# Properties

 Label 2475.2.a.p.1.1 Level $2475$ Weight $2$ Character 2475.1 Self dual yes Analytic conductor $19.763$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2475 = 3^{2} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2475.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$19.7629745003$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$2.56155$$ of defining polynomial Character $$\chi$$ $$=$$ 2475.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-2.56155 q^{2} +4.56155 q^{4} -1.00000 q^{7} -6.56155 q^{8} +O(q^{10})$$ $$q-2.56155 q^{2} +4.56155 q^{4} -1.00000 q^{7} -6.56155 q^{8} -1.00000 q^{11} +0.438447 q^{13} +2.56155 q^{14} +7.68466 q^{16} -1.43845 q^{17} +3.00000 q^{19} +2.56155 q^{22} -6.56155 q^{23} -1.12311 q^{26} -4.56155 q^{28} +5.12311 q^{29} +4.68466 q^{31} -6.56155 q^{32} +3.68466 q^{34} +10.8078 q^{37} -7.68466 q^{38} -7.68466 q^{41} -4.68466 q^{43} -4.56155 q^{44} +16.8078 q^{46} -5.43845 q^{47} -6.00000 q^{49} +2.00000 q^{52} +1.12311 q^{53} +6.56155 q^{56} -13.1231 q^{58} +10.5616 q^{59} -7.56155 q^{61} -12.0000 q^{62} +1.43845 q^{64} +6.68466 q^{67} -6.56155 q^{68} +8.80776 q^{71} -16.2462 q^{73} -27.6847 q^{74} +13.6847 q^{76} +1.00000 q^{77} -15.6847 q^{79} +19.6847 q^{82} +12.0000 q^{86} +6.56155 q^{88} -17.1231 q^{89} -0.438447 q^{91} -29.9309 q^{92} +13.9309 q^{94} -2.12311 q^{97} +15.3693 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} + 5 q^{4} - 2 q^{7} - 9 q^{8}+O(q^{10})$$ 2 * q - q^2 + 5 * q^4 - 2 * q^7 - 9 * q^8 $$2 q - q^{2} + 5 q^{4} - 2 q^{7} - 9 q^{8} - 2 q^{11} + 5 q^{13} + q^{14} + 3 q^{16} - 7 q^{17} + 6 q^{19} + q^{22} - 9 q^{23} + 6 q^{26} - 5 q^{28} + 2 q^{29} - 3 q^{31} - 9 q^{32} - 5 q^{34} + q^{37} - 3 q^{38} - 3 q^{41} + 3 q^{43} - 5 q^{44} + 13 q^{46} - 15 q^{47} - 12 q^{49} + 4 q^{52} - 6 q^{53} + 9 q^{56} - 18 q^{58} + 17 q^{59} - 11 q^{61} - 24 q^{62} + 7 q^{64} + q^{67} - 9 q^{68} - 3 q^{71} - 16 q^{73} - 43 q^{74} + 15 q^{76} + 2 q^{77} - 19 q^{79} + 27 q^{82} + 24 q^{86} + 9 q^{88} - 26 q^{89} - 5 q^{91} - 31 q^{92} - q^{94} + 4 q^{97} + 6 q^{98}+O(q^{100})$$ 2 * q - q^2 + 5 * q^4 - 2 * q^7 - 9 * q^8 - 2 * q^11 + 5 * q^13 + q^14 + 3 * q^16 - 7 * q^17 + 6 * q^19 + q^22 - 9 * q^23 + 6 * q^26 - 5 * q^28 + 2 * q^29 - 3 * q^31 - 9 * q^32 - 5 * q^34 + q^37 - 3 * q^38 - 3 * q^41 + 3 * q^43 - 5 * q^44 + 13 * q^46 - 15 * q^47 - 12 * q^49 + 4 * q^52 - 6 * q^53 + 9 * q^56 - 18 * q^58 + 17 * q^59 - 11 * q^61 - 24 * q^62 + 7 * q^64 + q^67 - 9 * q^68 - 3 * q^71 - 16 * q^73 - 43 * q^74 + 15 * q^76 + 2 * q^77 - 19 * q^79 + 27 * q^82 + 24 * q^86 + 9 * q^88 - 26 * q^89 - 5 * q^91 - 31 * q^92 - q^94 + 4 * q^97 + 6 * q^98

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.56155 −1.81129 −0.905646 0.424035i $$-0.860613\pi$$
−0.905646 + 0.424035i $$0.860613\pi$$
$$3$$ 0 0
$$4$$ 4.56155 2.28078
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −1.00000 −0.377964 −0.188982 0.981981i $$-0.560519\pi$$
−0.188982 + 0.981981i $$0.560519\pi$$
$$8$$ −6.56155 −2.31986
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −1.00000 −0.301511
$$12$$ 0 0
$$13$$ 0.438447 0.121603 0.0608017 0.998150i $$-0.480634\pi$$
0.0608017 + 0.998150i $$0.480634\pi$$
$$14$$ 2.56155 0.684604
$$15$$ 0 0
$$16$$ 7.68466 1.92116
$$17$$ −1.43845 −0.348875 −0.174437 0.984668i $$-0.555811\pi$$
−0.174437 + 0.984668i $$0.555811\pi$$
$$18$$ 0 0
$$19$$ 3.00000 0.688247 0.344124 0.938924i $$-0.388176\pi$$
0.344124 + 0.938924i $$0.388176\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 2.56155 0.546125
$$23$$ −6.56155 −1.36818 −0.684089 0.729398i $$-0.739800\pi$$
−0.684089 + 0.729398i $$0.739800\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −1.12311 −0.220259
$$27$$ 0 0
$$28$$ −4.56155 −0.862052
$$29$$ 5.12311 0.951337 0.475668 0.879625i $$-0.342206\pi$$
0.475668 + 0.879625i $$0.342206\pi$$
$$30$$ 0 0
$$31$$ 4.68466 0.841389 0.420695 0.907202i $$-0.361786\pi$$
0.420695 + 0.907202i $$0.361786\pi$$
$$32$$ −6.56155 −1.15993
$$33$$ 0 0
$$34$$ 3.68466 0.631914
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 10.8078 1.77679 0.888393 0.459084i $$-0.151822\pi$$
0.888393 + 0.459084i $$0.151822\pi$$
$$38$$ −7.68466 −1.24662
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −7.68466 −1.20014 −0.600071 0.799947i $$-0.704861\pi$$
−0.600071 + 0.799947i $$0.704861\pi$$
$$42$$ 0 0
$$43$$ −4.68466 −0.714404 −0.357202 0.934027i $$-0.616269\pi$$
−0.357202 + 0.934027i $$0.616269\pi$$
$$44$$ −4.56155 −0.687680
$$45$$ 0 0
$$46$$ 16.8078 2.47817
$$47$$ −5.43845 −0.793279 −0.396640 0.917974i $$-0.629824\pi$$
−0.396640 + 0.917974i $$0.629824\pi$$
$$48$$ 0 0
$$49$$ −6.00000 −0.857143
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 2.00000 0.277350
