pi$$
−0.739171 + 0.673517i $$0.764783\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −4.00000 −0.163846
$$597$$ 0 0
$$598$$ 32.0000 1.30858
$$599$$ −32.0000 −1.30748 −0.653742 0.756717i $$-0.726802\pi$$
−0.653742 + 0.756717i $$0.726802\pi$$
$$600$$ 0 0
$$601$$ −47.0000 −1.91717 −0.958585 0.284807i $$-0.908071\pi$$
−0.958585 + 0.284807i $$0.908071\pi$$
$$602$$ 11.0000 0.448327
$$603$$ 12.0000 0.488678
$$604$$ −14.0000 −0.569652
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −42.0000 −1.70473 −0.852364 0.522949i $$-0.824832\pi$$
−0.852364 + 0.522949i $$0.824832\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −16.0000 −0.647291
$$612$$ 9.00000 0.363803
$$613$$ −15.0000 −0.605844 −0.302922 0.953015i $$-0.597962\pi$$
−0.302922 + 0.953015i $$0.597962\pi$$
$$614$$ −8.00000 −0.322854
$$615$$ 0 0
$$616$$ −3.00000 −0.120873
$$617$$ −42.0000 −1.69086 −0.845428 0.534089i $$-0.820655\pi$$
−0.845428 + 0.534089i $$0.820655\pi$$
$$618$$ 0 0
$$619$$ −35.0000 −1.40677 −0.703384 0.710810i $$-0.748329\pi$$
−0.703384 + 0.710810i $$0.748329\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 3.00000 0.120289
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −14.0000 −0.559553
$$627$$ 0 0
$$628$$ −5.00000 −0.199522
$$629$$ 3.00000 0.119618
$$630$$ 0 0
$$631$$ 37.0000 1.47295 0.736473 0.676467i $$-0.236490\pi$$
0.736473 + 0.676467i $$0.236490\pi$$
$$632$$ −8.00000 −0.318223
$$633$$ 0 0
$$634$$ −21.0000 −0.834017
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 24.0000 0.950915
$$638$$ 9.00000 0.356313
$$639$$ −18.0000 −0.712069
$$640$$ 0 0
$$641$$ −25.0000 −0.987441 −0.493720 0.869621i $$-0.664363\pi$$
−0.493720 + 0.869621i $$0.664363\pi$$
$$642$$ 0 0
$$643$$ 7.00000 0.276053 0.138027 0.990429i $$-0.455924\pi$$
0.138027 + 0.990429i $$0.455924\pi$$
$$644$$ −8.00000 −0.315244
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 2.00000 0.0786281 0.0393141 0.999227i $$-0.487483\pi$$
0.0393141 + 0.999227i $$0.487483\pi$$
$$648$$ 9.00000 0.353553
$$649$$ 36.0000 1.41312
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 3.00000 0.117489
$$653$$ 38.0000 1.48705 0.743527 0.668705i $$-0.233151\pi$$
0.743527 + 0.668705i $$0.233151\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 11.0000 0.429478
$$657$$ −6.00000 −0.234082
$$658$$ 4.00000 0.155936
$$659$$ 48.0000 1.86981 0.934907 0.354892i $$-0.115482\pi$$
0.934907 + 0.354892i $$0.115482\pi$$
$$660$$ 0 0
$$661$$ −25.0000 −0.972387 −0.486194 0.873851i $$-0.661615\pi$$
−0.486194 + 0.873851i $$0.661615\pi$$
$$662$$ 2.00000 0.0777322
$$663$$ 0 0
$$664$$ 12.0000 0.465690
$$665$$ 0 0
$$666$$ 3.00000 0.116248
$$667$$ 24.0000 0.929284
$$668$$ −14.0000 −0.541676
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 45.0000 1.73721
$$672$$ 0 0
$$673$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$674$$ −28.0000 −1.07852
$$675$$ 0 0
$$676$$ 3.00000 0.115385
$$677$$ −42.0000 −1.61419 −0.807096 0.590421i $$-0.798962\pi$$
−0.807096 + 0.590421i $$0.798962\pi$$
$$678$$ 0 0
$$679$$ −1.00000 −0.0383765
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 21.0000 0.804132
$$683$$ 13.0000 0.497431 0.248716 0.968577i $$-0.419992\pi$$
0.248716 + 0.968577i $$0.419992\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −13.0000 −0.496342
$$687$$ 0 0
$$688$$ 11.0000 0.419371
$$689$$ −44.0000 −1.67627
$$690$$ 0 0
$$691$$ −41.0000 −1.55971 −0.779857 0.625958i $$-0.784708\pi$$
−0.779857 + 0.625958i $$0.784708\pi$$
$$692$$ −13.0000 −0.494186
