# Properties

 Label 1575.2.a.n.1.1 Level $1575$ Weight $2$ Character 1575.1 Self dual yes Analytic conductor $12.576$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1575 = 3^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1575.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$12.5764383184$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{10})^+$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 175) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$1.61803$$ of defining polynomial Character $$\chi$$ $$=$$ 1575.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.61803 q^{2} +0.618034 q^{4} +1.00000 q^{7} +2.23607 q^{8} +O(q^{10})$$ $$q-1.61803 q^{2} +0.618034 q^{4} +1.00000 q^{7} +2.23607 q^{8} -4.23607 q^{11} -3.23607 q^{13} -1.61803 q^{14} -4.85410 q^{16} +6.47214 q^{17} +4.47214 q^{19} +6.85410 q^{22} -1.76393 q^{23} +5.23607 q^{26} +0.618034 q^{28} -5.00000 q^{29} -9.70820 q^{31} +3.38197 q^{32} -10.4721 q^{34} +3.00000 q^{37} -7.23607 q^{38} -9.23607 q^{41} +6.23607 q^{43} -2.61803 q^{44} +2.85410 q^{46} +2.00000 q^{47} +1.00000 q^{49} -2.00000 q^{52} +0.472136 q^{53} +2.23607 q^{56} +8.09017 q^{58} +1.70820 q^{59} +3.70820 q^{61} +15.7082 q^{62} +4.23607 q^{64} +0.236068 q^{67} +4.00000 q^{68} +4.70820 q^{71} -13.2361 q^{73} -4.85410 q^{74} +2.76393 q^{76} -4.23607 q^{77} +11.1803 q^{79} +14.9443 q^{82} -5.70820 q^{83} -10.0902 q^{86} -9.47214 q^{88} -12.7639 q^{89} -3.23607 q^{91} -1.09017 q^{92} -3.23607 q^{94} +0.763932 q^{97} -1.61803 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} - q^{4} + 2 q^{7} + O(q^{10})$$ $$2 q - q^{2} - q^{4} + 2 q^{7} - 4 q^{11} - 2 q^{13} - q^{14} - 3 q^{16} + 4 q^{17} + 7 q^{22} - 8 q^{23} + 6 q^{26} - q^{28} - 10 q^{29} - 6 q^{31} + 9 q^{32} - 12 q^{34} + 6 q^{37} - 10 q^{38} - 14 q^{41} + 8 q^{43} - 3 q^{44} - q^{46} + 4 q^{47} + 2 q^{49} - 4 q^{52} - 8 q^{53} + 5 q^{58} - 10 q^{59} - 6 q^{61} + 18 q^{62} + 4 q^{64} - 4 q^{67} + 8 q^{68} - 4 q^{71} - 22 q^{73} - 3 q^{74} + 10 q^{76} - 4 q^{77} + 12 q^{82} + 2 q^{83} - 9 q^{86} - 10 q^{88} - 30 q^{89} - 2 q^{91} + 9 q^{92} - 2 q^{94} + 6 q^{97} - q^{98} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.61803 −1.14412 −0.572061 0.820211i $$-0.693856\pi$$
−0.572061 + 0.820211i $$0.693856\pi$$
$$3$$ 0 0
$$4$$ 0.618034 0.309017
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 1.00000 0.377964
$$8$$ 2.23607 0.790569
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −4.23607 −1.27722 −0.638611 0.769529i $$-0.720491\pi$$
−0.638611 + 0.769529i $$0.720491\pi$$
$$12$$ 0 0
$$13$$ −3.23607 −0.897524 −0.448762 0.893651i $$-0.648135\pi$$
−0.448762 + 0.893651i $$0.648135\pi$$
$$14$$ −1.61803 −0.432438
$$15$$ 0 0
$$16$$ −4.85410 −1.21353
$$17$$ 6.47214 1.56972 0.784862 0.619671i $$-0.212734\pi$$
0.784862 + 0.619671i $$0.212734\pi$$
$$18$$ 0 0
$$19$$ 4.47214 1.02598 0.512989 0.858395i $$-0.328538\pi$$
0.512989 + 0.858395i $$0.328538\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 6.85410 1.46130
$$23$$ −1.76393 −0.367805 −0.183903 0.982944i $$-0.558873\pi$$
−0.183903 + 0.982944i $$0.558873\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 5.23607 1.02688
$$27$$ 0 0
$$28$$ 0.618034 0.116797
$$29$$ −5.00000 −0.928477 −0.464238 0.885710i $$-0.653672\pi$$
−0.464238 + 0.885710i $$0.653672\pi$$
$$30$$ 0 0
$$31$$ −9.70820 −1.74364 −0.871822 0.489822i $$-0.837062\pi$$
−0.871822 + 0.489822i $$0.837062\pi$$
$$32$$ 3.38197 0.597853
$$33$$ 0 0
$$34$$ −10.4721 −1.79596
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 3.00000 0.493197 0.246598 0.969118i $$-0.420687\pi$$
0.246598 + 0.969118i $$0.420687\pi$$
$$38$$ −7.23607 −1.17385
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −9.23607 −1.44243 −0.721216 0.692711i $$-0.756416\pi$$
−0.721216 + 0.692711i $$0.756416\pi$$
$$42$$ 0 0
$$43$$ 6.23607 0.950991 0.475496 0.879718i $$-0.342269\pi$$
0.475496 + 0.879718i $$0.342269\pi$$
$$44$$ −2.61803 −0.394683
$$45$$ 0 0
$$46$$ 2.85410 0.420814
$$47$$ 2.00000 0.291730 0.145865 0.989305i $$-0.453403\pi$$
0.145865 + 0.989305i $$0.453403\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 0 0
$$52$$ −2.00000 −0.277350
$$53$$ 0.472136 0.0648529 0.0324264 0.999474i $$-0.489677\pi$$
0.0324264 + 0.999474i $$0.489677\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 2.23607 0.298807
$$57$$ 0 0
$$58$$ 8.09017 1.06229
$$59$$ 1.70820 0.222389 0.111195 0.993799i $$-0.464532\pi$$
0.111195 + 0.993799i $$0.464532\pi$$
$$60$$ 0 0
$$61$$ 3.70820 0.474787 0.237393 0.971414i $$-0.423707\pi$$
0.237393 + 0.971414i $$0.423707\pi$$
$$62$$ 15.7082 1.99494
$$63$$ 0 0
$$64$$ 4.23607 0.529508
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 0.236068 0.0288403 0.0144201 0.999896i $$-0.495410\pi$$
