# Properties

 Label 567.2.a.d.1.2 Level $567$ Weight $2$ Character 567.1 Self dual yes Analytic conductor $4.528$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [567,2,Mod(1,567)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(567, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("567.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$567 = 3^{4} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 567.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$4.52751779461$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.321.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 4x + 1$$ x^3 - x^2 - 4*x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 63) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$0.239123$$ of defining polynomial Character $$\chi$$ $$=$$ 567.1

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-0.239123 q^{2} -1.94282 q^{4} +1.18194 q^{5} -1.00000 q^{7} +0.942820 q^{8} +O(q^{10})$$ $$q-0.239123 q^{2} -1.94282 q^{4} +1.18194 q^{5} -1.00000 q^{7} +0.942820 q^{8} -0.282630 q^{10} -3.70370 q^{11} +1.00000 q^{13} +0.239123 q^{14} +3.66019 q^{16} -6.94282 q^{17} +1.94282 q^{19} -2.29630 q^{20} +0.885640 q^{22} -5.60301 q^{23} -3.60301 q^{25} -0.239123 q^{26} +1.94282 q^{28} +0.239123 q^{29} +1.66019 q^{31} -2.76088 q^{32} +1.66019 q^{34} -1.18194 q^{35} -9.54583 q^{37} -0.464574 q^{38} +1.11436 q^{40} -10.1819 q^{41} +2.22545 q^{43} +7.19562 q^{44} +1.33981 q^{46} +5.82846 q^{47} +1.00000 q^{49} +0.861564 q^{50} -1.94282 q^{52} -11.6030 q^{53} -4.37756 q^{55} -0.942820 q^{56} -0.0571799 q^{58} +2.60301 q^{59} -7.60301 q^{61} -0.396990 q^{62} -6.66019 q^{64} +1.18194 q^{65} +3.50808 q^{67} +13.4887 q^{68} +0.282630 q^{70} +8.60301 q^{71} +15.1488 q^{73} +2.28263 q^{74} -3.77455 q^{76} +3.70370 q^{77} +7.37756 q^{79} +4.32614 q^{80} +2.43474 q^{82} -6.94282 q^{83} -8.20602 q^{85} -0.532157 q^{86} -3.49192 q^{88} +2.74720 q^{89} -1.00000 q^{91} +10.8856 q^{92} -1.39372 q^{94} +2.29630 q^{95} +7.16827 q^{97} -0.239123 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - q^{2} + 3 q^{4} - 5 q^{5} - 3 q^{7} - 6 q^{8}+O(q^{10})$$ 3 * q - q^2 + 3 * q^4 - 5 * q^5 - 3 * q^7 - 6 * q^8 $$3 q - q^{2} + 3 q^{4} - 5 q^{5} - 3 q^{7} - 6 q^{8} - 2 q^{11} + 3 q^{13} + q^{14} + 3 q^{16} - 12 q^{17} - 3 q^{19} - 16 q^{20} - 15 q^{22} + 6 q^{25} - q^{26} - 3 q^{28} + q^{29} - 3 q^{31} - 8 q^{32} - 3 q^{34} + 5 q^{35} - 3 q^{37} + 8 q^{38} + 21 q^{40} - 22 q^{41} - 3 q^{43} + 23 q^{44} + 12 q^{46} - 9 q^{47} + 3 q^{49} + 10 q^{50} + 3 q^{52} - 18 q^{53} - 6 q^{55} + 6 q^{56} - 9 q^{58} - 9 q^{59} - 6 q^{61} - 18 q^{62} - 12 q^{64} - 5 q^{65} + 6 q^{68} + 9 q^{71} + 3 q^{73} + 6 q^{74} - 21 q^{76} + 2 q^{77} + 15 q^{79} + 11 q^{80} + 9 q^{82} - 12 q^{83} + 9 q^{85} + 34 q^{86} - 21 q^{88} - 2 q^{89} - 3 q^{91} + 15 q^{92} + 24 q^{94} + 16 q^{95} + 3 q^{97} - q^{98}+O(q^{100})$$ 3 * q - q^2 + 3 * q^4 - 5 * q^5 - 3 * q^7 - 6 * q^8 - 2 * q^11 + 3 * q^13 + q^14 + 3 * q^16 - 12 * q^17 - 3 * q^19 - 16 * q^20 - 15 * q^22 + 6 * q^25 - q^26 - 3 * q^28 + q^29 - 3 * q^31 - 8 * q^32 - 3 * q^34 + 5 * q^35 - 3 * q^37 + 8 * q^38 + 21 * q^40 - 22 * q^41 - 3 * q^43 + 23 * q^44 + 12 * q^46 - 9 * q^47 + 3 * q^49 + 10 * q^50 + 3 * q^52 - 18 * q^53 - 6 * q^55 + 6 * q^56 - 9 * q^58 - 9 * q^59 - 6 * q^61 - 18 * q^62 - 12 * q^64 - 5 * q^65 + 6 * q^68 + 9 * q^71 + 3 * q^73 + 6 * q^74 - 21 * q^76 + 2 * q^77 + 15 * q^79 + 11 * q^80 + 9 * q^82 - 12 * q^83 + 9 * q^85 + 34 * q^86 - 21 * q^88 - 2 * q^89 - 3 * q^91 + 15 * q^92 + 24 * q^94 + 16 * q^95 + 3 * q^97 - 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −0.239123 −0.169086 −0.0845428 0.996420i $$-0.526943\pi$$
−0.0845428 + 0.996420i $$0.526943\pi$$
$$3$$ 0 0
$$4$$ −1.94282 −0.971410
$$5$$ 1.18194 0.528581 0.264291 0.964443i $$-0.414862\pi$$
0.264291 + 0.964443i $$0.414862\pi$$
$$6$$ 0 0
$$7$$ −1.00000 −0.377964
$$8$$ 0.942820 0.333337
$$9$$ 0 0
$$10$$ −0.282630 −0.0893755
$$11$$ −3.70370 −1.11671 −0.558353 0.829603i $$-0.688567\pi$$
−0.558353 + 0.829603i $$0.688567\pi$$
$$12$$ 0 0
$$13$$ 1.00000 0.277350 0.138675 0.990338i $$-0.455716\pi$$
0.138675 + 0.990338i $$0.455716\pi$$
$$14$$ 0.239123 0.0639084
$$15$$ 0 0
$$16$$ 3.66019 0.915047
$$17$$ −6.94282 −1.68388 −0.841941 0.539570i $$-0.818587\pi$$
−0.841941 + 0.539570i $$0.818587\pi$$
$$18$$ 0 0
$$19$$ 1.94282 0.445713 0.222857 0.974851i $$-0.428462\pi$$
0.222857 + 0.974851i $$0.428462\pi$$
$$20$$ −2.29630 −0.513469
$$21$$ 0 0
$$22$$ 0.885640 0.188819
$$23$$ −5.60301 −1.16831 −0.584154 0.811643i $$-0.698574\pi$$
−0.584154 + 0.811643i $$0.698574\pi$$
$$24$$ 0 0
$$25$$ −3.60301 −0.720602
$$26$$ −0.239123 −0.0468959
$$27$$ 0 0
$$28$$ 1.94282 0.367158
$$29$$ 0.239123 0.0444041 0.0222020 0.999754i $$-0.492932\pi$$
0.0222020 + 0.999754i $$0.492932\pi$$
$$30$$ 0 0
$$31$$ 1.66019 0.298179 0.149089 0.988824i $$-0.452366\pi$$
0.149089 + 0.988824i $$0.452366\pi$$
$$32$$ −2.76088 −0.488059
$$33$$ 0 0
$$34$$ 1.66019 0.284720
$$35$$ −1.18194 −0.199785
