# Properties

 Label 1875.2.a.c.1.2 Level $1875$ Weight $2$ Character 1875.1 Self dual yes Analytic conductor $14.972$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1875,2,Mod(1,1875)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1875, 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("1875.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1875 = 3 \cdot 5^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1875.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: $$14.9719503790$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{10})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.61803$$ of defining polynomial Character $$\chi$$ $$=$$ 1875.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.61803 q^{2} +1.00000 q^{3} +0.618034 q^{4} +1.61803 q^{6} -2.00000 q^{7} -2.23607 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.61803 q^{2} +1.00000 q^{3} +0.618034 q^{4} +1.61803 q^{6} -2.00000 q^{7} -2.23607 q^{8} +1.00000 q^{9} -3.00000 q^{11} +0.618034 q^{12} -1.00000 q^{13} -3.23607 q^{14} -4.85410 q^{16} -4.23607 q^{17} +1.61803 q^{18} -6.70820 q^{19} -2.00000 q^{21} -4.85410 q^{22} +5.38197 q^{23} -2.23607 q^{24} -1.61803 q^{26} +1.00000 q^{27} -1.23607 q^{28} -3.61803 q^{29} +8.70820 q^{31} -3.38197 q^{32} -3.00000 q^{33} -6.85410 q^{34} +0.618034 q^{36} -2.00000 q^{37} -10.8541 q^{38} -1.00000 q^{39} -9.38197 q^{41} -3.23607 q^{42} -7.38197 q^{43} -1.85410 q^{44} +8.70820 q^{46} -4.76393 q^{47} -4.85410 q^{48} -3.00000 q^{49} -4.23607 q^{51} -0.618034 q^{52} +11.2361 q^{53} +1.61803 q^{54} +4.47214 q^{56} -6.70820 q^{57} -5.85410 q^{58} +3.94427 q^{59} +8.70820 q^{61} +14.0902 q^{62} -2.00000 q^{63} +4.23607 q^{64} -4.85410 q^{66} -13.1803 q^{67} -2.61803 q^{68} +5.38197 q^{69} +10.0902 q^{71} -2.23607 q^{72} +15.7082 q^{73} -3.23607 q^{74} -4.14590 q^{76} +6.00000 q^{77} -1.61803 q^{78} -9.14590 q^{79} +1.00000 q^{81} -15.1803 q^{82} +9.00000 q^{83} -1.23607 q^{84} -11.9443 q^{86} -3.61803 q^{87} +6.70820 q^{88} -11.1803 q^{89} +2.00000 q^{91} +3.32624 q^{92} +8.70820 q^{93} -7.70820 q^{94} -3.38197 q^{96} +3.85410 q^{97} -4.85410 q^{98} -3.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} + 2 q^{3} - q^{4} + q^{6} - 4 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q + q^2 + 2 * q^3 - q^4 + q^6 - 4 * q^7 + 2 * q^9 $$2 q + q^{2} + 2 q^{3} - q^{4} + q^{6} - 4 q^{7} + 2 q^{9} - 6 q^{11} - q^{12} - 2 q^{13} - 2 q^{14} - 3 q^{16} - 4 q^{17} + q^{18} - 4 q^{21} - 3 q^{22} + 13 q^{23} - q^{26} + 2 q^{27} + 2 q^{28} - 5 q^{29} + 4 q^{31} - 9 q^{32} - 6 q^{33} - 7 q^{34} - q^{36} - 4 q^{37} - 15 q^{38} - 2 q^{39} - 21 q^{41} - 2 q^{42} - 17 q^{43} + 3 q^{44} + 4 q^{46} - 14 q^{47} - 3 q^{48} - 6 q^{49} - 4 q^{51} + q^{52} + 18 q^{53} + q^{54} - 5 q^{58} - 10 q^{59} + 4 q^{61} + 17 q^{62} - 4 q^{63} + 4 q^{64} - 3 q^{66} - 4 q^{67} - 3 q^{68} + 13 q^{69} + 9 q^{71} + 18 q^{73} - 2 q^{74} - 15 q^{76} + 12 q^{77} - q^{78} - 25 q^{79} + 2 q^{81} - 8 q^{82} + 18 q^{83} + 2 q^{84} - 6 q^{86} - 5 q^{87} + 4 q^{91} - 9 q^{92} + 4 q^{93} - 2 q^{94} - 9 q^{96} + q^{97} - 3 q^{98} - 6 q^{99}+O(q^{100})$$ 2 * q + q^2 + 2 * q^3 - q^4 + q^6 - 4 * q^7 + 2 * q^9 - 6 * q^11 - q^12 - 2 * q^13 - 2 * q^14 - 3 * q^16 - 4 * q^17 + q^18 - 4 * q^21 - 3 * q^22 + 13 * q^23 - q^26 + 2 * q^27 + 2 * q^28 - 5 * q^29 + 4 * q^31 - 9 * q^32 - 6 * q^33 - 7 * q^34 - q^36 - 4 * q^37 - 15 * q^38 - 2 * q^39 - 21 * q^41 - 2 * q^42 - 17 * q^43 + 3 * q^44 + 4 * q^46 - 14 * q^47 - 3 * q^48 - 6 * q^49 - 4 * q^51 + q^52 + 18 * q^53 + q^54 - 5 * q^58 - 10 * q^59 + 4 * q^61 + 17 * q^62 - 4 * q^63 + 4 * q^64 - 3 * q^66 - 4 * q^67 - 3 * q^68 + 13 * q^69 + 9 * q^71 + 18 * q^73 - 2 * q^74 - 15 * q^76 + 12 * q^77 - q^78 - 25 * q^79 + 2 * q^81 - 8 * q^82 + 18 * q^83 + 2 * q^84 - 6 * q^86 - 5 * q^87 + 4 * q^91 - 9 * q^92 + 4 * q^93 - 2 * q^94 - 9 * q^96 + q^97 - 3 * q^98 - 6 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.61803 1.14412 0.572061 0.820211i $$-0.306144\pi$$
0.572061 + 0.820211i $$0.306144\pi$$
$$3$$ 1.00000 0.577350
$$4$$ 0.618034 0.309017
$$5$$ 0 0
$$6$$ 1.61803 0.660560
$$7$$ −2.00000 −0.755929 −0.377964 0.925820i $$-0.623376\pi$$
−0.377964 + 0.925820i $$0.623376\pi$$
$$8$$ −2.23607 −0.790569
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −3.00000 −0.904534 −0.452267 0.891883i $$-0.649385\pi$$
−0.452267 + 0.891883i $$0.649385\pi$$
$$12$$ 0.618034 0.178411
$$13$$ −1.00000 −0.277350 −0.138675 0.990338i $$-0.544284\pi$$
−0.138675 + 0.990338i $$0.544284\pi$$
$$14$$ −3.23607 −0.864876
$$15$$ 0 0
$$16$$ −4.85410 −1.21353
$$17$$ −4.23607 −1.02740 −0.513699 0.857971i $$-0.671725\pi$$
−0.513699 + 0.857971i $$0.671725\pi$$
$$18$$ 1.61803 0.381374
$$19$$ −6.70820 −1.53897 −0.769484 0.638666i $$-0.779486\pi$$
−0.769484 + 0.638666i $$0.779486\pi$$
$$20$$ 0 0
$$21$$ −2.00000 −0.436436
$$22$$ −4.85410 −1.03490
$$23$$ 5.38197 1.12222 0.561109 0.827742i $$-0.310375\pi$$
0.561109 + 0.827742i $$0.310375\pi$$
$$24$$ −2.23607 −0.456435
$$25$$ 0 0
$$26$$ −1.61803 −0.317323