0.0144201 + 0.999896i $$0.495410\pi$$
$$68$$ 4.00000 0.485071
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 4.70820 0.558761 0.279381 0.960180i $$-0.409871\pi$$
0.279381 + 0.960180i $$0.409871\pi$$
$$72$$ 0 0
$$73$$ −13.2361 −1.54916 −0.774582 0.632473i $$-0.782040\pi$$
−0.774582 + 0.632473i $$0.782040\pi$$
$$74$$ −4.85410 −0.564278
$$75$$ 0 0
$$76$$ 2.76393 0.317045
$$77$$ −4.23607 −0.482745
$$78$$ 0 0
$$79$$ 11.1803 1.25789 0.628943 0.777451i $$-0.283488\pi$$
0.628943 + 0.777451i $$0.283488\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 14.9443 1.65032
$$83$$ −5.70820 −0.626557 −0.313278 0.949661i $$-0.601427\pi$$
−0.313278 + 0.949661i $$0.601427\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −10.0902 −1.08805
$$87$$ 0 0
$$88$$ −9.47214 −1.00973
$$89$$ −12.7639 −1.35297 −0.676487 0.736455i $$-0.736499\pi$$
−0.676487 + 0.736455i $$0.736499\pi$$
$$90$$ 0 0
$$91$$ −3.23607 −0.339232
$$92$$ −1.09017 −0.113658
$$93$$ 0 0
$$94$$ −3.23607 −0.333775
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 0.763932 0.0775655 0.0387828 0.999248i $$-0.487652\pi$$
0.0387828 + 0.999248i $$0.487652\pi$$
$$98$$ −1.61803 −0.163446
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −9.23607 −0.919023 −0.459512 0.888172i $$-0.651976\pi$$
−0.459512 + 0.888172i $$0.651976\pi$$
$$102$$ 0 0
$$103$$ −0.472136 −0.0465209 −0.0232605 0.999729i $$-0.507405\pi$$
−0.0232605 + 0.999729i $$0.507405\pi$$
$$104$$ −7.23607 −0.709555
$$105$$ 0 0
$$106$$ −0.763932 −0.0741996
$$107$$ −8.00000 −0.773389 −0.386695 0.922208i $$-0.626383\pi$$
−0.386695 + 0.922208i $$0.626383\pi$$
$$108$$ 0 0
$$109$$ −18.4164 −1.76397 −0.881986 0.471276i $$-0.843794\pi$$
−0.881986 + 0.471276i $$0.843794\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −4.85410 −0.458670
$$113$$ −12.4164 −1.16804 −0.584019 0.811740i $$-0.698521\pi$$
−0.584019 + 0.811740i $$0.698521\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −3.09017 −0.286915
$$117$$ 0 0
$$118$$ −2.76393 −0.254441
$$119$$ 6.47214 0.593300
$$120$$ 0 0
$$121$$ 6.94427 0.631297
$$122$$ −6.00000 −0.543214
$$123$$ 0 0
$$124$$ −6.00000 −0.538816
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −17.6525 −1.56640 −0.783202 0.621767i $$-0.786415\pi$$
−0.783202 + 0.621767i $$0.786415\pi$$
$$128$$ −13.6180 −1.20368
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −0.944272 −0.0825014 −0.0412507 0.999149i $$-0.513134\pi$$
−0.0412507 + 0.999149i $$0.513134\pi$$
$$132$$ 0 0
$$133$$ 4.47214 0.387783
$$134$$ −0.381966 −0.0329968
$$135$$ 0 0
$$136$$ 14.4721 1.24098
$$137$$ −6.94427 −0.593289 −0.296645 0.954988i $$-0.595868\pi$$
−0.296645 + 0.954988i $$0.595868\pi$$
$$138$$ 0 0
$$139$$ −20.6525 −1.75172 −0.875860 0.482565i $$-0.839705\pi$$
−0.875860 + 0.482565i $$0.839705\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −7.61803 −0.639291
$$143$$ 13.7082 1.14634
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 21.4164 1.77243
$$147$$ 0 0
$$148$$ 1.85410 0.152406
$$149$$ 13.9443 1.14236 0.571180 0.820825i $$-0.306486\pi$$
0.571180 + 0.820825i $$0.306486\pi$$
$$150$$ 0 0
$$151$$ −15.7639 −1.28285 −0.641425 0.767185i $$-0.721657\pi$$
−0.641425 + 0.767185i $$0.721657\pi$$
$$152$$ 10.0000 0.811107
$$153$$ 0 0
$$154$$ 6.85410 0.552319
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 5.23607 0.417884 0.208942 0.977928i $$-0.432998\pi$$
0.208942 + 0.977928i $$0.432998\pi$$
$$158$$ −18.0902 −1.43918
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −1.76393 −0.139017
$$162$$ 0 0
$$163$$ −10.4721 −0.820241 −0.410120 0.912031i $$-0.634513\pi$$
−0.410120 + 0.912031i $$0.634513\pi$$
$$164$$ −5.70820 −0.445736
$$165$$ 0 0
$$166$$ 9.23607 0.716858
$$167$$ −0.763932 −0.0591148 −0.0295574 0.999563i $$-0.509410\pi$$
−0.0295574 + 0.999563i $$0.509410\pi$$
$$168$$ 0 0
$$169$$ −2.52786 −0.194451
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 3.85410 0.293873
$$173$$ 20.4721 1.55647 0.778234 0.627975i $$-0.216116\pi$$
0.778234 + 0.627975i $$0.216116\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 20.5623 1.54994
$$177$$ 0 0
$$178$$ 20.6525 1.54797
$$179$$ −3.41641 −0.255354 −0.127677 0.991816i $$-0.540752\pi$$
−0.127677 + 0.991816i $$0.540752\pi$$
$$180$$ 0 0
$$181$$ −14.1803 −1.05402 −0.527008 0.849860i $$-0.676686\pi$$
−0.527008 + 0.849860i $$0.676686\pi$$
$$182$$ 5.23607 0.388123
$$183$$ 0 0
$$184$$ −3.94427 −0.290776
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −27.4164 −2.00489
$$188$$ 1.23607 0.0901495
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 2.47214 0.178877 0.0894387 0.995992i $$-0.471493\pi$$
0.0894387 + 0.995992i $$0.471493\pi$$
$$192$$ 0 0
$$193$$ −14.4164 −1.03772 −0.518858 0.854861i $$-0.673643\pi$$
−0.518858 + 0.854861i $$0.673643\pi$$