$$36$$ 0 0
$$37$$ −9.54583 −1.56932 −0.784662 0.619923i $$-0.787164\pi$$
−0.784662 + 0.619923i $$0.787164\pi$$
$$38$$ −0.464574 −0.0753638
$$39$$ 0 0
$$40$$ 1.11436 0.176196
$$41$$ −10.1819 −1.59015 −0.795076 0.606510i $$-0.792569\pi$$
−0.795076 + 0.606510i $$0.792569\pi$$
$$42$$ 0 0
$$43$$ 2.22545 0.339378 0.169689 0.985498i $$-0.445724\pi$$
0.169689 + 0.985498i $$0.445724\pi$$
$$44$$ 7.19562 1.08478
$$45$$ 0 0
$$46$$ 1.33981 0.197544
$$47$$ 5.82846 0.850168 0.425084 0.905154i $$-0.360245\pi$$
0.425084 + 0.905154i $$0.360245\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0.861564 0.121843
$$51$$ 0 0
$$52$$ −1.94282 −0.269421
$$53$$ −11.6030 −1.59380 −0.796898 0.604114i $$-0.793527\pi$$
−0.796898 + 0.604114i $$0.793527\pi$$
$$54$$ 0 0
$$55$$ −4.37756 −0.590270
$$56$$ −0.942820 −0.125990
$$57$$ 0 0
$$58$$ −0.0571799 −0.00750809
$$59$$ 2.60301 0.338883 0.169442 0.985540i $$-0.445804\pi$$
0.169442 + 0.985540i $$0.445804\pi$$
$$60$$ 0 0
$$61$$ −7.60301 −0.973466 −0.486733 0.873551i $$-0.661811\pi$$
−0.486733 + 0.873551i $$0.661811\pi$$
$$62$$ −0.396990 −0.0504178
$$63$$ 0 0
$$64$$ −6.66019 −0.832524
$$65$$ 1.18194 0.146602
$$66$$ 0 0
$$67$$ 3.50808 0.428580 0.214290 0.976770i $$-0.431256\pi$$
0.214290 + 0.976770i $$0.431256\pi$$
$$68$$ 13.4887 1.63574
$$69$$ 0 0
$$70$$ 0.282630 0.0337808
$$71$$ 8.60301 1.02099 0.510495 0.859881i $$-0.329462\pi$$
0.510495 + 0.859881i $$0.329462\pi$$
$$72$$ 0 0
$$73$$ 15.1488 1.77304 0.886519 0.462693i $$-0.153117\pi$$
0.886519 + 0.462693i $$0.153117\pi$$
$$74$$ 2.28263 0.265350
$$75$$ 0 0
$$76$$ −3.77455 −0.432971
$$77$$ 3.70370 0.422075
$$78$$ 0 0
$$79$$ 7.37756 0.830040 0.415020 0.909812i $$-0.363775\pi$$
0.415020 + 0.909812i $$0.363775\pi$$
$$80$$ 4.32614 0.483677
$$81$$ 0 0
$$82$$ 2.43474 0.268872
$$83$$ −6.94282 −0.762074 −0.381037 0.924560i $$-0.624433\pi$$
−0.381037 + 0.924560i $$0.624433\pi$$
$$84$$ 0 0
$$85$$ −8.20602 −0.890068
$$86$$ −0.532157 −0.0573840
$$87$$ 0 0
$$88$$ −3.49192 −0.372240
$$89$$ 2.74720 0.291203 0.145602 0.989343i $$-0.453488\pi$$
0.145602 + 0.989343i $$0.453488\pi$$
$$90$$ 0 0
$$91$$ −1.00000 −0.104828
$$92$$ 10.8856 1.13491
$$93$$ 0 0
$$94$$ −1.39372 −0.143751
$$95$$ 2.29630 0.235596
$$96$$ 0 0
$$97$$ 7.16827 0.727828 0.363914 0.931433i $$-0.381440\pi$$
0.363914 + 0.931433i $$0.381440\pi$$
$$98$$ −0.239123 −0.0241551
$$99$$ 0 0
$$100$$ 7.00000 0.700000
$$101$$ −12.7850 −1.27215 −0.636075 0.771627i $$-0.719443\pi$$
−0.636075 + 0.771627i $$0.719443\pi$$
$$102$$ 0 0
$$103$$ −4.39699 −0.433248 −0.216624 0.976255i $$-0.569505\pi$$
−0.216624 + 0.976255i $$0.569505\pi$$
$$104$$ 0.942820 0.0924511
$$105$$ 0 0
$$106$$ 2.77455 0.269488
$$107$$ 13.7278 1.32711 0.663557 0.748126i $$-0.269046\pi$$
0.663557 + 0.748126i $$0.269046\pi$$
$$108$$ 0 0
$$109$$ 1.26320 0.120993 0.0604963 0.998168i $$-0.480732\pi$$
0.0604963 + 0.998168i $$0.480732\pi$$
$$110$$ 1.04678 0.0998062
$$111$$ 0 0
$$112$$ −3.66019 −0.345855
$$113$$ 12.1625 1.14415 0.572076 0.820200i $$-0.306138\pi$$
0.572076 + 0.820200i $$0.306138\pi$$
$$114$$ 0 0
$$115$$ −6.62244 −0.617546
$$116$$ −0.464574 −0.0431346
$$117$$ 0 0
$$118$$ −0.622440 −0.0573003
$$119$$ 6.94282 0.636447
$$120$$ 0 0
$$121$$ 2.71737 0.247034
$$122$$ 1.81806 0.164599
$$123$$ 0 0
$$124$$ −3.22545 −0.289654
$$125$$ −10.1683 −0.909478
$$126$$ 0 0
$$127$$ 1.33981 0.118889 0.0594445 0.998232i $$-0.481067\pi$$
0.0594445 + 0.998232i $$0.481067\pi$$
$$128$$ 7.11436 0.628827
$$129$$ 0 0
$$130$$ −0.282630 −0.0247883
$$131$$ −4.96690 −0.433960 −0.216980 0.976176i $$-0.569621\pi$$
−0.216980 + 0.976176i $$0.569621\pi$$
$$132$$ 0 0
$$133$$ −1.94282 −0.168464
$$134$$ −0.838864 −0.0724668
$$135$$ 0 0
$$136$$ −6.54583 −0.561300
$$137$$ −4.33981 −0.370775 −0.185387 0.982665i $$-0.559354\pi$$
−0.185387 + 0.982665i $$0.559354\pi$$
$$138$$ 0 0
$$139$$ 3.94282 0.334426 0.167213 0.985921i $$-0.446523\pi$$
0.167213 + 0.985921i $$0.446523\pi$$
$$140$$ 2.29630 0.194073
$$141$$ 0 0
$$142$$ −2.05718 −0.172635
$$143$$ −3.70370 −0.309719
$$144$$ 0 0
$$145$$ 0.282630 0.0234712
$$146$$ −3.62244 −0.299795
$$147$$ 0 0
$$148$$ 18.5458 1.52446
$$149$$ 11.1111 0.910256 0.455128 0.890426i $$-0.349594\pi$$
0.455128 + 0.890426i $$0.349594\pi$$
$$150$$ 0 0
$$151$$ 13.9234 1.13307 0.566535 0.824038i $$-0.308284\pi$$
0.566535 + 0.824038i $$0.308284\pi$$
$$152$$ 1.83173 0.148573
$$153$$ 0 0
$$154$$ −0.885640 −0.0713669
$$155$$ 1.96225 0.157612
$$156$$ 0 0
$$157$$ 0.0571799 0.00456346 0.00228173 0.999997i $$-0.499274\pi$$
0.00228173 + 0.999997i $$0.499274\pi$$
$$158$$ −1.76415 −0.140348
$$159$$ 0 0
$$160$$ −3.26320 −0.257979
$$161$$ 5.60301 0.441579
$$162$$ 0 0
$$163$$ −1.50808 −0.118122 −0.0590610 0.998254i $$-0.518811\pi$$
−0.0590610 + 0.998254i $$0.518811\pi$$
$$164$$ 19.7817 1.54469
$$165$$ 0 0
$$166$$ 1.66019 0.128856
$$167$$ −14.6843 −1.13630 −0.568151 0.822924i $$-0.692341\pi$$
−0.568151 + 0.822924i $$0.692341\pi$$
$$168$$ 0 0
$$169$$ −12.0000 −0.923077
$$170$$ 1.96225 0.150498
