# Properties

 Label 2175.2.a.z.1.1 Level $2175$ Weight $2$ Character 2175.1 Self dual yes Analytic conductor $17.367$ Analytic rank $0$ Dimension $5$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2175,2,Mod(1,2175)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2175, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2175.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2175 = 3 \cdot 5^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2175.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: $$17.3674624396$$ Analytic rank: $$0$$ Dimension: $$5$$ Coefficient field: 5.5.246832.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{5} - 2x^{4} - 5x^{3} + 6x^{2} + 7x - 2$$ x^5 - 2*x^4 - 5*x^3 + 6*x^2 + 7*x - 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 435) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$2.71457$$ of defining polynomial Character $$\chi$$ $$=$$ 2175.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.71457 q^{2} +1.00000 q^{3} +0.939748 q^{4} -1.71457 q^{6} -0.654317 q^{7} +1.81788 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.71457 q^{2} +1.00000 q^{3} +0.939748 q^{4} -1.71457 q^{6} -0.654317 q^{7} +1.81788 q^{8} +1.00000 q^{9} +0.163559 q^{11} +0.939748 q^{12} +2.65432 q^{13} +1.12187 q^{14} -4.99637 q^{16} -3.86328 q^{17} -1.71457 q^{18} -3.47954 q^{19} -0.654317 q^{21} -0.280433 q^{22} +7.69972 q^{23} +1.81788 q^{24} -4.55101 q^{26} +1.00000 q^{27} -0.614893 q^{28} +1.00000 q^{29} +5.05274 q^{31} +4.93087 q^{32} +0.163559 q^{33} +6.62385 q^{34} +0.939748 q^{36} +10.5904 q^{37} +5.96591 q^{38} +2.65432 q^{39} -6.17417 q^{41} +1.12187 q^{42} -10.5547 q^{43} +0.153704 q^{44} -13.2017 q^{46} +10.3036 q^{47} -4.99637 q^{48} -6.57187 q^{49} -3.86328 q^{51} +2.49439 q^{52} +4.04766 q^{53} -1.71457 q^{54} -1.18947 q^{56} -3.47954 q^{57} -1.71457 q^{58} -0.328734 q^{59} -5.72054 q^{61} -8.66328 q^{62} -0.654317 q^{63} +1.53842 q^{64} -0.280433 q^{66} +3.51985 q^{67} -3.63050 q^{68} +7.69972 q^{69} +11.7457 q^{71} +1.81788 q^{72} -1.12143 q^{73} -18.1580 q^{74} -3.26989 q^{76} -0.107019 q^{77} -4.55101 q^{78} -12.4074 q^{79} +1.00000 q^{81} +10.5860 q^{82} +7.89306 q^{83} -0.614893 q^{84} +18.0968 q^{86} +1.00000 q^{87} +0.297329 q^{88} +5.04702 q^{89} -1.73677 q^{91} +7.23579 q^{92} +5.05274 q^{93} -17.6663 q^{94} +4.93087 q^{96} +8.49076 q^{97} +11.2679 q^{98} +0.163559 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q + 3 q^{2} + 5 q^{3} + 5 q^{4} + 3 q^{6} + 8 q^{7} + 9 q^{8} + 5 q^{9}+O(q^{10})$$ 5 * q + 3 * q^2 + 5 * q^3 + 5 * q^4 + 3 * q^6 + 8 * q^7 + 9 * q^8 + 5 * q^9 $$5 q + 3 q^{2} + 5 q^{3} + 5 q^{4} + 3 q^{6} + 8 q^{7} + 9 q^{8} + 5 q^{9} + 12 q^{11} + 5 q^{12} + 2 q^{13} + 6 q^{14} + q^{16} + 3 q^{18} - 2 q^{19} + 8 q^{21} + 14 q^{22} + 8 q^{23} + 9 q^{24} + 5 q^{27} - 6 q^{28} + 5 q^{29} + 2 q^{31} + q^{32} + 12 q^{33} + 4 q^{34} + 5 q^{36} + 16 q^{37} - 14 q^{38} + 2 q^{39} - 14 q^{41} + 6 q^{42} + 20 q^{44} - 6 q^{46} + 2 q^{47} + q^{48} - 7 q^{49} + 16 q^{52} + 26 q^{53} + 3 q^{54} - 2 q^{56} - 2 q^{57} + 3 q^{58} + 4 q^{59} - 12 q^{61} + 8 q^{63} - 9 q^{64} + 14 q^{66} + 12 q^{67} - 20 q^{68} + 8 q^{69} + 30 q^{71} + 9 q^{72} - 12 q^{73} + 2 q^{74} - 44 q^{76} + 18 q^{77} - 18 q^{79} + 5 q^{81} + 10 q^{82} + 2 q^{83} - 6 q^{84} + 30 q^{86} + 5 q^{87} + 42 q^{88} - 22 q^{89} - 12 q^{91} + 20 q^{92} + 2 q^{93} - 50 q^{94} + q^{96} + 20 q^{97} - 9 q^{98} + 12 q^{99}+O(q^{100})$$ 5 * q + 3 * q^2 + 5 * q^3 + 5 * q^4 + 3 * q^6 + 8 * q^7 + 9 * q^8 + 5 * q^9 + 12 * q^11 + 5 * q^12 + 2 * q^13 + 6 * q^14 + q^16 + 3 * q^18 - 2 * q^19 + 8 * q^21 + 14 * q^22 + 8 * q^23 + 9 * q^24 + 5 * q^27 - 6 * q^28 + 5 * q^29 + 2 * q^31 + q^32 + 12 * q^33 + 4 * q^34 + 5 * q^36 + 16 * q^37 - 14 * q^38 + 2 * q^39 - 14 * q^41 + 6 * q^42 + 20 * q^44 - 6 * q^46 + 2 * q^47 + q^48 - 7 * q^49 + 16 * q^52 + 26 * q^53 + 3 * q^54 - 2 * q^56 - 2 * q^57 + 3 * q^58 + 4 * q^59 - 12 * q^61 + 8 * q^63 - 9 * q^64 + 14 * q^66 + 12 * q^67 - 20 * q^68 + 8 * q^69 + 30 * q^71 + 9 * q^72 - 12 * q^73 + 2 * q^74 - 44 * q^76 + 18 * q^77 - 18 * q^79 + 5 * q^81 + 10 * q^82 + 2 * q^83 - 6 * q^84 + 30 * q^86 + 5 * q^87 + 42 * q^88 - 22 * q^89 - 12 * q^91 + 20 * q^92 + 2 * q^93 - 50 * q^94 + q^96 + 20 * q^97 - 9 * q^98 + 12 * q^99

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.71457 −1.21238 −0.606192 0.795319i $$-0.707304\pi$$
−0.606192 + 0.795319i $$0.707304\pi$$
$$3$$ 1.00000 0.577350
$$4$$ 0.939748 0.469874
$$5$$ 0 0
$$6$$ −1.71457 −0.699970
$$7$$ −0.654317 −0.247309 −0.123654 0.992325i $$-0.539461\pi$$
−0.123654 + 0.992325i $$0.539461\pi$$
$$8$$ 1.81788 0.642716
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 0.163559 0.0493148 0.0246574 0.999696i $$-0.492151\pi$$
0.0246574 + 0.999696i $$0.492151\pi$$
$$12$$ 0.939748 0.271282
$$13$$ 2.65432 0.736175 0.368088 0.929791i $$-0.380013\pi$$
0.368088 + 0.929791i $$0.380013\pi$$
$$14$$ 1.12187 0.299833
$$15$$ 0 0
$$16$$ −4.99637 −1.24909
$$17$$ −3.86328 −0.936982 −0.468491 0.883468i $$-0.655202\pi$$
−0.468491 + 0.883468i $$0.655202\pi$$
$$18$$ −1.71457 −0.404128