$$194$$ −1.23607 −0.0887445
$$195$$ 0 0
$$196$$ 0.618034 0.0441453
$$197$$ −7.47214 −0.532368 −0.266184 0.963922i $$-0.585763\pi$$
−0.266184 + 0.963922i $$0.585763\pi$$
$$198$$ 0 0
$$199$$ 2.76393 0.195930 0.0979650 0.995190i $$-0.468767\pi$$
0.0979650 + 0.995190i $$0.468767\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 14.9443 1.05148
$$203$$ −5.00000 −0.350931
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0.763932 0.0532257
$$207$$ 0 0
$$208$$ 15.7082 1.08917
$$209$$ −18.9443 −1.31040
$$210$$ 0 0
$$211$$ 12.0000 0.826114 0.413057 0.910705i $$-0.364461\pi$$
0.413057 + 0.910705i $$0.364461\pi$$
$$212$$ 0.291796 0.0200406
$$213$$ 0 0
$$214$$ 12.9443 0.884852
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −9.70820 −0.659036
$$218$$ 29.7984 2.01820
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −20.9443 −1.40886
$$222$$ 0 0
$$223$$ −2.18034 −0.146006 −0.0730032 0.997332i $$-0.523258\pi$$
−0.0730032 + 0.997332i $$0.523258\pi$$
$$224$$ 3.38197 0.225967
$$225$$ 0 0
$$226$$ 20.0902 1.33638
$$227$$ 5.41641 0.359500 0.179750 0.983712i $$-0.442471\pi$$
0.179750 + 0.983712i $$0.442471\pi$$
$$228$$ 0 0
$$229$$ 4.47214 0.295527 0.147764 0.989023i $$-0.452793\pi$$
0.147764 + 0.989023i $$0.452793\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −11.1803 −0.734025
$$233$$ 9.94427 0.651471 0.325735 0.945461i $$-0.394388\pi$$
0.325735 + 0.945461i $$0.394388\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 1.05573 0.0687220
$$237$$ 0 0
$$238$$ −10.4721 −0.678808
$$239$$ 14.4721 0.936125 0.468062 0.883695i $$-0.344952\pi$$
0.468062 + 0.883695i $$0.344952\pi$$
$$240$$ 0 0
$$241$$ −12.4721 −0.803401 −0.401700 0.915771i $$-0.631581\pi$$
−0.401700 + 0.915771i $$0.631581\pi$$
$$242$$ −11.2361 −0.722282
$$243$$ 0 0
$$244$$ 2.29180 0.146717
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −14.4721 −0.920840
$$248$$ −21.7082 −1.37847
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 2.47214 0.156040 0.0780199 0.996952i $$-0.475140\pi$$
0.0780199 + 0.996952i $$0.475140\pi$$
$$252$$ 0 0
$$253$$ 7.47214 0.469769
$$254$$ 28.5623 1.79216
$$255$$ 0 0
$$256$$ 13.5623 0.847644
$$257$$ −18.6525 −1.16351 −0.581755 0.813364i $$-0.697634\pi$$
−0.581755 + 0.813364i $$0.697634\pi$$
$$258$$ 0 0
$$259$$ 3.00000 0.186411
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 1.52786 0.0943918
$$263$$ −11.7639 −0.725395 −0.362698 0.931907i $$-0.618144\pi$$
−0.362698 + 0.931907i $$0.618144\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −7.23607 −0.443672
$$267$$ 0 0
$$268$$ 0.145898 0.00891214
$$269$$ −1.70820 −0.104151 −0.0520755 0.998643i $$-0.516584\pi$$
−0.0520755 + 0.998643i $$0.516584\pi$$
$$270$$ 0 0
$$271$$ 10.2918 0.625182 0.312591 0.949888i $$-0.398803\pi$$
0.312591 + 0.949888i $$0.398803\pi$$
$$272$$ −31.4164 −1.90490
$$273$$ 0 0
$$274$$ 11.2361 0.678796
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 15.8885 0.954650 0.477325 0.878727i $$-0.341606\pi$$
0.477325 + 0.878727i $$0.341606\pi$$
$$278$$ 33.4164 2.00418
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −29.3607 −1.75151 −0.875756 0.482755i $$-0.839636\pi$$
−0.875756 + 0.482755i $$0.839636\pi$$
$$282$$ 0 0
$$283$$ −9.41641 −0.559747 −0.279874 0.960037i $$-0.590293\pi$$
−0.279874 + 0.960037i $$0.590293\pi$$
$$284$$ 2.90983 0.172667
$$285$$ 0 0
$$286$$ −22.1803 −1.31155
$$287$$ −9.23607 −0.545188
$$288$$ 0 0
$$289$$ 24.8885 1.46403
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −8.18034 −0.478718
$$293$$ −9.12461 −0.533066 −0.266533 0.963826i $$-0.585878\pi$$
−0.266533 + 0.963826i $$0.585878\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 6.70820 0.389906
$$297$$ 0 0
$$298$$ −22.5623 −1.30700
$$299$$ 5.70820 0.330114
$$300$$ 0 0
$$301$$ 6.23607 0.359441
$$302$$ 25.5066 1.46774
$$303$$ 0 0
$$304$$ −21.7082 −1.24505
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 31.4164 1.79303 0.896515 0.443014i $$-0.146091\pi$$
0.896515 + 0.443014i $$0.146091\pi$$
$$308$$ −2.61803 −0.149176
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 20.3607 1.15455 0.577274 0.816550i $$-0.304116\pi$$
0.577274 + 0.816550i $$0.304116\pi$$
$$312$$ 0 0
$$313$$ 28.4721 1.60934 0.804670 0.593722i $$-0.202342\pi$$
0.804670 + 0.593722i $$0.202342\pi$$
$$314$$ −8.47214 −0.478110
$$315$$ 0 0
$$316$$ 6.90983 0.388708
$$317$$ 19.3607 1.08740 0.543702 0.839278i $$-0.317022\pi$$
0.543702 + 0.839278i $$0.317022\pi$$
$$318$$ 0 0
$$319$$ 21.1803 1.18587
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 2.85410 0.159053
$$323$$ 28.9443 1.61050
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 16.9443 0.938456
$$327$$ 0 0
$$328$$ −20.6525 −1.14034
$$329$$ 2.00000 0.110264
$$330$$ 0 0
$$331$$ −11.2918 −0.620653 −0.310327 0.950630i $$-0.600438\pi$$