$$171$$ 0 0
$$172$$ −4.32365 −0.329675
$$173$$ 0.252796 0.0192197 0.00960987 0.999954i $$-0.496941\pi$$
0.00960987 + 0.999954i $$0.496941\pi$$
$$174$$ 0 0
$$175$$ 3.60301 0.272362
$$176$$ −13.5562 −1.02184
$$177$$ 0 0
$$178$$ −0.656920 −0.0492383
$$179$$ 14.1923 1.06079 0.530393 0.847752i $$-0.322044\pi$$
0.530393 + 0.847752i $$0.322044\pi$$
$$180$$ 0 0
$$181$$ −1.43147 −0.106400 −0.0532002 0.998584i $$-0.516942\pi$$
−0.0532002 + 0.998584i $$0.516942\pi$$
$$182$$ 0.239123 0.0177250
$$183$$ 0 0
$$184$$ −5.28263 −0.389441
$$185$$ −11.2826 −0.829515
$$186$$ 0 0
$$187$$ 25.7141 1.88040
$$188$$ −11.3236 −0.825862
$$189$$ 0 0
$$190$$ −0.549100 −0.0398359
$$191$$ 15.0676 1.09025 0.545126 0.838354i $$-0.316482\pi$$
0.545126 + 0.838354i $$0.316482\pi$$
$$192$$ 0 0
$$193$$ −7.84789 −0.564904 −0.282452 0.959282i $$-0.591148\pi$$
−0.282452 + 0.959282i $$0.591148\pi$$
$$194$$ −1.71410 −0.123065
$$195$$ 0 0
$$196$$ −1.94282 −0.138773
$$197$$ 6.69002 0.476644 0.238322 0.971186i $$-0.423403\pi$$
0.238322 + 0.971186i $$0.423403\pi$$
$$198$$ 0 0
$$199$$ −19.9396 −1.41348 −0.706739 0.707475i $$-0.749834\pi$$
−0.706739 + 0.707475i $$0.749834\pi$$
$$200$$ −3.39699 −0.240203
$$201$$ 0 0
$$202$$ 3.05718 0.215102
$$203$$ −0.239123 −0.0167832
$$204$$ 0 0
$$205$$ −12.0345 −0.840525
$$206$$ 1.05142 0.0732561
$$207$$ 0 0
$$208$$ 3.66019 0.253789
$$209$$ −7.19562 −0.497731
$$210$$ 0 0
$$211$$ −18.0917 −1.24548 −0.622741 0.782428i $$-0.713981\pi$$
−0.622741 + 0.782428i $$0.713981\pi$$
$$212$$ 22.5426 1.54823
$$213$$ 0 0
$$214$$ −3.28263 −0.224396
$$215$$ 2.63036 0.179389
$$216$$ 0 0
$$217$$ −1.66019 −0.112701
$$218$$ −0.302060 −0.0204581
$$219$$ 0 0
$$220$$ 8.50481 0.573394
$$221$$ −6.94282 −0.467025
$$222$$ 0 0
$$223$$ −22.6569 −1.51722 −0.758610 0.651545i $$-0.774121\pi$$
−0.758610 + 0.651545i $$0.774121\pi$$
$$224$$ 2.76088 0.184469
$$225$$ 0 0
$$226$$ −2.90834 −0.193460
$$227$$ −5.28263 −0.350620 −0.175310 0.984513i $$-0.556093\pi$$
−0.175310 + 0.984513i $$0.556093\pi$$
$$228$$ 0 0
$$229$$ 19.3365 1.27779 0.638897 0.769292i $$-0.279391\pi$$
0.638897 + 0.769292i $$0.279391\pi$$
$$230$$ 1.58358 0.104418
$$231$$ 0 0
$$232$$ 0.225450 0.0148015
$$233$$ −16.9806 −1.11243 −0.556217 0.831037i $$-0.687748\pi$$
−0.556217 + 0.831037i $$0.687748\pi$$
$$234$$ 0 0
$$235$$ 6.88891 0.449383
$$236$$ −5.05718 −0.329194
$$237$$ 0 0
$$238$$ −1.66019 −0.107614
$$239$$ 16.8856 1.09224 0.546121 0.837707i $$-0.316104\pi$$
0.546121 + 0.837707i $$0.316104\pi$$
$$240$$ 0 0
$$241$$ 27.1456 1.74860 0.874300 0.485386i $$-0.161321\pi$$
0.874300 + 0.485386i $$0.161321\pi$$
$$242$$ −0.649786 −0.0417699
$$243$$ 0 0
$$244$$ 14.7713 0.945634
$$245$$ 1.18194 0.0755116
$$246$$ 0 0
$$247$$ 1.94282 0.123619
$$248$$ 1.56526 0.0993941
$$249$$ 0 0
$$250$$ 2.43147 0.153780
$$251$$ −19.0780 −1.20419 −0.602096 0.798424i $$-0.705668\pi$$
−0.602096 + 0.798424i $$0.705668\pi$$
$$252$$ 0 0
$$253$$ 20.7518 1.30466
$$254$$ −0.320380 −0.0201024
$$255$$ 0 0
$$256$$ 11.6192 0.726198
$$257$$ −14.8421 −0.925827 −0.462913 0.886404i $$-0.653196\pi$$
−0.462913 + 0.886404i $$0.653196\pi$$
$$258$$ 0 0
$$259$$ 9.54583 0.593149
$$260$$ −2.29630 −0.142411
$$261$$ 0 0
$$262$$ 1.18770 0.0733764
$$263$$ −7.74145 −0.477358 −0.238679 0.971099i $$-0.576714\pi$$
−0.238679 + 0.971099i $$0.576714\pi$$
$$264$$ 0 0
$$265$$ −13.7141 −0.842450
$$266$$ 0.464574 0.0284848
$$267$$ 0 0
$$268$$ −6.81557 −0.416327
$$269$$ −1.51135 −0.0921486 −0.0460743 0.998938i $$-0.514671\pi$$
−0.0460743 + 0.998938i $$0.514671\pi$$
$$270$$ 0 0
$$271$$ −21.9806 −1.33522 −0.667612 0.744509i $$-0.732684\pi$$
−0.667612 + 0.744509i $$0.732684\pi$$
$$272$$ −25.4120 −1.54083
$$273$$ 0 0
$$274$$ 1.03775 0.0626927
$$275$$ 13.3445 0.804701
$$276$$ 0 0
$$277$$ −10.8285 −0.650619 −0.325310 0.945608i $$-0.605469\pi$$
−0.325310 + 0.945608i $$0.605469\pi$$
$$278$$ −0.942820 −0.0565466
$$279$$ 0 0
$$280$$ −1.11436 −0.0665957
$$281$$ −16.8766 −1.00677 −0.503387 0.864061i $$-0.667913\pi$$
−0.503387 + 0.864061i $$0.667913\pi$$
$$282$$ 0 0
$$283$$ −15.3171 −0.910508 −0.455254 0.890362i $$-0.650451\pi$$
−0.455254 + 0.890362i $$0.650451\pi$$
$$284$$ −16.7141 −0.991799
$$285$$ 0 0
$$286$$ 0.885640 0.0523690
$$287$$ 10.1819 0.601021
$$288$$ 0 0
$$289$$ 31.2028 1.83546
$$290$$ −0.0675835 −0.00396864
$$291$$ 0 0
$$292$$ −29.4315 −1.72235
$$293$$ −9.36964 −0.547380 −0.273690 0.961818i $$-0.588244\pi$$
−0.273690 + 0.961818i $$0.588244\pi$$
$$294$$ 0 0
$$295$$ 3.07661 0.179127
$$296$$ −9.00000 −0.523114
$$297$$ 0 0
$$298$$ −2.65692 −0.153911
$$299$$ −5.60301 −0.324030
$$300$$ 0 0
$$301$$ −2.22545 −0.128273
$$302$$ −3.32941 −0.191586
$$303$$ 0 0
$$304$$ 7.11109 0.407849
$$305$$ −8.98633 −0.514556
$$306$$ 0 0
$$307$$ 2.71410 0.154902 0.0774509 0.996996i $$-0.475322\pi$$
0.0774509 + 0.996996i $$0.475322\pi$$
$$308$$ −7.19562 −0.410008
$$309$$ 0 0
$$310$$ −0.469220 −0.0266499
$$311$$ −13.9806 −0.792765 −0.396383 0.918085i $$-0.629735\pi$$
−0.396383 + 0.918085i $$0.629735\pi$$
$$312$$ 0 0