$$19$$ −3.47954 −0.798260 −0.399130 0.916894i $$-0.630688\pi$$
−0.399130 + 0.916894i $$0.630688\pi$$
$$20$$ 0 0
$$21$$ −0.654317 −0.142784
$$22$$ −0.280433 −0.0597884
$$23$$ 7.69972 1.60550 0.802751 0.596314i $$-0.203369\pi$$
0.802751 + 0.596314i $$0.203369\pi$$
$$24$$ 1.81788 0.371072
$$25$$ 0 0
$$26$$ −4.55101 −0.892527
$$27$$ 1.00000 0.192450
$$28$$ −0.614893 −0.116204
$$29$$ 1.00000 0.185695
$$30$$ 0 0
$$31$$ 5.05274 0.907499 0.453750 0.891129i $$-0.350086\pi$$
0.453750 + 0.891129i $$0.350086\pi$$
$$32$$ 4.93087 0.871663
$$33$$ 0.163559 0.0284719
$$34$$ 6.62385 1.13598
$$35$$ 0 0
$$36$$ 0.939748 0.156625
$$37$$ 10.5904 1.74106 0.870528 0.492118i $$-0.163777\pi$$
0.870528 + 0.492118i $$0.163777\pi$$
$$38$$ 5.96591 0.967798
$$39$$ 2.65432 0.425031
$$40$$ 0 0
$$41$$ −6.17417 −0.964244 −0.482122 0.876104i $$-0.660134\pi$$
−0.482122 + 0.876104i $$0.660134\pi$$
$$42$$ 1.12187 0.173109
$$43$$ −10.5547 −1.60958 −0.804790 0.593559i $$-0.797722\pi$$
−0.804790 + 0.593559i $$0.797722\pi$$
$$44$$ 0.153704 0.0231717
$$45$$ 0 0
$$46$$ −13.2017 −1.94648
$$47$$ 10.3036 1.50294 0.751470 0.659767i $$-0.229345\pi$$
0.751470 + 0.659767i $$0.229345\pi$$
$$48$$ −4.99637 −0.721164
$$49$$ −6.57187 −0.938838
$$50$$ 0 0
$$51$$ −3.86328 −0.540967
$$52$$ 2.49439 0.345909
$$53$$ 4.04766 0.555989 0.277995 0.960583i $$-0.410330\pi$$
0.277995 + 0.960583i $$0.410330\pi$$
$$54$$ −1.71457 −0.233323
$$55$$ 0 0
$$56$$ −1.18947 −0.158949
$$57$$ −3.47954 −0.460876
$$58$$ −1.71457 −0.225134
$$59$$ −0.328734 −0.0427975 −0.0213988 0.999771i $$-0.506812\pi$$
−0.0213988 + 0.999771i $$0.506812\pi$$
$$60$$ 0 0
$$61$$ −5.72054 −0.732441 −0.366220 0.930528i $$-0.619348\pi$$
−0.366220 + 0.930528i $$0.619348\pi$$
$$62$$ −8.66328 −1.10024
$$63$$ −0.654317 −0.0824362
$$64$$ 1.53842 0.192303
$$65$$ 0 0
$$66$$ −0.280433 −0.0345189
$$67$$ 3.51985 0.430019 0.215009 0.976612i $$-0.431022\pi$$
0.215009 + 0.976612i $$0.431022\pi$$
$$68$$ −3.63050 −0.440263
$$69$$ 7.69972 0.926937
$$70$$ 0 0
$$71$$ 11.7457 1.39396 0.696981 0.717090i $$-0.254526\pi$$
0.696981 + 0.717090i $$0.254526\pi$$
$$72$$ 1.81788 0.214239
$$73$$ −1.12143 −0.131253 −0.0656267 0.997844i $$-0.520905\pi$$
−0.0656267 + 0.997844i $$0.520905\pi$$
$$74$$ −18.1580 −2.11083
$$75$$ 0 0
$$76$$ −3.26989 −0.375082
$$77$$ −0.107019 −0.0121960
$$78$$ −4.55101 −0.515300
$$79$$ −12.4074 −1.39594 −0.697970 0.716127i $$-0.745913\pi$$
−0.697970 + 0.716127i $$0.745913\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 10.5860 1.16903
$$83$$ 7.89306 0.866376 0.433188 0.901304i $$-0.357389\pi$$
0.433188 + 0.901304i $$0.357389\pi$$
$$84$$ −0.614893 −0.0670903
$$85$$ 0 0
$$86$$ 18.0968 1.95143
$$87$$ 1.00000 0.107211
$$88$$ 0.297329 0.0316954
$$89$$ 5.04702 0.534983 0.267491 0.963560i $$-0.413805\pi$$
0.267491 + 0.963560i $$0.413805\pi$$
$$90$$ 0 0
$$91$$ −1.73677 −0.182062
$$92$$ 7.23579 0.754383
$$93$$ 5.05274 0.523945
$$94$$ −17.6663 −1.82214
$$95$$ 0 0
$$96$$ 4.93087 0.503255
$$97$$ 8.49076 0.862106 0.431053 0.902327i $$-0.358142\pi$$
0.431053 + 0.902327i $$0.358142\pi$$
$$98$$ 11.2679 1.13823
$$99$$ 0.163559 0.0164383
$$100$$ 0 0
$$101$$ 1.81126 0.180227 0.0901136 0.995931i $$-0.471277\pi$$
0.0901136 + 0.995931i $$0.471277\pi$$
$$102$$ 6.62385 0.655859
$$103$$ 5.80807 0.572286 0.286143 0.958187i $$-0.407627\pi$$
0.286143 + 0.958187i $$0.407627\pi$$
$$104$$ 4.82522 0.473152
$$105$$ 0 0
$$106$$ −6.94000 −0.674072
$$107$$ 3.48568 0.336973 0.168487 0.985704i $$-0.446112\pi$$
0.168487 + 0.985704i $$0.446112\pi$$
$$108$$ 0.939748 0.0904273
$$109$$ 8.11029 0.776825 0.388412 0.921486i $$-0.373024\pi$$
0.388412 + 0.921486i $$0.373024\pi$$
$$110$$ 0 0
$$111$$ 10.5904 1.00520
$$112$$ 3.26921 0.308911
$$113$$ 2.06715 0.194461 0.0972307 0.995262i $$-0.469002\pi$$
0.0972307 + 0.995262i $$0.469002\pi$$
$$114$$ 5.96591 0.558758
$$115$$ 0 0
$$116$$ 0.939748 0.0872534
$$117$$ 2.65432 0.245392
$$118$$ 0.563637 0.0518870
$$119$$ 2.52781 0.231724
$$120$$ 0 0
$$121$$ −10.9732 −0.997568
$$122$$ 9.80827 0.887999
$$123$$ −6.17417 −0.556706
$$124$$ 4.74830 0.426410
$$125$$ 0 0
$$126$$ 1.12187 0.0999443
$$127$$ −12.9962 −1.15323 −0.576613 0.817017i $$-0.695626\pi$$
−0.576613 + 0.817017i $$0.695626\pi$$
$$128$$ −12.4995 −1.10481
$$129$$ −10.5547 −0.929292
$$130$$ 0 0
$$131$$ 8.69003 0.759251 0.379626 0.925140i $$-0.376053\pi$$
0.379626 + 0.925140i $$0.376053\pi$$
$$132$$ 0.153704 0.0133782
$$133$$ 2.27672 0.197417
$$134$$ −6.03504 −0.521348
$$135$$ 0 0
$$136$$ −7.02295 −0.602213
$$137$$ 14.4550 1.23498 0.617489 0.786580i $$-0.288150\pi$$
0.617489 + 0.786580i $$0.288150\pi$$
$$138$$ −13.2017 −1.12380
$$139$$ −1.72208 −0.146065 −0.0730324 0.997330i $$-0.523268\pi$$
−0.0730324 + 0.997330i $$0.523268\pi$$
$$140$$ 0 0
$$141$$ 10.3036 0.867723
$$142$$ −20.1389 −1.69002
$$143$$ 0.434137 0.0363043
$$144$$ −4.99637 −0.416364
$$145$$ 0 0
$$146$$ 1.92277 0.159129
$$147$$ −6.57187 −0.542039
$$148$$ 9.95234 0.818077
$$149$$ 16.1461 1.32274 0.661369 0.750060i $$-0.269976\pi$$
0.661369 + 0.750060i $$0.269976\pi$$
$$150$$ 0 0
$$151$$ −0.733083 −0.0596574 −0.0298287 0.999555i $$-0.509496\pi$$