−0.310327 + 0.950630i $$0.600438\pi$$
$$332$$ −3.52786 −0.193617
$$333$$ 0 0
$$334$$ 1.23607 0.0676346
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −7.52786 −0.410069 −0.205034 0.978755i $$-0.565731\pi$$
−0.205034 + 0.978755i $$0.565731\pi$$
$$338$$ 4.09017 0.222476
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 41.1246 2.22702
$$342$$ 0 0
$$343$$ 1.00000 0.0539949
$$344$$ 13.9443 0.751825
$$345$$ 0 0
$$346$$ −33.1246 −1.78079
$$347$$ −15.7639 −0.846252 −0.423126 0.906071i $$-0.639067\pi$$
−0.423126 + 0.906071i $$0.639067\pi$$
$$348$$ 0 0
$$349$$ −4.47214 −0.239388 −0.119694 0.992811i $$-0.538191\pi$$
−0.119694 + 0.992811i $$0.538191\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −14.3262 −0.763591
$$353$$ −20.1803 −1.07409 −0.537046 0.843553i $$-0.680460\pi$$
−0.537046 + 0.843553i $$0.680460\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −7.88854 −0.418092
$$357$$ 0 0
$$358$$ 5.52786 0.292157
$$359$$ −10.1246 −0.534357 −0.267178 0.963647i $$-0.586091\pi$$
−0.267178 + 0.963647i $$0.586091\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 22.9443 1.20592
$$363$$ 0 0
$$364$$ −2.00000 −0.104828
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 3.12461 0.163103 0.0815517 0.996669i $$-0.474012\pi$$
0.0815517 + 0.996669i $$0.474012\pi$$
$$368$$ 8.56231 0.446341
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0.472136 0.0245121
$$372$$ 0 0
$$373$$ 15.8328 0.819792 0.409896 0.912132i $$-0.365565\pi$$
0.409896 + 0.912132i $$0.365565\pi$$
$$374$$ 44.3607 2.29384
$$375$$ 0 0
$$376$$ 4.47214 0.230633
$$377$$ 16.1803 0.833330
$$378$$ 0 0
$$379$$ 11.1803 0.574295 0.287148 0.957886i $$-0.407293\pi$$
0.287148 + 0.957886i $$0.407293\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ −4.00000 −0.204658
$$383$$ 28.7639 1.46977 0.734884 0.678193i $$-0.237237\pi$$
0.734884 + 0.678193i $$0.237237\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 23.3262 1.18727
$$387$$ 0 0
$$388$$ 0.472136 0.0239691
$$389$$ 32.8885 1.66752 0.833758 0.552131i $$-0.186185\pi$$
0.833758 + 0.552131i $$0.186185\pi$$
$$390$$ 0 0
$$391$$ −11.4164 −0.577353
$$392$$ 2.23607 0.112938
$$393$$ 0 0
$$394$$ 12.0902 0.609094
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 26.9443 1.35229 0.676147 0.736767i $$-0.263648\pi$$
0.676147 + 0.736767i $$0.263648\pi$$
$$398$$ −4.47214 −0.224168
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −11.4721 −0.572891 −0.286446 0.958097i $$-0.592474\pi$$
−0.286446 + 0.958097i $$0.592474\pi$$
$$402$$ 0 0
$$403$$ 31.4164 1.56496
$$404$$ −5.70820 −0.283994
$$405$$ 0 0
$$406$$ 8.09017 0.401508
$$407$$ −12.7082 −0.629922
$$408$$ 0 0
$$409$$ 15.5279 0.767803 0.383902 0.923374i $$-0.374580\pi$$
0.383902 + 0.923374i $$0.374580\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ −0.291796 −0.0143758
$$413$$ 1.70820 0.0840552
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −10.9443 −0.536587
$$417$$ 0 0
$$418$$ 30.6525 1.49926
$$419$$ 3.81966 0.186603 0.0933013 0.995638i $$-0.470258\pi$$
0.0933013 + 0.995638i $$0.470258\pi$$
$$420$$ 0 0
$$421$$ −13.0000 −0.633581 −0.316791 0.948495i $$-0.602605\pi$$
−0.316791 + 0.948495i $$0.602605\pi$$
$$422$$ −19.4164 −0.945176
$$423$$ 0 0
$$424$$ 1.05573 0.0512707
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 3.70820 0.179453
$$428$$ −4.94427 −0.238990
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −26.4721 −1.27512 −0.637559 0.770402i $$-0.720056\pi$$
−0.637559 + 0.770402i $$0.720056\pi$$
$$432$$ 0 0
$$433$$ 16.3607 0.786244 0.393122 0.919486i $$-0.371395\pi$$
0.393122 + 0.919486i $$0.371395\pi$$
$$434$$ 15.7082 0.754018
$$435$$ 0 0
$$436$$ −11.3820 −0.545097
$$437$$ −7.88854 −0.377360
$$438$$ 0 0
$$439$$ −21.7082 −1.03608 −0.518038 0.855358i $$-0.673337\pi$$
−0.518038 + 0.855358i $$0.673337\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 33.8885 1.61191
$$443$$ −7.41641 −0.352364 −0.176182 0.984358i $$-0.556375\pi$$
−0.176182 + 0.984358i $$0.556375\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 3.52786 0.167049
$$447$$ 0 0
$$448$$ 4.23607 0.200135
$$449$$ −29.4721 −1.39088 −0.695438 0.718586i $$-0.744790\pi$$
−0.695438 + 0.718586i $$0.744790\pi$$
$$450$$ 0 0
$$451$$ 39.1246 1.84231
$$452$$ −7.67376 −0.360943
$$453$$ 0 0
$$454$$ −8.76393 −0.411312
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −21.4721 −1.00442 −0.502212 0.864744i $$-0.667480\pi$$
−0.502212 + 0.864744i $$0.667480\pi$$
$$458$$ −7.23607 −0.338119
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −8.18034 −0.380996 −0.190498 0.981688i $$-0.561010\pi$$
−0.190498 + 0.981688i $$0.561010\pi$$
$$462$$ 0 0
$$463$$ 21.8885 1.01725 0.508623 0.860989i $$-0.330155\pi$$
0.508623 + 0.860989i $$0.330155\pi$$