$$53$$ 1.12311 0.154270 0.0771352 0.997021i $$-0.475423\pi$$
0.0771352 + 0.997021i $$0.475423\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 6.56155 0.876824
$$57$$ 0 0
$$58$$ −13.1231 −1.72315
$$59$$ 10.5616 1.37500 0.687499 0.726186i $$-0.258709\pi$$
0.687499 + 0.726186i $$0.258709\pi$$
$$60$$ 0 0
$$61$$ −7.56155 −0.968158 −0.484079 0.875024i $$-0.660845\pi$$
−0.484079 + 0.875024i $$0.660845\pi$$
$$62$$ −12.0000 −1.52400
$$63$$ 0 0
$$64$$ 1.43845 0.179806
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 6.68466 0.816661 0.408331 0.912834i $$-0.366111\pi$$
0.408331 + 0.912834i $$0.366111\pi$$
$$68$$ −6.56155 −0.795705
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 8.80776 1.04529 0.522645 0.852551i $$-0.324945\pi$$
0.522645 + 0.852551i $$0.324945\pi$$
$$72$$ 0 0
$$73$$ −16.2462 −1.90148 −0.950738 0.309997i $$-0.899672\pi$$
−0.950738 + 0.309997i $$0.899672\pi$$
$$74$$ −27.6847 −3.21828
$$75$$ 0 0
$$76$$ 13.6847 1.56974
$$77$$ 1.00000 0.113961
$$78$$ 0 0
$$79$$ −15.6847 −1.76466 −0.882331 0.470629i $$-0.844027\pi$$
−0.882331 + 0.470629i $$0.844027\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 19.6847 2.17381
$$83$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 12.0000 1.29399
$$87$$ 0 0
$$88$$ 6.56155 0.699464
$$89$$ −17.1231 −1.81505 −0.907523 0.420003i $$-0.862029\pi$$
−0.907523 + 0.420003i $$0.862029\pi$$
$$90$$ 0 0
$$91$$ −0.438447 −0.0459618
$$92$$ −29.9309 −3.12051
$$93$$ 0 0
$$94$$ 13.9309 1.43686
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −2.12311 −0.215569 −0.107784 0.994174i $$-0.534376\pi$$
−0.107784 + 0.994174i $$0.534376\pi$$
$$98$$ 15.3693 1.55254
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 1.43845 0.143131 0.0715654 0.997436i $$-0.477201\pi$$
0.0715654 + 0.997436i $$0.477201\pi$$
$$102$$ 0 0
$$103$$ 9.12311 0.898926 0.449463 0.893299i $$-0.351615\pi$$
0.449463 + 0.893299i $$0.351615\pi$$
$$104$$ −2.87689 −0.282103
$$105$$ 0 0
$$106$$ −2.87689 −0.279429
$$107$$ 9.12311 0.881964 0.440982 0.897516i $$-0.354630\pi$$
0.440982 + 0.897516i $$0.354630\pi$$
$$108$$ 0 0
$$109$$ −7.80776 −0.747848 −0.373924 0.927459i $$-0.621988\pi$$
−0.373924 + 0.927459i $$0.621988\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −7.68466 −0.726132
$$113$$ 9.12311 0.858230 0.429115 0.903250i $$-0.358826\pi$$
0.429115 + 0.903250i $$0.358826\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 23.3693 2.16979
$$117$$ 0 0
$$118$$ −27.0540 −2.49052
$$119$$ 1.43845 0.131862
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 19.3693 1.75362
$$123$$ 0 0
$$124$$ 21.3693 1.91902
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 4.80776 0.426620 0.213310 0.976985i $$-0.431576\pi$$
0.213310 + 0.976985i $$0.431576\pi$$
$$128$$ 9.43845 0.834249
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −16.4924 −1.44095 −0.720475 0.693481i $$-0.756076\pi$$
−0.720475 + 0.693481i $$0.756076\pi$$
$$132$$ 0 0
$$133$$ −3.00000 −0.260133
$$134$$ −17.1231 −1.47921
$$135$$ 0 0
$$136$$ 9.43845 0.809340
$$137$$ 3.36932 0.287860 0.143930 0.989588i $$-0.454026\pi$$
0.143930 + 0.989588i $$0.454026\pi$$
$$138$$ 0 0
$$139$$ 8.00000 0.678551 0.339276 0.940687i $$-0.389818\pi$$
0.339276 + 0.940687i $$0.389818\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −22.5616 −1.89332
$$143$$ −0.438447 −0.0366648
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 41.6155 3.44413
$$147$$ 0 0
$$148$$ 49.3002 4.05245
$$149$$ 8.80776 0.721560 0.360780 0.932651i $$-0.382510\pi$$
0.360780 + 0.932651i $$0.382510\pi$$
$$150$$ 0 0
$$151$$ −16.6847 −1.35778 −0.678889 0.734241i $$-0.737538\pi$$
−0.678889 + 0.734241i $$0.737538\pi$$
$$152$$ −19.6847 −1.59664
$$153$$ 0 0
$$154$$ −2.56155 −0.206416
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −2.68466 −0.214259 −0.107130 0.994245i $$-0.534166\pi$$
−0.107130 + 0.994245i $$0.534166\pi$$
$$158$$ 40.1771 3.19632
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 6.56155 0.517123
$$162$$ 0 0
$$163$$ 5.31534 0.416330 0.208165 0.978094i $$-0.433251\pi$$
0.208165 + 0.978094i $$0.433251\pi$$
$$164$$ −35.0540 −2.73726
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 15.3693 1.18931 0.594657 0.803980i $$-0.297288\pi$$
0.594657 + 0.803980i $$0.297288\pi$$
$$168$$ 0 0
$$169$$ −12.8078 −0.985213
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −21.3693 −1.62940
$$173$$ −13.4384 −1.02171 −0.510853 0.859668i $$-0.670670\pi$$
−0.510853 + 0.859668i $$0.670670\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −7.68466 −0.579253
$$177$$ 0 0
$$178$$ 43.8617 3.28758
$$179$$ 14.5616 1.08838 0.544191 0.838961i $$-0.316837\pi$$
0.544191 + 0.838961i $$0.316837\pi$$
$$180$$ 0 0