$$693$$ 9.00000 0.341882
$$694$$ 12.0000 0.455514
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −33.0000 −1.24996
$$698$$ −6.00000 −0.227103
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −22.0000 −0.830929 −0.415464 0.909610i $$-0.636381\pi$$
−0.415464 + 0.909610i $$0.636381\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ −3.00000 −0.113067
$$705$$ 0 0
$$706$$ −3.00000 −0.112906
$$707$$ 14.0000 0.526524
$$708$$ 0 0
$$709$$ 29.0000 1.08912 0.544559 0.838723i $$-0.316697\pi$$
0.544559 + 0.838723i $$0.316697\pi$$
$$710$$ 0 0
$$711$$ 24.0000 0.900070
$$712$$ 0 0
$$713$$ 56.0000 2.09722
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 10.0000 0.373718
$$717$$ 0 0
$$718$$ −14.0000 −0.522475
$$719$$ 20.0000 0.745874 0.372937 0.927857i $$-0.378351\pi$$
0.372937 + 0.927857i $$0.378351\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ −19.0000 −0.707107
$$723$$ 0 0
$$724$$ −16.0000 −0.594635
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −14.0000 −0.519231 −0.259616 0.965712i $$-0.583596\pi$$
−0.259616 + 0.965712i $$0.583596\pi$$
$$728$$ −4.00000 −0.148250
$$729$$ −27.0000 −1.00000
$$730$$ 0 0
$$731$$ −33.0000 −1.22055
$$732$$ 0 0
$$733$$ 1.00000 0.0369358 0.0184679 0.999829i $$-0.494121\pi$$
0.0184679 + 0.999829i $$0.494121\pi$$
$$734$$ 13.0000 0.479839
$$735$$ 0 0
$$736$$ −8.00000 −0.294884
$$737$$ 12.0000 0.442026
$$738$$ −33.0000 −1.21475
$$739$$ −15.0000 −0.551784 −0.275892 0.961189i $$-0.588973\pi$$
−0.275892 + 0.961189i $$0.588973\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 11.0000 0.403823
$$743$$ −13.0000 −0.476924 −0.238462 0.971152i $$-0.576643\pi$$
−0.238462 + 0.971152i $$0.576643\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 38.0000 1.39128
$$747$$ −36.0000 −1.31717
$$748$$ 9.00000 0.329073
$$749$$ −10.0000 −0.365392
$$750$$ 0 0
$$751$$ −32.0000 −1.16770 −0.583848 0.811863i $$-0.698454\pi$$
−0.583848 + 0.811863i $$0.698454\pi$$
$$752$$ 4.00000 0.145865
$$753$$ 0 0
$$754$$ 12.0000 0.437014
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −28.0000 −1.01768 −0.508839 0.860862i $$-0.669925\pi$$
−0.508839 + 0.860862i $$0.669925\pi$$
$$758$$ 12.0000 0.435860
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 7.00000 0.253750 0.126875 0.991919i $$-0.459505\pi$$
0.126875 + 0.991919i $$0.459505\pi$$
$$762$$ 0 0
$$763$$ −11.0000 −0.398227
$$764$$ −17.0000 −0.615038
$$765$$ 0 0
$$766$$ −16.0000 −0.578103
$$767$$ 48.0000 1.73318
$$768$$ 0 0
$$769$$ −44.0000 −1.58668 −0.793340 0.608778i $$-0.791660\pi$$
−0.793340 + 0.608778i $$0.791660\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −14.0000 −0.503871
$$773$$ −51.0000 −1.83434 −0.917171 0.398493i $$-0.869533\pi$$
−0.917171 + 0.398493i $$0.869533\pi$$
$$774$$ −33.0000 −1.18616
$$775$$ 0 0
$$776$$ −1.00000 −0.0358979
$$777$$ 0 0
$$778$$ 23.0000 0.824590
$$779$$ 0 0
$$780$$ 0 0
$$781$$ −18.0000 −0.644091
$$782$$ 24.0000 0.858238
$$783$$ 0 0
$$784$$ −6.00000 −0.214286
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 22.0000 0.784215 0.392108 0.919919i $$-0.371746\pi$$
0.392108 + 0.919919i $$0.371746\pi$$
$$788$$ 2.00000 0.0712470
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −1.00000 −0.0355559
$$792$$ 9.00000 0.319801
$$793$$ 60.0000 2.13066
$$794$$ 14.0000 0.496841
$$795$$ 0 0
$$796$$ 16.0000 0.567105
$$797$$ 12.0000 0.425062 0.212531 0.977154i $$-0.431829\pi$$
0.212531 + 0.977154i $$0.431829\pi$$
$$798$$ 0 0
$$799$$ −12.0000 −0.424529
