# Properties

 Label 2535.2.a.bc.1.2 Level $2535$ Weight $2$ Character 2535.1 Self dual yes Analytic conductor $20.242$ Analytic rank $0$ 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] = [2535,2,Mod(1,2535)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2535, 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("2535.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2535 = 3 \cdot 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2535.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: $$20.2420769124$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.316.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 4x + 2$$ x^3 - x^2 - 4*x + 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 195) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-1.81361$$ of defining polynomial Character $$\chi$$ $$=$$ 2535.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+0.289169 q^{2} -1.00000 q^{3} -1.91638 q^{4} +1.00000 q^{5} -0.289169 q^{6} -4.91638 q^{7} -1.13249 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+0.289169 q^{2} -1.00000 q^{3} -1.91638 q^{4} +1.00000 q^{5} -0.289169 q^{6} -4.91638 q^{7} -1.13249 q^{8} +1.00000 q^{9} +0.289169 q^{10} -4.91638 q^{11} +1.91638 q^{12} -1.42166 q^{14} -1.00000 q^{15} +3.50528 q^{16} -4.33804 q^{17} +0.289169 q^{18} -2.57834 q^{19} -1.91638 q^{20} +4.91638 q^{21} -1.42166 q^{22} -6.33804 q^{23} +1.13249 q^{24} +1.00000 q^{25} -1.00000 q^{27} +9.42166 q^{28} +6.00000 q^{29} -0.289169 q^{30} -1.42166 q^{31} +3.27861 q^{32} +4.91638 q^{33} -1.25443 q^{34} -4.91638 q^{35} -1.91638 q^{36} -9.49472 q^{37} -0.745574 q^{38} -1.13249 q^{40} -4.33804 q^{41} +1.42166 q^{42} -1.15667 q^{43} +9.42166 q^{44} +1.00000 q^{45} -1.83276 q^{46} +5.42166 q^{47} -3.50528 q^{48} +17.1708 q^{49} +0.289169 q^{50} +4.33804 q^{51} -0.338044 q^{53} -0.289169 q^{54} -4.91638 q^{55} +5.56777 q^{56} +2.57834 q^{57} +1.73501 q^{58} +11.2544 q^{59} +1.91638 q^{60} -10.1708 q^{61} -0.411100 q^{62} -4.91638 q^{63} -6.06249 q^{64} +1.42166 q^{66} +7.25443 q^{67} +8.31335 q^{68} +6.33804 q^{69} -1.42166 q^{70} -0.916382 q^{71} -1.13249 q^{72} +3.15667 q^{73} -2.74557 q^{74} -1.00000 q^{75} +4.94108 q^{76} +24.1708 q^{77} -3.49472 q^{79} +3.50528 q^{80} +1.00000 q^{81} -1.25443 q^{82} -11.2544 q^{83} -9.42166 q^{84} -4.33804 q^{85} -0.334474 q^{86} -6.00000 q^{87} +5.56777 q^{88} +0.338044 q^{89} +0.289169 q^{90} +12.1461 q^{92} +1.42166 q^{93} +1.56777 q^{94} -2.57834 q^{95} -3.27861 q^{96} +12.3380 q^{97} +4.96526 q^{98} -4.91638 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{3} + 8 q^{4} + 3 q^{5} - q^{7} - 6 q^{8} + 3 q^{9}+O(q^{10})$$ 3 * q - 3 * q^3 + 8 * q^4 + 3 * q^5 - q^7 - 6 * q^8 + 3 * q^9 $$3 q - 3 q^{3} + 8 q^{4} + 3 q^{5} - q^{7} - 6 q^{8} + 3 q^{9} - q^{11} - 8 q^{12} - 6 q^{14} - 3 q^{15} + 26 q^{16} - q^{17} - 6 q^{19} + 8 q^{20} + q^{21} - 6 q^{22} - 7 q^{23} + 6 q^{24} + 3 q^{25} - 3 q^{27} + 30 q^{28} + 18 q^{29} - 6 q^{31} - 22 q^{32} + q^{33} + 22 q^{34} - q^{35} + 8 q^{36} - 13 q^{37} - 28 q^{38} - 6 q^{40} - q^{41} + 6 q^{42} + 30 q^{44} + 3 q^{45} + 22 q^{46} + 18 q^{47} - 26 q^{48} + 12 q^{49} + q^{51} + 11 q^{53} - q^{55} - 16 q^{56} + 6 q^{57} + 8 q^{59} - 8 q^{60} + 9 q^{61} + 28 q^{62} - q^{63} + 30 q^{64} + 6 q^{66} - 4 q^{67} + 18 q^{68} + 7 q^{69} - 6 q^{70} + 11 q^{71} - 6 q^{72} + 6 q^{73} - 34 q^{74} - 3 q^{75} - 4 q^{76} + 33 q^{77} + 5 q^{79} + 26 q^{80} + 3 q^{81} + 22 q^{82} - 8 q^{83} - 30 q^{84} - q^{85} - 56 q^{86} - 18 q^{87} - 16 q^{88} - 11 q^{89} + 2 q^{92} + 6 q^{93} - 28 q^{94} - 6 q^{95} + 22 q^{96} + 25 q^{97} - 10 q^{98} - q^{99}+O(q^{100})$$ 3 * q - 3 * q^3 + 8 * q^4 + 3 * q^5 - q^7 - 6 * q^8 + 3 * q^9 - q^11 - 8 * q^12 - 6 * q^14 - 3 * q^15 + 26 * q^16 - q^17 - 6 * q^19 + 8 * q^20 + q^21 - 6 * q^22 - 7 * q^23 + 6 * q^24 + 3 * q^25 - 3 * q^27 + 30 * q^28 + 18 * q^29 - 6 * q^31 - 22 * q^32 + q^33 + 22 * q^34 - q^35 + 8 * q^36 - 13 * q^37 - 28 * q^38 - 6 * q^40 - q^41 + 6 * q^42 + 30 * q^44 + 3 * q^45 + 22 * q^46 + 18 * q^47 - 26 * q^48 + 12 * q^49 + q^51 + 11 * q^53 - q^55 - 16 * q^56 + 6 * q^57 + 8 * q^59 - 8 * q^60 + 9 * q^61 + 28 * q^62 - q^63 + 30 * q^64 + 6 * q^66 - 4 * q^67 + 18 * q^68 + 7 * q^69 - 6 * q^70 + 11 * q^71 - 6 * q^72 + 6 * q^73 - 34 * q^74 - 3 * q^75 - 4 * q^76 + 33 * q^77 + 5 * q^79 + 26 * q^80 + 3 * q^81 + 22 * q^82 - 8 * q^83 - 30 * q^84 - q^85 - 56 * q^86 - 18 * q^87 - 16 * q^88 - 11 * q^89 + 2 * q^92 + 6 * q^93 - 28 * q^94 - 6 * q^95 + 22 * q^96 + 25 * q^97 - 10 * q^98 - 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0.289169 0.204473 0.102237 0.994760i $$-0.467400\pi$$
0.102237 + 0.994760i $$0.467400\pi$$
$$3$$ −1.00000 −0.577350
$$4$$ −1.91638 −0.958191
$$5$$ 1.00000 0.447214
$$6$$ −0.289169 −0.118053
$$7$$ −4.91638 −1.85822 −0.929109 0.369807i $$-0.879424\pi$$
−0.929109 + 0.369807i $$0.879424\pi$$
$$8$$ −1.13249 −0.400397
$$9$$ 1.00000 0.333333
$$10$$ 0.289169 0.0914431
$$11$$ −4.91638 −1.48234 −0.741172 0.671315i $$-0.765730\pi$$
−0.741172 + 0.671315i $$0.765730\pi$$
$$12$$ 1.91638 0.553212
$$13$$ 0 0
$$14$$ −1.42166 −0.379955
$$15$$ −1.00000 −0.258199
$$16$$ 3.50528 0.876320
$$17$$ −4.33804 −1.05213 −0.526065 0.850444i $$-0.676333\pi$$
−0.526065 + 0.850444i $$0.676333\pi$$
$$18$$ 0.289169 0.0681577
$$19$$ −2.57834 −0.591511 −0.295756 0.955264i $$-0.595571\pi$$
−0.295756 + 0.955264i $$0.595571\pi$$
$$20$$ −1.91638 −0.428516
$$21$$ 4.91638 1.07284
$$22$$ −1.42166 −0.303100
$$23$$ −6.33804 −1.32157 −0.660787 0.750574i $$-0.729777\pi$$
−0.660787 + 0.750574i $$0.729777\pi$$
$$24$$ 1.13249 0.231169
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 9.42166 1.78053
$$29$$ 6.00000 1.11417 0.557086 0.830455i $$-0.311919\pi$$
0.557086 + 0.830455i $$0.311919\pi$$
$$30$$ −0.289169 −0.0527947
$$31$$ −1.42166 −0.255338 −0.127669 0.991817i $$-0.540750\pi$$
−0.127669 + 0.991817i $$0.540750\pi$$
$$32$$ 3.27861 0.579581
$$33$$ 4.91638 0.855832
$$34$$ −1.25443 −0.215132
$$35$$ −4.91638 −0.831020
$$36$$ −1.91638 −0.319397
$$37$$ −9.49472 −1.56092 −0.780461 0.625204i $$-0.785016\pi$$
−0.780461 + 0.625204i $$0.785016\pi$$
$$38$$ −0.745574 −0.120948
$$39$$ 0 0
$$40$$ −1.13249 −0.179063
$$41$$ −4.33804 −0.677489 −0.338744 0.940878i $$-0.610002\pi$$
−0.338744 + 0.940878i $$0.610002\pi$$
$$42$$ 1.42166 0.219367
$$43$$ −1.15667 −0.176391 −0.0881956 0.996103i $$-0.528110\pi$$
−0.0881956 + 0.996103i $$0.528110\pi$$
$$44$$ 9.42166 1.42037
$$45$$ 1.00000 0.149071
$$46$$ −1.83276 −0.270226
$$47$$ 5.42166 0.790831 0.395415 0.918502i $$-0.370601\pi$$
0.395415 + 0.918502i $$0.370601\pi$$
$$48$$ −3.50528 −0.505944
$$49$$ 17.1708 2.45297
$$50$$ 0.289169 0.0408946
$$51$$ 4.33804 0.607448
$$52$$ 0 0
$$53$$ −0.338044 −0.0464340 −0.0232170 0.999730i $$-0.507391\pi$$
−0.0232170 + 0.999730i $$0.507391\pi$$
$$54$$ −0.289169 −0.0393509
$$55$$ −4.91638 −0.662925
$$56$$ 5.56777 0.744025
$$57$$ 2.57834 0.341509
$$58$$ 1.73501 0.227818
$$59$$ 11.2544 1.46520 0.732601 0.680659i $$-0.238306\pi$$
0.732601 + 0.680659i $$0.238306\pi$$
$$60$$ 1.91638 0.247404
$$61$$ −10.1708 −1.30224 −0.651119 0.758975i $$-0.725700\pi$$
−0.651119 + 0.758975i $$0.725700\pi$$
$$62$$ −0.411100 −0.0522098
$$63$$ −4.91638 −0.619406
$$64$$ −6.06249 −0.757812
$$65$$ 0 0
$$66$$ 1.42166 0.174995
$$67$$ 7.25443 0.886269 0.443135 0.896455i $$-0.353866\pi$$
0.443135 + 0.896455i $$0.353866\pi$$
$$68$$ 8.31335 1.00814
$$69$$ 6.33804 0.763011
$$70$$ −1.42166 −0.169921
$$71$$ −0.916382 −0.108754 −0.0543772 0.998520i $$-0.517317\pi$$
−0.0543772 + 0.998520i $$0.517317\pi$$
$$72$$ −1.13249 −0.133466
$$73$$ 3.15667 0.369461 0.184730 0.982789i $$-0.440859\pi$$
0.184730 + 0.982789i $$0.440859\pi$$
$$74$$ −2.74557 −0.319166
$$75$$ −1.00000 −0.115470
$$76$$ 4.94108 0.566780
$$77$$ 24.1708 2.75452
$$78$$ 0 0
$$79$$ −3.49472 −0.393187 −0.196593 0.980485i $$-0.562988\pi$$
−0.196593 + 0.980485i $$0.562988\pi$$
$$80$$ 3.50528 0.391902
$$81$$ 1.00000 0.111111
$$82$$ −1.25443 −0.138528