$$464$$ 24.2705 1.12673
$$465$$ 0 0
$$466$$ −16.0902 −0.745363
$$467$$ 10.9443 0.506441 0.253220 0.967409i $$-0.418510\pi$$
0.253220 + 0.967409i $$0.418510\pi$$
$$468$$ 0 0
$$469$$ 0.236068 0.0109006
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 3.81966 0.175814
$$473$$ −26.4164 −1.21463
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 4.00000 0.183340
$$477$$ 0 0
$$478$$ −23.4164 −1.07104
$$479$$ 3.81966 0.174525 0.0872624 0.996185i $$-0.472188\pi$$
0.0872624 + 0.996185i $$0.472188\pi$$
$$480$$ 0 0
$$481$$ −9.70820 −0.442656
$$482$$ 20.1803 0.919189
$$483$$ 0 0
$$484$$ 4.29180 0.195082
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 10.2361 0.463841 0.231920 0.972735i $$-0.425499\pi$$
0.231920 + 0.972735i $$0.425499\pi$$
$$488$$ 8.29180 0.375352
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 10.2361 0.461947 0.230974 0.972960i $$-0.425809\pi$$
0.230974 + 0.972960i $$0.425809\pi$$
$$492$$ 0 0
$$493$$ −32.3607 −1.45745
$$494$$ 23.4164 1.05355
$$495$$ 0 0
$$496$$ 47.1246 2.11596
$$497$$ 4.70820 0.211192
$$498$$ 0 0
$$499$$ 28.9443 1.29572 0.647862 0.761758i $$-0.275663\pi$$
0.647862 + 0.761758i $$0.275663\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ −4.00000 −0.178529
$$503$$ 43.8885 1.95689 0.978447 0.206499i $$-0.0662072\pi$$
0.978447 + 0.206499i $$0.0662072\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −12.0902 −0.537474
$$507$$ 0 0
$$508$$ −10.9098 −0.484045
$$509$$ −9.34752 −0.414322 −0.207161 0.978307i $$-0.566422\pi$$
−0.207161 + 0.978307i $$0.566422\pi$$
$$510$$ 0 0
$$511$$ −13.2361 −0.585529
$$512$$ 5.29180 0.233867
$$513$$ 0 0
$$514$$ 30.1803 1.33120
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −8.47214 −0.372604
$$518$$ −4.85410 −0.213277
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −22.0000 −0.963837 −0.481919 0.876216i $$-0.660060\pi$$
−0.481919 + 0.876216i $$0.660060\pi$$
$$522$$ 0 0
$$523$$ −28.3607 −1.24013 −0.620063 0.784552i $$-0.712893\pi$$
−0.620063 + 0.784552i $$0.712893\pi$$
$$524$$ −0.583592 −0.0254943
$$525$$ 0 0
$$526$$ 19.0344 0.829941
$$527$$ −62.8328 −2.73704
$$528$$ 0 0
$$529$$ −19.8885 −0.864719
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 2.76393 0.119832
$$533$$ 29.8885 1.29462
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0.527864 0.0228003
$$537$$ 0 0
$$538$$ 2.76393 0.119162
$$539$$ −4.23607 −0.182460
$$540$$ 0 0
$$541$$ −1.94427 −0.0835908 −0.0417954 0.999126i $$-0.513308\pi$$
−0.0417954 + 0.999126i $$0.513308\pi$$
$$542$$ −16.6525 −0.715285
$$543$$ 0 0
$$544$$ 21.8885 0.938464
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −14.2361 −0.608690 −0.304345 0.952562i $$-0.598438\pi$$
−0.304345 + 0.952562i $$0.598438\pi$$
$$548$$ −4.29180 −0.183336
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −22.3607 −0.952597
$$552$$ 0 0
$$553$$ 11.1803 0.475436
$$554$$ −25.7082 −1.09224
$$555$$ 0 0
$$556$$ −12.7639 −0.541311
$$557$$ 44.8885 1.90199 0.950994 0.309208i $$-0.100064\pi$$
0.950994 + 0.309208i $$0.100064\pi$$
$$558$$ 0 0
$$559$$ −20.1803 −0.853537
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 47.5066 2.00394
$$563$$ 9.41641 0.396854 0.198427 0.980116i $$-0.436417\pi$$
0.198427 + 0.980116i $$0.436417\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 15.2361 0.640420
$$567$$ 0 0
$$568$$ 10.5279 0.441739
$$569$$ −13.9443 −0.584574 −0.292287 0.956331i $$-0.594416\pi$$
−0.292287 + 0.956331i $$0.594416\pi$$
$$570$$ 0 0
$$571$$ −12.5967 −0.527157 −0.263579 0.964638i $$-0.584903\pi$$
−0.263579 + 0.964638i $$0.584903\pi$$
$$572$$ 8.47214 0.354238
$$573$$ 0 0
$$574$$ 14.9443 0.623762
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −2.00000 −0.0832611 −0.0416305 0.999133i $$-0.513255\pi$$
−0.0416305 + 0.999133i $$0.513255\pi$$
$$578$$ −40.2705 −1.67503
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −5.70820 −0.236816
$$582$$ 0 0
$$583$$ −2.00000 −0.0828315
$$584$$ −29.5967 −1.22472
$$585$$ 0 0
$$586$$ 14.7639 0.609892
$$587$$ 29.2361 1.20670 0.603351 0.797476i $$-0.293832\pi$$
0.603351 + 0.797476i $$0.293832\pi$$
$$588$$ 0 0
$$589$$ −43.4164 −1.78894
$$590$$ 0 0
$$591$$ 0 0
$$592$$ −14.5623 −0.598507
$$593$$ −25.3050 −1.03915 −0.519575 0.854425i $$-0.673910\pi$$
−0.519575 + 0.854425i $$0.673910\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 8.61803 0.353008
$$597$$ 0 0
$$598$$ −9.23607 −0.377691
$$599$$ −11.1803 −0.456816 −0.228408 0.973565i $$-0.573352\pi$$
−0.228408 + 0.973565i $$0.573352\pi$$
$$600$$ 0 0
$$601$$ −19.0557 −0.777299 −0.388650 0.921386i $$-0.627058\pi$$
−0.388650 + 0.921386i $$0.627058\pi$$
$$602$$ −10.0902 −0.411245
$$603$$ 0 0
$$604$$ −9.74265 −0.396423
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 33.1246 1.34449 0.672243 0.740330i $$-0.265331\pi$$