$$313$$ −19.0539 −1.07699 −0.538495 0.842628i $$-0.681007\pi$$
−0.538495 + 0.842628i $$0.681007\pi$$
$$314$$ −0.0136731 −0.000771615 0
$$315$$ 0 0
$$316$$ −14.3333 −0.806309
$$317$$ −4.01943 −0.225754 −0.112877 0.993609i $$-0.536007\pi$$
−0.112877 + 0.993609i $$0.536007\pi$$
$$318$$ 0 0
$$319$$ −0.885640 −0.0495863
$$320$$ −7.87197 −0.440056
$$321$$ 0 0
$$322$$ −1.33981 −0.0746647
$$323$$ −13.4887 −0.750529
$$324$$ 0 0
$$325$$ −3.60301 −0.199859
$$326$$ 0.360617 0.0199727
$$327$$ 0 0
$$328$$ −9.59974 −0.530057
$$329$$ −5.82846 −0.321333
$$330$$ 0 0
$$331$$ −12.3776 −0.680332 −0.340166 0.940365i $$-0.610483\pi$$
−0.340166 + 0.940365i $$0.610483\pi$$
$$332$$ 13.4887 0.740286
$$333$$ 0 0
$$334$$ 3.51135 0.192133
$$335$$ 4.14635 0.226539
$$336$$ 0 0
$$337$$ 12.2599 0.667841 0.333920 0.942601i $$-0.391628\pi$$
0.333920 + 0.942601i $$0.391628\pi$$
$$338$$ 2.86948 0.156079
$$339$$ 0 0
$$340$$ 15.9428 0.864621
$$341$$ −6.14884 −0.332978
$$342$$ 0 0
$$343$$ −1.00000 −0.0539949
$$344$$ 2.09820 0.113127
$$345$$ 0 0
$$346$$ −0.0604495 −0.00324978
$$347$$ 6.64979 0.356979 0.178490 0.983942i $$-0.442879\pi$$
0.178490 + 0.983942i $$0.442879\pi$$
$$348$$ 0 0
$$349$$ 11.4347 0.612088 0.306044 0.952017i $$-0.400995\pi$$
0.306044 + 0.952017i $$0.400995\pi$$
$$350$$ −0.861564 −0.0460525
$$351$$ 0 0
$$352$$ 10.2255 0.545018
$$353$$ −22.1956 −1.18135 −0.590677 0.806908i $$-0.701139\pi$$
−0.590677 + 0.806908i $$0.701139\pi$$
$$354$$ 0 0
$$355$$ 10.1683 0.539676
$$356$$ −5.33732 −0.282878
$$357$$ 0 0
$$358$$ −3.39372 −0.179364
$$359$$ −7.55623 −0.398803 −0.199401 0.979918i $$-0.563900\pi$$
−0.199401 + 0.979918i $$0.563900\pi$$
$$360$$ 0 0
$$361$$ −15.2255 −0.801339
$$362$$ 0.342298 0.0179908
$$363$$ 0 0
$$364$$ 1.94282 0.101831
$$365$$ 17.9051 0.937194
$$366$$ 0 0
$$367$$ −18.5231 −0.966900 −0.483450 0.875372i $$-0.660616\pi$$
−0.483450 + 0.875372i $$0.660616\pi$$
$$368$$ −20.5081 −1.06906
$$369$$ 0 0
$$370$$ 2.69794 0.140259
$$371$$ 11.6030 0.602398
$$372$$ 0 0
$$373$$ 15.6602 0.810854 0.405427 0.914127i $$-0.367123\pi$$
0.405427 + 0.914127i $$0.367123\pi$$
$$374$$ −6.14884 −0.317949
$$375$$ 0 0
$$376$$ 5.49519 0.283393
$$377$$ 0.239123 0.0123155
$$378$$ 0 0
$$379$$ 4.03775 0.207405 0.103703 0.994608i $$-0.466931\pi$$
0.103703 + 0.994608i $$0.466931\pi$$
$$380$$ −4.46130 −0.228860
$$381$$ 0 0
$$382$$ −3.60301 −0.184346
$$383$$ 0.225450 0.0115200 0.00575998 0.999983i $$-0.498167\pi$$
0.00575998 + 0.999983i $$0.498167\pi$$
$$384$$ 0 0
$$385$$ 4.37756 0.223101
$$386$$ 1.87661 0.0955171
$$387$$ 0 0
$$388$$ −13.9267 −0.707019
$$389$$ 25.2632 1.28090 0.640448 0.768002i $$-0.278749\pi$$
0.640448 + 0.768002i $$0.278749\pi$$
$$390$$ 0 0
$$391$$ 38.9007 1.96729
$$392$$ 0.942820 0.0476196
$$393$$ 0 0
$$394$$ −1.59974 −0.0805938
$$395$$ 8.71986 0.438744
$$396$$ 0 0
$$397$$ −20.3009 −1.01888 −0.509438 0.860508i $$-0.670147\pi$$
−0.509438 + 0.860508i $$0.670147\pi$$
$$398$$ 4.76801 0.238999
$$399$$ 0 0
$$400$$ −13.1877 −0.659385
$$401$$ −15.2255 −0.760323 −0.380161 0.924920i $$-0.624132\pi$$
−0.380161 + 0.924920i $$0.624132\pi$$
$$402$$ 0 0
$$403$$ 1.66019 0.0826999
$$404$$ 24.8389 1.23578
$$405$$ 0 0
$$406$$ 0.0571799 0.00283779
$$407$$ 35.3549 1.75248
$$408$$ 0 0
$$409$$ 1.65692 0.0819294 0.0409647 0.999161i $$-0.486957\pi$$
0.0409647 + 0.999161i $$0.486957\pi$$
$$410$$ 2.87772 0.142121
$$411$$ 0 0
$$412$$ 8.54256 0.420862
$$413$$ −2.60301 −0.128086
$$414$$ 0 0
$$415$$ −8.20602 −0.402818
$$416$$ −2.76088 −0.135363
$$417$$ 0 0
$$418$$ 1.72064 0.0841592
$$419$$ 33.3743 1.63044 0.815220 0.579151i $$-0.196616\pi$$
0.815220 + 0.579151i $$0.196616\pi$$
$$420$$ 0 0
$$421$$ 18.2405 0.888988 0.444494 0.895782i $$-0.353384\pi$$
0.444494 + 0.895782i $$0.353384\pi$$
$$422$$ 4.32614 0.210593
$$423$$ 0 0
$$424$$ −10.9396 −0.531272
$$425$$ 25.0150 1.21341
$$426$$ 0 0
$$427$$ 7.60301 0.367935
$$428$$ −26.6706 −1.28917
$$429$$ 0 0
$$430$$ −0.628979 −0.0303321
$$431$$ 29.2826 1.41049 0.705247 0.708961i $$-0.250836\pi$$
0.705247 + 0.708961i $$0.250836\pi$$
$$432$$ 0 0
$$433$$ −12.2449 −0.588451 −0.294226 0.955736i $$-0.595062\pi$$
−0.294226 + 0.955736i $$0.595062\pi$$
$$434$$ 0.396990 0.0190561
$$435$$ 0 0
$$436$$ −2.45417 −0.117533
$$437$$ −10.8856 −0.520731
$$438$$ 0 0
$$439$$ −4.83173 −0.230606 −0.115303 0.993330i $$-0.536784\pi$$
−0.115303 + 0.993330i $$0.536784\pi$$
$$440$$ −4.12725 −0.196759
$$441$$ 0 0
$$442$$ 1.66019 0.0789672
$$443$$ 1.24488 0.0591461 0.0295730 0.999563i $$-0.490585\pi$$
0.0295730 + 0.999563i $$0.490585\pi$$
$$444$$ 0 0
$$445$$ 3.24704 0.153924
$$446$$ 5.41780 0.256540
$$447$$ 0 0
$$448$$ 6.66019 0.314664
$$449$$ 8.82846 0.416641 0.208320 0.978061i $$-0.433200\pi$$
0.208320 + 0.978061i $$0.433200\pi$$
$$450$$ 0 0
$$451$$ 37.7108 1.77573
$$452$$ −23.6296 −1.11144
$$453$$ 0 0
$$454$$ 1.26320 0.0592849
$$455$$ −1.18194 −0.0554104
$$456$$ 0 0
$$457$$ −10.5081 −0.491547 −0.245774 0.969327i $$-0.579042\pi$$
−0.245774 + 0.969327i $$0.579042\pi$$
$$458$$ −4.62382 −0.216057