−0.0298287 + 0.999555i $$0.509496\pi$$
$$152$$ −6.32536 −0.513055
$$153$$ −3.86328 −0.312327
$$154$$ 0.183492 0.0147862
$$155$$ 0 0
$$156$$ 2.49439 0.199711
$$157$$ 10.8270 0.864085 0.432043 0.901853i $$-0.357793\pi$$
0.432043 + 0.901853i $$0.357793\pi$$
$$158$$ 21.2733 1.69241
$$159$$ 4.04766 0.321000
$$160$$ 0 0
$$161$$ −5.03806 −0.397054
$$162$$ −1.71457 −0.134709
$$163$$ −13.5528 −1.06154 −0.530768 0.847517i $$-0.678096\pi$$
−0.530768 + 0.847517i $$0.678096\pi$$
$$164$$ −5.80216 −0.453073
$$165$$ 0 0
$$166$$ −13.5332 −1.05038
$$167$$ 18.4671 1.42902 0.714512 0.699623i $$-0.246649\pi$$
0.714512 + 0.699623i $$0.246649\pi$$
$$168$$ −1.18947 −0.0917694
$$169$$ −5.95460 −0.458046
$$170$$ 0 0
$$171$$ −3.47954 −0.266087
$$172$$ −9.91878 −0.756300
$$173$$ −8.92785 −0.678772 −0.339386 0.940647i $$-0.610219\pi$$
−0.339386 + 0.940647i $$0.610219\pi$$
$$174$$ −1.71457 −0.129981
$$175$$ 0 0
$$176$$ −0.817200 −0.0615987
$$177$$ −0.328734 −0.0247092
$$178$$ −8.65346 −0.648604
$$179$$ −1.86896 −0.139693 −0.0698465 0.997558i $$-0.522251\pi$$
−0.0698465 + 0.997558i $$0.522251\pi$$
$$180$$ 0 0
$$181$$ 16.8852 1.25507 0.627533 0.778590i $$-0.284065\pi$$
0.627533 + 0.778590i $$0.284065\pi$$
$$182$$ 2.97780 0.220730
$$183$$ −5.72054 −0.422875
$$184$$ 13.9971 1.03188
$$185$$ 0 0
$$186$$ −8.66328 −0.635222
$$187$$ −0.631872 −0.0462071
$$188$$ 9.68282 0.706192
$$189$$ −0.654317 −0.0475946
$$190$$ 0 0
$$191$$ 7.58677 0.548960 0.274480 0.961593i $$-0.411494\pi$$
0.274480 + 0.961593i $$0.411494\pi$$
$$192$$ 1.53842 0.111026
$$193$$ 27.2794 1.96362 0.981808 0.189876i $$-0.0608086\pi$$
0.981808 + 0.189876i $$0.0608086\pi$$
$$194$$ −14.5580 −1.04520
$$195$$ 0 0
$$196$$ −6.17590 −0.441136
$$197$$ −9.70626 −0.691542 −0.345771 0.938319i $$-0.612383\pi$$
−0.345771 + 0.938319i $$0.612383\pi$$
$$198$$ −0.280433 −0.0199295
$$199$$ 22.9520 1.62703 0.813513 0.581547i $$-0.197552\pi$$
0.813513 + 0.581547i $$0.197552\pi$$
$$200$$ 0 0
$$201$$ 3.51985 0.248271
$$202$$ −3.10553 −0.218504
$$203$$ −0.654317 −0.0459241
$$204$$ −3.63050 −0.254186
$$205$$ 0 0
$$206$$ −9.95834 −0.693831
$$207$$ 7.69972 0.535167
$$208$$ −13.2619 −0.919551
$$209$$ −0.569108 −0.0393660
$$210$$ 0 0
$$211$$ 18.0020 1.23931 0.619655 0.784874i $$-0.287273\pi$$
0.619655 + 0.784874i $$0.287273\pi$$
$$212$$ 3.80378 0.261245
$$213$$ 11.7457 0.804804
$$214$$ −5.97644 −0.408541
$$215$$ 0 0
$$216$$ 1.81788 0.123691
$$217$$ −3.30610 −0.224432
$$218$$ −13.9057 −0.941810
$$219$$ −1.12143 −0.0757792
$$220$$ 0 0
$$221$$ −10.2544 −0.689783
$$222$$ −18.1580 −1.21869
$$223$$ −19.1383 −1.28160 −0.640799 0.767708i $$-0.721397\pi$$
−0.640799 + 0.767708i $$0.721397\pi$$
$$224$$ −3.22635 −0.215570
$$225$$ 0 0
$$226$$ −3.54428 −0.235762
$$227$$ −13.3621 −0.886872 −0.443436 0.896306i $$-0.646241\pi$$
−0.443436 + 0.896306i $$0.646241\pi$$
$$228$$ −3.26989 −0.216554
$$229$$ 7.45145 0.492405 0.246203 0.969218i $$-0.420817\pi$$
0.246203 + 0.969218i $$0.420817\pi$$
$$230$$ 0 0
$$231$$ −0.107019 −0.00704135
$$232$$ 1.81788 0.119349
$$233$$ 21.5080 1.40904 0.704518 0.709687i $$-0.251163\pi$$
0.704518 + 0.709687i $$0.251163\pi$$
$$234$$ −4.55101 −0.297509
$$235$$ 0 0
$$236$$ −0.308927 −0.0201094
$$237$$ −12.4074 −0.805946
$$238$$ −4.33410 −0.280938
$$239$$ 24.1169 1.55999 0.779997 0.625784i $$-0.215221\pi$$
0.779997 + 0.625784i $$0.215221\pi$$
$$240$$ 0 0
$$241$$ −7.22805 −0.465600 −0.232800 0.972525i $$-0.574789\pi$$
−0.232800 + 0.972525i $$0.574789\pi$$
$$242$$ 18.8144 1.20944
$$243$$ 1.00000 0.0641500
$$244$$ −5.37587 −0.344155
$$245$$ 0 0
$$246$$ 10.5860 0.674942
$$247$$ −9.23579 −0.587659
$$248$$ 9.18526 0.583264
$$249$$ 7.89306 0.500203
$$250$$ 0 0
$$251$$ −8.74458 −0.551953 −0.275977 0.961164i $$-0.589001\pi$$
−0.275977 + 0.961164i $$0.589001\pi$$
$$252$$ −0.614893 −0.0387346
$$253$$ 1.25936 0.0791750
$$254$$ 22.2829 1.39815
$$255$$ 0 0
$$256$$ 18.3544 1.14715
$$257$$ −10.1877 −0.635494 −0.317747 0.948176i $$-0.602926\pi$$
−0.317747 + 0.948176i $$0.602926\pi$$
$$258$$ 18.0968 1.12666
$$259$$ −6.92950 −0.430578
$$260$$ 0 0
$$261$$ 1.00000 0.0618984
$$262$$ −14.8997 −0.920504
$$263$$ 11.6794 0.720184 0.360092 0.932917i $$-0.382745\pi$$
0.360092 + 0.932917i $$0.382745\pi$$
$$264$$ 0.297329 0.0182994
$$265$$ 0 0
$$266$$ −3.90359 −0.239345
$$267$$ 5.04702 0.308872
$$268$$ 3.30778 0.202055
$$269$$ 23.0385 1.40468 0.702342 0.711839i $$-0.252138\pi$$
0.702342 + 0.711839i $$0.252138\pi$$
$$270$$ 0 0
$$271$$ −4.45505 −0.270625 −0.135312 0.990803i $$-0.543204\pi$$
−0.135312 + 0.990803i $$0.543204\pi$$
$$272$$ 19.3024 1.17038
$$273$$ −1.73677 −0.105114
$$274$$ −24.7842 −1.49727
$$275$$ 0 0
$$276$$ 7.23579 0.435543
$$277$$ 15.7000 0.943321 0.471661 0.881780i $$-0.343655\pi$$
0.471661 + 0.881780i $$0.343655\pi$$
$$278$$ 2.95262 0.177087
$$279$$ 5.05274 0.302500
$$280$$ 0 0
$$281$$ 13.4721 0.803677 0.401838 0.915711i $$-0.368371\pi$$
0.401838 + 0.915711i $$0.368371\pi$$
$$282$$ −17.6663 −1.05201
$$283$$ 5.86481 0.348627 0.174313 0.984690i $$-0.444229\pi$$
0.174313 + 0.984690i $$0.444229\pi$$
$$284$$ 11.0380 0.654986
$$285$$ 0 0