0.672243 + 0.740330i $$0.265331\pi$$
$$608$$ 15.1246 0.613384
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −6.47214 −0.261835
$$612$$ 0 0
$$613$$ −17.5836 −0.710195 −0.355097 0.934829i $$-0.615552\pi$$
−0.355097 + 0.934829i $$0.615552\pi$$
$$614$$ −50.8328 −2.05145
$$615$$ 0 0
$$616$$ −9.47214 −0.381643
$$617$$ −11.9443 −0.480858 −0.240429 0.970667i $$-0.577288\pi$$
−0.240429 + 0.970667i $$0.577288\pi$$
$$618$$ 0 0
$$619$$ 1.70820 0.0686585 0.0343293 0.999411i $$-0.489071\pi$$
0.0343293 + 0.999411i $$0.489071\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ −32.9443 −1.32094
$$623$$ −12.7639 −0.511376
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −46.0689 −1.84128
$$627$$ 0 0
$$628$$ 3.23607 0.129133
$$629$$ 19.4164 0.774183
$$630$$ 0 0
$$631$$ −3.65248 −0.145403 −0.0727014 0.997354i $$-0.523162\pi$$
−0.0727014 + 0.997354i $$0.523162\pi$$
$$632$$ 25.0000 0.994447
$$633$$ 0 0
$$634$$ −31.3262 −1.24412
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −3.23607 −0.128218
$$638$$ −34.2705 −1.35678
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 9.83282 0.388373 0.194186 0.980965i $$-0.437793\pi$$
0.194186 + 0.980965i $$0.437793\pi$$
$$642$$ 0 0
$$643$$ 9.52786 0.375742 0.187871 0.982194i $$-0.439841\pi$$
0.187871 + 0.982194i $$0.439841\pi$$
$$644$$ −1.09017 −0.0429587
$$645$$ 0 0
$$646$$ −46.8328 −1.84261
$$647$$ −15.8885 −0.624643 −0.312322 0.949976i $$-0.601107\pi$$
−0.312322 + 0.949976i $$0.601107\pi$$
$$648$$ 0 0
$$649$$ −7.23607 −0.284041
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −6.47214 −0.253468
$$653$$ −42.9443 −1.68054 −0.840270 0.542169i $$-0.817603\pi$$
−0.840270 + 0.542169i $$0.817603\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 44.8328 1.75043
$$657$$ 0 0
$$658$$ −3.23607 −0.126155
$$659$$ 17.8885 0.696839 0.348419 0.937339i $$-0.386719\pi$$
0.348419 + 0.937339i $$0.386719\pi$$
$$660$$ 0 0
$$661$$ 46.7214 1.81725 0.908625 0.417613i $$-0.137133\pi$$
0.908625 + 0.417613i $$0.137133\pi$$
$$662$$ 18.2705 0.710104
$$663$$ 0 0
$$664$$ −12.7639 −0.495337
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 8.81966 0.341499
$$668$$ −0.472136 −0.0182675
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −15.7082 −0.606408
$$672$$ 0 0
$$673$$ 28.4721 1.09752 0.548760 0.835980i $$-0.315100\pi$$
0.548760 + 0.835980i $$0.315100\pi$$
$$674$$ 12.1803 0.469169
$$675$$ 0 0
$$676$$ −1.56231 −0.0600887
$$677$$ −30.3607 −1.16686 −0.583428 0.812165i $$-0.698289\pi$$
−0.583428 + 0.812165i $$0.698289\pi$$
$$678$$ 0 0
$$679$$ 0.763932 0.0293170
$$680$$ 0 0
$$681$$ 0 0
$$682$$ −66.5410 −2.54799
$$683$$ 26.1246 0.999630 0.499815 0.866132i $$-0.333401\pi$$
0.499815 + 0.866132i $$0.333401\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −1.61803 −0.0617768
$$687$$ 0 0
$$688$$ −30.2705 −1.15405
$$689$$ −1.52786 −0.0582070
$$690$$ 0 0
$$691$$ 18.1803 0.691613 0.345806 0.938306i $$-0.387605\pi$$
0.345806 + 0.938306i $$0.387605\pi$$
$$692$$ 12.6525 0.480975
$$693$$ 0 0
$$694$$ 25.5066 0.968216
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −59.7771 −2.26422
$$698$$ 7.23607 0.273889
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 46.9443 1.77306 0.886530 0.462670i $$-0.153109\pi$$
0.886530 + 0.462670i $$0.153109\pi$$
$$702$$ 0 0
$$703$$ 13.4164 0.506009
$$704$$ −17.9443 −0.676300
$$705$$ 0 0
$$706$$ 32.6525 1.22889
$$707$$ −9.23607 −0.347358
$$708$$ 0 0
$$709$$ −47.8885 −1.79849 −0.899246 0.437443i $$-0.855884\pi$$
−0.899246 + 0.437443i $$0.855884\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ −28.5410 −1.06962
$$713$$ 17.1246 0.641322
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −2.11146 −0.0789088
$$717$$ 0 0
$$718$$ 16.3820 0.611370
$$719$$ 6.18034 0.230488 0.115244 0.993337i $$-0.463235\pi$$
0.115244 + 0.993337i $$0.463235\pi$$
$$720$$ 0 0
$$721$$ −0.472136 −0.0175833
$$722$$ −1.61803 −0.0602170
$$723$$ 0 0
$$724$$ −8.76393 −0.325709
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −20.9443 −0.776780 −0.388390 0.921495i $$-0.626969\pi$$
−0.388390 + 0.921495i $$0.626969\pi$$
$$728$$ −7.23607 −0.268187
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 40.3607 1.49279
$$732$$ 0 0
$$733$$ 4.00000 0.147743 0.0738717 0.997268i $$-0.476464\pi$$
0.0738717 + 0.997268i $$0.476464\pi$$
$$734$$ −5.05573 −0.186610
$$735$$ 0 0
$$736$$ −5.96556 −0.219893
$$737$$ −1.00000 −0.0368355
$$738$$ 0 0
$$739$$ −5.65248 −0.207930 −0.103965 0.994581i $$-0.533153\pi$$
−0.103965 + 0.994581i $$0.533153\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −0.763932 −0.0280448
$$743$$ 1.52786 0.0560519 0.0280259 0.999607i $$-0.491078\pi$$
0.0280259 + 0.999607i $$0.491078\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −25.6180 −0.937943