$$459$$ 0 0
$$460$$ 12.8662 0.599890
$$461$$ 22.5516 1.05033 0.525166 0.851000i $$-0.324003\pi$$
0.525166 + 0.851000i $$0.324003\pi$$
$$462$$ 0 0
$$463$$ 10.3970 0.483189 0.241595 0.970377i $$-0.422330\pi$$
0.241595 + 0.970377i $$0.422330\pi$$
$$464$$ 0.875237 0.0406318
$$465$$ 0 0
$$466$$ 4.06045 0.188097
$$467$$ 13.3171 0.616242 0.308121 0.951347i $$-0.400300\pi$$
0.308121 + 0.951347i $$0.400300\pi$$
$$468$$ 0 0
$$469$$ −3.50808 −0.161988
$$470$$ −1.64730 −0.0759842
$$471$$ 0 0
$$472$$ 2.45417 0.112962
$$473$$ −8.24239 −0.378986
$$474$$ 0 0
$$475$$ −7.00000 −0.321182
$$476$$ −13.4887 −0.618251
$$477$$ 0 0
$$478$$ −4.03775 −0.184682
$$479$$ 14.5354 0.664141 0.332070 0.943255i $$-0.392253\pi$$
0.332070 + 0.943255i $$0.392253\pi$$
$$480$$ 0 0
$$481$$ −9.54583 −0.435252
$$482$$ −6.49114 −0.295663
$$483$$ 0 0
$$484$$ −5.27936 −0.239971
$$485$$ 8.47249 0.384716
$$486$$ 0 0
$$487$$ 13.0539 0.591529 0.295765 0.955261i $$-0.404426\pi$$
0.295765 + 0.955261i $$0.404426\pi$$
$$488$$ −7.16827 −0.324492
$$489$$ 0 0
$$490$$ −0.282630 −0.0127679
$$491$$ 19.3445 0.873003 0.436502 0.899704i $$-0.356217\pi$$
0.436502 + 0.899704i $$0.356217\pi$$
$$492$$ 0 0
$$493$$ −1.66019 −0.0747712
$$494$$ −0.464574 −0.0209022
$$495$$ 0 0
$$496$$ 6.07661 0.272848
$$497$$ −8.60301 −0.385898
$$498$$ 0 0
$$499$$ −36.2222 −1.62153 −0.810764 0.585374i $$-0.800948\pi$$
−0.810764 + 0.585374i $$0.800948\pi$$
$$500$$ 19.7551 0.883476
$$501$$ 0 0
$$502$$ 4.56199 0.203612
$$503$$ 15.6764 0.698974 0.349487 0.936941i $$-0.386356\pi$$
0.349487 + 0.936941i $$0.386356\pi$$
$$504$$ 0 0
$$505$$ −15.1111 −0.672435
$$506$$ −4.96225 −0.220599
$$507$$ 0 0
$$508$$ −2.60301 −0.115490
$$509$$ 34.3034 1.52047 0.760237 0.649646i $$-0.225083\pi$$
0.760237 + 0.649646i $$0.225083\pi$$
$$510$$ 0 0
$$511$$ −15.1488 −0.670145
$$512$$ −17.0071 −0.751616
$$513$$ 0 0
$$514$$ 3.54910 0.156544
$$515$$ −5.19699 −0.229007
$$516$$ 0 0
$$517$$ −21.5868 −0.949389
$$518$$ −2.28263 −0.100293
$$519$$ 0 0
$$520$$ 1.11436 0.0488679
$$521$$ −10.2449 −0.448836 −0.224418 0.974493i $$-0.572048\pi$$
−0.224418 + 0.974493i $$0.572048\pi$$
$$522$$ 0 0
$$523$$ 30.6030 1.33818 0.669088 0.743183i $$-0.266685\pi$$
0.669088 + 0.743183i $$0.266685\pi$$
$$524$$ 9.64979 0.421553
$$525$$ 0 0
$$526$$ 1.85116 0.0807144
$$527$$ −11.5264 −0.502098
$$528$$ 0 0
$$529$$ 8.39372 0.364944
$$530$$ 3.27936 0.142446
$$531$$ 0 0
$$532$$ 3.77455 0.163647
$$533$$ −10.1819 −0.441029
$$534$$ 0 0
$$535$$ 16.2255 0.701487
$$536$$ 3.30749 0.142862
$$537$$ 0 0
$$538$$ 0.361399 0.0155810
$$539$$ −3.70370 −0.159530
$$540$$ 0 0
$$541$$ −26.0917 −1.12177 −0.560884 0.827894i $$-0.689539\pi$$
−0.560884 + 0.827894i $$0.689539\pi$$
$$542$$ 5.25607 0.225767
$$543$$ 0 0
$$544$$ 19.1683 0.821833
$$545$$ 1.49303 0.0639544
$$546$$ 0 0
$$547$$ −10.9234 −0.467050 −0.233525 0.972351i $$-0.575026\pi$$
−0.233525 + 0.972351i $$0.575026\pi$$
$$548$$ 8.43147 0.360175
$$549$$ 0 0
$$550$$ −3.19097 −0.136063
$$551$$ 0.464574 0.0197915
$$552$$ 0 0
$$553$$ −7.37756 −0.313726
$$554$$ 2.58934 0.110010
$$555$$ 0 0
$$556$$ −7.66019 −0.324864
$$557$$ −13.9442 −0.590835 −0.295417 0.955368i $$-0.595459\pi$$
−0.295417 + 0.955368i $$0.595459\pi$$
$$558$$ 0 0
$$559$$ 2.22545 0.0941265
$$560$$ −4.32614 −0.182813
$$561$$ 0 0
$$562$$ 4.03559 0.170231
$$563$$ 30.2574 1.27520 0.637600 0.770368i $$-0.279927\pi$$
0.637600 + 0.770368i $$0.279927\pi$$
$$564$$ 0 0
$$565$$ 14.3754 0.604778
$$566$$ 3.66268 0.153954
$$567$$ 0 0
$$568$$ 8.11109 0.340334
$$569$$ −21.1352 −0.886032 −0.443016 0.896514i $$-0.646092\pi$$
−0.443016 + 0.896514i $$0.646092\pi$$
$$570$$ 0 0
$$571$$ −32.7863 −1.37207 −0.686033 0.727571i $$-0.740649\pi$$
−0.686033 + 0.727571i $$0.740649\pi$$
$$572$$ 7.19562 0.300864
$$573$$ 0 0
$$574$$ −2.43474 −0.101624
$$575$$ 20.1877 0.841885
$$576$$ 0 0
$$577$$ −17.3743 −0.723301 −0.361651 0.932314i $$-0.617787\pi$$
−0.361651 + 0.932314i $$0.617787\pi$$
$$578$$ −7.46130 −0.310349
$$579$$ 0 0
$$580$$ −0.549100 −0.0228001
$$581$$ 6.94282 0.288037
$$582$$ 0 0
$$583$$ 42.9740 1.77980
$$584$$ 14.2826 0.591019
$$585$$ 0 0
$$586$$ 2.24050 0.0925542
$$587$$ 16.9759 0.700671 0.350336 0.936624i $$-0.386068\pi$$
0.350336 + 0.936624i $$0.386068\pi$$
$$588$$ 0 0
$$589$$ 3.22545 0.132902
$$590$$ −0.735689 −0.0302878
$$591$$ 0 0
$$592$$ −34.9396 −1.43601
$$593$$ −13.0733 −0.536858 −0.268429 0.963300i $$-0.586504\pi$$
−0.268429 + 0.963300i $$0.586504\pi$$
$$594$$ 0 0
$$595$$ 8.20602 0.336414
$$596$$ −21.5868 −0.884232
$$597$$ 0 0
$$598$$ 1.33981 0.0547889
$$599$$ 29.2060 1.19333 0.596663 0.802492i $$-0.296493\pi$$
0.596663 + 0.802492i $$0.296493\pi$$
$$600$$ 0 0
$$601$$ 7.79071 0.317790 0.158895 0.987296i $$-0.449207\pi$$
0.158895 + 0.987296i $$0.449207\pi$$
$$602$$ 0.532157 0.0216891
$$603$$ 0 0
$$604$$ −27.0506 −1.10067
$$605$$ 3.21178 0.130577
$$606$$ 0 0
$$607$$ −19.6408 −0.797194 −0.398597 0.917126i $$-0.630503\pi$$
−0.398597 + 0.917126i $$0.630503\pi$$
$$608$$ −5.36389 −0.217534
$$609$$ 0 0