$$286$$ −0.744357 −0.0440148
$$287$$ 4.03987 0.238466
$$288$$ 4.93087 0.290554
$$289$$ −2.07510 −0.122065
$$290$$ 0 0
$$291$$ 8.49076 0.497737
$$292$$ −1.05386 −0.0616726
$$293$$ −10.3642 −0.605485 −0.302742 0.953072i $$-0.597902\pi$$
−0.302742 + 0.953072i $$0.597902\pi$$
$$294$$ 11.2679 0.657159
$$295$$ 0 0
$$296$$ 19.2521 1.11901
$$297$$ 0.163559 0.00949064
$$298$$ −27.6836 −1.60367
$$299$$ 20.4375 1.18193
$$300$$ 0 0
$$301$$ 6.90614 0.398063
$$302$$ 1.25692 0.0723277
$$303$$ 1.81126 0.104054
$$304$$ 17.3850 0.997101
$$305$$ 0 0
$$306$$ 6.62385 0.378660
$$307$$ −16.0760 −0.917504 −0.458752 0.888564i $$-0.651703\pi$$
−0.458752 + 0.888564i $$0.651703\pi$$
$$308$$ −0.100571 −0.00573057
$$309$$ 5.80807 0.330410
$$310$$ 0 0
$$311$$ 16.9976 0.963846 0.481923 0.876214i $$-0.339939\pi$$
0.481923 + 0.876214i $$0.339939\pi$$
$$312$$ 4.82522 0.273174
$$313$$ 25.2391 1.42660 0.713300 0.700858i $$-0.247200\pi$$
0.713300 + 0.700858i $$0.247200\pi$$
$$314$$ −18.5636 −1.04760
$$315$$ 0 0
$$316$$ −11.6598 −0.655916
$$317$$ 3.06731 0.172277 0.0861386 0.996283i $$-0.472547\pi$$
0.0861386 + 0.996283i $$0.472547\pi$$
$$318$$ −6.94000 −0.389176
$$319$$ 0.163559 0.00915753
$$320$$ 0 0
$$321$$ 3.48568 0.194552
$$322$$ 8.63810 0.481382
$$323$$ 13.4424 0.747955
$$324$$ 0.939748 0.0522082
$$325$$ 0 0
$$326$$ 23.2372 1.28699
$$327$$ 8.11029 0.448500
$$328$$ −11.2239 −0.619735
$$329$$ −6.74185 −0.371690
$$330$$ 0 0
$$331$$ −19.9971 −1.09914 −0.549571 0.835447i $$-0.685209\pi$$
−0.549571 + 0.835447i $$0.685209\pi$$
$$332$$ 7.41749 0.407088
$$333$$ 10.5904 0.580352
$$334$$ −31.6631 −1.73253
$$335$$ 0 0
$$336$$ 3.26921 0.178350
$$337$$ −6.45613 −0.351688 −0.175844 0.984418i $$-0.556265\pi$$
−0.175844 + 0.984418i $$0.556265\pi$$
$$338$$ 10.2096 0.555328
$$339$$ 2.06715 0.112272
$$340$$ 0 0
$$341$$ 0.826420 0.0447531
$$342$$ 5.96591 0.322599
$$343$$ 8.88031 0.479491
$$344$$ −19.1872 −1.03450
$$345$$ 0 0
$$346$$ 15.3074 0.822932
$$347$$ −14.9415 −0.802099 −0.401050 0.916056i $$-0.631355\pi$$
−0.401050 + 0.916056i $$0.631355\pi$$
$$348$$ 0.939748 0.0503758
$$349$$ −12.1055 −0.647992 −0.323996 0.946058i $$-0.605026\pi$$
−0.323996 + 0.946058i $$0.605026\pi$$
$$350$$ 0 0
$$351$$ 2.65432 0.141677
$$352$$ 0.806487 0.0429859
$$353$$ −18.1937 −0.968355 −0.484178 0.874970i $$-0.660881\pi$$
−0.484178 + 0.874970i $$0.660881\pi$$
$$354$$ 0.563637 0.0299570
$$355$$ 0 0
$$356$$ 4.74292 0.251374
$$357$$ 2.52781 0.133786
$$358$$ 3.20447 0.169361
$$359$$ −35.1010 −1.85256 −0.926279 0.376838i $$-0.877011\pi$$
−0.926279 + 0.376838i $$0.877011\pi$$
$$360$$ 0 0
$$361$$ −6.89283 −0.362780
$$362$$ −28.9508 −1.52162
$$363$$ −10.9732 −0.575946
$$364$$ −1.63212 −0.0855464
$$365$$ 0 0
$$366$$ 9.80827 0.512686
$$367$$ 23.1943 1.21073 0.605367 0.795946i $$-0.293026\pi$$
0.605367 + 0.795946i $$0.293026\pi$$
$$368$$ −38.4706 −2.00542
$$369$$ −6.17417 −0.321415
$$370$$ 0 0
$$371$$ −2.64845 −0.137501
$$372$$ 4.74830 0.246188
$$373$$ −4.99469 −0.258615 −0.129308 0.991605i $$-0.541275\pi$$
−0.129308 + 0.991605i $$0.541275\pi$$
$$374$$ 1.08339 0.0560207
$$375$$ 0 0
$$376$$ 18.7307 0.965964
$$377$$ 2.65432 0.136704
$$378$$ 1.12187 0.0577029
$$379$$ −28.7738 −1.47801 −0.739006 0.673699i $$-0.764704\pi$$
−0.739006 + 0.673699i $$0.764704\pi$$
$$380$$ 0 0
$$381$$ −12.9962 −0.665816
$$382$$ −13.0080 −0.665550
$$383$$ −24.9482 −1.27479 −0.637396 0.770537i $$-0.719988\pi$$
−0.637396 + 0.770537i $$0.719988\pi$$
$$384$$ −12.4995 −0.637861
$$385$$ 0 0
$$386$$ −46.7725 −2.38066
$$387$$ −10.5547 −0.536527
$$388$$ 7.97917 0.405081
$$389$$ 31.0949 1.57657 0.788286 0.615308i $$-0.210969\pi$$
0.788286 + 0.615308i $$0.210969\pi$$
$$390$$ 0 0
$$391$$ −29.7461 −1.50433
$$392$$ −11.9468 −0.603407
$$393$$ 8.69003 0.438354
$$394$$ 16.6421 0.838414
$$395$$ 0 0
$$396$$ 0.153704 0.00772391
$$397$$ −12.1948 −0.612038 −0.306019 0.952025i $$-0.598997\pi$$
−0.306019 + 0.952025i $$0.598997\pi$$
$$398$$ −39.3528 −1.97258
$$399$$ 2.27672 0.113979
$$400$$ 0 0
$$401$$ 4.33039 0.216249 0.108125 0.994137i $$-0.465515\pi$$
0.108125 + 0.994137i $$0.465515\pi$$
$$402$$ −6.03504 −0.301000
$$403$$ 13.4116 0.668078
$$404$$ 1.70213 0.0846841
$$405$$ 0 0
$$406$$ 1.12187 0.0556776
$$407$$ 1.73216 0.0858599
$$408$$ −7.02295 −0.347688
$$409$$ 5.12178 0.253256 0.126628 0.991950i $$-0.459585\pi$$
0.126628 + 0.991950i $$0.459585\pi$$
$$410$$ 0 0
$$411$$ 14.4550 0.713015
$$412$$ 5.45812 0.268902
$$413$$ 0.215096 0.0105842
$$414$$ −13.2017 −0.648828
$$415$$ 0 0
$$416$$ 13.0881 0.641697
$$417$$ −1.72208 −0.0843306
$$418$$ 0.975776 0.0477267
$$419$$ 14.1945 0.693445 0.346723 0.937968i $$-0.387295\pi$$
0.346723 + 0.937968i $$0.387295\pi$$
$$420$$ 0 0
$$421$$ 4.83364 0.235577 0.117789 0.993039i $$-0.462419\pi$$
0.117789 + 0.993039i $$0.462419\pi$$
$$422$$ −30.8657 −1.50252
$$423$$ 10.3036 0.500980
$$424$$ 7.35815 0.357343
$$425$$ 0 0
$$426$$ −20.1389 −0.975731
$$427$$ 3.74305 0.181139
$$428$$ 3.27566 0.158335
$$429$$ 0.434137 0.0209603
$$430$$ 0 0
$$431$$ −18.0910 −0.871411 −0.435706 0.900089i $$-0.643501\pi$$
−0.435706 + 0.900089i $$0.643501\pi$$
$$432$$ −4.99637 −0.240388