$$83$$ −11.2544 −1.23533 −0.617667 0.786440i $$-0.711922\pi$$
−0.617667 + 0.786440i $$0.711922\pi$$
$$84$$ −9.42166 −1.02799
$$85$$ −4.33804 −0.470527
$$86$$ −0.334474 −0.0360672
$$87$$ −6.00000 −0.643268
$$88$$ 5.56777 0.593527
$$89$$ 0.338044 0.0358326 0.0179163 0.999839i $$-0.494297\pi$$
0.0179163 + 0.999839i $$0.494297\pi$$
$$90$$ 0.289169 0.0304810
$$91$$ 0 0
$$92$$ 12.1461 1.26632
$$93$$ 1.42166 0.147420
$$94$$ 1.56777 0.161704
$$95$$ −2.57834 −0.264532
$$96$$ −3.27861 −0.334621
$$97$$ 12.3380 1.25274 0.626369 0.779526i $$-0.284540\pi$$
0.626369 + 0.779526i $$0.284540\pi$$
$$98$$ 4.96526 0.501567
$$99$$ −4.91638 −0.494115
$$100$$ −1.91638 −0.191638
$$101$$ 10.6761 1.06231 0.531155 0.847274i $$-0.321758\pi$$
0.531155 + 0.847274i $$0.321758\pi$$
$$102$$ 1.25443 0.124207
$$103$$ −14.5089 −1.42960 −0.714800 0.699329i $$-0.753482\pi$$
−0.714800 + 0.699329i $$0.753482\pi$$
$$104$$ 0 0
$$105$$ 4.91638 0.479790
$$106$$ −0.0977518 −0.00949450
$$107$$ 4.17081 0.403207 0.201604 0.979467i $$-0.435385\pi$$
0.201604 + 0.979467i $$0.435385\pi$$
$$108$$ 1.91638 0.184404
$$109$$ 3.83276 0.367112 0.183556 0.983009i $$-0.441239\pi$$
0.183556 + 0.983009i $$0.441239\pi$$
$$110$$ −1.42166 −0.135550
$$111$$ 9.49472 0.901199
$$112$$ −17.2333 −1.62839
$$113$$ −0.843326 −0.0793334 −0.0396667 0.999213i $$-0.512630\pi$$
−0.0396667 + 0.999213i $$0.512630\pi$$
$$114$$ 0.745574 0.0698294
$$115$$ −6.33804 −0.591026
$$116$$ −11.4983 −1.06759
$$117$$ 0 0
$$118$$ 3.25443 0.299594
$$119$$ 21.3275 1.95509
$$120$$ 1.13249 0.103382
$$121$$ 13.1708 1.19735
$$122$$ −2.94108 −0.266273
$$123$$ 4.33804 0.391148
$$124$$ 2.72445 0.244663
$$125$$ 1.00000 0.0894427
$$126$$ −1.42166 −0.126652
$$127$$ 1.83276 0.162631 0.0813157 0.996688i $$-0.474088\pi$$
0.0813157 + 0.996688i $$0.474088\pi$$
$$128$$ −8.31029 −0.734533
$$129$$ 1.15667 0.101839
$$130$$ 0 0
$$131$$ 5.83276 0.509611 0.254805 0.966992i $$-0.417989\pi$$
0.254805 + 0.966992i $$0.417989\pi$$
$$132$$ −9.42166 −0.820050
$$133$$ 12.6761 1.09916
$$134$$ 2.09775 0.181218
$$135$$ −1.00000 −0.0860663
$$136$$ 4.91281 0.421270
$$137$$ 16.5089 1.41045 0.705223 0.708985i $$-0.250847\pi$$
0.705223 + 0.708985i $$0.250847\pi$$
$$138$$ 1.83276 0.156015
$$139$$ 7.49472 0.635694 0.317847 0.948142i $$-0.397040\pi$$
0.317847 + 0.948142i $$0.397040\pi$$
$$140$$ 9.42166 0.796276
$$141$$ −5.42166 −0.456586
$$142$$ −0.264989 −0.0222374
$$143$$ 0 0
$$144$$ 3.50528 0.292107
$$145$$ 6.00000 0.498273
$$146$$ 0.912811 0.0755448
$$147$$ −17.1708 −1.41622
$$148$$ 18.1955 1.49566
$$149$$ −20.4842 −1.67813 −0.839064 0.544033i $$-0.816897\pi$$
−0.839064 + 0.544033i $$0.816897\pi$$
$$150$$ −0.289169 −0.0236105
$$151$$ 16.4111 1.33552 0.667758 0.744378i $$-0.267254\pi$$
0.667758 + 0.744378i $$0.267254\pi$$
$$152$$ 2.91995 0.236839
$$153$$ −4.33804 −0.350710
$$154$$ 6.98944 0.563225
$$155$$ −1.42166 −0.114191
$$156$$ 0 0
$$157$$ −21.6655 −1.72910 −0.864549 0.502549i $$-0.832396\pi$$
−0.864549 + 0.502549i $$0.832396\pi$$
$$158$$ −1.01056 −0.0803961
$$159$$ 0.338044 0.0268087
$$160$$ 3.27861 0.259197
$$161$$ 31.1602 2.45577
$$162$$ 0.289169 0.0227192
$$163$$ −6.07306 −0.475678 −0.237839 0.971305i $$-0.576439\pi$$
−0.237839 + 0.971305i $$0.576439\pi$$
$$164$$ 8.31335 0.649163
$$165$$ 4.91638 0.382740
$$166$$ −3.25443 −0.252592
$$167$$ 0.745574 0.0576942 0.0288471 0.999584i $$-0.490816\pi$$
0.0288471 + 0.999584i $$0.490816\pi$$
$$168$$ −5.56777 −0.429563
$$169$$ 0 0
$$170$$ −1.25443 −0.0962101
$$171$$ −2.57834 −0.197170
$$172$$ 2.21663 0.169016
$$173$$ 0.843326 0.0641169 0.0320584 0.999486i $$-0.489794\pi$$
0.0320584 + 0.999486i $$0.489794\pi$$
$$174$$ −1.73501 −0.131531
$$175$$ −4.91638 −0.371644
$$176$$ −17.2333 −1.29901
$$177$$ −11.2544 −0.845934
$$178$$ 0.0977518 0.00732681
$$179$$ −18.9894 −1.41934 −0.709669 0.704536i $$-0.751155\pi$$
−0.709669 + 0.704536i $$0.751155\pi$$
$$180$$ −1.91638 −0.142839
$$181$$ 17.4947 1.30037 0.650186 0.759775i $$-0.274691\pi$$