$$181$$ 11.8769 0.882803 0.441401 0.897310i $$-0.354482\pi$$
0.441401 + 0.897310i $$0.354482\pi$$
$$182$$ 1.12311 0.0832501
$$183$$ 0 0
$$184$$ 43.0540 3.17398
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 1.43845 0.105190
$$188$$ −24.8078 −1.80929
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −1.93087 −0.139713 −0.0698564 0.997557i $$-0.522254\pi$$
−0.0698564 + 0.997557i $$0.522254\pi$$
$$192$$ 0 0
$$193$$ −17.5616 −1.26411 −0.632054 0.774924i $$-0.717788\pi$$
−0.632054 + 0.774924i $$0.717788\pi$$
$$194$$ 5.43845 0.390458
$$195$$ 0 0
$$196$$ −27.3693 −1.95495
$$197$$ −17.9309 −1.27752 −0.638761 0.769405i $$-0.720553\pi$$
−0.638761 + 0.769405i $$0.720553\pi$$
$$198$$ 0 0
$$199$$ −13.8078 −0.978806 −0.489403 0.872058i $$-0.662785\pi$$
−0.489403 + 0.872058i $$0.662785\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ −3.68466 −0.259252
$$203$$ −5.12311 −0.359572
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −23.3693 −1.62822
$$207$$ 0 0
$$208$$ 3.36932 0.233620
$$209$$ −3.00000 −0.207514
$$210$$ 0 0
$$211$$ 3.31534 0.228238 0.114119 0.993467i $$-0.463596\pi$$
0.114119 + 0.993467i $$0.463596\pi$$
$$212$$ 5.12311 0.351856
$$213$$ 0 0
$$214$$ −23.3693 −1.59749
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −4.68466 −0.318015
$$218$$ 20.0000 1.35457
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −0.630683 −0.0424243
$$222$$ 0 0
$$223$$ −21.8078 −1.46036 −0.730178 0.683257i $$-0.760563\pi$$
−0.730178 + 0.683257i $$0.760563\pi$$
$$224$$ 6.56155 0.438412
$$225$$ 0 0
$$226$$ −23.3693 −1.55450
$$227$$ −20.4924 −1.36013 −0.680065 0.733152i $$-0.738048\pi$$
−0.680065 + 0.733152i $$0.738048\pi$$
$$228$$ 0 0
$$229$$ −8.12311 −0.536790 −0.268395 0.963309i $$-0.586493\pi$$
−0.268395 + 0.963309i $$0.586493\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −33.6155 −2.20697
$$233$$ −24.8078 −1.62521 −0.812605 0.582814i $$-0.801951\pi$$
−0.812605 + 0.582814i $$0.801951\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 48.1771 3.13606
$$237$$ 0 0
$$238$$ −3.68466 −0.238841
$$239$$ −16.4924 −1.06681 −0.533403 0.845861i $$-0.679087\pi$$
−0.533403 + 0.845861i $$0.679087\pi$$
$$240$$ 0 0
$$241$$ −14.0540 −0.905296 −0.452648 0.891689i $$-0.649521\pi$$
−0.452648 + 0.891689i $$0.649521\pi$$
$$242$$ −2.56155 −0.164663
$$243$$ 0 0
$$244$$ −34.4924 −2.20815
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 1.31534 0.0836932
$$248$$ −30.7386 −1.95191
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −12.0000 −0.757433 −0.378717 0.925513i $$-0.623635\pi$$
−0.378717 + 0.925513i $$0.623635\pi$$
$$252$$ 0 0
$$253$$ 6.56155 0.412521
$$254$$ −12.3153 −0.772733
$$255$$ 0 0
$$256$$ −27.0540 −1.69087
$$257$$ −18.2462 −1.13817 −0.569084 0.822280i $$-0.692702\pi$$
−0.569084 + 0.822280i $$0.692702\pi$$
$$258$$ 0 0
$$259$$ −10.8078 −0.671562
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 42.2462 2.60998
$$263$$ 3.36932 0.207761 0.103880 0.994590i $$-0.466874\pi$$
0.103880 + 0.994590i $$0.466874\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 7.68466 0.471177
$$267$$ 0 0
$$268$$ 30.4924 1.86262
$$269$$ 28.4924 1.73721 0.868607 0.495502i $$-0.165016\pi$$
0.868607 + 0.495502i $$0.165016\pi$$
$$270$$ 0 0
$$271$$ −28.1771 −1.71164 −0.855818 0.517277i $$-0.826946\pi$$
−0.855818 + 0.517277i $$0.826946\pi$$
$$272$$ −11.0540 −0.670246
$$273$$ 0 0
$$274$$ −8.63068 −0.521399
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 23.1771 1.39258 0.696288 0.717763i $$-0.254834\pi$$
0.696288 + 0.717763i $$0.254834\pi$$
$$278$$ −20.4924 −1.22905
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −3.19224 −0.190433 −0.0952164 0.995457i $$-0.530354\pi$$
−0.0952164 + 0.995457i $$0.530354\pi$$
$$282$$ 0 0
$$283$$ −32.1231 −1.90952 −0.954760 0.297377i $$-0.903888\pi$$
−0.954760 + 0.297377i $$0.903888\pi$$
$$284$$ 40.1771 2.38407
$$285$$ 0 0
$$286$$ 1.12311 0.0664106
$$287$$ 7.68466 0.453611
$$288$$ 0 0
$$289$$ −14.9309 −0.878286
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −74.1080 −4.33684
$$293$$ −2.06913 −0.120880 −0.0604399 0.998172i $$-0.519250\pi$$
−0.0604399 + 0.998172i $$0.519250\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −70.9157 −4.12189
$$297$$ 0 0
$$298$$ −22.5616 −1.30696
$$299$$ −2.87689 −0.166375
$$300$$ 0 0
$$301$$ 4.68466 0.270019
$$302$$ 42.7386 2.45933
$$303$$ 0 0
$$304$$ 23.0540 1.32224
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 22.9309 1.30873 0.654367 0.756177i $$-0.272935\pi$$
0.654367 + 0.756177i $$0.272935\pi$$
$$308$$ 4.56155 0.259919
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −13.7538 −0.779906 −0.389953 0.920835i $$-0.627509\pi$$
−0.389953 + 0.920835i $$0.627509\pi$$
$$312$$ 0 0