$$27$$ 1.00000 0.192450
$$28$$ −1.23607 −0.233595
$$29$$ −3.61803 −0.671852 −0.335926 0.941888i $$-0.609049\pi$$
−0.335926 + 0.941888i $$0.609049\pi$$
$$30$$ 0 0
$$31$$ 8.70820 1.56404 0.782020 0.623254i $$-0.214190\pi$$
0.782020 + 0.623254i $$0.214190\pi$$
$$32$$ −3.38197 −0.597853
$$33$$ −3.00000 −0.522233
$$34$$ −6.85410 −1.17547
$$35$$ 0 0
$$36$$ 0.618034 0.103006
$$37$$ −2.00000 −0.328798 −0.164399 0.986394i $$-0.552568\pi$$
−0.164399 + 0.986394i $$0.552568\pi$$
$$38$$ −10.8541 −1.76077
$$39$$ −1.00000 −0.160128
$$40$$ 0 0
$$41$$ −9.38197 −1.46522 −0.732608 0.680650i $$-0.761697\pi$$
−0.732608 + 0.680650i $$0.761697\pi$$
$$42$$ −3.23607 −0.499336
$$43$$ −7.38197 −1.12574 −0.562870 0.826546i $$-0.690303\pi$$
−0.562870 + 0.826546i $$0.690303\pi$$
$$44$$ −1.85410 −0.279516
$$45$$ 0 0
$$46$$ 8.70820 1.28395
$$47$$ −4.76393 −0.694891 −0.347445 0.937700i $$-0.612951\pi$$
−0.347445 + 0.937700i $$0.612951\pi$$
$$48$$ −4.85410 −0.700629
$$49$$ −3.00000 −0.428571
$$50$$ 0 0
$$51$$ −4.23607 −0.593168
$$52$$ −0.618034 −0.0857059
$$53$$ 11.2361 1.54339 0.771696 0.635991i $$-0.219409\pi$$
0.771696 + 0.635991i $$0.219409\pi$$
$$54$$ 1.61803 0.220187
$$55$$ 0 0
$$56$$ 4.47214 0.597614
$$57$$ −6.70820 −0.888523
$$58$$ −5.85410 −0.768681
$$59$$ 3.94427 0.513500 0.256750 0.966478i $$-0.417348\pi$$
0.256750 + 0.966478i $$0.417348\pi$$
$$60$$ 0 0
$$61$$ 8.70820 1.11497 0.557486 0.830187i $$-0.311766\pi$$
0.557486 + 0.830187i $$0.311766\pi$$
$$62$$ 14.0902 1.78945
$$63$$ −2.00000 −0.251976
$$64$$ 4.23607 0.529508
$$65$$ 0 0
$$66$$ −4.85410 −0.597499
$$67$$ −13.1803 −1.61023 −0.805117 0.593115i $$-0.797898\pi$$
−0.805117 + 0.593115i $$0.797898\pi$$
$$68$$ −2.61803 −0.317483
$$69$$ 5.38197 0.647913
$$70$$ 0 0
$$71$$ 10.0902 1.19748 0.598741 0.800942i $$-0.295668\pi$$
0.598741 + 0.800942i $$0.295668\pi$$
$$72$$ −2.23607 −0.263523
$$73$$ 15.7082 1.83851 0.919253 0.393667i $$-0.128794\pi$$
0.919253 + 0.393667i $$0.128794\pi$$
$$74$$ −3.23607 −0.376185
$$75$$ 0 0
$$76$$ −4.14590 −0.475567
$$77$$ 6.00000 0.683763
$$78$$ −1.61803 −0.183206
$$79$$ −9.14590 −1.02899 −0.514497 0.857492i $$-0.672021\pi$$
−0.514497 + 0.857492i $$0.672021\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ −15.1803 −1.67639
$$83$$ 9.00000 0.987878 0.493939 0.869496i $$-0.335557\pi$$
0.493939 + 0.869496i $$0.335557\pi$$
$$84$$ −1.23607 −0.134866
$$85$$ 0 0
$$86$$ −11.9443 −1.28798
$$87$$ −3.61803 −0.387894
$$88$$ 6.70820 0.715097
$$89$$ −11.1803 −1.18511 −0.592557 0.805529i $$-0.701881\pi$$
−0.592557 + 0.805529i $$0.701881\pi$$
$$90$$ 0 0
$$91$$ 2.00000 0.209657
$$92$$ 3.32624 0.346784
$$93$$ 8.70820 0.902999
$$94$$ −7.70820 −0.795041
$$95$$ 0 0
$$96$$ −3.38197 −0.345170
$$97$$ 3.85410 0.391325 0.195662 0.980671i $$-0.437314\pi$$
0.195662 + 0.980671i $$0.437314\pi$$
$$98$$ −4.85410 −0.490338
$$99$$ −3.00000 −0.301511
$$100$$ 0 0
$$101$$ −9.38197 −0.933541 −0.466770 0.884379i $$-0.654583\pi$$
−0.466770 + 0.884379i $$0.654583\pi$$
$$102$$ −6.85410 −0.678657
$$103$$ −14.4164 −1.42049 −0.710245 0.703954i $$-0.751416\pi$$
−0.710245 + 0.703954i $$0.751416\pi$$
$$104$$ 2.23607 0.219265
$$105$$ 0 0
$$106$$ 18.1803 1.76583
$$107$$ −1.14590 −0.110778 −0.0553891 0.998465i $$-0.517640\pi$$
−0.0553891 + 0.998465i $$0.517640\pi$$
$$108$$ 0.618034 0.0594703
$$109$$ −4.14590 −0.397105 −0.198553 0.980090i $$-0.563624\pi$$
−0.198553 + 0.980090i $$0.563624\pi$$
$$110$$ 0 0
$$111$$ −2.00000 −0.189832
$$112$$ 9.70820 0.917339
$$113$$ −3.76393 −0.354081 −0.177040 0.984204i $$-0.556652\pi$$
−0.177040 + 0.984204i $$0.556652\pi$$
$$114$$ −10.8541 −1.01658
$$115$$ 0 0
$$116$$ −2.23607 −0.207614
$$117$$ −1.00000 −0.0924500
$$118$$ 6.38197 0.587508
$$119$$ 8.47214 0.776639
$$120$$ 0 0
$$121$$ −2.00000 −0.181818
$$122$$ 14.0902 1.27566
$$123$$ −9.38197 −0.845943
$$124$$ 5.38197 0.483315
$$125$$ 0 0
$$126$$ −3.23607 −0.288292
$$127$$ 13.6525 1.21146 0.605731 0.795670i $$-0.292881\pi$$
0.605731 + 0.795670i $$0.292881\pi$$
$$128$$ 13.6180 1.20368
$$129$$ −7.38197 −0.649946
$$130$$ 0 0
$$131$$ −14.1803 −1.23894 −0.619471 0.785020i $$-0.712653\pi$$
−0.619471 + 0.785020i $$0.712653\pi$$
$$132$$ −1.85410 −0.161379
$$133$$ 13.4164 1.16335
$$134$$ −21.3262 −1.84231
$$135$$ 0 0
$$136$$ 9.47214 0.812229
$$137$$ 0.437694 0.0373947 0.0186974 0.999825i $$-0.494048\pi$$
0.0186974 + 0.999825i $$0.494048\pi$$
$$138$$ 8.70820 0.741292
$$139$$ 13.4164 1.13796 0.568982 0.822350i $$-0.307337\pi$$
0.568982 + 0.822350i $$0.307337\pi$$
$$140$$ 0 0
$$141$$ −4.76393 −0.401195
$$142$$ 16.3262 1.37007
$$143$$ 3.00000 0.250873
$$144$$ −4.85410 −0.404508
$$145$$ 0 0
$$146$$ 25.4164 2.10348
$$147$$ −3.00000 −0.247436
$$148$$ −1.23607 −0.101604
$$149$$ −13.0902 −1.07239 −0.536194 0.844095i $$-0.680139\pi$$
−0.536194 + 0.844095i $$0.680139\pi$$
$$150$$ 0 0
$$151$$ −6.61803 −0.538568 −0.269284 0.963061i $$-0.586787\pi$$
−0.269284 + 0.963061i $$0.586787\pi$$
$$152$$ 15.0000 1.21666
$$153$$ −4.23607 −0.342466
$$154$$ 9.70820 0.782309
$$155$$ 0 0
$$156$$ −0.618034 −0.0494823