$$433$$ −1.71005 −0.0821798 −0.0410899 0.999155i $$-0.513083\pi$$
−0.0410899 + 0.999155i $$0.513083\pi$$
$$434$$ 5.66853 0.272098
$$435$$ 0 0
$$436$$ 7.62163 0.365010
$$437$$ −26.7914 −1.28161
$$438$$ 1.92277 0.0918734
$$439$$ 24.3533 1.16232 0.581160 0.813789i $$-0.302599\pi$$
0.581160 + 0.813789i $$0.302599\pi$$
$$440$$ 0 0
$$441$$ −6.57187 −0.312946
$$442$$ 17.5818 0.836281
$$443$$ 18.9066 0.898280 0.449140 0.893461i $$-0.351730\pi$$
0.449140 + 0.893461i $$0.351730\pi$$
$$444$$ 9.95234 0.472317
$$445$$ 0 0
$$446$$ 32.8140 1.55379
$$447$$ 16.1461 0.763684
$$448$$ −1.00661 −0.0475581
$$449$$ 25.2808 1.19307 0.596537 0.802586i $$-0.296543\pi$$
0.596537 + 0.802586i $$0.296543\pi$$
$$450$$ 0 0
$$451$$ −1.00984 −0.0475515
$$452$$ 1.94260 0.0913723
$$453$$ −0.733083 −0.0344432
$$454$$ 22.9102 1.07523
$$455$$ 0 0
$$456$$ −6.32536 −0.296212
$$457$$ 4.25835 0.199197 0.0995986 0.995028i $$-0.468244\pi$$
0.0995986 + 0.995028i $$0.468244\pi$$
$$458$$ −12.7760 −0.596984
$$459$$ −3.86328 −0.180322
$$460$$ 0 0
$$461$$ 2.48353 0.115670 0.0578349 0.998326i $$-0.481580\pi$$
0.0578349 + 0.998326i $$0.481580\pi$$
$$462$$ 0.183492 0.00853682
$$463$$ 27.4876 1.27746 0.638728 0.769433i $$-0.279461\pi$$
0.638728 + 0.769433i $$0.279461\pi$$
$$464$$ −4.99637 −0.231951
$$465$$ 0 0
$$466$$ −36.8769 −1.70829
$$467$$ −6.24698 −0.289076 −0.144538 0.989499i $$-0.546170\pi$$
−0.144538 + 0.989499i $$0.546170\pi$$
$$468$$ 2.49439 0.115303
$$469$$ −2.30310 −0.106347
$$470$$ 0 0
$$471$$ 10.8270 0.498880
$$472$$ −0.597598 −0.0275067
$$473$$ −1.72632 −0.0793761
$$474$$ 21.2733 0.977116
$$475$$ 0 0
$$476$$ 2.37550 0.108881
$$477$$ 4.04766 0.185330
$$478$$ −41.3501 −1.89131
$$479$$ −11.5736 −0.528812 −0.264406 0.964412i $$-0.585176\pi$$
−0.264406 + 0.964412i $$0.585176\pi$$
$$480$$ 0 0
$$481$$ 28.1104 1.28172
$$482$$ 12.3930 0.564485
$$483$$ −5.03806 −0.229240
$$484$$ −10.3121 −0.468731
$$485$$ 0 0
$$486$$ −1.71457 −0.0777744
$$487$$ −31.1947 −1.41357 −0.706783 0.707431i $$-0.749854\pi$$
−0.706783 + 0.707431i $$0.749854\pi$$
$$488$$ −10.3992 −0.470751
$$489$$ −13.5528 −0.612878
$$490$$ 0 0
$$491$$ −7.93512 −0.358107 −0.179053 0.983839i $$-0.557303\pi$$
−0.179053 + 0.983839i $$0.557303\pi$$
$$492$$ −5.80216 −0.261582
$$493$$ −3.86328 −0.173993
$$494$$ 15.8354 0.712469
$$495$$ 0 0
$$496$$ −25.2454 −1.13355
$$497$$ −7.68543 −0.344739
$$498$$ −13.5332 −0.606437
$$499$$ −32.8984 −1.47274 −0.736368 0.676581i $$-0.763461\pi$$
−0.736368 + 0.676581i $$0.763461\pi$$
$$500$$ 0 0
$$501$$ 18.4671 0.825047
$$502$$ 14.9932 0.669179
$$503$$ −40.7038 −1.81489 −0.907446 0.420168i $$-0.861971\pi$$
−0.907446 + 0.420168i $$0.861971\pi$$
$$504$$ −1.18947 −0.0529831
$$505$$ 0 0
$$506$$ −2.15925 −0.0959905
$$507$$ −5.95460 −0.264453
$$508$$ −12.2132 −0.541871
$$509$$ −36.9407 −1.63737 −0.818684 0.574245i $$-0.805296\pi$$
−0.818684 + 0.574245i $$0.805296\pi$$
$$510$$ 0 0
$$511$$ 0.733771 0.0324601
$$512$$ −6.47089 −0.285976
$$513$$ −3.47954 −0.153625
$$514$$ 17.4676 0.770462
$$515$$ 0 0
$$516$$ −9.91878 −0.436650
$$517$$ 1.68525 0.0741172
$$518$$ 11.8811 0.522026
$$519$$ −8.92785 −0.391889
$$520$$ 0 0
$$521$$ −33.2555 −1.45695 −0.728476 0.685072i $$-0.759771\pi$$
−0.728476 + 0.685072i $$0.759771\pi$$
$$522$$ −1.71457 −0.0750447
$$523$$ −0.341001 −0.0149109 −0.00745547 0.999972i $$-0.502373\pi$$
−0.00745547 + 0.999972i $$0.502373\pi$$
$$524$$ 8.16644 0.356752
$$525$$ 0 0
$$526$$ −20.0252 −0.873139
$$527$$ −19.5201 −0.850310
$$528$$ −0.817200 −0.0355640
$$529$$ 36.2856 1.57764
$$530$$ 0 0
$$531$$ −0.328734 −0.0142658
$$532$$ 2.13954 0.0927609
$$533$$ −16.3882 −0.709852
$$534$$ −8.65346 −0.374472
$$535$$ 0 0
$$536$$ 6.39866 0.276380
$$537$$ −1.86896 −0.0806518
$$538$$ −39.5012 −1.70302
$$539$$ −1.07489 −0.0462986
$$540$$ 0 0
$$541$$ −18.4196 −0.791919 −0.395960 0.918268i $$-0.629588\pi$$
−0.395960 + 0.918268i $$0.629588\pi$$
$$542$$ 7.63849 0.328101
$$543$$ 16.8852 0.724613
$$544$$ −19.0493 −0.816732
$$545$$ 0 0
$$546$$ 2.97780 0.127438
$$547$$ 42.5936 1.82117 0.910586 0.413320i $$-0.135631\pi$$
0.910586 + 0.413320i $$0.135631\pi$$
$$548$$ 13.5841 0.580284
$$549$$ −5.72054 −0.244147
$$550$$ 0 0
$$551$$ −3.47954 −0.148233
$$552$$ 13.9971 0.595757
$$553$$ 8.11836 0.345228
$$554$$ −26.9187 −1.14367
$$555$$ 0 0
$$556$$ −1.61832 −0.0686321
$$557$$ 17.5805 0.744908 0.372454 0.928051i $$-0.378516\pi$$
0.372454 + 0.928051i $$0.378516\pi$$
$$558$$ −8.66328 −0.366746
$$559$$ −28.0156 −1.18493
$$560$$ 0 0
$$561$$ −0.631872 −0.0266777
$$562$$ −23.0988 −0.974365
$$563$$ 6.29621 0.265354 0.132677 0.991159i $$-0.457643\pi$$
0.132677 + 0.991159i $$0.457643\pi$$
$$564$$ 9.68282 0.407720
$$565$$ 0 0
$$566$$ −10.0556 −0.422669
$$567$$ −0.654317 −0.0274787
$$568$$ 21.3523 0.895921
$$569$$ −42.3981 −1.77742 −0.888710 0.458469i $$-0.848398\pi$$
−0.888710 + 0.458469i $$0.848398\pi$$
$$570$$ 0 0
$$571$$ −18.4882 −0.773708 −0.386854 0.922141i $$-0.626438\pi$$
−0.386854 + 0.922141i $$0.626438\pi$$
$$572$$ 0.407979 0.0170585
$$573$$ 7.58677 0.316942
$$574$$ −6.92663 −0.289112
$$575$$ 0 0
$$576$$ 1.53842 0.0641009