$$313$$ 31.2462 1.76614 0.883070 0.469241i $$-0.155472\pi$$
0.883070 + 0.469241i $$0.155472\pi$$
$$314$$ 6.87689 0.388086
$$315$$ 0 0
$$316$$ −71.5464 −4.02480
$$317$$ −6.87689 −0.386245 −0.193122 0.981175i $$-0.561861\pi$$
−0.193122 + 0.981175i $$0.561861\pi$$
$$318$$ 0 0
$$319$$ −5.12311 −0.286839
$$320$$ 0 0
$$321$$ 0 0
$$322$$ −16.8078 −0.936660
$$323$$ −4.31534 −0.240112
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −13.6155 −0.754094
$$327$$ 0 0
$$328$$ 50.4233 2.78416
$$329$$ 5.43845 0.299831
$$330$$ 0 0
$$331$$ −10.2462 −0.563183 −0.281591 0.959534i $$-0.590862\pi$$
−0.281591 + 0.959534i $$0.590862\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ −39.3693 −2.15419
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 10.6847 0.582030 0.291015 0.956718i $$-0.406007\pi$$
0.291015 + 0.956718i $$0.406007\pi$$
$$338$$ 32.8078 1.78451
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −4.68466 −0.253688
$$342$$ 0 0
$$343$$ 13.0000 0.701934
$$344$$ 30.7386 1.65732
$$345$$ 0 0
$$346$$ 34.4233 1.85061
$$347$$ 27.3693 1.46926 0.734631 0.678467i $$-0.237355\pi$$
0.734631 + 0.678467i $$0.237355\pi$$
$$348$$ 0 0
$$349$$ 6.63068 0.354932 0.177466 0.984127i $$-0.443210\pi$$
0.177466 + 0.984127i $$0.443210\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 6.56155 0.349732
$$353$$ −27.8617 −1.48293 −0.741465 0.670991i $$-0.765869\pi$$
−0.741465 + 0.670991i $$0.765869\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −78.1080 −4.13971
$$357$$ 0 0
$$358$$ −37.3002 −1.97138
$$359$$ 26.2462 1.38522 0.692611 0.721311i $$-0.256460\pi$$
0.692611 + 0.721311i $$0.256460\pi$$
$$360$$ 0 0
$$361$$ −10.0000 −0.526316
$$362$$ −30.4233 −1.59901
$$363$$ 0 0
$$364$$ −2.00000 −0.104828
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −14.0540 −0.733612 −0.366806 0.930298i $$-0.619549\pi$$
−0.366806 + 0.930298i $$0.619549\pi$$
$$368$$ −50.4233 −2.62850
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −1.12311 −0.0583087
$$372$$ 0 0
$$373$$ 13.8078 0.714939 0.357469 0.933925i $$-0.383640\pi$$
0.357469 + 0.933925i $$0.383640\pi$$
$$374$$ −3.68466 −0.190529
$$375$$ 0 0
$$376$$ 35.6847 1.84030
$$377$$ 2.24621 0.115686
$$378$$ 0 0
$$379$$ −24.3002 −1.24822 −0.624108 0.781338i $$-0.714538\pi$$
−0.624108 + 0.781338i $$0.714538\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 4.94602 0.253061
$$383$$ −22.2462 −1.13673 −0.568364 0.822777i $$-0.692424\pi$$
−0.568364 + 0.822777i $$0.692424\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 44.9848 2.28967
$$387$$ 0 0
$$388$$ −9.68466 −0.491664
$$389$$ −23.8617 −1.20984 −0.604919 0.796287i $$-0.706795\pi$$
−0.604919 + 0.796287i $$0.706795\pi$$
$$390$$ 0 0
$$391$$ 9.43845 0.477323
$$392$$ 39.3693 1.98845
$$393$$ 0 0
$$394$$ 45.9309 2.31396
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −28.4384 −1.42728 −0.713642 0.700510i $$-0.752956\pi$$
−0.713642 + 0.700510i $$0.752956\pi$$
$$398$$ 35.3693 1.77290
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 5.61553 0.280426 0.140213 0.990121i $$-0.455221\pi$$
0.140213 + 0.990121i $$0.455221\pi$$
$$402$$ 0 0
$$403$$ 2.05398 0.102316
$$404$$ 6.56155 0.326449
$$405$$ 0 0
$$406$$ 13.1231 0.651289
$$407$$ −10.8078 −0.535721
$$408$$ 0 0
$$409$$ 8.05398 0.398243 0.199122 0.979975i $$-0.436191\pi$$
0.199122 + 0.979975i $$0.436191\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 41.6155 2.05025
$$413$$ −10.5616 −0.519700
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −2.87689 −0.141051
$$417$$ 0 0
$$418$$ 7.68466 0.375869
$$419$$ 25.4384 1.24275 0.621375 0.783514i $$-0.286574\pi$$
0.621375 + 0.783514i $$0.286574\pi$$
$$420$$ 0 0
$$421$$ −17.0540 −0.831160 −0.415580 0.909557i $$-0.636421\pi$$
−0.415580 + 0.909557i $$0.636421\pi$$
$$422$$ −8.49242 −0.413405
$$423$$ 0 0
$$424$$ −7.36932 −0.357886
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 7.56155 0.365929
$$428$$ 41.6155 2.01156
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −21.7538 −1.04784 −0.523922 0.851767i $$-0.675532\pi$$
−0.523922 + 0.851767i $$0.675532\pi$$
$$432$$ 0 0
$$433$$ −16.9309 −0.813646 −0.406823 0.913507i $$-0.633363\pi$$
−0.406823 + 0.913507i $$0.633363\pi$$
$$434$$ 12.0000 0.576018
$$435$$ 0 0
$$436$$ −35.6155 −1.70567
$$437$$ −19.6847 −0.941645
$$438$$ 0 0
$$439$$ 11.7386 0.560254 0.280127 0.959963i $$-0.409623\pi$$
0.280127 + 0.959963i $$0.409623\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 1.61553 0.0768428
$$443$$ −25.3002 −1.20205 −0.601024 0.799231i $$-0.705240\pi$$
−0.601024 + 0.799231i $$0.705240\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 55.8617 2.64513
$$447$$ 0 0
$$448$$ −1.43845 −0.0679602
$$449$$ −28.4924 −1.34464 −0.672320 0.740260i $$-0.734702\pi$$