0.650186 + 0.759775i $$0.274691\pi$$
$$182$$ 0 0
$$183$$ 10.1708 0.751848
$$184$$ 7.17780 0.529154
$$185$$ −9.49472 −0.698066
$$186$$ 0.411100 0.0301433
$$187$$ 21.3275 1.55962
$$188$$ −10.3900 −0.757767
$$189$$ 4.91638 0.357614
$$190$$ −0.745574 −0.0540896
$$191$$ −22.5089 −1.62868 −0.814342 0.580386i $$-0.802902\pi$$
−0.814342 + 0.580386i $$0.802902\pi$$
$$192$$ 6.06249 0.437523
$$193$$ −2.65139 −0.190851 −0.0954257 0.995437i $$-0.530421\pi$$
−0.0954257 + 0.995437i $$0.530421\pi$$
$$194$$ 3.56777 0.256151
$$195$$ 0 0
$$196$$ −32.9058 −2.35042
$$197$$ 12.9894 0.925459 0.462730 0.886500i $$-0.346870\pi$$
0.462730 + 0.886500i $$0.346870\pi$$
$$198$$ −1.42166 −0.101033
$$199$$ −2.84333 −0.201558 −0.100779 0.994909i $$-0.532134\pi$$
−0.100779 + 0.994909i $$0.532134\pi$$
$$200$$ −1.13249 −0.0800794
$$201$$ −7.25443 −0.511688
$$202$$ 3.08719 0.217214
$$203$$ −29.4983 −2.07037
$$204$$ −8.31335 −0.582051
$$205$$ −4.33804 −0.302982
$$206$$ −4.19550 −0.292315
$$207$$ −6.33804 −0.440525
$$208$$ 0 0
$$209$$ 12.6761 0.876823
$$210$$ 1.42166 0.0981041
$$211$$ 6.31335 0.434629 0.217314 0.976102i $$-0.430270\pi$$
0.217314 + 0.976102i $$0.430270\pi$$
$$212$$ 0.647822 0.0444926
$$213$$ 0.916382 0.0627894
$$214$$ 1.20607 0.0824450
$$215$$ −1.15667 −0.0788845
$$216$$ 1.13249 0.0770565
$$217$$ 6.98944 0.474474
$$218$$ 1.10831 0.0750645
$$219$$ −3.15667 −0.213308
$$220$$ 9.42166 0.635208
$$221$$ 0 0
$$222$$ 2.74557 0.184271
$$223$$ −19.2544 −1.28937 −0.644686 0.764448i $$-0.723012\pi$$
−0.644686 + 0.764448i $$0.723012\pi$$
$$224$$ −16.1189 −1.07699
$$225$$ 1.00000 0.0666667
$$226$$ −0.243863 −0.0162215
$$227$$ 13.0872 0.868627 0.434314 0.900762i $$-0.356991\pi$$
0.434314 + 0.900762i $$0.356991\pi$$
$$228$$ −4.94108 −0.327231
$$229$$ 24.5089 1.61959 0.809795 0.586713i $$-0.199578\pi$$
0.809795 + 0.586713i $$0.199578\pi$$
$$230$$ −1.83276 −0.120849
$$231$$ −24.1708 −1.59032
$$232$$ −6.79497 −0.446111
$$233$$ 8.33804 0.546243 0.273122 0.961979i $$-0.411944\pi$$
0.273122 + 0.961979i $$0.411944\pi$$
$$234$$ 0 0
$$235$$ 5.42166 0.353670
$$236$$ −21.5678 −1.40394
$$237$$ 3.49472 0.227006
$$238$$ 6.16724 0.399763
$$239$$ −8.91638 −0.576753 −0.288376 0.957517i $$-0.593115\pi$$
−0.288376 + 0.957517i $$0.593115\pi$$
$$240$$ −3.50528 −0.226265
$$241$$ 6.00000 0.386494 0.193247 0.981150i $$-0.438098\pi$$
0.193247 + 0.981150i $$0.438098\pi$$
$$242$$ 3.80858 0.244825
$$243$$ −1.00000 −0.0641500
$$244$$ 19.4911 1.24779
$$245$$ 17.1708 1.09700
$$246$$ 1.25443 0.0799793
$$247$$ 0 0
$$248$$ 1.61003 0.102237
$$249$$ 11.2544 0.713220
$$250$$ 0.289169 0.0182886
$$251$$ −6.31335 −0.398495 −0.199248 0.979949i $$-0.563850\pi$$
−0.199248 + 0.979949i $$0.563850\pi$$
$$252$$ 9.42166 0.593509
$$253$$ 31.1602 1.95903
$$254$$ 0.529977 0.0332537
$$255$$ 4.33804 0.271659
$$256$$ 9.72191 0.607619
$$257$$ −11.1567 −0.695934 −0.347967 0.937507i $$-0.613128\pi$$
−0.347967 + 0.937507i $$0.613128\pi$$
$$258$$ 0.334474 0.0208234
$$259$$ 46.6797 2.90053
$$260$$ 0 0
$$261$$ 6.00000 0.371391
$$262$$ 1.68665 0.104202
$$263$$ 8.00000 0.493301 0.246651 0.969104i $$-0.420670\pi$$
0.246651 + 0.969104i $$0.420670\pi$$
$$264$$ −5.56777 −0.342673
$$265$$ −0.338044 −0.0207659
$$266$$ 3.66553 0.224748
$$267$$ −0.338044 −0.0206880
$$268$$ −13.9022 −0.849215
$$269$$ 18.6761 1.13870 0.569351 0.822095i $$-0.307195\pi$$
0.569351 + 0.822095i $$0.307195\pi$$
$$270$$ −0.289169 −0.0175982
$$271$$ −6.57834 −0.399606 −0.199803 0.979836i $$-0.564030\pi$$
−0.199803 + 0.979836i $$0.564030\pi$$
$$272$$ −15.2061 −0.922003
$$273$$ 0 0
$$274$$ 4.77384 0.288398
$$275$$ −4.91638 −0.296469
$$276$$ −12.1461 −0.731110
$$277$$ 25.6655 1.54209 0.771046 0.636779i $$-0.219734\pi$$
0.771046 + 0.636779i $$0.219734\pi$$
$$278$$ 2.16724 0.129982
$$279$$ −1.42166 −0.0851127
$$280$$ 5.56777 0.332738
$$281$$ −3.15667 −0.188311 −0.0941557 0.995557i $$-0.530015\pi$$
−0.0941557 + 0.995557i $$0.530015\pi$$
$$282$$ −1.56777 −0.0933596
$$283$$ −3.47002 −0.206271 −0.103136 0.994667i $$-0.532888\pi$$