$$577$$ −22.0727 −0.918897 −0.459448 0.888204i $$-0.651953\pi$$
−0.459448 + 0.888204i $$0.651953\pi$$
$$578$$ 3.55791 0.147990
$$579$$ 27.2794 1.13369
$$580$$ 0 0
$$581$$ −5.16457 −0.214262
$$582$$ −14.5580 −0.603448
$$583$$ 0.662030 0.0274185
$$584$$ −2.03862 −0.0843587
$$585$$ 0 0
$$586$$ 17.7702 0.734080
$$587$$ −44.1932 −1.82405 −0.912024 0.410136i $$-0.865481\pi$$
−0.912024 + 0.410136i $$0.865481\pi$$
$$588$$ −6.17590 −0.254690
$$589$$ −17.5812 −0.724421
$$590$$ 0 0
$$591$$ −9.70626 −0.399262
$$592$$ −52.9137 −2.17474
$$593$$ 17.4159 0.715184 0.357592 0.933878i $$-0.383598\pi$$
0.357592 + 0.933878i $$0.383598\pi$$
$$594$$ −0.280433 −0.0115063
$$595$$ 0 0
$$596$$ 15.1732 0.621520
$$597$$ 22.9520 0.939364
$$598$$ −35.0415 −1.43295
$$599$$ 7.70544 0.314836 0.157418 0.987532i $$-0.449683\pi$$
0.157418 + 0.987532i $$0.449683\pi$$
$$600$$ 0 0
$$601$$ −31.5500 −1.28695 −0.643476 0.765466i $$-0.722508\pi$$
−0.643476 + 0.765466i $$0.722508\pi$$
$$602$$ −11.8410 −0.482605
$$603$$ 3.51985 0.143340
$$604$$ −0.688913 −0.0280315
$$605$$ 0 0
$$606$$ −3.10553 −0.126154
$$607$$ −13.2516 −0.537864 −0.268932 0.963159i $$-0.586671\pi$$
−0.268932 + 0.963159i $$0.586671\pi$$
$$608$$ −17.1571 −0.695814
$$609$$ −0.654317 −0.0265143
$$610$$ 0 0
$$611$$ 27.3491 1.10643
$$612$$ −3.63050 −0.146754
$$613$$ 29.4609 1.18991 0.594957 0.803758i $$-0.297169\pi$$
0.594957 + 0.803758i $$0.297169\pi$$
$$614$$ 27.5634 1.11237
$$615$$ 0 0
$$616$$ −0.194548 −0.00783855
$$617$$ −5.61068 −0.225877 −0.112939 0.993602i $$-0.536026\pi$$
−0.112939 + 0.993602i $$0.536026\pi$$
$$618$$ −9.95834 −0.400583
$$619$$ 40.5248 1.62883 0.814415 0.580282i $$-0.197058\pi$$
0.814415 + 0.580282i $$0.197058\pi$$
$$620$$ 0 0
$$621$$ 7.69972 0.308979
$$622$$ −29.1436 −1.16855
$$623$$ −3.30235 −0.132306
$$624$$ −13.2619 −0.530903
$$625$$ 0 0
$$626$$ −43.2743 −1.72959
$$627$$ −0.569108 −0.0227280
$$628$$ 10.1746 0.406011
$$629$$ −40.9138 −1.63134
$$630$$ 0 0
$$631$$ −20.2197 −0.804935 −0.402467 0.915434i $$-0.631847\pi$$
−0.402467 + 0.915434i $$0.631847\pi$$
$$632$$ −22.5551 −0.897193
$$633$$ 18.0020 0.715516
$$634$$ −5.25911 −0.208866
$$635$$ 0 0
$$636$$ 3.80378 0.150830
$$637$$ −17.4438 −0.691149
$$638$$ −0.280433 −0.0111024
$$639$$ 11.7457 0.464654
$$640$$ 0 0
$$641$$ −44.0804 −1.74107 −0.870535 0.492107i $$-0.836227\pi$$
−0.870535 + 0.492107i $$0.836227\pi$$
$$642$$ −5.97644 −0.235871
$$643$$ 3.56303 0.140512 0.0702561 0.997529i $$-0.477618\pi$$
0.0702561 + 0.997529i $$0.477618\pi$$
$$644$$ −4.73450 −0.186566
$$645$$ 0 0
$$646$$ −23.0479 −0.906809
$$647$$ −40.7523 −1.60214 −0.801069 0.598573i $$-0.795735\pi$$
−0.801069 + 0.598573i $$0.795735\pi$$
$$648$$ 1.81788 0.0714129
$$649$$ −0.0537673 −0.00211055
$$650$$ 0 0
$$651$$ −3.30610 −0.129576
$$652$$ −12.7362 −0.498788
$$653$$ −11.6902 −0.457472 −0.228736 0.973489i $$-0.573459\pi$$
−0.228736 + 0.973489i $$0.573459\pi$$
$$654$$ −13.9057 −0.543754
$$655$$ 0 0
$$656$$ 30.8484 1.20443
$$657$$ −1.12143 −0.0437511
$$658$$ 11.5594 0.450631
$$659$$ −28.1795 −1.09772 −0.548858 0.835915i $$-0.684937\pi$$
−0.548858 + 0.835915i $$0.684937\pi$$
$$660$$ 0 0
$$661$$ −20.6736 −0.804108 −0.402054 0.915616i $$-0.631704\pi$$
−0.402054 + 0.915616i $$0.631704\pi$$
$$662$$ 34.2865 1.33258
$$663$$ −10.2544 −0.398246
$$664$$ 14.3486 0.556834
$$665$$ 0 0
$$666$$ −18.1580 −0.703610
$$667$$ 7.69972 0.298134
$$668$$ 17.3544 0.671461
$$669$$ −19.1383 −0.739931
$$670$$ 0 0
$$671$$ −0.935645 −0.0361202
$$672$$ −3.22635 −0.124459
$$673$$ 44.7915 1.72659 0.863293 0.504703i $$-0.168398\pi$$
0.863293 + 0.504703i $$0.168398\pi$$
$$674$$ 11.0695 0.426380
$$675$$ 0 0
$$676$$ −5.59582 −0.215224
$$677$$ 16.8746 0.648542 0.324271 0.945964i $$-0.394881\pi$$
0.324271 + 0.945964i $$0.394881\pi$$
$$678$$ −3.54428 −0.136117
$$679$$ −5.55565 −0.213206
$$680$$ 0 0
$$681$$ −13.3621 −0.512036
$$682$$ −1.41695 −0.0542580
$$683$$ −28.6207 −1.09514 −0.547571 0.836759i $$-0.684447\pi$$
−0.547571 + 0.836759i $$0.684447\pi$$
$$684$$ −3.26989 −0.125027
$$685$$ 0 0
$$686$$ −15.2259 −0.581328
$$687$$ 7.45145 0.284290
$$688$$ 52.7353 2.01051
$$689$$ 10.7438 0.409305
$$690$$ 0 0
$$691$$ −28.5928 −1.08772 −0.543861 0.839175i $$-0.683038\pi$$
−0.543861 + 0.839175i $$0.683038\pi$$
$$692$$ −8.38993 −0.318937
$$693$$ −0.107019 −0.00406532
$$694$$ 25.6182 0.972452
$$695$$ 0 0
$$696$$ 1.81788 0.0689064
$$697$$ 23.8525 0.903479
$$698$$ 20.7557 0.785615
$$699$$ 21.5080 0.813507
$$700$$ 0 0
$$701$$ 49.6636 1.87577 0.937885 0.346946i $$-0.112781\pi$$
0.937885 + 0.346946i $$0.112781\pi$$
$$702$$ −4.55101 −0.171767
$$703$$ −36.8498 −1.38982
$$704$$ 0.251622 0.00948336
$$705$$ 0 0
$$706$$ 31.1944 1.17402
$$707$$ −1.18514 −0.0445717
$$708$$ −0.308927 −0.0116102
$$709$$ −19.4755 −0.731419 −0.365710 0.930729i $$-0.619174\pi$$
−0.365710 + 0.930729i $$0.619174\pi$$
$$710$$ 0 0
$$711$$ −12.4074 −0.465313
$$712$$ 9.17485 0.343842
$$713$$ 38.9047 1.45699
$$714$$ −4.33410 −0.162200
$$715$$ 0 0
$$716$$ −1.75636 −0.0656381
$$717$$ 24.1169 0.900663
$$718$$ 60.1830 2.24601
$$719$$ −34.3195 −1.27990 −0.639951 0.768415i $$-0.721046\pi$$