$$610$$ 2.14884 0.0870040
$$611$$ 5.82846 0.235794
$$612$$ 0 0
$$613$$ 23.5653 0.951792 0.475896 0.879502i $$-0.342124\pi$$
0.475896 + 0.879502i $$0.342124\pi$$
$$614$$ −0.649005 −0.0261917
$$615$$ 0 0
$$616$$ 3.49192 0.140693
$$617$$ −10.6602 −0.429163 −0.214582 0.976706i $$-0.568839\pi$$
−0.214582 + 0.976706i $$0.568839\pi$$
$$618$$ 0 0
$$619$$ −18.0150 −0.724086 −0.362043 0.932161i $$-0.617921\pi$$
−0.362043 + 0.932161i $$0.617921\pi$$
$$620$$ −3.81230 −0.153106
$$621$$ 0 0
$$622$$ 3.34308 0.134045
$$623$$ −2.74720 −0.110064
$$624$$ 0 0
$$625$$ 5.99673 0.239869
$$626$$ 4.55623 0.182104
$$627$$ 0 0
$$628$$ −0.111090 −0.00443299
$$629$$ 66.2750 2.64256
$$630$$ 0 0
$$631$$ 12.4703 0.496436 0.248218 0.968704i $$-0.420155\pi$$
0.248218 + 0.968704i $$0.420155\pi$$
$$632$$ 6.95571 0.276683
$$633$$ 0 0
$$634$$ 0.961139 0.0381717
$$635$$ 1.58358 0.0628424
$$636$$ 0 0
$$637$$ 1.00000 0.0396214
$$638$$ 0.211777 0.00838434
$$639$$ 0 0
$$640$$ 8.40877 0.332386
$$641$$ 19.1456 0.756205 0.378102 0.925764i $$-0.376577\pi$$
0.378102 + 0.925764i $$0.376577\pi$$
$$642$$ 0 0
$$643$$ 6.48865 0.255887 0.127944 0.991781i $$-0.459162\pi$$
0.127944 + 0.991781i $$0.459162\pi$$
$$644$$ −10.8856 −0.428954
$$645$$ 0 0
$$646$$ 3.22545 0.126904
$$647$$ −48.0988 −1.89096 −0.945479 0.325682i $$-0.894406\pi$$
−0.945479 + 0.325682i $$0.894406\pi$$
$$648$$ 0 0
$$649$$ −9.64076 −0.378433
$$650$$ 0.861564 0.0337933
$$651$$ 0 0
$$652$$ 2.92993 0.114745
$$653$$ −43.2405 −1.69213 −0.846066 0.533079i $$-0.821035\pi$$
−0.846066 + 0.533079i $$0.821035\pi$$
$$654$$ 0 0
$$655$$ −5.87059 −0.229383
$$656$$ −37.2678 −1.45506
$$657$$ 0 0
$$658$$ 1.39372 0.0543329
$$659$$ −2.50808 −0.0977009 −0.0488505 0.998806i $$-0.515556\pi$$
−0.0488505 + 0.998806i $$0.515556\pi$$
$$660$$ 0 0
$$661$$ −42.3354 −1.64666 −0.823329 0.567565i $$-0.807886\pi$$
−0.823329 + 0.567565i $$0.807886\pi$$
$$662$$ 2.95976 0.115034
$$663$$ 0 0
$$664$$ −6.54583 −0.254027
$$665$$ −2.29630 −0.0890468
$$666$$ 0 0
$$667$$ −1.33981 −0.0518777
$$668$$ 28.5289 1.10382
$$669$$ 0 0
$$670$$ −0.991489 −0.0383046
$$671$$ 28.1592 1.08708
$$672$$ 0 0
$$673$$ 13.4153 0.517122 0.258561 0.965995i $$-0.416752\pi$$
0.258561 + 0.965995i $$0.416752\pi$$
$$674$$ −2.93163 −0.112922
$$675$$ 0 0
$$676$$ 23.3138 0.896686
$$677$$ 1.96225 0.0754154 0.0377077 0.999289i $$-0.487994\pi$$
0.0377077 + 0.999289i $$0.487994\pi$$
$$678$$ 0 0
$$679$$ −7.16827 −0.275093
$$680$$ −7.73680 −0.296693
$$681$$ 0 0
$$682$$ 1.47033 0.0563019
$$683$$ 27.1672 1.03952 0.519761 0.854312i $$-0.326021\pi$$
0.519761 + 0.854312i $$0.326021\pi$$
$$684$$ 0 0
$$685$$ −5.12941 −0.195985
$$686$$ 0.239123 0.00912977
$$687$$ 0 0
$$688$$ 8.14557 0.310547
$$689$$ −11.6030 −0.442039
$$690$$ 0 0
$$691$$ −50.3171 −1.91415 −0.957077 0.289835i $$-0.906399\pi$$
−0.957077 + 0.289835i $$0.906399\pi$$
$$692$$ −0.491138 −0.0186703
$$693$$ 0 0
$$694$$ −1.59012 −0.0603601
$$695$$ 4.66019 0.176771
$$696$$ 0 0
$$697$$ 70.6914 2.67763
$$698$$ −2.73431 −0.103495
$$699$$ 0 0
$$700$$ −7.00000 −0.264575
$$701$$ 45.1672 1.70594 0.852970 0.521960i $$-0.174799\pi$$
0.852970 + 0.521960i $$0.174799\pi$$
$$702$$ 0 0
$$703$$ −18.5458 −0.699469
$$704$$ 24.6673 0.929685
$$705$$ 0 0
$$706$$ 5.30749 0.199750
$$707$$ 12.7850 0.480828
$$708$$ 0 0
$$709$$ 39.6181 1.48789 0.743944 0.668242i $$-0.232953\pi$$
0.743944 + 0.668242i $$0.232953\pi$$
$$710$$ −2.43147 −0.0912514
$$711$$ 0 0
$$712$$ 2.59012 0.0970688
$$713$$ −9.30206 −0.348365
$$714$$ 0 0
$$715$$ −4.37756 −0.163711
$$716$$ −27.5732 −1.03046
$$717$$ 0 0
$$718$$ 1.80687 0.0674318
$$719$$ −22.0377 −0.821869 −0.410935 0.911665i $$-0.634798\pi$$
−0.410935 + 0.911665i $$0.634798\pi$$
$$720$$ 0 0
$$721$$ 4.39699 0.163752
$$722$$ 3.64076 0.135495
$$723$$ 0 0
$$724$$ 2.78109 0.103358
$$725$$ −0.861564 −0.0319977
$$726$$ 0 0
$$727$$ 28.1111 1.04258 0.521291 0.853379i $$-0.325450\pi$$
0.521291 + 0.853379i $$0.325450\pi$$
$$728$$ −0.942820 −0.0349432
$$729$$ 0 0
$$730$$ −4.28152 −0.158466
$$731$$ −15.4509 −0.571472
$$732$$ 0 0
$$733$$ 11.8695 0.438409 0.219205 0.975679i $$-0.429654\pi$$
0.219205 + 0.975679i $$0.429654\pi$$
$$734$$ 4.42931 0.163489
$$735$$ 0 0
$$736$$ 15.4692 0.570203
$$737$$ −12.9929 −0.478598
$$738$$ 0 0
$$739$$ −12.1844 −0.448212 −0.224106 0.974565i $$-0.571946\pi$$
−0.224106 + 0.974565i $$0.571946\pi$$
$$740$$ 21.9201 0.805800
$$741$$ 0 0
$$742$$ −2.77455 −0.101857
$$743$$ −44.4854 −1.63201 −0.816005 0.578045i $$-0.803816\pi$$
−0.816005 + 0.578045i $$0.803816\pi$$
$$744$$ 0 0
$$745$$ 13.1327 0.481144
$$746$$ −3.74472 −0.137104
$$747$$ 0 0
$$748$$ −49.9579 −1.82664
$$749$$ −13.7278 −0.501602
$$750$$ 0 0
$$751$$ 42.8058 1.56200 0.781002 0.624528i $$-0.214709\pi$$
0.781002 + 0.624528i $$0.214709\pi$$
$$752$$ 21.3333 0.777944
$$753$$ 0 0
$$754$$ −0.0571799 −0.00208237
$$755$$ 16.4567 0.598919
$$756$$ 0 0
$$757$$ −22.4919 −0.817483 −0.408741 0.912650i $$-0.634032\pi$$
−0.408741 + 0.912650i $$0.634032\pi$$
$$758$$ −0.965520 −0.0350693
$$759$$ 0 0