−0.103136 + 0.994667i $$0.532888\pi$$
$$284$$ 1.75614 0.104208
$$285$$ 2.57834 0.152728
$$286$$ 0 0
$$287$$ 21.3275 1.25892
$$288$$ 3.27861 0.193194
$$289$$ 1.81863 0.106978
$$290$$ 1.73501 0.101883
$$291$$ −12.3380 −0.723269
$$292$$ −6.04939 −0.354014
$$293$$ −28.6550 −1.67404 −0.837020 0.547172i $$-0.815704\pi$$
−0.837020 + 0.547172i $$0.815704\pi$$
$$294$$ −4.96526 −0.289580
$$295$$ 11.2544 0.655258
$$296$$ 10.7527 0.624989
$$297$$ 4.91638 0.285277
$$298$$ −5.92337 −0.343132
$$299$$ 0 0
$$300$$ 1.91638 0.110642
$$301$$ 5.68665 0.327773
$$302$$ 4.74557 0.273077
$$303$$ −10.6761 −0.613325
$$304$$ −9.03780 −0.518353
$$305$$ −10.1708 −0.582379
$$306$$ −1.25443 −0.0717108
$$307$$ 1.92694 0.109977 0.0549883 0.998487i $$-0.482488\pi$$
0.0549883 + 0.998487i $$0.482488\pi$$
$$308$$ −46.3205 −2.63935
$$309$$ 14.5089 0.825380
$$310$$ −0.411100 −0.0233489
$$311$$ −29.9789 −1.69995 −0.849973 0.526826i $$-0.823382\pi$$
−0.849973 + 0.526826i $$0.823382\pi$$
$$312$$ 0 0
$$313$$ −16.3133 −0.922085 −0.461042 0.887378i $$-0.652524\pi$$
−0.461042 + 0.887378i $$0.652524\pi$$
$$314$$ −6.26499 −0.353554
$$315$$ −4.91638 −0.277007
$$316$$ 6.69721 0.376748
$$317$$ −30.6761 −1.72294 −0.861470 0.507808i $$-0.830456\pi$$
−0.861470 + 0.507808i $$0.830456\pi$$
$$318$$ 0.0977518 0.00548165
$$319$$ −29.4983 −1.65159
$$320$$ −6.06249 −0.338904
$$321$$ −4.17081 −0.232792
$$322$$ 9.01056 0.502139
$$323$$ 11.1849 0.622347
$$324$$ −1.91638 −0.106466
$$325$$ 0 0
$$326$$ −1.75614 −0.0972634
$$327$$ −3.83276 −0.211952
$$328$$ 4.91281 0.271265
$$329$$ −26.6550 −1.46954
$$330$$ 1.42166 0.0782600
$$331$$ 10.0978 0.555023 0.277511 0.960722i $$-0.410490\pi$$
0.277511 + 0.960722i $$0.410490\pi$$
$$332$$ 21.5678 1.18369
$$333$$ −9.49472 −0.520307
$$334$$ 0.215597 0.0117969
$$335$$ 7.25443 0.396352
$$336$$ 17.2333 0.940154
$$337$$ −1.32391 −0.0721180 −0.0360590 0.999350i $$-0.511480\pi$$
−0.0360590 + 0.999350i $$0.511480\pi$$
$$338$$ 0 0
$$339$$ 0.843326 0.0458032
$$340$$ 8.31335 0.450855
$$341$$ 6.98944 0.378499
$$342$$ −0.745574 −0.0403160
$$343$$ −50.0036 −2.69994
$$344$$ 1.30993 0.0706265
$$345$$ 6.33804 0.341229
$$346$$ 0.243863 0.0131102
$$347$$ −7.49472 −0.402338 −0.201169 0.979557i $$-0.564474\pi$$
−0.201169 + 0.979557i $$0.564474\pi$$
$$348$$ 11.4983 0.616373
$$349$$ 22.1461 1.18545 0.592727 0.805403i $$-0.298051\pi$$
0.592727 + 0.805403i $$0.298051\pi$$
$$350$$ −1.42166 −0.0759911
$$351$$ 0 0
$$352$$ −16.1189 −0.859139
$$353$$ −4.50885 −0.239982 −0.119991 0.992775i $$-0.538287\pi$$
−0.119991 + 0.992775i $$0.538287\pi$$
$$354$$ −3.25443 −0.172971
$$355$$ −0.916382 −0.0486365
$$356$$ −0.647822 −0.0343345
$$357$$ −21.3275 −1.12877
$$358$$ −5.49115 −0.290216
$$359$$ 20.4111 1.07726 0.538628 0.842543i $$-0.318943\pi$$
0.538628 + 0.842543i $$0.318943\pi$$
$$360$$ −1.13249 −0.0596877
$$361$$ −12.3522 −0.650115
$$362$$ 5.05892 0.265891
$$363$$ −13.1708 −0.691288
$$364$$ 0 0
$$365$$ 3.15667 0.165228
$$366$$ 2.94108 0.153733
$$367$$ −10.3133 −0.538352 −0.269176 0.963091i $$-0.586751\pi$$
−0.269176 + 0.963091i $$0.586751\pi$$
$$368$$ −22.2166 −1.15812
$$369$$ −4.33804 −0.225830
$$370$$ −2.74557 −0.142736
$$371$$ 1.66196 0.0862844
$$372$$ −2.72445 −0.141256
$$373$$ 18.6761 0.967011 0.483506 0.875341i $$-0.339363\pi$$
0.483506 + 0.875341i $$0.339363\pi$$
$$374$$ 6.16724 0.318900
$$375$$ −1.00000 −0.0516398
$$376$$ −6.14000 −0.316646
$$377$$ 0 0
$$378$$ 1.42166 0.0731224
$$379$$ 28.7527 1.47693 0.738464 0.674293i $$-0.235552\pi$$
0.738464 + 0.674293i $$0.235552\pi$$
$$380$$ 4.94108 0.253472
$$381$$ −1.83276 −0.0938953
$$382$$ −6.50885 −0.333022
$$383$$ 14.2439 0.727827 0.363914 0.931433i $$-0.381440\pi$$
0.363914 + 0.931433i $$0.381440\pi$$
$$384$$ 8.31029 0.424083
$$385$$ 24.1708 1.23186
$$386$$ −0.766699 −0.0390240
$$387$$ −1.15667 −0.0587971
$$388$$ −23.6444 −1.20036
$$389$$ 34.6761 1.75815 0.879074 0.476686i $$-0.158162\pi$$
0.879074 + 0.476686i $$0.158162\pi$$
$$390$$ 0 0