−0.672320 + 0.740260i $$0.734702\pi$$
$$450$$ 0 0
$$451$$ 7.68466 0.361856
$$452$$ 41.6155 1.95743
$$453$$ 0 0
$$454$$ 52.4924 2.46359
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −13.3693 −0.625390 −0.312695 0.949854i $$-0.601232\pi$$
−0.312695 + 0.949854i $$0.601232\pi$$
$$458$$ 20.8078 0.972283
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −15.8617 −0.738755 −0.369377 0.929279i $$-0.620429\pi$$
−0.369377 + 0.929279i $$0.620429\pi$$
$$462$$ 0 0
$$463$$ 1.75379 0.0815055 0.0407527 0.999169i $$-0.487024\pi$$
0.0407527 + 0.999169i $$0.487024\pi$$
$$464$$ 39.3693 1.82767
$$465$$ 0 0
$$466$$ 63.5464 2.94373
$$467$$ −2.24621 −0.103942 −0.0519711 0.998649i $$-0.516550\pi$$
−0.0519711 + 0.998649i $$0.516550\pi$$
$$468$$ 0 0
$$469$$ −6.68466 −0.308669
$$470$$ 0 0
$$471$$ 0 0
$$472$$ −69.3002 −3.18980
$$473$$ 4.68466 0.215401
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 6.56155 0.300748
$$477$$ 0 0
$$478$$ 42.2462 1.93230
$$479$$ 3.36932 0.153948 0.0769740 0.997033i $$-0.475474\pi$$
0.0769740 + 0.997033i $$0.475474\pi$$
$$480$$ 0 0
$$481$$ 4.73863 0.216063
$$482$$ 36.0000 1.63976
$$483$$ 0 0
$$484$$ 4.56155 0.207343
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 12.0540 0.546218 0.273109 0.961983i $$-0.411948\pi$$
0.273109 + 0.961983i $$0.411948\pi$$
$$488$$ 49.6155 2.24599
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 16.4924 0.744293 0.372146 0.928174i $$-0.378622\pi$$
0.372146 + 0.928174i $$0.378622\pi$$
$$492$$ 0 0
$$493$$ −7.36932 −0.331897
$$494$$ −3.36932 −0.151593
$$495$$ 0 0
$$496$$ 36.0000 1.61645
$$497$$ −8.80776 −0.395082
$$498$$ 0 0
$$499$$ 3.80776 0.170459 0.0852295 0.996361i $$-0.472838\pi$$
0.0852295 + 0.996361i $$0.472838\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 30.7386 1.37193
$$503$$ 16.4924 0.735361 0.367680 0.929952i $$-0.380152\pi$$
0.367680 + 0.929952i $$0.380152\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −16.8078 −0.747196
$$507$$ 0 0
$$508$$ 21.9309 0.973025
$$509$$ −17.1231 −0.758968 −0.379484 0.925198i $$-0.623899\pi$$
−0.379484 + 0.925198i $$0.623899\pi$$
$$510$$ 0 0
$$511$$ 16.2462 0.718690
$$512$$ 50.4233 2.22842
$$513$$ 0 0
$$514$$ 46.7386 2.06155
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 5.43845 0.239183
$$518$$ 27.6847 1.21639
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −25.6155 −1.12224 −0.561118 0.827736i $$-0.689629\pi$$
−0.561118 + 0.827736i $$0.689629\pi$$
$$522$$ 0 0
$$523$$ −9.24621 −0.404309 −0.202154 0.979354i $$-0.564794\pi$$
−0.202154 + 0.979354i $$0.564794\pi$$
$$524$$ −75.2311 −3.28648
$$525$$ 0 0
$$526$$ −8.63068 −0.376316
$$527$$ −6.73863 −0.293539
$$528$$ 0 0
$$529$$ 20.0540 0.871912
$$530$$ 0 0
$$531$$ 0 0
$$532$$ −13.6847 −0.593305
$$533$$ −3.36932 −0.145941
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −43.8617 −1.89454
$$537$$ 0 0
$$538$$ −72.9848 −3.14660
$$539$$ 6.00000 0.258438
$$540$$ 0 0
$$541$$ −7.31534 −0.314511 −0.157256 0.987558i $$-0.550265\pi$$
−0.157256 + 0.987558i $$0.550265\pi$$
$$542$$ 72.1771 3.10027
$$543$$ 0 0
$$544$$ 9.43845 0.404670
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −3.68466 −0.157545 −0.0787723 0.996893i $$-0.525100\pi$$
−0.0787723 + 0.996893i $$0.525100\pi$$
$$548$$ 15.3693 0.656545
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 15.3693 0.654755
$$552$$ 0 0
$$553$$ 15.6847 0.666980
$$554$$ −59.3693 −2.52236
$$555$$ 0 0
$$556$$ 36.4924 1.54762
$$557$$ −1.12311 −0.0475875 −0.0237938 0.999717i $$-0.507575\pi$$
−0.0237938 + 0.999717i $$0.507575\pi$$
$$558$$ 0 0
$$559$$ −2.05398 −0.0868739
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 8.17708 0.344929
$$563$$ 10.8769 0.458406 0.229203 0.973379i $$-0.426388\pi$$
0.229203 + 0.973379i $$0.426388\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 82.2850 3.45870
$$567$$ 0 0
$$568$$ −57.7926 −2.42492
$$569$$ 37.9309 1.59014 0.795072 0.606515i $$-0.207433\pi$$
0.795072 + 0.606515i $$0.207433\pi$$
$$570$$ 0 0
$$571$$ 38.0540 1.59251 0.796255 0.604962i $$-0.206812\pi$$
0.796255 + 0.604962i $$0.206812\pi$$
$$572$$ −2.00000 −0.0836242
$$573$$ 0 0
$$574$$ −19.6847 −0.821622
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 28.3693 1.18103 0.590515 0.807027i $$-0.298925\pi$$
0.590515 + 0.807027i $$0.298925\pi$$
$$578$$ 38.2462 1.59083
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −1.12311 −0.0465143
$$584$$ 106.600 4.41115
$$585$$ 0 0
$$586$$ 5.30019 0.218949
$$587$$ 27.1922 1.12234 0.561172 0.827699i $$-0.310351\pi$$
0.561172 + 0.827699i $$0.310351\pi$$
$$588$$ 0 0
$$589$$ 14.0540 0.579084
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 83.0540 3.41350
$$593$$ 34.1080 1.40065 0.700323 0.713826i $$-0.253039\pi$$