$$760$$ 2.16500 0.0785328
$$761$$ −14.3365 −0.519699 −0.259850 0.965649i $$-0.583673\pi$$
−0.259850 + 0.965649i $$0.583673\pi$$
$$762$$ 0 0
$$763$$ −1.26320 −0.0457309
$$764$$ −29.2736 −1.05908
$$765$$ 0 0
$$766$$ −0.0539104 −0.00194786
$$767$$ 2.60301 0.0939892
$$768$$ 0 0
$$769$$ −31.2211 −1.12586 −0.562930 0.826504i $$-0.690326\pi$$
−0.562930 + 0.826504i $$0.690326\pi$$
$$770$$ −1.04678 −0.0377232
$$771$$ 0 0
$$772$$ 15.2470 0.548753
$$773$$ 4.38005 0.157539 0.0787697 0.996893i $$-0.474901\pi$$
0.0787697 + 0.996893i $$0.474901\pi$$
$$774$$ 0 0
$$775$$ −5.98168 −0.214868
$$776$$ 6.75839 0.242612
$$777$$ 0 0
$$778$$ −6.04102 −0.216581
$$779$$ −19.7817 −0.708752
$$780$$ 0 0
$$781$$ −31.8629 −1.14015
$$782$$ −9.30206 −0.332641
$$783$$ 0 0
$$784$$ 3.66019 0.130721
$$785$$ 0.0675835 0.00241216
$$786$$ 0 0
$$787$$ 27.6213 0.984594 0.492297 0.870427i $$-0.336157\pi$$
0.492297 + 0.870427i $$0.336157\pi$$
$$788$$ −12.9975 −0.463017
$$789$$ 0 0
$$790$$ −2.08512 −0.0741853
$$791$$ −12.1625 −0.432449
$$792$$ 0 0
$$793$$ −7.60301 −0.269991
$$794$$ 4.85443 0.172277
$$795$$ 0 0
$$796$$ 38.7390 1.37307
$$797$$ −2.96363 −0.104977 −0.0524885 0.998622i $$-0.516715\pi$$
−0.0524885 + 0.998622i $$0.516715\pi$$
$$798$$ 0 0
$$799$$ −40.4660 −1.43158
$$800$$ 9.94747 0.351696
$$801$$ 0 0
$$802$$ 3.64076 0.128560
$$803$$ −56.1067 −1.97996
$$804$$ 0 0
$$805$$ 6.62244 0.233410
$$806$$ −0.396990 −0.0139834
$$807$$ 0 0
$$808$$ −12.0539 −0.424055
$$809$$ −24.7896 −0.871556 −0.435778 0.900054i $$-0.643527\pi$$
−0.435778 + 0.900054i $$0.643527\pi$$
$$810$$ 0 0
$$811$$ 8.24377 0.289478 0.144739 0.989470i $$-0.453766\pi$$
0.144739 + 0.989470i $$0.453766\pi$$
$$812$$ 0.464574 0.0163033
$$813$$ 0 0
$$814$$ −8.45417 −0.296319
$$815$$ −1.78247 −0.0624370
$$816$$ 0 0
$$817$$ 4.32365 0.151265
$$818$$ −0.396208 −0.0138531
$$819$$ 0 0
$$820$$ 23.3808 0.816494
$$821$$ −28.8993 −1.00859 −0.504296 0.863531i $$-0.668248\pi$$
−0.504296 + 0.863531i $$0.668248\pi$$
$$822$$ 0 0
$$823$$ −36.0000 −1.25488 −0.627441 0.778664i $$-0.715897\pi$$
−0.627441 + 0.778664i $$0.715897\pi$$
$$824$$ −4.14557 −0.144418
$$825$$ 0 0
$$826$$ 0.622440 0.0216575
$$827$$ −50.7108 −1.76339 −0.881694 0.471821i $$-0.843597\pi$$
−0.881694 + 0.471821i $$0.843597\pi$$
$$828$$ 0 0
$$829$$ 14.8123 0.514452 0.257226 0.966351i $$-0.417191\pi$$
0.257226 + 0.966351i $$0.417191\pi$$
$$830$$ 1.96225 0.0681107
$$831$$ 0 0
$$832$$ −6.66019 −0.230901
$$833$$ −6.94282 −0.240554
$$834$$ 0 0
$$835$$ −17.3560 −0.600628
$$836$$ 13.9798 0.483501
$$837$$ 0 0
$$838$$ −7.98057 −0.275684
$$839$$ 33.7212 1.16419 0.582093 0.813122i $$-0.302234\pi$$
0.582093 + 0.813122i $$0.302234\pi$$
$$840$$ 0 0
$$841$$ −28.9428 −0.998028
$$842$$ −4.36173 −0.150315
$$843$$ 0 0
$$844$$ 35.1488 1.20987
$$845$$ −14.1833 −0.487921
$$846$$ 0 0
$$847$$ −2.71737 −0.0933699
$$848$$ −42.4692 −1.45840
$$849$$ 0 0
$$850$$ −5.98168 −0.205170
$$851$$ 53.4854 1.83346
$$852$$ 0 0
$$853$$ 11.7896 0.403668 0.201834 0.979420i $$-0.435310\pi$$
0.201834 + 0.979420i $$0.435310\pi$$
$$854$$ −1.81806 −0.0622126
$$855$$ 0 0
$$856$$ 12.9428 0.442376
$$857$$ −31.3261 −1.07008 −0.535040 0.844827i $$-0.679704\pi$$
−0.535040 + 0.844827i $$0.679704\pi$$
$$858$$ 0 0
$$859$$ −50.3893 −1.71926 −0.859631 0.510915i $$-0.829307\pi$$
−0.859631 + 0.510915i $$0.829307\pi$$
$$860$$ −5.11031 −0.174260
$$861$$ 0 0
$$862$$ −7.00216 −0.238494
$$863$$ 1.13268 0.0385568 0.0192784 0.999814i $$-0.493863\pi$$
0.0192784 + 0.999814i $$0.493863\pi$$
$$864$$ 0 0
$$865$$ 0.298791 0.0101592
$$866$$ 2.92804 0.0994987
$$867$$ 0 0
$$868$$ 3.22545 0.109479
$$869$$ −27.3242 −0.926911
$$870$$ 0 0
$$871$$ 3.50808 0.118867
$$872$$ 1.19097 0.0403313
$$873$$ 0 0
$$874$$ 2.60301 0.0880481
$$875$$ 10.1683 0.343750
$$876$$ 0 0
$$877$$ −27.3937 −0.925020 −0.462510 0.886614i $$-0.653051\pi$$
−0.462510 + 0.886614i $$0.653051\pi$$
$$878$$ 1.15538 0.0389922
$$879$$ 0 0
$$880$$ −16.0227 −0.540125
$$881$$ 1.20929 0.0407420 0.0203710 0.999792i $$-0.493515\pi$$
0.0203710 + 0.999792i $$0.493515\pi$$
$$882$$ 0 0
$$883$$ −51.0884 −1.71926 −0.859631 0.510916i $$-0.829306\pi$$
−0.859631 + 0.510916i $$0.829306\pi$$
$$884$$ 13.4887 0.453672
$$885$$ 0 0
$$886$$ −0.297680 −0.0100008
$$887$$ −41.5757 −1.39597 −0.697987 0.716110i $$-0.745921\pi$$
−0.697987 + 0.716110i $$0.745921\pi$$
$$888$$ 0 0
$$889$$ −1.33981 −0.0449358
$$890$$ −0.776443 −0.0260264
$$891$$ 0 0
$$892$$ 44.0183 1.47384
$$893$$ 11.3236 0.378931
$$894$$ 0 0
$$895$$ 16.7745 0.560711
$$896$$ −7.11436 −0.237674
$$897$$ 0 0
$$898$$ −2.11109 −0.0704480
$$899$$ 0.396990 0.0132404
$$900$$ 0 0
$$901$$ 80.5576 2.68376
$$902$$ −9.01754 −0.300251
$$903$$ 0 0
$$904$$ 11.4671 0.381389
$$905$$ −1.69192 −0.0562412
$$906$$ 0 0
$$907$$ 35.4509 1.17713 0.588564 0.808451i $$-0.299694\pi$$
0.588564 + 0.808451i $$0.299694\pi$$
$$908$$ 10.2632 0.340596
$$909$$ 0 0
$$910$$ 0.282630 0.00936910
$$911$$ −20.7108 −0.686180 −0.343090 0.939302i $$-0.611474\pi$$
−0.343090 + 0.939302i $$0.611474\pi$$