$$391$$ 27.4947 1.39047
$$392$$ −19.4458 −0.982163
$$393$$ −5.83276 −0.294224
$$394$$ 3.75614 0.189231
$$395$$ −3.49472 −0.175838
$$396$$ 9.42166 0.473456
$$397$$ −7.18137 −0.360423 −0.180211 0.983628i $$-0.557678\pi$$
−0.180211 + 0.983628i $$0.557678\pi$$
$$398$$ −0.822200 −0.0412132
$$399$$ −12.6761 −0.634598
$$400$$ 3.50528 0.175264
$$401$$ −37.8610 −1.89069 −0.945345 0.326072i $$-0.894275\pi$$
−0.945345 + 0.326072i $$0.894275\pi$$
$$402$$ −2.09775 −0.104626
$$403$$ 0 0
$$404$$ −20.4595 −1.01790
$$405$$ 1.00000 0.0496904
$$406$$ −8.52998 −0.423336
$$407$$ 46.6797 2.31382
$$408$$ −4.91281 −0.243220
$$409$$ 14.0000 0.692255 0.346128 0.938187i $$-0.387496\pi$$
0.346128 + 0.938187i $$0.387496\pi$$
$$410$$ −1.25443 −0.0619517
$$411$$ −16.5089 −0.814322
$$412$$ 27.8045 1.36983
$$413$$ −55.3311 −2.72266
$$414$$ −1.83276 −0.0900754
$$415$$ −11.2544 −0.552458
$$416$$ 0 0
$$417$$ −7.49472 −0.367018
$$418$$ 3.66553 0.179287
$$419$$ −33.4983 −1.63650 −0.818249 0.574864i $$-0.805055\pi$$
−0.818249 + 0.574864i $$0.805055\pi$$
$$420$$ −9.42166 −0.459730
$$421$$ −13.5194 −0.658896 −0.329448 0.944174i $$-0.606863\pi$$
−0.329448 + 0.944174i $$0.606863\pi$$
$$422$$ 1.82562 0.0888699
$$423$$ 5.42166 0.263610
$$424$$ 0.382833 0.0185920
$$425$$ −4.33804 −0.210426
$$426$$ 0.264989 0.0128387
$$427$$ 50.0036 2.41984
$$428$$ −7.99286 −0.386349
$$429$$ 0 0
$$430$$ −0.334474 −0.0161298
$$431$$ −12.4111 −0.597822 −0.298911 0.954281i $$-0.596623\pi$$
−0.298911 + 0.954281i $$0.596623\pi$$
$$432$$ −3.50528 −0.168648
$$433$$ −17.3239 −0.832534 −0.416267 0.909242i $$-0.636662\pi$$
−0.416267 + 0.909242i $$0.636662\pi$$
$$434$$ 2.02113 0.0970171
$$435$$ −6.00000 −0.287678
$$436$$ −7.34504 −0.351763
$$437$$ 16.3416 0.781725
$$438$$ −0.912811 −0.0436158
$$439$$ 0.651393 0.0310893 0.0155446 0.999879i $$-0.495052\pi$$
0.0155446 + 0.999879i $$0.495052\pi$$
$$440$$ 5.56777 0.265433
$$441$$ 17.1708 0.817658
$$442$$ 0 0
$$443$$ −8.84690 −0.420329 −0.210164 0.977666i $$-0.567400\pi$$
−0.210164 + 0.977666i $$0.567400\pi$$
$$444$$ −18.1955 −0.863520
$$445$$ 0.338044 0.0160248
$$446$$ −5.56777 −0.263642
$$447$$ 20.4842 0.968867
$$448$$ 29.8055 1.40818
$$449$$ −4.33804 −0.204725 −0.102362 0.994747i $$-0.532640\pi$$
−0.102362 + 0.994747i $$0.532640\pi$$
$$450$$ 0.289169 0.0136315
$$451$$ 21.3275 1.00427
$$452$$ 1.61613 0.0760166
$$453$$ −16.4111 −0.771061
$$454$$ 3.78440 0.177611
$$455$$ 0 0
$$456$$ −2.91995 −0.136739
$$457$$ −15.3275 −0.716989 −0.358495 0.933532i $$-0.616710\pi$$
−0.358495 + 0.933532i $$0.616710\pi$$
$$458$$ 7.08719 0.331163
$$459$$ 4.33804 0.202483
$$460$$ 12.1461 0.566315
$$461$$ 11.8575 0.552257 0.276128 0.961121i $$-0.410948\pi$$
0.276128 + 0.961121i $$0.410948\pi$$
$$462$$ −6.98944 −0.325178
$$463$$ 26.4147 1.22759 0.613797 0.789464i $$-0.289641\pi$$
0.613797 + 0.789464i $$0.289641\pi$$
$$464$$ 21.0317 0.976372
$$465$$ 1.42166 0.0659280
$$466$$ 2.41110 0.111692
$$467$$ −33.6691 −1.55802 −0.779010 0.627012i $$-0.784278\pi$$
−0.779010 + 0.627012i $$0.784278\pi$$
$$468$$ 0 0
$$469$$ −35.6655 −1.64688
$$470$$ 1.56777 0.0723160
$$471$$ 21.6655 0.998295
$$472$$ −12.7456 −0.586663
$$473$$ 5.68665 0.261473
$$474$$ 1.01056 0.0464167
$$475$$ −2.57834 −0.118302
$$476$$ −40.8716 −1.87335
$$477$$ −0.338044 −0.0154780
$$478$$ −2.57834 −0.117930
$$479$$ −10.7491 −0.491141 −0.245570 0.969379i $$-0.578975\pi$$
−0.245570 + 0.969379i $$0.578975\pi$$
$$480$$ −3.27861 −0.149647
$$481$$ 0 0
$$482$$ 1.73501 0.0790276
$$483$$ −31.1602 −1.41784
$$484$$ −25.2403 −1.14729
$$485$$ 12.3380 0.560242
$$486$$ −0.289169 −0.0131170
$$487$$ −22.7491 −1.03086 −0.515431 0.856931i $$-0.672368\pi$$
−0.515431 + 0.856931i $$0.672368\pi$$
$$488$$ 11.5184 0.521413
$$489$$ 6.07306 0.274633
$$490$$ 4.96526 0.224307
$$491$$ −17.6867 −0.798187 −0.399094 0.916910i $$-0.630675\pi$$
−0.399094 + 0.916910i $$0.630675\pi$$
$$492$$ −8.31335 −0.374795
$$493$$ −26.0283 −1.17225
$$494$$ 0 0
$$495$$ −4.91638 −0.220975
$$496$$ −4.98333 −0.223758