−0.639951 + 0.768415i $$0.721046\pi$$
$$720$$ 0 0
$$721$$ −3.80032 −0.141531
$$722$$ 11.8182 0.439829
$$723$$ −7.22805 −0.268814
$$724$$ 15.8678 0.589723
$$725$$ 0 0
$$726$$ 18.8144 0.698268
$$727$$ 19.3206 0.716560 0.358280 0.933614i $$-0.383363\pi$$
0.358280 + 0.933614i $$0.383363\pi$$
$$728$$ −3.15722 −0.117014
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 40.7758 1.50815
$$732$$ −5.37587 −0.198698
$$733$$ 3.10588 0.114718 0.0573591 0.998354i $$-0.481732\pi$$
0.0573591 + 0.998354i $$0.481732\pi$$
$$734$$ −39.7683 −1.46787
$$735$$ 0 0
$$736$$ 37.9663 1.39946
$$737$$ 0.575703 0.0212063
$$738$$ 10.5860 0.389678
$$739$$ 38.7106 1.42399 0.711997 0.702183i $$-0.247791\pi$$
0.711997 + 0.702183i $$0.247791\pi$$
$$740$$ 0 0
$$741$$ −9.23579 −0.339285
$$742$$ 4.54096 0.166704
$$743$$ −8.41130 −0.308581 −0.154290 0.988026i $$-0.549309\pi$$
−0.154290 + 0.988026i $$0.549309\pi$$
$$744$$ 9.18526 0.336748
$$745$$ 0 0
$$746$$ 8.56373 0.313541
$$747$$ 7.89306 0.288792
$$748$$ −0.593801 −0.0217115
$$749$$ −2.28074 −0.0833364
$$750$$ 0 0
$$751$$ −30.5930 −1.11635 −0.558177 0.829722i $$-0.688499\pi$$
−0.558177 + 0.829722i $$0.688499\pi$$
$$752$$ −51.4808 −1.87731
$$753$$ −8.74458 −0.318670
$$754$$ −4.55101 −0.165738
$$755$$ 0 0
$$756$$ −0.614893 −0.0223634
$$757$$ −41.6526 −1.51389 −0.756945 0.653479i $$-0.773309\pi$$
−0.756945 + 0.653479i $$0.773309\pi$$
$$758$$ 49.3347 1.79192
$$759$$ 1.25936 0.0457117
$$760$$ 0 0
$$761$$ −28.3957 −1.02934 −0.514671 0.857388i $$-0.672086\pi$$
−0.514671 + 0.857388i $$0.672086\pi$$
$$762$$ 22.2829 0.807224
$$763$$ −5.30670 −0.192115
$$764$$ 7.12965 0.257942
$$765$$ 0 0
$$766$$ 42.7753 1.54554
$$767$$ −0.872565 −0.0315065
$$768$$ 18.3544 0.662306
$$769$$ −48.6782 −1.75538 −0.877690 0.479229i $$-0.840916\pi$$
−0.877690 + 0.479229i $$0.840916\pi$$
$$770$$ 0 0
$$771$$ −10.1877 −0.366902
$$772$$ 25.6358 0.922652
$$773$$ −54.9974 −1.97812 −0.989060 0.147513i $$-0.952873\pi$$
−0.989060 + 0.147513i $$0.952873\pi$$
$$774$$ 18.0968 0.650476
$$775$$ 0 0
$$776$$ 15.4351 0.554089
$$777$$ −6.92950 −0.248595
$$778$$ −53.3143 −1.91141
$$779$$ 21.4833 0.769717
$$780$$ 0 0
$$781$$ 1.92112 0.0687429
$$782$$ 51.0018 1.82382
$$783$$ 1.00000 0.0357371
$$784$$ 32.8355 1.17270
$$785$$ 0 0
$$786$$ −14.8997 −0.531453
$$787$$ 32.7472 1.16731 0.583655 0.812002i $$-0.301622\pi$$
0.583655 + 0.812002i $$0.301622\pi$$
$$788$$ −9.12143 −0.324938
$$789$$ 11.6794 0.415798
$$790$$ 0 0
$$791$$ −1.35257 −0.0480920
$$792$$ 0.297329 0.0105651
$$793$$ −15.1841 −0.539205
$$794$$ 20.9088 0.742025
$$795$$ 0 0
$$796$$ 21.5691 0.764497
$$797$$ 6.54714 0.231912 0.115956 0.993254i $$-0.463007\pi$$
0.115956 + 0.993254i $$0.463007\pi$$
$$798$$ −3.90359 −0.138186
$$799$$ −39.8058 −1.40823
$$800$$ 0 0
$$801$$ 5.04702 0.178328
$$802$$ −7.42475 −0.262177
$$803$$ −0.183420 −0.00647274
$$804$$ 3.30778 0.116656
$$805$$ 0 0
$$806$$ −22.9951 −0.809967
$$807$$ 23.0385 0.810995
$$808$$ 3.29265 0.115835
$$809$$ 40.5849 1.42689 0.713444 0.700712i $$-0.247134\pi$$
0.713444 + 0.700712i $$0.247134\pi$$
$$810$$ 0 0
$$811$$ 15.4588 0.542832 0.271416 0.962462i $$-0.412508\pi$$
0.271416 + 0.962462i $$0.412508\pi$$
$$812$$ −0.614893 −0.0215785
$$813$$ −4.45505 −0.156245
$$814$$ −2.96990 −0.104095
$$815$$ 0 0
$$816$$ 19.3024 0.675717
$$817$$ 36.7255 1.28486
$$818$$ −8.78165 −0.307043
$$819$$ −1.73677 −0.0606875
$$820$$ 0 0
$$821$$ −22.4828 −0.784656 −0.392328 0.919825i $$-0.628330\pi$$
−0.392328 + 0.919825i $$0.628330\pi$$
$$822$$ −24.7842 −0.864448
$$823$$ 45.1129 1.57254 0.786268 0.617886i $$-0.212011\pi$$
0.786268 + 0.617886i $$0.212011\pi$$
$$824$$ 10.5584 0.367818
$$825$$ 0 0
$$826$$ −0.368798 −0.0128321
$$827$$ 9.44115 0.328301 0.164150 0.986435i $$-0.447512\pi$$
0.164150 + 0.986435i $$0.447512\pi$$
$$828$$ 7.23579 0.251461
$$829$$ −55.4844 −1.92705 −0.963527 0.267612i $$-0.913765\pi$$
−0.963527 + 0.267612i $$0.913765\pi$$
$$830$$ 0 0
$$831$$ 15.7000 0.544627
$$832$$ 4.08346 0.141568
$$833$$ 25.3889 0.879675
$$834$$ 2.95262 0.102241
$$835$$ 0 0
$$836$$ −0.534818 −0.0184971
$$837$$ 5.05274 0.174648
$$838$$ −24.3374 −0.840721
$$839$$ 36.6797 1.26632 0.633162 0.774019i $$-0.281757\pi$$
0.633162 + 0.774019i $$0.281757\pi$$
$$840$$ 0 0
$$841$$ 1.00000 0.0344828
$$842$$ −8.28762 −0.285610
$$843$$ 13.4721 0.464003
$$844$$ 16.9174 0.582319
$$845$$ 0 0
$$846$$ −17.6663 −0.607380
$$847$$ 7.17998 0.246707
$$848$$ −20.2236 −0.694482
$$849$$ 5.86481 0.201280
$$850$$ 0 0
$$851$$ 81.5433 2.79527
$$852$$ 11.0380 0.378156
$$853$$ −16.2128 −0.555114 −0.277557 0.960709i $$-0.589525\pi$$
−0.277557 + 0.960709i $$0.589525\pi$$
$$854$$ −6.41772 −0.219610
$$855$$ 0 0
$$856$$ 6.33653 0.216578
$$857$$ 52.9150 1.80754 0.903772 0.428015i $$-0.140787\pi$$
0.903772 + 0.428015i $$0.140787\pi$$
$$858$$ −0.744357 −0.0254119
$$859$$ −13.7986 −0.470803 −0.235401 0.971898i $$-0.575641\pi$$
−0.235401 + 0.971898i $$0.575641\pi$$
$$860$$ 0 0
$$861$$ 4.03987 0.137678
$$862$$ 31.0182 1.05648
$$863$$ −4.25394 −0.144806 −0.0724029 0.997375i $$-0.523067\pi$$
−0.0724029 + 0.997375i $$0.523067\pi$$
$$864$$ 4.93087 0.167752