$$912$$ 0 0
$$913$$ 25.7141 0.851013
$$914$$ 2.51273 0.0831136
$$915$$ 0 0
$$916$$ −37.5674 −1.24126
$$917$$ 4.96690 0.164021
$$918$$ 0 0
$$919$$ 14.3926 0.474768 0.237384 0.971416i $$-0.423710\pi$$
0.237384 + 0.971416i $$0.423710\pi$$
$$920$$ −6.24377 −0.205851
$$921$$ 0 0
$$922$$ −5.39261 −0.177596
$$923$$ 8.60301 0.283172
$$924$$ 0 0
$$925$$ 34.3937 1.13086
$$926$$ −2.48616 −0.0817004
$$927$$ 0 0
$$928$$ −0.660190 −0.0216718
$$929$$ −41.7428 −1.36954 −0.684769 0.728760i $$-0.740097\pi$$
−0.684769 + 0.728760i $$0.740097\pi$$
$$930$$ 0 0
$$931$$ 1.94282 0.0636734
$$932$$ 32.9902 1.08063
$$933$$ 0 0
$$934$$ −3.18443 −0.104198
$$935$$ 30.3926 0.993945
$$936$$ 0 0
$$937$$ 3.17154 0.103610 0.0518048 0.998657i $$-0.483503\pi$$
0.0518048 + 0.998657i $$0.483503\pi$$
$$938$$ 0.838864 0.0273899
$$939$$ 0 0
$$940$$ −13.3839 −0.436535
$$941$$ −3.22080 −0.104995 −0.0524976 0.998621i $$-0.516718\pi$$
−0.0524976 + 0.998621i $$0.516718\pi$$
$$942$$ 0 0
$$943$$ 57.0495 1.85779
$$944$$ 9.52751 0.310094
$$945$$ 0 0
$$946$$ 1.97095 0.0640810
$$947$$ −45.3469 −1.47358 −0.736789 0.676123i $$-0.763659\pi$$
−0.736789 + 0.676123i $$0.763659\pi$$
$$948$$ 0 0
$$949$$ 15.1488 0.491752
$$950$$ 1.67386 0.0543073
$$951$$ 0 0
$$952$$ 6.54583 0.212152
$$953$$ −54.2703 −1.75799 −0.878994 0.476832i $$-0.841785\pi$$
−0.878994 + 0.476832i $$0.841785\pi$$
$$954$$ 0 0
$$955$$ 17.8090 0.576287
$$956$$ −32.8058 −1.06101
$$957$$ 0 0
$$958$$ −3.47576 −0.112297
$$959$$ 4.33981 0.140140
$$960$$ 0 0
$$961$$ −28.2438 −0.911089
$$962$$ 2.28263 0.0735950
$$963$$ 0 0
$$964$$ −52.7390 −1.69861
$$965$$ −9.27576 −0.298597
$$966$$ 0 0
$$967$$ 25.6591 0.825140 0.412570 0.910926i $$-0.364631\pi$$
0.412570 + 0.910926i $$0.364631\pi$$
$$968$$ 2.56199 0.0823455
$$969$$ 0 0
$$970$$ −2.02597 −0.0650500
$$971$$ −21.0183 −0.674510 −0.337255 0.941413i $$-0.609498\pi$$
−0.337255 + 0.941413i $$0.609498\pi$$
$$972$$ 0 0
$$973$$ −3.94282 −0.126401
$$974$$ −3.12149 −0.100019
$$975$$ 0 0
$$976$$ −27.8285 −0.890767
$$977$$ −2.09820 −0.0671273 −0.0335637 0.999437i $$-0.510686\pi$$
−0.0335637 + 0.999437i $$0.510686\pi$$
$$978$$ 0 0
$$979$$ −10.1748 −0.325188
$$980$$ −2.29630 −0.0733527
$$981$$ 0 0
$$982$$ −4.62571 −0.147612
$$983$$ 42.9923 1.37124 0.685622 0.727958i $$-0.259531\pi$$
0.685622 + 0.727958i $$0.259531\pi$$
$$984$$ 0 0
$$985$$ 7.90723 0.251945
$$986$$ 0.396990 0.0126427
$$987$$ 0 0
$$988$$ −3.77455 −0.120084
$$989$$ −12.4692 −0.396498
$$990$$ 0 0
$$991$$ −17.2632 −0.548384 −0.274192 0.961675i $$-0.588410\pi$$
−0.274192 + 0.961675i $$0.588410\pi$$
$$992$$ −4.58358 −0.145529
$$993$$ 0 0
$$994$$ 2.05718 0.0652498
$$995$$ −23.5674 −0.747137
$$996$$ 0 0
$$997$$ −38.9018 −1.23203 −0.616016 0.787733i $$-0.711254\pi$$
−0.616016 + 0.787733i $$0.711254\pi$$
$$998$$ 8.66157 0.274177
$$999$$ 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 567.2.a.d.1.2 3
3.2 odd 2 567.2.a.g.1.2 3
4.3 odd 2 9072.2.a.bq.1.3 3
7.6 odd 2 3969.2.a.m.1.2 3
9.2 odd 6 189.2.f.a.64.2 6
9.4 even 3 63.2.f.b.43.2 yes 6
9.5 odd 6 189.2.f.a.127.2 6
9.7 even 3 63.2.f.b.22.2 6
12.11 even 2 9072.2.a.cd.1.1 3
21.20 even 2 3969.2.a.p.1.2 3
36.7 odd 6 1008.2.r.k.337.2 6
36.11 even 6 3024.2.r.g.1009.3 6
36.23 even 6 3024.2.r.g.2017.3 6
36.31 odd 6 1008.2.r.k.673.2 6
63.2 odd 6 1323.2.g.c.361.2 6
63.4 even 3 441.2.g.e.79.2 6
63.5 even 6 1323.2.h.e.802.2 6
63.11 odd 6 1323.2.h.d.226.2 6
63.13 odd 6 441.2.f.d.295.2 6
63.16 even 3 441.2.g.e.67.2 6
63.20 even 6 1323.2.f.c.442.2 6
63.23 odd 6 1323.2.h.d.802.2 6
63.25 even 3 441.2.h.c.373.2 6
63.31 odd 6 441.2.g.d.79.2 6
63.32 odd 6 1323.2.g.c.667.2 6
63.34 odd 6 441.2.f.d.148.2 6
63.38 even 6 1323.2.h.e.226.2 6
63.40 odd 6 441.2.h.b.214.2 6
63.41 even 6 1323.2.f.c.883.2 6
63.47 even 6 1323.2.g.b.361.2 6
63.52 odd 6 441.2.h.b.373.2 6
63.58 even 3 441.2.h.c.214.2 6
63.59 even 6 1323.2.g.b.667.2 6
63.61 odd 6 441.2.g.d.67.2 6

By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.f.b.22.2 6 9.7 even 3
63.2.f.b.43.2 yes 6 9.4 even 3
189.2.f.a.64.2 6 9.2 odd 6
189.2.f.a.127.2 6 9.5 odd 6
441.2.f.d.148.2 6 63.34 odd 6
441.2.f.d.295.2 6 63.13 odd 6
441.2.g.d.67.2 6 63.61 odd 6
441.2.g.d.79.2 6 63.31 odd 6
441.2.g.e.67.2 6 63.16 even 3
441.2.g.e.79.2 6 63.4 even 3
441.2.h.b.214.2 6 63.40 odd 6
441.2.h.b.373.2 6 63.52 odd 6
441.2.h.c.214.2 6 63.58 even 3
441.2.h.c.373.2 6 63.25 even 3
567.2.a.d.1.2 3 1.1 even 1 trivial
567.2.a.g.1.2 3 3.2 odd 2
1008.2.r.k.337.2 6 36.7 odd 6
1008.2.r.k.673.2 6 36.31 odd 6
1323.2.f.c.442.2 6 63.20 even 6
1323.2.f.c.883.2 6 63.41 even 6
1323.2.g.b.361.2 6 63.47 even 6
1323.2.g.b.667.2 6 63.59 even 6
1323.2.g.c.361.2 6 63.2 odd 6
1323.2.g.c.667.2 6 63.32 odd 6
1323.2.h.d.226.2 6 63.11 odd 6
1323.2.h.d.802.2 6 63.23 odd 6
1323.2.h.e.226.2 6 63.38 even 6
1323.2.h.e.802.2 6 63.5 even 6
3024.2.r.g.1009.3 6 36.11 even 6
3024.2.r.g.2017.3 6 36.23 even 6
3969.2.a.m.1.2 3 7.6 odd 2
3969.2.a.p.1.2 3 21.20 even 2
9072.2.a.bq.1.3 3 4.3 odd 2
9072.2.a.cd.1.1 3 12.11 even 2