$$157$$ −2.85410 −0.227782 −0.113891 0.993493i $$-0.536331\pi$$
−0.113891 + 0.993493i $$0.536331\pi$$
$$158$$ −14.7984 −1.17730
$$159$$ 11.2361 0.891078
$$160$$ 0 0
$$161$$ −10.7639 −0.848317
$$162$$ 1.61803 0.127125
$$163$$ 18.2705 1.43106 0.715528 0.698584i $$-0.246186\pi$$
0.715528 + 0.698584i $$0.246186\pi$$
$$164$$ −5.79837 −0.452777
$$165$$ 0 0
$$166$$ 14.5623 1.13025
$$167$$ 17.7984 1.37728 0.688640 0.725104i $$-0.258208\pi$$
0.688640 + 0.725104i $$0.258208\pi$$
$$168$$ 4.47214 0.345033
$$169$$ −12.0000 −0.923077
$$170$$ 0 0
$$171$$ −6.70820 −0.512989
$$172$$ −4.56231 −0.347873
$$173$$ 0.909830 0.0691731 0.0345865 0.999402i $$-0.488989\pi$$
0.0345865 + 0.999402i $$0.488989\pi$$
$$174$$ −5.85410 −0.443798
$$175$$ 0 0
$$176$$ 14.5623 1.09768
$$177$$ 3.94427 0.296470
$$178$$ −18.0902 −1.35592
$$179$$ −15.6525 −1.16992 −0.584960 0.811062i $$-0.698890\pi$$
−0.584960 + 0.811062i $$0.698890\pi$$
$$180$$ 0 0
$$181$$ −12.4721 −0.927047 −0.463523 0.886085i $$-0.653415\pi$$
−0.463523 + 0.886085i $$0.653415\pi$$
$$182$$ 3.23607 0.239873
$$183$$ 8.70820 0.643729
$$184$$ −12.0344 −0.887191
$$185$$ 0 0
$$186$$ 14.0902 1.03314
$$187$$ 12.7082 0.929316
$$188$$ −2.94427 −0.214733
$$189$$ −2.00000 −0.145479
$$190$$ 0 0
$$191$$ 17.3262 1.25368 0.626841 0.779147i $$-0.284347\pi$$
0.626841 + 0.779147i $$0.284347\pi$$
$$192$$ 4.23607 0.305712
$$193$$ −11.0000 −0.791797 −0.395899 0.918294i $$-0.629567\pi$$
−0.395899 + 0.918294i $$0.629567\pi$$
$$194$$ 6.23607 0.447724
$$195$$ 0 0
$$196$$ −1.85410 −0.132436
$$197$$ −0.0901699 −0.00642434 −0.00321217 0.999995i $$-0.501022\pi$$
−0.00321217 + 0.999995i $$0.501022\pi$$
$$198$$ −4.85410 −0.344966
$$199$$ 11.7082 0.829973 0.414986 0.909828i $$-0.363786\pi$$
0.414986 + 0.909828i $$0.363786\pi$$
$$200$$ 0 0
$$201$$ −13.1803 −0.929669
$$202$$ −15.1803 −1.06808
$$203$$ 7.23607 0.507872
$$204$$ −2.61803 −0.183299
$$205$$ 0 0
$$206$$ −23.3262 −1.62522
$$207$$ 5.38197 0.374072
$$208$$ 4.85410 0.336571
$$209$$ 20.1246 1.39205
$$210$$ 0 0
$$211$$ −3.00000 −0.206529 −0.103264 0.994654i $$-0.532929\pi$$
−0.103264 + 0.994654i $$0.532929\pi$$
$$212$$ 6.94427 0.476935
$$213$$ 10.0902 0.691367
$$214$$ −1.85410 −0.126744
$$215$$ 0 0
$$216$$ −2.23607 −0.152145
$$217$$ −17.4164 −1.18230
$$218$$ −6.70820 −0.454337
$$219$$ 15.7082 1.06146
$$220$$ 0 0
$$221$$ 4.23607 0.284949
$$222$$ −3.23607 −0.217191
$$223$$ −10.1459 −0.679420 −0.339710 0.940530i $$-0.610329\pi$$
−0.339710 + 0.940530i $$0.610329\pi$$
$$224$$ 6.76393 0.451934
$$225$$ 0 0
$$226$$ −6.09017 −0.405112
$$227$$ 5.76393 0.382566 0.191283 0.981535i $$-0.438735\pi$$
0.191283 + 0.981535i $$0.438735\pi$$
$$228$$ −4.14590 −0.274569
$$229$$ −16.1803 −1.06923 −0.534613 0.845097i $$-0.679543\pi$$
−0.534613 + 0.845097i $$0.679543\pi$$
$$230$$ 0 0
$$231$$ 6.00000 0.394771
$$232$$ 8.09017 0.531146
$$233$$ 10.1803 0.666936 0.333468 0.942761i $$-0.391781\pi$$
0.333468 + 0.942761i $$0.391781\pi$$
$$234$$ −1.61803 −0.105774
$$235$$ 0 0
$$236$$ 2.43769 0.158680
$$237$$ −9.14590 −0.594090
$$238$$ 13.7082 0.888571
$$239$$ 21.3820 1.38308 0.691542 0.722336i $$-0.256932\pi$$
0.691542 + 0.722336i $$0.256932\pi$$
$$240$$ 0 0
$$241$$ 7.32624 0.471924 0.235962 0.971762i $$-0.424176\pi$$
0.235962 + 0.971762i $$0.424176\pi$$
$$242$$ −3.23607 −0.208022
$$243$$ 1.00000 0.0641500
$$244$$ 5.38197 0.344545
$$245$$ 0 0
$$246$$ −15.1803 −0.967863
$$247$$ 6.70820 0.426833
$$248$$ −19.4721 −1.23648
$$249$$ 9.00000 0.570352
$$250$$ 0 0
$$251$$ −18.9787 −1.19793 −0.598963 0.800777i $$-0.704420\pi$$
−0.598963 + 0.800777i $$0.704420\pi$$
$$252$$ −1.23607 −0.0778650
$$253$$ −16.1459 −1.01508
$$254$$ 22.0902 1.38606
$$255$$ 0 0
$$256$$ 13.5623 0.847644
$$257$$ 31.2148 1.94712 0.973562 0.228422i $$-0.0733565\pi$$
0.973562 + 0.228422i $$0.0733565\pi$$
$$258$$ −11.9443 −0.743618
$$259$$ 4.00000 0.248548
$$260$$ 0 0
$$261$$ −3.61803 −0.223951
$$262$$ −22.9443 −1.41750
$$263$$ −12.5066 −0.771189 −0.385594 0.922668i $$-0.626004\pi$$
−0.385594 + 0.922668i $$0.626004\pi$$
$$264$$ 6.70820 0.412861
$$265$$ 0 0
$$266$$ 21.7082 1.33102
$$267$$ −11.1803 −0.684226
$$268$$ −8.14590 −0.497590
$$269$$ −20.5279 −1.25161 −0.625803 0.779981i $$-0.715229\pi$$
−0.625803 + 0.779981i $$0.715229\pi$$
$$270$$ 0 0
$$271$$ −11.4164 −0.693497 −0.346749 0.937958i $$-0.612714\pi$$
−0.346749 + 0.937958i $$0.612714\pi$$
$$272$$ 20.5623 1.24677
$$273$$ 2.00000 0.121046
$$274$$ 0.708204 0.0427842
$$275$$ 0 0
$$276$$ 3.32624 0.200216
$$277$$ −13.0557 −0.784443 −0.392221 0.919871i $$-0.628293\pi$$
−0.392221 + 0.919871i $$0.628293\pi$$
$$278$$ 21.7082 1.30197
$$279$$ 8.70820 0.521347
$$280$$ 0 0
$$281$$ −14.1803 −0.845928 −0.422964 0.906146i $$-0.639010\pi$$
−0.422964 + 0.906146i $$0.639010\pi$$
$$282$$ −7.70820 −0.459017
$$283$$ 2.29180 0.136233 0.0681166 0.997677i $$-0.478301\pi$$
0.0681166 + 0.997677i $$0.478301\pi$$
$$284$$ 6.23607 0.370043
$$285$$ 0 0
$$286$$ 4.85410 0.287029
$$287$$ 18.7639 1.10760
$$288$$ −3.38197 −0.199284
$$289$$ 0.944272 0.0555454
$$290$$ 0 0
$$291$$ 3.85410 0.225931