$$747$$ 0 0
$$748$$ −16.9443 −0.619544
$$749$$ −8.00000 −0.292314
$$750$$ 0 0
$$751$$ 20.9443 0.764267 0.382134 0.924107i $$-0.375189\pi$$
0.382134 + 0.924107i $$0.375189\pi$$
$$752$$ −9.70820 −0.354022
$$753$$ 0 0
$$754$$ −26.1803 −0.953432
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 46.4164 1.68703 0.843517 0.537103i $$-0.180481\pi$$
0.843517 + 0.537103i $$0.180481\pi$$
$$758$$ −18.0902 −0.657065
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 43.7771 1.58692 0.793459 0.608624i $$-0.208278\pi$$
0.793459 + 0.608624i $$0.208278\pi$$
$$762$$ 0 0
$$763$$ −18.4164 −0.666719
$$764$$ 1.52786 0.0552762
$$765$$ 0 0
$$766$$ −46.5410 −1.68160
$$767$$ −5.52786 −0.199600
$$768$$ 0 0
$$769$$ 33.0132 1.19048 0.595242 0.803546i $$-0.297056\pi$$
0.595242 + 0.803546i $$0.297056\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −8.90983 −0.320672
$$773$$ −27.8197 −1.00060 −0.500302 0.865851i $$-0.666778\pi$$
−0.500302 + 0.865851i $$0.666778\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 1.70820 0.0613209
$$777$$ 0 0
$$778$$ −53.2148 −1.90784
$$779$$ −41.3050 −1.47990
$$780$$ 0 0
$$781$$ −19.9443 −0.713662
$$782$$ 18.4721 0.660562
$$783$$ 0 0
$$784$$ −4.85410 −0.173361
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 45.2361 1.61249 0.806246 0.591581i $$-0.201496\pi$$
0.806246 + 0.591581i $$0.201496\pi$$
$$788$$ −4.61803 −0.164511
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −12.4164 −0.441477
$$792$$ 0 0
$$793$$ −12.0000 −0.426132
$$794$$ −43.5967 −1.54719
$$795$$ 0 0
$$796$$ 1.70820 0.0605457
$$797$$ 8.58359 0.304046 0.152023 0.988377i $$-0.451421\pi$$
0.152023 + 0.988377i $$0.451421\pi$$
$$798$$ 0 0
$$799$$ 12.9443 0.457935
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 18.5623 0.655458
$$803$$ 56.0689 1.97863
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −50.8328 −1.79051
$$807$$ 0 0
$$808$$ −20.6525 −0.726552
$$809$$ −20.5279 −0.721721 −0.360861 0.932620i $$-0.617517\pi$$
−0.360861 + 0.932620i $$0.617517\pi$$
$$810$$ 0 0
$$811$$ 46.7214 1.64061 0.820304 0.571927i $$-0.193804\pi$$
0.820304 + 0.571927i $$0.193804\pi$$
$$812$$ −3.09017 −0.108444
$$813$$ 0 0
$$814$$ 20.5623 0.720708
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 27.8885 0.975697
$$818$$ −25.1246 −0.878461
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 24.8328 0.866671 0.433336 0.901233i $$-0.357336\pi$$
0.433336 + 0.901233i $$0.357336\pi$$
$$822$$ 0 0
$$823$$ −0.347524 −0.0121139 −0.00605697 0.999982i $$-0.501928\pi$$
−0.00605697 + 0.999982i $$0.501928\pi$$
$$824$$ −1.05573 −0.0367780
$$825$$ 0 0
$$826$$ −2.76393 −0.0961695
$$827$$ 25.5410 0.888148 0.444074 0.895990i $$-0.353533\pi$$
0.444074 + 0.895990i $$0.353533\pi$$
$$828$$ 0 0
$$829$$ −52.3607 −1.81856 −0.909281 0.416183i $$-0.863368\pi$$
−0.909281 + 0.416183i $$0.863368\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ −13.7082 −0.475246
$$833$$ 6.47214 0.224246
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −11.7082 −0.404937
$$837$$ 0 0
$$838$$ −6.18034 −0.213496
$$839$$ −0.652476 −0.0225260 −0.0112630 0.999937i $$-0.503585\pi$$
−0.0112630 + 0.999937i $$0.503585\pi$$
$$840$$ 0 0
$$841$$ −4.00000 −0.137931
$$842$$ 21.0344 0.724895
$$843$$ 0 0
$$844$$ 7.41641 0.255283
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 6.94427 0.238608
$$848$$ −2.29180 −0.0787006
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −5.29180 −0.181400
$$852$$ 0 0
$$853$$ 0.583592 0.0199818 0.00999091 0.999950i $$-0.496820\pi$$
0.00999091 + 0.999950i $$0.496820\pi$$
$$854$$ −6.00000 −0.205316
$$855$$ 0 0
$$856$$ −17.8885 −0.611418
$$857$$ 38.1803 1.30422 0.652108 0.758126i $$-0.273885\pi$$
0.652108 + 0.758126i $$0.273885\pi$$
$$858$$ 0 0
$$859$$ −22.3607 −0.762937 −0.381468 0.924382i $$-0.624581\pi$$
−0.381468 + 0.924382i $$0.624581\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 42.8328 1.45889
$$863$$ −49.6525 −1.69019 −0.845095 0.534616i $$-0.820456\pi$$
−0.845095 + 0.534616i $$0.820456\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −26.4721 −0.899560
$$867$$ 0 0
$$868$$ −6.00000 −0.203653
$$869$$ −47.3607 −1.60660
$$870$$ 0 0
$$871$$ −0.763932 −0.0258848
$$872$$ −41.1803 −1.39454
$$873$$ 0 0
$$874$$ 12.7639 0.431746
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −14.3607 −0.484926 −0.242463 0.970161i $$-0.577955\pi$$
−0.242463 + 0.970161i $$0.577955\pi$$
$$878$$ 35.1246 1.18540
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −28.1803 −0.949420 −0.474710 0.880142i $$-0.657447\pi$$
−0.474710 + 0.880142i $$0.657447\pi$$
$$882$$ 0 0
$$883$$ −50.5967 −1.70272 −0.851358 0.524585i $$-0.824220\pi$$
−0.851358 + 0.524585i $$0.824220\pi$$
$$884$$ −12.9443 −0.435363
$$885$$ 0 0
$$886$$ 12.0000 0.403148