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 10.0000 0.353112
$$803$$ −6.00000 −0.211735
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 28.0000 0.986258
$$807$$ 0 0
$$808$$ 14.0000 0.492518
$$809$$ −12.0000 −0.421898 −0.210949 0.977497i $$-0.567655\pi$$
−0.210949 + 0.977497i $$0.567655\pi$$
$$810$$ 0 0
$$811$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$812$$ −3.00000 −0.105279
$$813$$ 0 0
$$814$$ 3.00000 0.105150
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 0 0
$$818$$ −4.00000 −0.139857
$$819$$ 12.0000 0.419314
$$820$$ 0 0
$$821$$ −14.0000 −0.488603 −0.244302 0.969699i $$-0.578559\pi$$
−0.244302 + 0.969699i $$0.578559\pi$$
$$822$$ 0 0
$$823$$ 40.0000 1.39431 0.697156 0.716919i $$-0.254448\pi$$
0.697156 + 0.716919i $$0.254448\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ −12.0000 −0.417533
$$827$$ −3.00000 −0.104320 −0.0521601 0.998639i $$-0.516611\pi$$
−0.0521601 + 0.998639i $$0.516611\pi$$
$$828$$ 24.0000 0.834058
$$829$$ −23.0000 −0.798823 −0.399412 0.916772i $$-0.630786\pi$$
−0.399412 + 0.916772i $$0.630786\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ −4.00000 −0.138675
$$833$$ 18.0000 0.623663
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 20.0000 0.690889
$$839$$ −12.0000 −0.414286 −0.207143 0.978311i $$-0.566417\pi$$
−0.207143 + 0.978311i $$0.566417\pi$$
$$840$$ 0 0
$$841$$ −20.0000 −0.689655
$$842$$ −30.0000 −1.03387
$$843$$ 0 0
$$844$$ −1.00000 −0.0344214
$$845$$ 0 0
$$846$$ −12.0000 −0.412568
$$847$$ −2.00000 −0.0687208
$$848$$ 11.0000 0.377742
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 8.00000 0.274236
$$852$$ 0 0
$$853$$ −30.0000 −1.02718 −0.513590 0.858036i $$-0.671685\pi$$
−0.513590 + 0.858036i $$0.671685\pi$$
$$854$$ −15.0000 −0.513289
$$855$$ 0 0
$$856$$ −10.0000 −0.341793
$$857$$ −17.0000 −0.580709 −0.290354 0.956919i $$-0.593773\pi$$
−0.290354 + 0.956919i $$0.593773\pi$$
$$858$$ 0 0
$$859$$ −20.0000 −0.682391 −0.341196 0.939992i $$-0.610832\pi$$
−0.341196 + 0.939992i $$0.610832\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 23.0000 0.783383
$$863$$ −39.0000 −1.32758 −0.663788 0.747921i $$-0.731052\pi$$
−0.663788 + 0.747921i $$0.731052\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −16.0000 −0.543702
$$867$$ 0 0
$$868$$ −7.00000 −0.237595
$$869$$ 24.0000 0.814144
$$870$$ 0 0
$$871$$ 16.0000 0.542139
$$872$$ −11.0000 −0.372507
$$873$$ 3.00000 0.101535
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −1.00000 −0.0337676 −0.0168838 0.999857i $$-0.505375\pi$$
−0.0168838 + 0.999857i $$0.505375\pi$$
$$878$$ 7.00000 0.236239
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 39.0000 1.31394 0.656972 0.753915i $$-0.271837\pi$$
0.656972 + 0.753915i $$0.271837\pi$$
$$882$$ 18.0000 0.606092
$$883$$ 29.0000 0.975928 0.487964 0.872864i $$-0.337740\pi$$
0.487964 + 0.872864i $$0.337740\pi$$
$$884$$ 12.0000 0.403604
$$885$$ 0 0
$$886$$ 34.0000 1.14225
$$887$$ 7.00000 0.235037 0.117518 0.993071i $$-0.462506\pi$$
0.117518 + 0.993071i $$0.462506\pi$$
$$888$$ 0 0
$$889$$ 16.0000 0.536623
$$890$$ 0 0
$$891$$ −27.0000 −0.904534
$$892$$ 11.0000 0.368307
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 1.00000 0.0334077
$$897$$ 0 0
$$898$$ 36.0000 1.20134
$$899$$ 21.0000 0.700389
$$900$$ 0 0
$$901$$ −33.0000 −1.09939
$$902$$ −33.0000 −1.09878
$$903$$ 0 0
$$904$$ −1.00000 −0.0332595
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 28.0000 0.929725 0.464862 0.885383i $$-0.346104\pi$$
0.464862 + 0.885383i $$0.346104\pi$$