$$497$$ 4.50528 0.202089
$$498$$ 3.25443 0.145834
$$499$$ −19.9305 −0.892212 −0.446106 0.894980i $$-0.647190\pi$$
−0.446106 + 0.894980i $$0.647190\pi$$
$$500$$ −1.91638 −0.0857032
$$501$$ −0.745574 −0.0333098
$$502$$ −1.82562 −0.0814815
$$503$$ 33.3522 1.48710 0.743550 0.668680i $$-0.233140\pi$$
0.743550 + 0.668680i $$0.233140\pi$$
$$504$$ 5.56777 0.248008
$$505$$ 10.6761 0.475080
$$506$$ 9.01056 0.400568
$$507$$ 0 0
$$508$$ −3.51227 −0.155832
$$509$$ −13.8363 −0.613285 −0.306642 0.951825i $$-0.599206\pi$$
−0.306642 + 0.951825i $$0.599206\pi$$
$$510$$ 1.25443 0.0555469
$$511$$ −15.5194 −0.686538
$$512$$ 19.4319 0.858775
$$513$$ 2.57834 0.113836
$$514$$ −3.22616 −0.142300
$$515$$ −14.5089 −0.639336
$$516$$ −2.21663 −0.0975817
$$517$$ −26.6550 −1.17228
$$518$$ 13.4983 0.593081
$$519$$ −0.843326 −0.0370179
$$520$$ 0 0
$$521$$ −23.3522 −1.02308 −0.511539 0.859260i $$-0.670924\pi$$
−0.511539 + 0.859260i $$0.670924\pi$$
$$522$$ 1.73501 0.0759394
$$523$$ −4.00000 −0.174908 −0.0874539 0.996169i $$-0.527873\pi$$
−0.0874539 + 0.996169i $$0.527873\pi$$
$$524$$ −11.1778 −0.488304
$$525$$ 4.91638 0.214568
$$526$$ 2.31335 0.100867
$$527$$ 6.16724 0.268649
$$528$$ 17.2333 0.749983
$$529$$ 17.1708 0.746557
$$530$$ −0.0977518 −0.00424607
$$531$$ 11.2544 0.488400
$$532$$ −24.2922 −1.05320
$$533$$ 0 0
$$534$$ −0.0977518 −0.00423014
$$535$$ 4.17081 0.180320
$$536$$ −8.21560 −0.354860
$$537$$ 18.9894 0.819455
$$538$$ 5.40054 0.232834
$$539$$ −84.4182 −3.63615
$$540$$ 1.91638 0.0824679
$$541$$ −16.1744 −0.695391 −0.347695 0.937608i $$-0.613036\pi$$
−0.347695 + 0.937608i $$0.613036\pi$$
$$542$$ −1.90225 −0.0817086
$$543$$ −17.4947 −0.750770
$$544$$ −14.2227 −0.609795
$$545$$ 3.83276 0.164178
$$546$$ 0 0
$$547$$ 9.68665 0.414171 0.207086 0.978323i $$-0.433602\pi$$
0.207086 + 0.978323i $$0.433602\pi$$
$$548$$ −31.6373 −1.35148
$$549$$ −10.1708 −0.434079
$$550$$ −1.42166 −0.0606199
$$551$$ −15.4700 −0.659045
$$552$$ −7.17780 −0.305507
$$553$$ 17.1814 0.730626
$$554$$ 7.42166 0.315316
$$555$$ 9.49472 0.403028
$$556$$ −14.3627 −0.609116
$$557$$ −0.647822 −0.0274491 −0.0137246 0.999906i $$-0.504369\pi$$
−0.0137246 + 0.999906i $$0.504369\pi$$
$$558$$ −0.411100 −0.0174033
$$559$$ 0 0
$$560$$ −17.2333 −0.728240
$$561$$ −21.3275 −0.900447
$$562$$ −0.912811 −0.0385046
$$563$$ −16.3169 −0.687676 −0.343838 0.939029i $$-0.611727\pi$$
−0.343838 + 0.939029i $$0.611727\pi$$
$$564$$ 10.3900 0.437497
$$565$$ −0.843326 −0.0354790
$$566$$ −1.00342 −0.0421769
$$567$$ −4.91638 −0.206469
$$568$$ 1.03780 0.0435450
$$569$$ 44.6550 1.87203 0.936017 0.351956i $$-0.114483\pi$$
0.936017 + 0.351956i $$0.114483\pi$$
$$570$$ 0.745574 0.0312287
$$571$$ −6.67252 −0.279236 −0.139618 0.990205i $$-0.544587\pi$$
−0.139618 + 0.990205i $$0.544587\pi$$
$$572$$ 0 0
$$573$$ 22.5089 0.940321
$$574$$ 6.16724 0.257415
$$575$$ −6.33804 −0.264315
$$576$$ −6.06249 −0.252604
$$577$$ −15.3275 −0.638091 −0.319046 0.947739i $$-0.603362\pi$$
−0.319046 + 0.947739i $$0.603362\pi$$
$$578$$ 0.525891 0.0218742
$$579$$ 2.65139 0.110188
$$580$$ −11.4983 −0.477440
$$581$$ 55.3311 2.29552
$$582$$ −3.56777 −0.147889
$$583$$ 1.66196 0.0688312
$$584$$ −3.57492 −0.147931
$$585$$ 0 0
$$586$$ −8.28611 −0.342296
$$587$$ −12.2650 −0.506230 −0.253115 0.967436i $$-0.581455\pi$$
−0.253115 + 0.967436i $$0.581455\pi$$
$$588$$ 32.9058 1.35701
$$589$$ 3.66553 0.151035
$$590$$ 3.25443 0.133983
$$591$$ −12.9894 −0.534314
$$592$$ −33.2817 −1.36787
$$593$$ 5.85389 0.240390 0.120195 0.992750i $$-0.461648\pi$$
0.120195 + 0.992750i $$0.461648\pi$$
$$594$$ 1.42166 0.0583315
$$595$$ 21.3275 0.874342
$$596$$ 39.2555 1.60797
$$597$$ 2.84333 0.116370
$$598$$ 0 0
$$599$$ 27.1355 1.10873 0.554364 0.832274i $$-0.312961\pi$$
0.554364 + 0.832274i $$0.312961\pi$$
$$600$$ 1.13249 0.0462339
$$601$$ 34.1708 1.39386 0.696928 0.717141i $$-0.254550\pi$$
0.696928 + 0.717141i $$0.254550\pi$$
$$602$$ 1.64440 0.0670208
$$603$$ 7.25443 0.295423
$$604$$ −31.4499 −1.27968