0.700323 + 0.713826i $$0.253039\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 40.1771 1.64572
$$597$$ 0 0
$$598$$ 7.36932 0.301354
$$599$$ 24.1771 0.987849 0.493924 0.869505i $$-0.335562\pi$$
0.493924 + 0.869505i $$0.335562\pi$$
$$600$$ 0 0
$$601$$ 37.4233 1.52653 0.763264 0.646087i $$-0.223596\pi$$
0.763264 + 0.646087i $$0.223596\pi$$
$$602$$ −12.0000 −0.489083
$$603$$ 0 0
$$604$$ −76.1080 −3.09679
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −6.24621 −0.253526 −0.126763 0.991933i $$-0.540459\pi$$
−0.126763 + 0.991933i $$0.540459\pi$$
$$608$$ −19.6847 −0.798318
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −2.38447 −0.0964654
$$612$$ 0 0
$$613$$ −28.1080 −1.13527 −0.567635 0.823281i $$-0.692141\pi$$
−0.567635 + 0.823281i $$0.692141\pi$$
$$614$$ −58.7386 −2.37050
$$615$$ 0 0
$$616$$ −6.56155 −0.264372
$$617$$ −21.7538 −0.875775 −0.437887 0.899030i $$-0.644273\pi$$
−0.437887 + 0.899030i $$0.644273\pi$$
$$618$$ 0 0
$$619$$ 7.17708 0.288471 0.144236 0.989543i $$-0.453928\pi$$
0.144236 + 0.989543i $$0.453928\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 35.2311 1.41264
$$623$$ 17.1231 0.686023
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −80.0388 −3.19899
$$627$$ 0 0
$$628$$ −12.2462 −0.488677
$$629$$ −15.5464 −0.619875
$$630$$ 0 0
$$631$$ −22.6847 −0.903062 −0.451531 0.892255i $$-0.649122\pi$$
−0.451531 + 0.892255i $$0.649122\pi$$
$$632$$ 102.916 4.09377
$$633$$ 0 0
$$634$$ 17.6155 0.699602
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −2.63068 −0.104231
$$638$$ 13.1231 0.519549
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 42.7386 1.68807 0.844037 0.536285i $$-0.180173\pi$$
0.844037 + 0.536285i $$0.180173\pi$$
$$642$$ 0 0
$$643$$ −15.3693 −0.606107 −0.303053 0.952974i $$-0.598006\pi$$
−0.303053 + 0.952974i $$0.598006\pi$$
$$644$$ 29.9309 1.17944
$$645$$ 0 0
$$646$$ 11.0540 0.434913
$$647$$ 40.8078 1.60432 0.802159 0.597110i $$-0.203684\pi$$
0.802159 + 0.597110i $$0.203684\pi$$
$$648$$ 0 0
$$649$$ −10.5616 −0.414577
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 24.2462 0.949555
$$653$$ −7.50758 −0.293794 −0.146897 0.989152i $$-0.546929\pi$$
−0.146897 + 0.989152i $$0.546929\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −59.0540 −2.30567
$$657$$ 0 0
$$658$$ −13.9309 −0.543082
$$659$$ 8.63068 0.336204 0.168102 0.985770i $$-0.446236\pi$$
0.168102 + 0.985770i $$0.446236\pi$$
$$660$$ 0 0
$$661$$ 6.80776 0.264791 0.132396 0.991197i $$-0.457733\pi$$
0.132396 + 0.991197i $$0.457733\pi$$
$$662$$ 26.2462 1.02009
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −33.6155 −1.30160
$$668$$ 70.1080 2.71256
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 7.56155 0.291911
$$672$$ 0 0
$$673$$ −10.6307 −0.409783 −0.204891 0.978785i $$-0.565684\pi$$
−0.204891 + 0.978785i $$0.565684\pi$$
$$674$$ −27.3693 −1.05423
$$675$$ 0 0
$$676$$ −58.4233 −2.24705
$$677$$ −29.1231 −1.11929 −0.559646 0.828732i $$-0.689063\pi$$
−0.559646 + 0.828732i $$0.689063\pi$$
$$678$$ 0 0
$$679$$ 2.12311 0.0814773
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 12.0000 0.459504
$$683$$ 20.8078 0.796187 0.398093 0.917345i $$-0.369672\pi$$
0.398093 + 0.917345i $$0.369672\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −33.3002 −1.27141
$$687$$ 0 0
$$688$$ −36.0000 −1.37249
$$689$$ 0.492423 0.0187598
$$690$$ 0 0
$$691$$ 12.4924 0.475234 0.237617 0.971359i $$-0.423634\pi$$
0.237617 + 0.971359i $$0.423634\pi$$
$$692$$ −61.3002 −2.33028
$$693$$ 0 0
$$694$$ −70.1080 −2.66126
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 11.0540 0.418699
$$698$$ −16.9848 −0.642886
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −1.93087 −0.0729279 −0.0364640 0.999335i $$-0.511609\pi$$
−0.0364640 + 0.999335i $$0.511609\pi$$
$$702$$ 0 0
$$703$$ 32.4233 1.22287
$$704$$ −1.43845 −0.0542135
$$705$$ 0 0
$$706$$ 71.3693 2.68602
$$707$$ −1.43845 −0.0540984
$$708$$ 0 0
$$709$$ 6.12311 0.229958 0.114979 0.993368i $$-0.463320\pi$$
0.114979 + 0.993368i $$0.463320\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 112.354 4.21065
$$713$$ −30.7386 −1.15117
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 66.4233 2.48235
$$717$$ 0 0
$$718$$ −67.2311 −2.50904
$$719$$ −32.9848 −1.23013 −0.615064 0.788478i $$-0.710870\pi$$
−0.615064 + 0.788478i $$0.710870\pi$$
$$720$$ 0 0
$$721$$ −9.12311 −0.339762
$$722$$ 25.6155 0.953311
$$723$$ 0 0
$$724$$ 54.1771 2.01348
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 30.0540 1.11464 0.557320 0.830298i $$-0.311830\pi$$
0.557320 + 0.830298i $$0.311830\pi$$
$$728$$ 2.87689 0.106625
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 6.73863 0.249237
$$732$$ 0 0
$$733$$ −2.00000 −0.0738717 −0.0369358 0.999318i $$-0.511760\pi$$