$$292$$ 9.70820 0.568130
$$293$$ −6.32624 −0.369583 −0.184791 0.982778i $$-0.559161\pi$$
−0.184791 + 0.982778i $$0.559161\pi$$
$$294$$ −4.85410 −0.283097
$$295$$ 0 0
$$296$$ 4.47214 0.259938
$$297$$ −3.00000 −0.174078
$$298$$ −21.1803 −1.22694
$$299$$ −5.38197 −0.311247
$$300$$ 0 0
$$301$$ 14.7639 0.850979
$$302$$ −10.7082 −0.616188
$$303$$ −9.38197 −0.538980
$$304$$ 32.5623 1.86758
$$305$$ 0 0
$$306$$ −6.85410 −0.391823
$$307$$ 8.85410 0.505330 0.252665 0.967554i $$-0.418693\pi$$
0.252665 + 0.967554i $$0.418693\pi$$
$$308$$ 3.70820 0.211295
$$309$$ −14.4164 −0.820121
$$310$$ 0 0
$$311$$ −13.5279 −0.767095 −0.383547 0.923521i $$-0.625298\pi$$
−0.383547 + 0.923521i $$0.625298\pi$$
$$312$$ 2.23607 0.126592
$$313$$ 2.29180 0.129540 0.0647700 0.997900i $$-0.479369\pi$$
0.0647700 + 0.997900i $$0.479369\pi$$
$$314$$ −4.61803 −0.260611
$$315$$ 0 0
$$316$$ −5.65248 −0.317977
$$317$$ 20.5623 1.15489 0.577447 0.816428i $$-0.304049\pi$$
0.577447 + 0.816428i $$0.304049\pi$$
$$318$$ 18.1803 1.01950
$$319$$ 10.8541 0.607713
$$320$$ 0 0
$$321$$ −1.14590 −0.0639578
$$322$$ −17.4164 −0.970578
$$323$$ 28.4164 1.58113
$$324$$ 0.618034 0.0343352
$$325$$ 0 0
$$326$$ 29.5623 1.63730
$$327$$ −4.14590 −0.229269
$$328$$ 20.9787 1.15836
$$329$$ 9.52786 0.525288
$$330$$ 0 0
$$331$$ −30.6869 −1.68671 −0.843353 0.537360i $$-0.819422\pi$$
−0.843353 + 0.537360i $$0.819422\pi$$
$$332$$ 5.56231 0.305271
$$333$$ −2.00000 −0.109599
$$334$$ 28.7984 1.57578
$$335$$ 0 0
$$336$$ 9.70820 0.529626
$$337$$ −33.1803 −1.80745 −0.903724 0.428115i $$-0.859178\pi$$
−0.903724 + 0.428115i $$0.859178\pi$$
$$338$$ −19.4164 −1.05611
$$339$$ −3.76393 −0.204429
$$340$$ 0 0
$$341$$ −26.1246 −1.41473
$$342$$ −10.8541 −0.586923
$$343$$ 20.0000 1.07990
$$344$$ 16.5066 0.889975
$$345$$ 0 0
$$346$$ 1.47214 0.0791425
$$347$$ 12.2705 0.658715 0.329358 0.944205i $$-0.393168\pi$$
0.329358 + 0.944205i $$0.393168\pi$$
$$348$$ −2.23607 −0.119866
$$349$$ 7.23607 0.387338 0.193669 0.981067i $$-0.437961\pi$$
0.193669 + 0.981067i $$0.437961\pi$$
$$350$$ 0 0
$$351$$ −1.00000 −0.0533761
$$352$$ 10.1459 0.540778
$$353$$ −12.3820 −0.659026 −0.329513 0.944151i $$-0.606884\pi$$
−0.329513 + 0.944151i $$0.606884\pi$$
$$354$$ 6.38197 0.339198
$$355$$ 0 0
$$356$$ −6.90983 −0.366220
$$357$$ 8.47214 0.448393
$$358$$ −25.3262 −1.33853
$$359$$ 23.9443 1.26373 0.631865 0.775078i $$-0.282290\pi$$
0.631865 + 0.775078i $$0.282290\pi$$
$$360$$ 0 0
$$361$$ 26.0000 1.36842
$$362$$ −20.1803 −1.06066
$$363$$ −2.00000 −0.104973
$$364$$ 1.23607 0.0647876
$$365$$ 0 0
$$366$$ 14.0902 0.736505
$$367$$ −12.5279 −0.653949 −0.326975 0.945033i $$-0.606029\pi$$
−0.326975 + 0.945033i $$0.606029\pi$$
$$368$$ −26.1246 −1.36184
$$369$$ −9.38197 −0.488406
$$370$$ 0 0
$$371$$ −22.4721 −1.16670
$$372$$ 5.38197 0.279042
$$373$$ 17.4164 0.901787 0.450894 0.892578i $$-0.351105\pi$$
0.450894 + 0.892578i $$0.351105\pi$$
$$374$$ 20.5623 1.06325
$$375$$ 0 0
$$376$$ 10.6525 0.549359
$$377$$ 3.61803 0.186338
$$378$$ −3.23607 −0.166445
$$379$$ −13.6180 −0.699511 −0.349756 0.936841i $$-0.613735\pi$$
−0.349756 + 0.936841i $$0.613735\pi$$
$$380$$ 0 0
$$381$$ 13.6525 0.699438
$$382$$ 28.0344 1.43437
$$383$$ 5.05573 0.258336 0.129168 0.991623i $$-0.458769\pi$$
0.129168 + 0.991623i $$0.458769\pi$$
$$384$$ 13.6180 0.694942
$$385$$ 0 0
$$386$$ −17.7984 −0.905913
$$387$$ −7.38197 −0.375246
$$388$$ 2.38197 0.120926
$$389$$ 0.652476 0.0330818 0.0165409 0.999863i $$-0.494735\pi$$
0.0165409 + 0.999863i $$0.494735\pi$$
$$390$$ 0 0
$$391$$ −22.7984 −1.15296
$$392$$ 6.70820 0.338815
$$393$$ −14.1803 −0.715304
$$394$$ −0.145898 −0.00735024
$$395$$ 0 0
$$396$$ −1.85410 −0.0931721
$$397$$ −2.52786 −0.126870 −0.0634349 0.997986i $$-0.520206\pi$$
−0.0634349 + 0.997986i $$0.520206\pi$$
$$398$$ 18.9443 0.949591
$$399$$ 13.4164 0.671660
$$400$$ 0 0
$$401$$ 36.2705 1.81126 0.905631 0.424066i $$-0.139397\pi$$
0.905631 + 0.424066i $$0.139397\pi$$
$$402$$ −21.3262 −1.06366
$$403$$ −8.70820 −0.433787
$$404$$ −5.79837 −0.288480
$$405$$ 0 0
$$406$$ 11.7082 0.581068
$$407$$ 6.00000 0.297409
$$408$$ 9.47214 0.468941
$$409$$ −5.12461 −0.253396 −0.126698 0.991941i $$-0.540438\pi$$
−0.126698 + 0.991941i $$0.540438\pi$$
$$410$$ 0 0
$$411$$ 0.437694 0.0215899
$$412$$ −8.90983 −0.438956
$$413$$ −7.88854 −0.388170
$$414$$ 8.70820 0.427985
$$415$$ 0 0
$$416$$ 3.38197 0.165815
$$417$$ 13.4164 0.657004
$$418$$ 32.5623 1.59267
$$419$$ 0.326238 0.0159378 0.00796888 0.999968i $$-0.497463\pi$$
0.00796888 + 0.999968i $$0.497463\pi$$
$$420$$ 0 0
$$421$$ −30.3607 −1.47969 −0.739844 0.672778i $$-0.765101\pi$$
−0.739844 + 0.672778i $$0.765101\pi$$
$$422$$ −4.85410 −0.236294
$$423$$ −4.76393 −0.231630
$$424$$ −25.1246 −1.22016
$$425$$ 0 0
$$426$$ 16.3262 0.791009
$$427$$ −17.4164 −0.842839
$$428$$ −0.708204 −0.0342323
$$429$$ 3.00000 0.144841
$$430$$ 0 0
$$431$$ 29.7639 1.43368 0.716839 0.697239i $$-0.245588\pi$$
0.716839 + 0.697239i $$0.245588\pi$$
$$432$$ −4.85410 −0.233543
$$433$$ 3.47214 0.166860 0.0834301 0.996514i $$-0.473412\pi$$