$$887$$ 52.6525 1.76790 0.883949 0.467584i $$-0.154876\pi$$
0.883949 + 0.467584i $$0.154876\pi$$
$$888$$ 0 0
$$889$$ −17.6525 −0.592045
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −1.34752 −0.0451184
$$893$$ 8.94427 0.299309
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −13.6180 −0.454947
$$897$$ 0 0
$$898$$ 47.6869 1.59133
$$899$$ 48.5410 1.61893
$$900$$ 0 0
$$901$$ 3.05573 0.101801
$$902$$ −63.3050 −2.10782
$$903$$ 0 0
$$904$$ −27.7639 −0.923415
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −18.8328 −0.625333 −0.312667 0.949863i $$-0.601222\pi$$
−0.312667 + 0.949863i $$0.601222\pi$$
$$908$$ 3.34752 0.111091
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −23.1803 −0.767999 −0.383999 0.923333i $$-0.625454\pi$$
−0.383999 + 0.923333i $$0.625454\pi$$
$$912$$ 0 0
$$913$$ 24.1803 0.800252
$$914$$ 34.7426 1.14918
$$915$$ 0 0
$$916$$ 2.76393 0.0913229
$$917$$ −0.944272 −0.0311826
$$918$$ 0 0
$$919$$ −32.2361 −1.06337 −0.531685 0.846942i $$-0.678441\pi$$
−0.531685 + 0.846942i $$0.678441\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 13.2361 0.435907
$$923$$ −15.2361 −0.501501
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −35.4164 −1.16386
$$927$$ 0 0
$$928$$ −16.9098 −0.555092
$$929$$ −51.7082 −1.69649 −0.848246 0.529603i $$-0.822341\pi$$
−0.848246 + 0.529603i $$0.822341\pi$$
$$930$$ 0 0
$$931$$ 4.47214 0.146568
$$932$$ 6.14590 0.201316
$$933$$ 0 0
$$934$$ −17.7082 −0.579430
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 30.7639 1.00501 0.502507 0.864573i $$-0.332411\pi$$
0.502507 + 0.864573i $$0.332411\pi$$
$$938$$ −0.381966 −0.0124716
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 0.763932 0.0249035 0.0124517 0.999922i $$-0.496036\pi$$
0.0124517 + 0.999922i $$0.496036\pi$$
$$942$$ 0 0
$$943$$ 16.2918 0.530534
$$944$$ −8.29180 −0.269875
$$945$$ 0 0
$$946$$ 42.7426 1.38968
$$947$$ 18.8328 0.611984 0.305992 0.952034i $$-0.401012\pi$$
0.305992 + 0.952034i $$0.401012\pi$$
$$948$$ 0 0
$$949$$ 42.8328 1.39041
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 14.4721 0.469045
$$953$$ 5.47214 0.177260 0.0886299 0.996065i $$-0.471751\pi$$
0.0886299 + 0.996065i $$0.471751\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 8.94427 0.289278
$$957$$ 0 0
$$958$$ −6.18034 −0.199678
$$959$$ −6.94427 −0.224242
$$960$$ 0 0
$$961$$ 63.2492 2.04030
$$962$$ 15.7082 0.506453
$$963$$ 0 0
$$964$$ −7.70820 −0.248265
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −49.8885 −1.60431 −0.802154 0.597118i $$-0.796313\pi$$
−0.802154 + 0.597118i $$0.796313\pi$$
$$968$$ 15.5279 0.499084
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 18.0000 0.577647 0.288824 0.957382i $$-0.406736\pi$$
0.288824 + 0.957382i $$0.406736\pi$$
$$972$$ 0 0
$$973$$ −20.6525 −0.662088
$$974$$ −16.5623 −0.530691
$$975$$ 0 0
$$976$$ −18.0000 −0.576166
$$977$$ 2.52786 0.0808735 0.0404368 0.999182i $$-0.487125\pi$$
0.0404368 + 0.999182i $$0.487125\pi$$
$$978$$ 0 0
$$979$$ 54.0689 1.72805
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −16.5623 −0.528524
$$983$$ −32.5410 −1.03790 −0.518949 0.854805i $$-0.673676\pi$$
−0.518949 + 0.854805i $$0.673676\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 52.3607 1.66750
$$987$$ 0 0
$$988$$ −8.94427 −0.284555
$$989$$ −11.0000 −0.349780
$$990$$ 0 0
$$991$$ −9.18034 −0.291623 −0.145812 0.989312i $$-0.546579\pi$$
−0.145812 + 0.989312i $$0.546579\pi$$
$$992$$ −32.8328 −1.04244
$$993$$ 0 0
$$994$$ −7.61803 −0.241629
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −18.5836 −0.588548 −0.294274 0.955721i $$-0.595078\pi$$
−0.294274 + 0.955721i $$0.595078\pi$$
$$998$$ −46.8328 −1.48247
$$999$$ 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 1575.2.a.n.1.1 2
3.2 odd 2 175.2.a.e.1.2 yes 2
5.2 odd 4 1575.2.d.k.1324.1 4
5.3 odd 4 1575.2.d.k.1324.4 4
5.4 even 2 1575.2.a.s.1.2 2
12.11 even 2 2800.2.a.bp.1.1 2
15.2 even 4 175.2.b.c.99.4 4
15.8 even 4 175.2.b.c.99.1 4
15.14 odd 2 175.2.a.d.1.1 2
21.20 even 2 1225.2.a.u.1.2 2
60.23 odd 4 2800.2.g.s.449.2 4
60.47 odd 4 2800.2.g.s.449.3 4
60.59 even 2 2800.2.a.bh.1.2 2
105.62 odd 4 1225.2.b.k.99.4 4
105.83 odd 4 1225.2.b.k.99.1 4
105.104 even 2 1225.2.a.n.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
175.2.a.d.1.1 2 15.14 odd 2
175.2.a.e.1.2 yes 2 3.2 odd 2
175.2.b.c.99.1 4 15.8 even 4
175.2.b.c.99.4 4 15.2 even 4
1225.2.a.n.1.1 2 105.104 even 2
1225.2.a.u.1.2 2 21.20 even 2
1225.2.b.k.99.1 4 105.83 odd 4
1225.2.b.k.99.4 4 105.62 odd 4
1575.2.a.n.1.1 2 1.1 even 1 trivial
1575.2.a.s.1.2 2 5.4 even 2
1575.2.d.k.1324.1 4 5.2 odd 4
1575.2.d.k.1324.4 4 5.3 odd 4
2800.2.a.bh.1.2 2 60.59 even 2
2800.2.a.bp.1.1 2 12.11 even 2
2800.2.g.s.449.2 4 60.23 odd 4
2800.2.g.s.449.3 4 60.47 odd 4