−0.0369358 + 0.999318i $$0.511760\pi$$
$$734$$ 36.0000 1.32878
$$735$$ 0 0
$$736$$ 43.0540 1.58699
$$737$$ −6.68466 −0.246233
$$738$$ 0 0
$$739$$ −11.6847 −0.429827 −0.214914 0.976633i $$-0.568947\pi$$
−0.214914 + 0.976633i $$0.568947\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 2.87689 0.105614
$$743$$ −47.8617 −1.75588 −0.877938 0.478773i $$-0.841082\pi$$
−0.877938 + 0.478773i $$0.841082\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −35.3693 −1.29496
$$747$$ 0 0
$$748$$ 6.56155 0.239914
$$749$$ −9.12311 −0.333351
$$750$$ 0 0
$$751$$ 26.7386 0.975707 0.487853 0.872926i $$-0.337780\pi$$
0.487853 + 0.872926i $$0.337780\pi$$
$$752$$ −41.7926 −1.52402
$$753$$ 0 0
$$754$$ −5.75379 −0.209541
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −6.30019 −0.228984 −0.114492 0.993424i $$-0.536524\pi$$
−0.114492 + 0.993424i $$0.536524\pi$$
$$758$$ 62.2462 2.26088
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 20.6307 0.747862 0.373931 0.927457i $$-0.378010\pi$$
0.373931 + 0.927457i $$0.378010\pi$$
$$762$$ 0 0
$$763$$ 7.80776 0.282660
$$764$$ −8.80776 −0.318654
$$765$$ 0 0
$$766$$ 56.9848 2.05895
$$767$$ 4.63068 0.167204
$$768$$ 0 0
$$769$$ 6.05398 0.218312 0.109156 0.994025i $$-0.465185\pi$$
0.109156 + 0.994025i $$0.465185\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −80.1080 −2.88315
$$773$$ −48.4924 −1.74415 −0.872076 0.489371i $$-0.837226\pi$$
−0.872076 + 0.489371i $$0.837226\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 13.9309 0.500089
$$777$$ 0 0
$$778$$ 61.1231 2.19137
$$779$$ −23.0540 −0.825994
$$780$$ 0 0
$$781$$ −8.80776 −0.315167
$$782$$ −24.1771 −0.864571
$$783$$ 0 0
$$784$$ −46.1080 −1.64671
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −4.26137 −0.151901 −0.0759507 0.997112i $$-0.524199\pi$$
−0.0759507 + 0.997112i $$0.524199\pi$$
$$788$$ −81.7926 −2.91374
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −9.12311 −0.324380
$$792$$ 0 0
$$793$$ −3.31534 −0.117731
$$794$$ 72.8466 2.58523
$$795$$ 0 0
$$796$$ −62.9848 −2.23244
$$797$$ −42.8769 −1.51878 −0.759389 0.650637i $$-0.774502\pi$$
−0.759389 + 0.650637i $$0.774502\pi$$
$$798$$ 0 0
$$799$$ 7.82292 0.276755
$$800$$ 0 0
$$801$$ 0 0
$$802$$ −14.3845 −0.507933
$$803$$ 16.2462 0.573316
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −5.26137 −0.185324
$$807$$ 0 0
$$808$$ −9.43845 −0.332043
$$809$$ 19.6847 0.692076 0.346038 0.938221i $$-0.387527\pi$$
0.346038 + 0.938221i $$0.387527\pi$$
$$810$$ 0 0
$$811$$ −1.00000 −0.0351147 −0.0175574 0.999846i $$-0.505589\pi$$
−0.0175574 + 0.999846i $$0.505589\pi$$
$$812$$ −23.3693 −0.820102
$$813$$ 0 0
$$814$$ 27.6847 0.970347
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −14.0540 −0.491686
$$818$$ −20.6307 −0.721335
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −25.1231 −0.876802 −0.438401 0.898779i $$-0.644455\pi$$
−0.438401 + 0.898779i $$0.644455\pi$$
$$822$$ 0 0
$$823$$ −0.0539753 −0.00188146 −0.000940731 1.00000i $$-0.500299\pi$$
−0.000940731 1.00000i $$0.500299\pi$$
$$824$$ −59.8617 −2.08538
$$825$$ 0 0
$$826$$ 27.0540 0.941328
$$827$$ −9.75379 −0.339172 −0.169586 0.985515i $$-0.554243\pi$$
−0.169586 + 0.985515i $$0.554243\pi$$
$$828$$ 0 0
$$829$$ 48.7386 1.69276 0.846381 0.532577i $$-0.178776\pi$$
0.846381 + 0.532577i $$0.178776\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0.630683 0.0218650
$$833$$ 8.63068 0.299035
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −13.6847 −0.473294
$$837$$ 0 0
$$838$$ −65.1619 −2.25098
$$839$$ 20.4924 0.707477 0.353738 0.935344i $$-0.384910\pi$$
0.353738 + 0.935344i $$0.384910\pi$$
$$840$$ 0 0
$$841$$ −2.75379 −0.0949582
$$842$$ 43.6847 1.50547
$$843$$ 0 0
$$844$$ 15.1231 0.520559
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −1.00000 −0.0343604
$$848$$ 8.63068 0.296379
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −70.9157 −2.43096
$$852$$ 0 0
$$853$$ 35.4233 1.21287 0.606435 0.795133i $$-0.292599\pi$$
0.606435 + 0.795133i $$0.292599\pi$$
$$854$$ −19.3693 −0.662804
$$855$$ 0 0
$$856$$ −59.8617 −2.04603
$$857$$ −5.43845 −0.185774 −0.0928869 0.995677i $$-0.529610\pi$$
−0.0928869 + 0.995677i $$0.529610\pi$$
$$858$$ 0 0
$$859$$ 56.3542 1.92278 0.961390 0.275191i $$-0.0887411\pi$$
0.961390 + 0.275191i $$0.0887411\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 55.7235 1.89795
$$863$$ 2.24621 0.0764619 0.0382310 0.999269i $$-0.487828\pi$$
0.0382310 + 0.999269i $$0.487828\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 43.3693 1.47375
$$867$$ 0 0
$$868$$ −21.3693 −0.725322
$$869$$ 15.6847 0.532066
$$870$$ 0 0
$$871$$ 2.93087 0.0993087
$$872$$ 51.2311 1.73490
$$873$$ 0 0
$$874$$ 50.4233 1.70559