0.0834301 + 0.996514i $$0.473412\pi$$
$$434$$ −28.1803 −1.35270
$$435$$ 0 0
$$436$$ −2.56231 −0.122712
$$437$$ −36.1033 −1.72706
$$438$$ 25.4164 1.21444
$$439$$ −32.0344 −1.52892 −0.764460 0.644671i $$-0.776994\pi$$
−0.764460 + 0.644671i $$0.776994\pi$$
$$440$$ 0 0
$$441$$ −3.00000 −0.142857
$$442$$ 6.85410 0.326016
$$443$$ −19.4164 −0.922501 −0.461251 0.887270i $$-0.652599\pi$$
−0.461251 + 0.887270i $$0.652599\pi$$
$$444$$ −1.23607 −0.0586612
$$445$$ 0 0
$$446$$ −16.4164 −0.777339
$$447$$ −13.0902 −0.619144
$$448$$ −8.47214 −0.400271
$$449$$ 16.5066 0.778994 0.389497 0.921028i $$-0.372649\pi$$
0.389497 + 0.921028i $$0.372649\pi$$
$$450$$ 0 0
$$451$$ 28.1459 1.32534
$$452$$ −2.32624 −0.109417
$$453$$ −6.61803 −0.310942
$$454$$ 9.32624 0.437702
$$455$$ 0 0
$$456$$ 15.0000 0.702439
$$457$$ −9.88854 −0.462567 −0.231283 0.972886i $$-0.574292\pi$$
−0.231283 + 0.972886i $$0.574292\pi$$
$$458$$ −26.1803 −1.22333
$$459$$ −4.23607 −0.197723
$$460$$ 0 0
$$461$$ −19.1803 −0.893317 −0.446659 0.894704i $$-0.647386\pi$$
−0.446659 + 0.894704i $$0.647386\pi$$
$$462$$ 9.70820 0.451667
$$463$$ −33.6869 −1.56556 −0.782782 0.622296i $$-0.786200\pi$$
−0.782782 + 0.622296i $$0.786200\pi$$
$$464$$ 17.5623 0.815310
$$465$$ 0 0
$$466$$ 16.4721 0.763057
$$467$$ −10.4164 −0.482014 −0.241007 0.970523i $$-0.577478\pi$$
−0.241007 + 0.970523i $$0.577478\pi$$
$$468$$ −0.618034 −0.0285686
$$469$$ 26.3607 1.21722
$$470$$ 0 0
$$471$$ −2.85410 −0.131510
$$472$$ −8.81966 −0.405958
$$473$$ 22.1459 1.01827
$$474$$ −14.7984 −0.679712
$$475$$ 0 0
$$476$$ 5.23607 0.239995
$$477$$ 11.2361 0.514464
$$478$$ 34.5967 1.58242
$$479$$ 4.79837 0.219243 0.109622 0.993973i $$-0.465036\pi$$
0.109622 + 0.993973i $$0.465036\pi$$
$$480$$ 0 0
$$481$$ 2.00000 0.0911922
$$482$$ 11.8541 0.539940
$$483$$ −10.7639 −0.489776
$$484$$ −1.23607 −0.0561849
$$485$$ 0 0
$$486$$ 1.61803 0.0733955
$$487$$ 16.6180 0.753035 0.376518 0.926410i $$-0.377121\pi$$
0.376518 + 0.926410i $$0.377121\pi$$
$$488$$ −19.4721 −0.881462
$$489$$ 18.2705 0.826221
$$490$$ 0 0
$$491$$ 22.3262 1.00757 0.503785 0.863829i $$-0.331941\pi$$
0.503785 + 0.863829i $$0.331941\pi$$
$$492$$ −5.79837 −0.261411
$$493$$ 15.3262 0.690259
$$494$$ 10.8541 0.488349
$$495$$ 0 0
$$496$$ −42.2705 −1.89800
$$497$$ −20.1803 −0.905212
$$498$$ 14.5623 0.652553
$$499$$ −15.0000 −0.671492 −0.335746 0.941953i $$-0.608988\pi$$
−0.335746 + 0.941953i $$0.608988\pi$$
$$500$$ 0 0
$$501$$ 17.7984 0.795173
$$502$$ −30.7082 −1.37057
$$503$$ −3.96556 −0.176815 −0.0884077 0.996084i $$-0.528178\pi$$
−0.0884077 + 0.996084i $$0.528178\pi$$
$$504$$ 4.47214 0.199205
$$505$$ 0 0
$$506$$ −26.1246 −1.16138
$$507$$ −12.0000 −0.532939
$$508$$ 8.43769 0.374362
$$509$$ 32.8885 1.45776 0.728880 0.684642i $$-0.240041\pi$$
0.728880 + 0.684642i $$0.240041\pi$$
$$510$$ 0 0
$$511$$ −31.4164 −1.38978
$$512$$ −5.29180 −0.233867
$$513$$ −6.70820 −0.296174
$$514$$ 50.5066 2.22775
$$515$$ 0 0
$$516$$ −4.56231 −0.200844
$$517$$ 14.2918 0.628552
$$518$$ 6.47214 0.284369
$$519$$ 0.909830 0.0399371
$$520$$ 0 0
$$521$$ 40.0902 1.75638 0.878191 0.478310i $$-0.158750\pi$$
0.878191 + 0.478310i $$0.158750\pi$$
$$522$$ −5.85410 −0.256227
$$523$$ 1.56231 0.0683149 0.0341574 0.999416i $$-0.489125\pi$$
0.0341574 + 0.999416i $$0.489125\pi$$
$$524$$ −8.76393 −0.382854
$$525$$ 0 0
$$526$$ −20.2361 −0.882334
$$527$$ −36.8885 −1.60689
$$528$$ 14.5623 0.633743
$$529$$ 5.96556 0.259372
$$530$$ 0 0
$$531$$ 3.94427 0.171167
$$532$$ 8.29180 0.359495
$$533$$ 9.38197 0.406378
$$534$$ −18.0902 −0.782838
$$535$$ 0 0
$$536$$ 29.4721 1.27300
$$537$$ −15.6525 −0.675454
$$538$$ −33.2148 −1.43199
$$539$$ 9.00000 0.387657
$$540$$ 0 0
$$541$$ −26.2918 −1.13037 −0.565186 0.824963i $$-0.691196\pi$$
−0.565186 + 0.824963i $$0.691196\pi$$
$$542$$ −18.4721 −0.793446
$$543$$ −12.4721 −0.535231
$$544$$ 14.3262 0.614232
$$545$$ 0 0
$$546$$ 3.23607 0.138491
$$547$$ 24.7082 1.05645 0.528223 0.849106i $$-0.322858\pi$$
0.528223 + 0.849106i $$0.322858\pi$$
$$548$$ 0.270510 0.0115556
$$549$$ 8.70820 0.371657
$$550$$ 0 0
$$551$$ 24.2705 1.03396
$$552$$ −12.0344 −0.512220
$$553$$ 18.2918 0.777846
$$554$$ −21.1246 −0.897499
$$555$$ 0 0
$$556$$ 8.29180 0.351650
$$557$$ −37.6525 −1.59539 −0.797693 0.603063i $$-0.793947\pi$$
−0.797693 + 0.603063i $$0.793947\pi$$
$$558$$ 14.0902 0.596484
$$559$$ 7.38197 0.312224
$$560$$ 0 0
$$561$$ 12.7082 0.536541
$$562$$ −22.9443 −0.967846
$$563$$ 9.00000 0.379305 0.189652 0.981851i $$-0.439264\pi$$
0.189652 + 0.981851i $$0.439264\pi$$
$$564$$ −2.94427 −0.123976
$$565$$ 0 0
$$566$$ 3.70820 0.155867
$$567$$ −2.00000 −0.0839921
$$568$$ −22.5623 −0.946693
$$569$$ −10.8541 −0.455028 −0.227514 0.973775i $$-0.573060\pi$$
−0.227514 + 0.973775i $$0.573060\pi$$
$$570$$ 0 0
$$571$$ −38.1246 −1.59547 −0.797733 0.603011i $$-0.793967\pi$$
−0.797733 + 0.603011i $$0.793967\pi$$
$$572$$ 1.85410 0.0775239
$$573$$ 17.3262 0.723814
$$574$$ 30.3607 1.26723
$$575$$ 0 0
$$576$$ 4.23607 0.176503
$$577$$ 3.72949 0.155261 0.0776304 0.996982i $$-0.475265\pi$$