$$908$$ 17.0000 0.564165
$$909$$ −42.0000 −1.39305
$$910$$ 0 0
$$911$$ 36.0000 1.19273 0.596367 0.802712i $$-0.296610\pi$$
0.596367 + 0.802712i $$0.296610\pi$$
$$912$$ 0 0
$$913$$ −36.0000 −1.19143
$$914$$ −37.0000 −1.22385
$$915$$ 0 0
$$916$$ 20.0000 0.660819
$$917$$ −10.0000 −0.330229
$$918$$ 0 0
$$919$$ −16.0000 −0.527791 −0.263896 0.964551i $$-0.585007\pi$$
−0.263896 + 0.964551i $$0.585007\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 3.00000 0.0987997
$$923$$ −24.0000 −0.789970
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −4.00000 −0.131448
$$927$$ 0 0
$$928$$ −3.00000 −0.0984798
$$929$$ −15.0000 −0.492134 −0.246067 0.969253i $$-0.579138\pi$$
−0.246067 + 0.969253i $$0.579138\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 6.00000 0.196537
$$933$$ 0 0
$$934$$ 3.00000 0.0981630
$$935$$ 0 0
$$936$$ 12.0000 0.392232
$$937$$ 28.0000 0.914720 0.457360 0.889282i $$-0.348795\pi$$
0.457360 + 0.889282i $$0.348795\pi$$
$$938$$ −4.00000 −0.130605
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 26.0000 0.847576 0.423788 0.905761i $$-0.360700\pi$$
0.423788 + 0.905761i $$0.360700\pi$$
$$942$$ 0 0
$$943$$ −88.0000 −2.86567
$$944$$ −12.0000 −0.390567
$$945$$ 0 0
$$946$$ −33.0000 −1.07292
$$947$$ 37.0000 1.20234 0.601169 0.799122i $$-0.294702\pi$$
0.601169 + 0.799122i $$0.294702\pi$$
$$948$$ 0 0
$$949$$ −8.00000 −0.259691
$$950$$ 0 0
$$951$$ 0 0
$$952$$ −3.00000 −0.0972306
$$953$$ 6.00000 0.194359 0.0971795 0.995267i $$-0.469018\pi$$
0.0971795 + 0.995267i $$0.469018\pi$$
$$954$$ −33.0000 −1.06841
$$955$$ 0 0
$$956$$ −17.0000 −0.549819
$$957$$ 0 0
$$958$$ −36.0000 −1.16311
$$959$$ 6.00000 0.193750
$$960$$ 0 0
$$961$$ 18.0000 0.580645
$$962$$ 4.00000 0.128965
$$963$$ 30.0000 0.966736
$$964$$ −14.0000 −0.450910
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −34.0000 −1.09337 −0.546683 0.837340i $$-0.684110\pi$$
−0.546683 + 0.837340i $$0.684110\pi$$
$$968$$ −2.00000 −0.0642824
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −1.00000 −0.0320915 −0.0160458 0.999871i $$-0.505108\pi$$
−0.0160458 + 0.999871i $$0.505108\pi$$
$$972$$ 0 0
$$973$$ 1.00000 0.0320585
$$974$$ 36.0000 1.15351
$$975$$ 0 0
$$976$$ −15.0000 −0.480138
$$977$$ 33.0000 1.05576 0.527882 0.849318i $$-0.322986\pi$$
0.527882 + 0.849318i $$0.322986\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 33.0000 1.05361
$$982$$ −36.0000 −1.14881
$$983$$ −11.0000 −0.350846 −0.175423 0.984493i $$-0.556129\pi$$
−0.175423 + 0.984493i $$0.556129\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 9.00000 0.286618
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −88.0000 −2.79824
$$990$$ 0 0
$$991$$ −45.0000 −1.42947 −0.714736 0.699394i $$-0.753453\pi$$
−0.714736 + 0.699394i $$0.753453\pi$$
$$992$$ −7.00000 −0.222250
$$993$$ 0 0
$$994$$ 6.00000 0.190308
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −44.0000 −1.39349 −0.696747 0.717317i $$-0.745370\pi$$
−0.696747 + 0.717317i $$0.745370\pi$$
$$998$$ −18.0000 −0.569780
$$999$$ 0 0
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 1850.2.a.l.1.1 1
5.2 odd 4 370.2.b.b.149.2 yes 2
5.3 odd 4 370.2.b.b.149.1 2
5.4 even 2 1850.2.a.d.1.1 1
15.2 even 4 3330.2.d.c.1999.1 2
15.8 even 4 3330.2.d.c.1999.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.b.b.149.1 2 5.3 odd 4
370.2.b.b.149.2 yes 2 5.2 odd 4
1850.2.a.d.1.1 1 5.4 even 2
1850.2.a.l.1.1 1 1.1 even 1 trivial
3330.2.d.c.1999.1 2 15.2 even 4
3330.2.d.c.1999.2 2 15.8 even 4