$$865$$ 0 0
$$866$$ 2.93200 0.0996334
$$867$$ −2.07510 −0.0704743
$$868$$ −3.10690 −0.105455
$$869$$ −2.02934 −0.0688405
$$870$$ 0 0
$$871$$ 9.34281 0.316569
$$872$$ 14.7435 0.499278
$$873$$ 8.49076 0.287369
$$874$$ 45.9358 1.55380
$$875$$ 0 0
$$876$$ −1.05386 −0.0356067
$$877$$ 49.9185 1.68563 0.842814 0.538205i $$-0.180897\pi$$
0.842814 + 0.538205i $$0.180897\pi$$
$$878$$ −41.7554 −1.40918
$$879$$ −10.3642 −0.349577
$$880$$ 0 0
$$881$$ −11.5561 −0.389334 −0.194667 0.980869i $$-0.562363\pi$$
−0.194667 + 0.980869i $$0.562363\pi$$
$$882$$ 11.2679 0.379411
$$883$$ 10.5007 0.353375 0.176688 0.984267i $$-0.443462\pi$$
0.176688 + 0.984267i $$0.443462\pi$$
$$884$$ −9.63651 −0.324111
$$885$$ 0 0
$$886$$ −32.4167 −1.08906
$$887$$ −13.1447 −0.441356 −0.220678 0.975347i $$-0.570827\pi$$
−0.220678 + 0.975347i $$0.570827\pi$$
$$888$$ 19.2521 0.646058
$$889$$ 8.50364 0.285203
$$890$$ 0 0
$$891$$ 0.163559 0.00547942
$$892$$ −17.9852 −0.602190
$$893$$ −35.8519 −1.19974
$$894$$ −27.6836 −0.925877
$$895$$ 0 0
$$896$$ 8.17862 0.273228
$$897$$ 20.4375 0.682388
$$898$$ −43.3456 −1.44646
$$899$$ 5.05274 0.168518
$$900$$ 0 0
$$901$$ −15.6372 −0.520952
$$902$$ 1.73144 0.0576506
$$903$$ 6.90614 0.229822
$$904$$ 3.75783 0.124983
$$905$$ 0 0
$$906$$ 1.25692 0.0417584
$$907$$ −8.45271 −0.280668 −0.140334 0.990104i $$-0.544818\pi$$
−0.140334 + 0.990104i $$0.544818\pi$$
$$908$$ −12.5570 −0.416718
$$909$$ 1.81126 0.0600757
$$910$$ 0 0
$$911$$ −23.3000 −0.771965 −0.385982 0.922506i $$-0.626137\pi$$
−0.385982 + 0.922506i $$0.626137\pi$$
$$912$$ 17.3850 0.575676
$$913$$ 1.29098 0.0427252
$$914$$ −7.30123 −0.241503
$$915$$ 0 0
$$916$$ 7.00248 0.231368
$$917$$ −5.68603 −0.187769
$$918$$ 6.62385 0.218620
$$919$$ −37.7491 −1.24523 −0.622614 0.782529i $$-0.713929\pi$$
−0.622614 + 0.782529i $$0.713929\pi$$
$$920$$ 0 0
$$921$$ −16.0760 −0.529721
$$922$$ −4.25819 −0.140236
$$923$$ 31.1769 1.02620
$$924$$ −0.100571 −0.00330855
$$925$$ 0 0
$$926$$ −47.1294 −1.54877
$$927$$ 5.80807 0.190762
$$928$$ 4.93087 0.161864
$$929$$ 34.3843 1.12811 0.564055 0.825737i $$-0.309240\pi$$
0.564055 + 0.825737i $$0.309240\pi$$
$$930$$ 0 0
$$931$$ 22.8671 0.749437
$$932$$ 20.2121 0.662069
$$933$$ 16.9976 0.556476
$$934$$ 10.7109 0.350471
$$935$$ 0 0
$$936$$ 4.82522 0.157717
$$937$$ −8.85110 −0.289153 −0.144576 0.989494i $$-0.546182\pi$$
−0.144576 + 0.989494i $$0.546182\pi$$
$$938$$ 3.94883 0.128934
$$939$$ 25.2391 0.823648
$$940$$ 0 0
$$941$$ −20.4890 −0.667921 −0.333960 0.942587i $$-0.608385\pi$$
−0.333960 + 0.942587i $$0.608385\pi$$
$$942$$ −18.5636 −0.604834
$$943$$ −47.5394 −1.54810
$$944$$ 1.64248 0.0534581
$$945$$ 0 0
$$946$$ 2.95989 0.0962343
$$947$$ −38.8206 −1.26150 −0.630750 0.775986i $$-0.717253\pi$$
−0.630750 + 0.775986i $$0.717253\pi$$
$$948$$ −11.6598 −0.378693
$$949$$ −2.97663 −0.0966255
$$950$$ 0 0
$$951$$ 3.06731 0.0994643
$$952$$ 4.59524 0.148933
$$953$$ 33.3755 1.08114 0.540570 0.841299i $$-0.318209\pi$$
0.540570 + 0.841299i $$0.318209\pi$$
$$954$$ −6.94000 −0.224691
$$955$$ 0 0
$$956$$ 22.6638 0.733000
$$957$$ 0.163559 0.00528710
$$958$$ 19.8437 0.641122
$$959$$ −9.45818 −0.305421
$$960$$ 0 0
$$961$$ −5.46979 −0.176445
$$962$$ −48.1972 −1.55394
$$963$$ 3.48568 0.112324
$$964$$ −6.79254 −0.218773
$$965$$ 0 0
$$966$$ 8.63810 0.277926
$$967$$ 14.2745 0.459036 0.229518 0.973304i $$-0.426285\pi$$
0.229518 + 0.973304i $$0.426285\pi$$
$$968$$ −19.9480 −0.641153
$$969$$ 13.4424 0.431832
$$970$$ 0 0
$$971$$ 58.2228 1.86846 0.934229 0.356675i $$-0.116090\pi$$
0.934229 + 0.356675i $$0.116090\pi$$
$$972$$ 0.939748 0.0301424
$$973$$ 1.12679 0.0361231
$$974$$ 53.4854 1.71378
$$975$$ 0 0
$$976$$ 28.5820 0.914886
$$977$$ 57.0456 1.82505 0.912526 0.409019i $$-0.134129\pi$$
0.912526 + 0.409019i $$0.134129\pi$$
$$978$$ 23.2372 0.743043
$$979$$ 0.825483 0.0263826
$$980$$ 0 0
$$981$$ 8.11029 0.258942
$$982$$ 13.6053 0.434163
$$983$$ 2.76691 0.0882506 0.0441253 0.999026i $$-0.485950\pi$$
0.0441253 + 0.999026i $$0.485950\pi$$
$$984$$ −11.2239 −0.357804
$$985$$ 0 0
$$986$$ 6.62385 0.210946
$$987$$ −6.74185 −0.214595
$$988$$ −8.67932 −0.276126
$$989$$ −81.2684 −2.58418
$$990$$ 0 0
$$991$$ 26.2090 0.832557 0.416278 0.909237i $$-0.363334\pi$$
0.416278 + 0.909237i $$0.363334\pi$$
$$992$$ 24.9144 0.791034
$$993$$ −19.9971 −0.634590
$$994$$ 13.1772 0.417955
$$995$$ 0 0
$$996$$ 7.41749 0.235032
$$997$$ 53.0937 1.68149 0.840747 0.541428i $$-0.182116\pi$$
0.840747 + 0.541428i $$0.182116\pi$$
$$998$$ 56.4066 1.78552
$$999$$ 10.5904 0.335067
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 2175.2.a.z.1.1 5
3.2 odd 2 6525.2.a.bl.1.5 5
5.2 odd 4 435.2.c.e.349.3 10
5.3 odd 4 435.2.c.e.349.8 yes 10
5.4 even 2 2175.2.a.w.1.5 5
15.2 even 4 1305.2.c.j.784.8 10
15.8 even 4 1305.2.c.j.784.3 10
15.14 odd 2 6525.2.a.bs.1.1 5

By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.c.e.349.3 10 5.2 odd 4
435.2.c.e.349.8 yes 10 5.3 odd 4
1305.2.c.j.784.3 10 15.8 even 4
1305.2.c.j.784.8 10 15.2 even 4
2175.2.a.w.1.5 5 5.4 even 2
2175.2.a.z.1.1 5 1.1 even 1 trivial
6525.2.a.bl.1.5 5 3.2 odd 2
6525.2.a.bs.1.1 5 15.14 odd 2