0.0776304 + 0.996982i $$0.475265\pi$$
$$578$$ 1.52786 0.0635508
$$579$$ −11.0000 −0.457144
$$580$$ 0 0
$$581$$ −18.0000 −0.746766
$$582$$ 6.23607 0.258493
$$583$$ −33.7082 −1.39605
$$584$$ −35.1246 −1.45347
$$585$$ 0 0
$$586$$ −10.2361 −0.422848
$$587$$ 39.3050 1.62229 0.811144 0.584846i $$-0.198845\pi$$
0.811144 + 0.584846i $$0.198845\pi$$
$$588$$ −1.85410 −0.0764619
$$589$$ −58.4164 −2.40701
$$590$$ 0 0
$$591$$ −0.0901699 −0.00370910
$$592$$ 9.70820 0.399005
$$593$$ 17.6180 0.723486 0.361743 0.932278i $$-0.382182\pi$$
0.361743 + 0.932278i $$0.382182\pi$$
$$594$$ −4.85410 −0.199166
$$595$$ 0 0
$$596$$ −8.09017 −0.331386
$$597$$ 11.7082 0.479185
$$598$$ −8.70820 −0.356105
$$599$$ 39.2705 1.60455 0.802275 0.596955i $$-0.203623\pi$$
0.802275 + 0.596955i $$0.203623\pi$$
$$600$$ 0 0
$$601$$ −24.7082 −1.00787 −0.503934 0.863742i $$-0.668115\pi$$
−0.503934 + 0.863742i $$0.668115\pi$$
$$602$$ 23.8885 0.973624
$$603$$ −13.1803 −0.536745
$$604$$ −4.09017 −0.166427
$$605$$ 0 0
$$606$$ −15.1803 −0.616659
$$607$$ −22.8541 −0.927619 −0.463810 0.885935i $$-0.653518\pi$$
−0.463810 + 0.885935i $$0.653518\pi$$
$$608$$ 22.6869 0.920076
$$609$$ 7.23607 0.293220
$$610$$ 0 0
$$611$$ 4.76393 0.192728
$$612$$ −2.61803 −0.105828
$$613$$ −5.87539 −0.237305 −0.118652 0.992936i $$-0.537857\pi$$
−0.118652 + 0.992936i $$0.537857\pi$$
$$614$$ 14.3262 0.578160
$$615$$ 0 0
$$616$$ −13.4164 −0.540562
$$617$$ 25.2361 1.01597 0.507983 0.861367i $$-0.330391\pi$$
0.507983 + 0.861367i $$0.330391\pi$$
$$618$$ −23.3262 −0.938319
$$619$$ 34.2705 1.37745 0.688724 0.725024i $$-0.258171\pi$$
0.688724 + 0.725024i $$0.258171\pi$$
$$620$$ 0 0
$$621$$ 5.38197 0.215971
$$622$$ −21.8885 −0.877651
$$623$$ 22.3607 0.895862
$$624$$ 4.85410 0.194320
$$625$$ 0 0
$$626$$ 3.70820 0.148210
$$627$$ 20.1246 0.803700
$$628$$ −1.76393 −0.0703886
$$629$$ 8.47214 0.337806
$$630$$ 0 0
$$631$$ −10.7639 −0.428505 −0.214253 0.976778i $$-0.568732\pi$$
−0.214253 + 0.976778i $$0.568732\pi$$
$$632$$ 20.4508 0.813491
$$633$$ −3.00000 −0.119239
$$634$$ 33.2705 1.32134
$$635$$ 0 0
$$636$$ 6.94427 0.275358
$$637$$ 3.00000 0.118864
$$638$$ 17.5623 0.695298
$$639$$ 10.0902 0.399161
$$640$$ 0 0
$$641$$ −23.3262 −0.921331 −0.460666 0.887574i $$-0.652389\pi$$
−0.460666 + 0.887574i $$0.652389\pi$$
$$642$$ −1.85410 −0.0731756
$$643$$ 5.90983 0.233061 0.116530 0.993187i $$-0.462823\pi$$
0.116530 + 0.993187i $$0.462823\pi$$
$$644$$ −6.65248 −0.262144
$$645$$ 0 0
$$646$$ 45.9787 1.80901
$$647$$ −19.0344 −0.748321 −0.374161 0.927364i $$-0.622069\pi$$
−0.374161 + 0.927364i $$0.622069\pi$$
$$648$$ −2.23607 −0.0878410
$$649$$ −11.8328 −0.464479
$$650$$ 0 0
$$651$$ −17.4164 −0.682603
$$652$$ 11.2918 0.442221
$$653$$ 29.6525 1.16039 0.580196 0.814477i $$-0.302976\pi$$
0.580196 + 0.814477i $$0.302976\pi$$
$$654$$ −6.70820 −0.262312
$$655$$ 0 0
$$656$$ 45.5410 1.77808
$$657$$ 15.7082 0.612835
$$658$$ 15.4164 0.600994
$$659$$ −2.23607 −0.0871048 −0.0435524 0.999051i $$-0.513868\pi$$
−0.0435524 + 0.999051i $$0.513868\pi$$
$$660$$ 0 0
$$661$$ 4.88854 0.190142 0.0950712 0.995470i $$-0.469692\pi$$
0.0950712 + 0.995470i $$0.469692\pi$$
$$662$$ −49.6525 −1.92980
$$663$$ 4.23607 0.164515
$$664$$ −20.1246 −0.780986
$$665$$ 0 0
$$666$$ −3.23607 −0.125395
$$667$$ −19.4721 −0.753964
$$668$$ 11.0000 0.425603
$$669$$ −10.1459 −0.392263
$$670$$ 0 0
$$671$$ −26.1246 −1.00853
$$672$$ 6.76393 0.260924
$$673$$ −46.7771 −1.80312 −0.901562 0.432650i $$-0.857579\pi$$
−0.901562 + 0.432650i $$0.857579\pi$$
$$674$$ −53.6869 −2.06794
$$675$$ 0 0
$$676$$ −7.41641 −0.285246
$$677$$ −44.8885 −1.72521 −0.862603 0.505881i $$-0.831168\pi$$
−0.862603 + 0.505881i $$0.831168\pi$$
$$678$$ −6.09017 −0.233892
$$679$$ −7.70820 −0.295814
$$680$$ 0 0
$$681$$ 5.76393 0.220874
$$682$$ −42.2705 −1.61862
$$683$$ −0.596748 −0.0228339 −0.0114170 0.999935i $$-0.503634\pi$$
−0.0114170 + 0.999935i $$0.503634\pi$$
$$684$$ −4.14590 −0.158522
$$685$$ 0 0
$$686$$ 32.3607 1.23554
$$687$$ −16.1803 −0.617318
$$688$$ 35.8328 1.36611
$$689$$ −11.2361 −0.428060
$$690$$ 0 0
$$691$$ 15.0902 0.574057 0.287029 0.957922i $$-0.407333\pi$$
0.287029 + 0.957922i $$0.407333\pi$$
$$692$$ 0.562306 0.0213757
$$693$$ 6.00000 0.227921
$$694$$ 19.8541 0.753651
$$695$$ 0 0
$$696$$ 8.09017 0.306657
$$697$$ 39.7426 1.50536
$$698$$ 11.7082 0.443162
$$699$$ 10.1803 0.385056
$$700$$ 0 0
$$701$$ −48.6525 −1.83758 −0.918789 0.394748i $$-0.870832\pi$$
−0.918789 + 0.394748i $$0.870832\pi$$
$$702$$ −1.61803 −0.0610688
$$703$$ 13.4164 0.506009
$$704$$ −12.7082 −0.478958
$$705$$ 0 0
$$706$$ −20.0344 −0.754006
$$707$$ 18.7639 0.705690
$$708$$ 2.43769 0.0916142
$$709$$ −5.20163 −0.195351 −0.0976756 0.995218i $$-0.531141\pi$$
−0.0976756 + 0.995218i $$0.531141\pi$$
$$710$$ 0 0
$$711$$ −9.14590 −0.342998
$$712$$ 25.0000 0.936915
$$713$$ 46.8673 1.75519
$$714$$ 13.7082 0.513017
$$715$$ 0 0
$$716$$ −9.67376 −0.361525
$$717$$ 21.3820 0.798524
$$718$$ 38.7426 1.44586
$$719$$ −35.1246 −1.30993 −0.654963 0.755661i $$-0.727316\pi$$
−0.654963 + 0.755661i $$0.727316\pi$$