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −14.0540 −0.474569 −0.237285 0.971440i $$-0.576257\pi$$
−0.237285 + 0.971440i $$0.576257\pi$$
$$878$$ −30.0691 −1.01478
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −9.75379 −0.328613 −0.164307 0.986409i $$-0.552539\pi$$
−0.164307 + 0.986409i $$0.552539\pi$$
$$882$$ 0 0
$$883$$ 44.9309 1.51204 0.756022 0.654546i $$-0.227140\pi$$
0.756022 + 0.654546i $$0.227140\pi$$
$$884$$ −2.87689 −0.0967604
$$885$$ 0 0
$$886$$ 64.8078 2.17726
$$887$$ −27.8617 −0.935506 −0.467753 0.883859i $$-0.654936\pi$$
−0.467753 + 0.883859i $$0.654936\pi$$
$$888$$ 0 0
$$889$$ −4.80776 −0.161247
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −99.4773 −3.33075
$$893$$ −16.3153 −0.545972
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −9.43845 −0.315316
$$897$$ 0 0
$$898$$ 72.9848 2.43554
$$899$$ 24.0000 0.800445
$$900$$ 0 0
$$901$$ −1.61553 −0.0538210
$$902$$ −19.6847 −0.655427
$$903$$ 0 0
$$904$$ −59.8617 −1.99097
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −46.1080 −1.53099 −0.765495 0.643442i $$-0.777506\pi$$
−0.765495 + 0.643442i $$0.777506\pi$$
$$908$$ −93.4773 −3.10215
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −57.1619 −1.89386 −0.946930 0.321441i $$-0.895833\pi$$
−0.946930 + 0.321441i $$0.895833\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 34.2462 1.13276
$$915$$ 0 0
$$916$$ −37.0540 −1.22430
$$917$$ 16.4924 0.544628
$$918$$ 0 0
$$919$$ 41.1080 1.35603 0.678013 0.735050i $$-0.262841\pi$$
0.678013 + 0.735050i $$0.262841\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 40.6307 1.33810
$$923$$ 3.86174 0.127111
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −4.49242 −0.147630
$$927$$ 0 0
$$928$$ −33.6155 −1.10348
$$929$$ 23.2311 0.762186 0.381093 0.924537i $$-0.375548\pi$$
0.381093 + 0.924537i $$0.375548\pi$$
$$930$$ 0 0
$$931$$ −18.0000 −0.589926
$$932$$ −113.162 −3.70674
$$933$$ 0 0
$$934$$ 5.75379 0.188270
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −34.6847 −1.13310 −0.566549 0.824028i $$-0.691722\pi$$
−0.566549 + 0.824028i $$0.691722\pi$$
$$938$$ 17.1231 0.559089
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 60.1771 1.96172 0.980858 0.194722i $$-0.0623806\pi$$
0.980858 + 0.194722i $$0.0623806\pi$$
$$942$$ 0 0
$$943$$ 50.4233 1.64201
$$944$$ 81.1619 2.64160
$$945$$ 0 0
$$946$$ −12.0000 −0.390154
$$947$$ 32.1771 1.04561 0.522807 0.852451i $$-0.324885\pi$$
0.522807 + 0.852451i $$0.324885\pi$$
$$948$$ 0 0
$$949$$ −7.12311 −0.231226
$$950$$ 0 0
$$951$$ 0 0
$$952$$ −9.43845 −0.305902
$$953$$ 4.80776 0.155739 0.0778694 0.996964i $$-0.475188\pi$$
0.0778694 + 0.996964i $$0.475188\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −75.2311 −2.43315
$$957$$ 0 0
$$958$$ −8.63068 −0.278845
$$959$$ −3.36932 −0.108801
$$960$$ 0 0
$$961$$ −9.05398 −0.292064
$$962$$ −12.1383 −0.391353
$$963$$ 0 0
$$964$$ −64.1080 −2.06478
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −44.0000 −1.41494 −0.707472 0.706741i $$-0.750165\pi$$
−0.707472 + 0.706741i $$0.750165\pi$$
$$968$$ −6.56155 −0.210896
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 8.17708 0.262415 0.131208 0.991355i $$-0.458115\pi$$
0.131208 + 0.991355i $$0.458115\pi$$
$$972$$ 0 0
$$973$$ −8.00000 −0.256468
$$974$$ −30.8769 −0.989360
$$975$$ 0 0
$$976$$ −58.1080 −1.85999
$$977$$ 25.6155 0.819513 0.409757 0.912195i $$-0.365614\pi$$
0.409757 + 0.912195i $$0.365614\pi$$
$$978$$ 0 0
$$979$$ 17.1231 0.547257
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −42.2462 −1.34813
$$983$$ −21.9309 −0.699486 −0.349743 0.936846i $$-0.613731\pi$$
−0.349743 + 0.936846i $$0.613731\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 18.8769 0.601163
$$987$$ 0 0
$$988$$ 6.00000 0.190885
$$989$$ 30.7386 0.977432
$$990$$ 0 0
$$991$$ 37.3153 1.18536 0.592680 0.805438i $$-0.298070\pi$$
0.592680 + 0.805438i $$0.298070\pi$$
$$992$$ −30.7386 −0.975953
$$993$$ 0 0
$$994$$ 22.5616 0.715609
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −20.7386 −0.656799 −0.328400 0.944539i $$-0.606509\pi$$
−0.328400 + 0.944539i $$0.606509\pi$$
$$998$$ −9.75379 −0.308751
$$999$$ 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 2475.2.a.p.1.1 2
3.2 odd 2 2475.2.a.u.1.2 yes 2
5.2 odd 4 2475.2.c.i.199.1 4
5.3 odd 4 2475.2.c.i.199.4 4
5.4 even 2 2475.2.a.v.1.2 yes 2
15.2 even 4 2475.2.c.j.199.4 4
15.8 even 4 2475.2.c.j.199.1 4
15.14 odd 2 2475.2.a.q.1.1 yes 2

By twisted newform
Twist Min Dim Char Parity Ord Type
2475.2.a.p.1.1 2 1.1 even 1 trivial
2475.2.a.q.1.1 yes 2 15.14 odd 2
2475.2.a.u.1.2 yes 2 3.2 odd 2
2475.2.a.v.1.2 yes 2 5.4 even 2
2475.2.c.i.199.1 4 5.2 odd 4
2475.2.c.i.199.4 4 5.3 odd 4
2475.2.c.j.199.1 4 15.8 even 4
2475.2.c.j.199.4 4 15.2 even 4