$$720$$ 0 0
$$721$$ 28.8328 1.07379
$$722$$ 42.0689 1.56564
$$723$$ 7.32624 0.272466
$$724$$ −7.70820 −0.286473
$$725$$ 0 0
$$726$$ −3.23607 −0.120102
$$727$$ −14.4377 −0.535464 −0.267732 0.963493i $$-0.586274\pi$$
−0.267732 + 0.963493i $$0.586274\pi$$
$$728$$ −4.47214 −0.165748
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 31.2705 1.15658
$$732$$ 5.38197 0.198923
$$733$$ −10.1459 −0.374747 −0.187374 0.982289i $$-0.559998\pi$$
−0.187374 + 0.982289i $$0.559998\pi$$
$$734$$ −20.2705 −0.748198
$$735$$ 0 0
$$736$$ −18.2016 −0.670921
$$737$$ 39.5410 1.45651
$$738$$ −15.1803 −0.558796
$$739$$ −1.70820 −0.0628373 −0.0314186 0.999506i $$-0.510003\pi$$
−0.0314186 + 0.999506i $$0.510003\pi$$
$$740$$ 0 0
$$741$$ 6.70820 0.246432
$$742$$ −36.3607 −1.33484
$$743$$ −16.5279 −0.606349 −0.303174 0.952935i $$-0.598046\pi$$
−0.303174 + 0.952935i $$0.598046\pi$$
$$744$$ −19.4721 −0.713883
$$745$$ 0 0
$$746$$ 28.1803 1.03176
$$747$$ 9.00000 0.329293
$$748$$ 7.85410 0.287174
$$749$$ 2.29180 0.0837404
$$750$$ 0 0
$$751$$ 15.2918 0.558006 0.279003 0.960290i $$-0.409996\pi$$
0.279003 + 0.960290i $$0.409996\pi$$
$$752$$ 23.1246 0.843268
$$753$$ −18.9787 −0.691623
$$754$$ 5.85410 0.213194
$$755$$ 0 0
$$756$$ −1.23607 −0.0449554
$$757$$ 32.2705 1.17289 0.586446 0.809988i $$-0.300527\pi$$
0.586446 + 0.809988i $$0.300527\pi$$
$$758$$ −22.0344 −0.800327
$$759$$ −16.1459 −0.586059
$$760$$ 0 0
$$761$$ 3.18034 0.115287 0.0576436 0.998337i $$-0.481641\pi$$
0.0576436 + 0.998337i $$0.481641\pi$$
$$762$$ 22.0902 0.800242
$$763$$ 8.29180 0.300183
$$764$$ 10.7082 0.387409
$$765$$ 0 0
$$766$$ 8.18034 0.295568
$$767$$ −3.94427 −0.142419
$$768$$ 13.5623 0.489388
$$769$$ −36.3050 −1.30919 −0.654595 0.755980i $$-0.727161\pi$$
−0.654595 + 0.755980i $$0.727161\pi$$
$$770$$ 0 0
$$771$$ 31.2148 1.12417
$$772$$ −6.79837 −0.244679
$$773$$ −41.7771 −1.50262 −0.751309 0.659951i $$-0.770577\pi$$
−0.751309 + 0.659951i $$0.770577\pi$$
$$774$$ −11.9443 −0.429328
$$775$$ 0 0
$$776$$ −8.61803 −0.309369
$$777$$ 4.00000 0.143499
$$778$$ 1.05573 0.0378497
$$779$$ 62.9361 2.25492
$$780$$ 0 0
$$781$$ −30.2705 −1.08316
$$782$$ −36.8885 −1.31913
$$783$$ −3.61803 −0.129298
$$784$$ 14.5623 0.520082
$$785$$ 0 0
$$786$$ −22.9443 −0.818395
$$787$$ 17.1459 0.611185 0.305593 0.952162i $$-0.401145\pi$$
0.305593 + 0.952162i $$0.401145\pi$$
$$788$$ −0.0557281 −0.00198523
$$789$$ −12.5066 −0.445246
$$790$$ 0 0
$$791$$ 7.52786 0.267660
$$792$$ 6.70820 0.238366
$$793$$ −8.70820 −0.309237
$$794$$ −4.09017 −0.145155
$$795$$ 0 0
$$796$$ 7.23607 0.256476
$$797$$ −30.0132 −1.06312 −0.531560 0.847020i $$-0.678394\pi$$
−0.531560 + 0.847020i $$0.678394\pi$$
$$798$$ 21.7082 0.768462
$$799$$ 20.1803 0.713929
$$800$$ 0 0
$$801$$ −11.1803 −0.395038
$$802$$ 58.6869 2.07231
$$803$$ −47.1246 −1.66299
$$804$$ −8.14590 −0.287284
$$805$$ 0 0
$$806$$ −14.0902 −0.496305
$$807$$ −20.5279 −0.722615
$$808$$ 20.9787 0.738029
$$809$$ −39.9230 −1.40362 −0.701809 0.712365i $$-0.747624\pi$$
−0.701809 + 0.712365i $$0.747624\pi$$
$$810$$ 0 0
$$811$$ −33.7771 −1.18607 −0.593037 0.805175i $$-0.702071\pi$$
−0.593037 + 0.805175i $$0.702071\pi$$
$$812$$ 4.47214 0.156941
$$813$$ −11.4164 −0.400391
$$814$$ 9.70820 0.340272
$$815$$ 0 0
$$816$$ 20.5623 0.719825
$$817$$ 49.5197 1.73248
$$818$$ −8.29180 −0.289916
$$819$$ 2.00000 0.0698857
$$820$$ 0 0
$$821$$ 5.94427 0.207457 0.103728 0.994606i $$-0.466923\pi$$
0.103728 + 0.994606i $$0.466923\pi$$
$$822$$ 0.708204 0.0247014
$$823$$ −28.5623 −0.995619 −0.497810 0.867286i $$-0.665862\pi$$
−0.497810 + 0.867286i $$0.665862\pi$$
$$824$$ 32.2361 1.12300
$$825$$ 0 0
$$826$$ −12.7639 −0.444114
$$827$$ 48.9787 1.70316 0.851578 0.524227i $$-0.175646\pi$$
0.851578 + 0.524227i $$0.175646\pi$$
$$828$$ 3.32624 0.115595
$$829$$ −50.1246 −1.74090 −0.870450 0.492257i $$-0.836172\pi$$
−0.870450 + 0.492257i $$0.836172\pi$$
$$830$$ 0 0
$$831$$ −13.0557 −0.452898
$$832$$ −4.23607 −0.146859
$$833$$ 12.7082 0.440313
$$834$$ 21.7082 0.751694
$$835$$ 0 0
$$836$$ 12.4377 0.430167
$$837$$ 8.70820 0.301000
$$838$$ 0.527864 0.0182348
$$839$$ −3.21478 −0.110987 −0.0554933 0.998459i $$-0.517673\pi$$
−0.0554933 + 0.998459i $$0.517673\pi$$
$$840$$ 0 0
$$841$$ −15.9098 −0.548615
$$842$$ −49.1246 −1.69295
$$843$$ −14.1803 −0.488397
$$844$$ −1.85410 −0.0638208
$$845$$ 0 0
$$846$$ −7.70820 −0.265014
$$847$$ 4.00000 0.137442
$$848$$ −54.5410 −1.87295
$$849$$ 2.29180 0.0786542
$$850$$ 0 0
$$851$$ −10.7639 −0.368983
$$852$$ 6.23607 0.213644
$$853$$ −20.3951 −0.698316 −0.349158 0.937064i $$-0.613532\pi$$
−0.349158 + 0.937064i $$0.613532\pi$$
$$854$$ −28.1803 −0.964311
$$855$$ 0 0
$$856$$ 2.56231 0.0875778
$$857$$ 9.05573 0.309338 0.154669 0.987966i $$-0.450569\pi$$
0.154669 + 0.987966i $$0.450569\pi$$
$$858$$ 4.85410 0.165716
$$859$$ 15.1246 0.516045 0.258023 0.966139i $$-0.416929\pi$$
0.258023 + 0.966139i $$0.416929\pi$$
$$860$$ 0 0
$$861$$ 18.7639 0.639473
$$862$$ 48.1591 1.64030
$$863$$ 13.0689 0.444870 0.222435 0.974948i $$-0.428599\pi$$
0.222435 + 0.974948i $$0.428599\pi$$
$$864$$ −3.38197 −0.115057
$$865$$ 0 0
$$866$$ 5.61803 0.190909
$$867$$ 0.944272 0.0320692
$$868$$ −10.7639 −0.365352
$$869$$ 27.4377 0.930760
$$870$$ 0 0
$$871$$ 13.1803 0.446599
$$872$$ 9.27051 0.313939
$$873$$ 3.85410 0.130442
$$874$$ −58.4164 −1.97596
$$875$$ 0 0
$$876$$ 9.70820 0.328010
$$877$$ 43.1246 1.45621 0.728107 0.685463i $$-0.240400\pi$$
0.728107 + 0.685463i $$0.240400\pi$$
$$878$$ −51.8328 −1.74927
$$879$$ −6.32624 −0.213379
$$880$$ 0 0
$$881$$ 15.0902 0.508401 0.254200 0.967152i $$-0.418188\pi$$
0.254200 + 0.967152i $$0.418188\pi$$
$$882$$ −4.85410 −0.163446
$$883$$ 29.2016 0.982713 0.491356 0.870959i $$-0.336501\pi$$
0.491356 + 0.870959i $$0.336501\pi$$
$$884$$ 2.61803 0.0880540
$$885$$ 0 0
$$886$$ −31.4164 −1.05545
$$887$$ 42.9230 1.44121 0.720606 0.693344i $$-0.243864\pi$$
0.720606 + 0.693344i $$0.243864\pi$$
$$888$$ 4.47214 0.150075
$$889$$ −27.3050 −0.915779
$$890$$ 0 0
$$891$$ −3.00000 −0.100504
$$892$$ −6.27051 −0.209952
$$893$$ 31.9574 1.06941
$$894$$ −21.1803 −0.708377
$$895$$ 0 0
$$896$$ −27.2361 −0.909893
$$897$$ −5.38197 −0.179699
$$898$$ 26.7082 0.891264
$$899$$ −31.5066 −1.05080
$$900$$ 0 0
$$901$$ −47.5967 −1.58568
$$902$$ 45.5410 1.51635
$$903$$ 14.7639 0.491313
$$904$$ 8.41641 0.279926
$$905$$ 0 0
$$906$$ −10.7082 −0.355756
$$907$$ −17.0000 −0.564476 −0.282238 0.959344i $$-0.591077\pi$$
−0.282238 + 0.959344i $$0.591077\pi$$
$$908$$ 3.56231 0.118219
$$909$$ −9.38197 −0.311180
$$910$$ 0 0
$$911$$ 14.8885 0.493279 0.246640 0.969107i $$-0.420674\pi$$
0.246640 + 0.969107i $$0.420674\pi$$
$$912$$ 32.5623 1.07825
$$913$$ −27.0000 −0.893570
$$914$$ −16.0000 −0.529233
$$915$$ 0 0
$$916$$ −10.0000 −0.330409
$$917$$ 28.3607 0.936552
$$918$$ −6.85410 −0.226219
$$919$$ −5.00000 −0.164935 −0.0824674 0.996594i $$-0.526280\pi$$
−0.0824674 + 0.996594i $$0.526280\pi$$
$$920$$ 0 0
$$921$$ 8.85410 0.291753
$$922$$ −31.0344 −1.02206
$$923$$ −10.0902 −0.332122
$$924$$ 3.70820 0.121991
$$925$$ 0 0
$$926$$ −54.5066 −1.79120
$$927$$ −14.4164 −0.473497
$$928$$ 12.2361 0.401669
$$929$$ −29.5967 −0.971038 −0.485519 0.874226i $$-0.661369\pi$$
−0.485519 + 0.874226i $$0.661369\pi$$
$$930$$ 0 0
$$931$$ 20.1246 0.659558
$$932$$ 6.29180 0.206095
$$933$$ −13.5279 −0.442882
$$934$$ −16.8541 −0.551483
$$935$$ 0 0
$$936$$ 2.23607 0.0730882
$$937$$ −10.4164 −0.340289 −0.170145 0.985419i $$-0.554423\pi$$
−0.170145 + 0.985419i $$0.554423\pi$$
$$938$$ 42.6525 1.39265
$$939$$ 2.29180 0.0747899
$$940$$ 0 0
$$941$$ 57.9787 1.89005 0.945026 0.326995i $$-0.106036\pi$$
0.945026 + 0.326995i $$0.106036\pi$$
$$942$$ −4.61803 −0.150464
$$943$$ −50.4934 −1.64429
$$944$$ −19.1459 −0.623146
$$945$$ 0 0
$$946$$ 35.8328 1.16503
$$947$$ −23.8328 −0.774462 −0.387231 0.921983i $$-0.626568\pi$$
−0.387231 + 0.921983i $$0.626568\pi$$
$$948$$ −5.65248 −0.183584
$$949$$ −15.7082 −0.509910
$$950$$ 0 0
$$951$$ 20.5623 0.666778
$$952$$ −18.9443 −0.613987
$$953$$ −42.0557 −1.36232 −0.681159 0.732135i $$-0.738524\pi$$
−0.681159 + 0.732135i $$0.738524\pi$$
$$954$$ 18.1803 0.588610
$$955$$ 0 0
$$956$$ 13.2148 0.427397
$$957$$ 10.8541 0.350863
$$958$$ 7.76393 0.250841
$$959$$ −0.875388 −0.0282678
$$960$$ 0 0
$$961$$ 44.8328 1.44622
$$962$$ 3.23607 0.104335
$$963$$ −1.14590 −0.0369260
$$964$$ 4.52786 0.145833
$$965$$ 0 0
$$966$$ −17.4164 −0.560364
$$967$$ −35.4164 −1.13891 −0.569457 0.822021i $$-0.692847\pi$$
−0.569457 + 0.822021i $$0.692847\pi$$
$$968$$ 4.47214 0.143740
$$969$$ 28.4164 0.912867
$$970$$ 0 0
$$971$$ 29.8885 0.959169 0.479585 0.877496i $$-0.340787\pi$$
0.479585 + 0.877496i $$0.340787\pi$$
$$972$$ 0.618034 0.0198234
$$973$$ −26.8328 −0.860221
$$974$$ 26.8885 0.861565
$$975$$ 0 0
$$976$$ −42.2705 −1.35305
$$977$$ −37.6525 −1.20461 −0.602305 0.798266i $$-0.705751\pi$$
−0.602305 + 0.798266i $$0.705751\pi$$
$$978$$ 29.5623 0.945298
$$979$$ 33.5410 1.07198
$$980$$ 0 0
$$981$$ −4.14590 −0.132368
$$982$$ 36.1246 1.15278
$$983$$ −24.6180 −0.785193 −0.392597 0.919711i $$-0.628423\pi$$
−0.392597 + 0.919711i $$0.628423\pi$$
$$984$$ 20.9787 0.668777
$$985$$ 0 0
$$986$$ 24.7984 0.789741
$$987$$ 9.52786 0.303275
$$988$$ 4.14590 0.131899
$$989$$ −39.7295 −1.26332
$$990$$ 0 0
$$991$$ 32.0000 1.01651 0.508257 0.861206i $$-0.330290\pi$$
0.508257 + 0.861206i $$0.330290\pi$$
$$992$$ −29.4508 −0.935065
$$993$$ −30.6869 −0.973820
$$994$$ −32.6525 −1.03567
$$995$$ 0 0
$$996$$ 5.56231 0.176248
$$997$$ −2.93112 −0.0928294 −0.0464147 0.998922i $$-0.514780\pi$$
−0.0464147 + 0.998922i $$0.514780\pi$$
$$998$$ −24.2705 −0.768270
$$999$$ −2.00000 −0.0632772
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 1875.2.a.c.1.2 yes 2
3.2 odd 2 5625.2.a.b.1.1 2
5.2 odd 4 1875.2.b.a.1249.4 4
5.3 odd 4 1875.2.b.a.1249.1 4
5.4 even 2 1875.2.a.b.1.1 2
15.14 odd 2 5625.2.a.g.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
1875.2.a.b.1.1 2 5.4 even 2
1875.2.a.c.1.2 yes 2 1.1 even 1 trivial
1875.2.b.a.1249.1 4 5.3 odd 4
1875.2.b.a.1249.4 4 5.2 odd 4
5625.2.a.b.1.1 2 3.2 odd 2
5625.2.a.g.1.2 2 15.14 odd 2