$$605$$ 13.1708 0.535469
$$606$$ −3.08719 −0.125408
$$607$$ 10.3133 0.418606 0.209303 0.977851i $$-0.432881\pi$$
0.209303 + 0.977851i $$0.432881\pi$$
$$608$$ −8.45335 −0.342829
$$609$$ 29.4983 1.19533
$$610$$ −2.94108 −0.119081
$$611$$ 0 0
$$612$$ 8.31335 0.336047
$$613$$ −0.484156 −0.0195549 −0.00977744 0.999952i $$-0.503112\pi$$
−0.00977744 + 0.999952i $$0.503112\pi$$
$$614$$ 0.557212 0.0224872
$$615$$ 4.33804 0.174927
$$616$$ −27.3733 −1.10290
$$617$$ 7.15667 0.288117 0.144058 0.989569i $$-0.453985\pi$$
0.144058 + 0.989569i $$0.453985\pi$$
$$618$$ 4.19550 0.168768
$$619$$ −5.42166 −0.217915 −0.108958 0.994046i $$-0.534751\pi$$
−0.108958 + 0.994046i $$0.534751\pi$$
$$620$$ 2.72445 0.109416
$$621$$ 6.33804 0.254337
$$622$$ −8.66895 −0.347593
$$623$$ −1.66196 −0.0665848
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ −4.71731 −0.188542
$$627$$ −12.6761 −0.506234
$$628$$ 41.5194 1.65681
$$629$$ 41.1885 1.64229
$$630$$ −1.42166 −0.0566404
$$631$$ 10.7244 0.426934 0.213467 0.976950i $$-0.431524\pi$$
0.213467 + 0.976950i $$0.431524\pi$$
$$632$$ 3.95775 0.157431
$$633$$ −6.31335 −0.250933
$$634$$ −8.87056 −0.352295
$$635$$ 1.83276 0.0727310
$$636$$ −0.647822 −0.0256878
$$637$$ 0 0
$$638$$ −8.52998 −0.337705
$$639$$ −0.916382 −0.0362515
$$640$$ −8.31029 −0.328493
$$641$$ −0.362741 −0.0143274 −0.00716370 0.999974i $$-0.502280\pi$$
−0.00716370 + 0.999974i $$0.502280\pi$$
$$642$$ −1.20607 −0.0475996
$$643$$ 9.39697 0.370580 0.185290 0.982684i $$-0.440678\pi$$
0.185290 + 0.982684i $$0.440678\pi$$
$$644$$ −59.7149 −2.35310
$$645$$ 1.15667 0.0455440
$$646$$ 3.23433 0.127253
$$647$$ 18.0036 0.707793 0.353897 0.935285i $$-0.384856\pi$$
0.353897 + 0.935285i $$0.384856\pi$$
$$648$$ −1.13249 −0.0444886
$$649$$ −55.3311 −2.17193
$$650$$ 0 0
$$651$$ −6.98944 −0.273938
$$652$$ 11.6383 0.455791
$$653$$ −34.8222 −1.36270 −0.681349 0.731959i $$-0.738606\pi$$
−0.681349 + 0.731959i $$0.738606\pi$$
$$654$$ −1.10831 −0.0433385
$$655$$ 5.83276 0.227905
$$656$$ −15.2061 −0.593697
$$657$$ 3.15667 0.123154
$$658$$ −7.70778 −0.300480
$$659$$ 11.4700 0.446809 0.223404 0.974726i $$-0.428283\pi$$
0.223404 + 0.974726i $$0.428283\pi$$
$$660$$ −9.42166 −0.366738
$$661$$ −12.1672 −0.473251 −0.236625 0.971601i $$-0.576041\pi$$
−0.236625 + 0.971601i $$0.576041\pi$$
$$662$$ 2.91995 0.113487
$$663$$ 0 0
$$664$$ 12.7456 0.494624
$$665$$ 12.6761 0.491558
$$666$$ −2.74557 −0.106389
$$667$$ −38.0283 −1.47246
$$668$$ −1.42880 −0.0552821
$$669$$ 19.2544 0.744419
$$670$$ 2.09775 0.0810432
$$671$$ 50.0036 1.93037
$$672$$ 16.1189 0.621799
$$673$$ −27.9789 −1.07851 −0.539253 0.842144i $$-0.681293\pi$$
−0.539253 + 0.842144i $$0.681293\pi$$
$$674$$ −0.382833 −0.0147462
$$675$$ −1.00000 −0.0384900
$$676$$ 0 0
$$677$$ 22.9930 0.883693 0.441847 0.897091i $$-0.354324\pi$$
0.441847 + 0.897091i $$0.354324\pi$$
$$678$$ 0.243863 0.00936551
$$679$$ −60.6585 −2.32786
$$680$$ 4.91281 0.188398
$$681$$ −13.0872 −0.501502
$$682$$ 2.02113 0.0773929
$$683$$ 28.6066 1.09460 0.547301 0.836936i $$-0.315655\pi$$
0.547301 + 0.836936i $$0.315655\pi$$
$$684$$ 4.94108 0.188927
$$685$$ 16.5089 0.630771
$$686$$ −14.4595 −0.552065
$$687$$ −24.5089 −0.935071
$$688$$ −4.05447 −0.154575
$$689$$ 0 0
$$690$$ 1.83276 0.0697721
$$691$$ 19.4005 0.738031 0.369016 0.929423i $$-0.379695\pi$$
0.369016 + 0.929423i $$0.379695\pi$$
$$692$$ −1.61613 −0.0614362
$$693$$ 24.1708 0.918173
$$694$$ −2.16724 −0.0822672
$$695$$ 7.49472 0.284291
$$696$$ 6.79497 0.257563
$$697$$ 18.8186 0.712806
$$698$$ 6.40396 0.242393
$$699$$ −8.33804 −0.315374
$$700$$ 9.42166 0.356105
$$701$$ −38.9683 −1.47181 −0.735906 0.677083i $$-0.763244\pi$$
−0.735906 + 0.677083i $$0.763244\pi$$
$$702$$ 0 0
$$703$$ 24.4806 0.923303
$$704$$ 29.8055 1.12334
$$705$$ −5.42166 −0.204192
$$706$$ −1.30382 −0.0490698
$$707$$ −52.4877 −1.97400
$$708$$ 21.5678 0.810567
$$709$$ 17.5194 0.657955 0.328978 0.944338i $$-0.393296\pi$$
0.328978 + 0.944338i $$0.393296\pi$$
$$710$$ −0.264989 −0.00994485
$$711$$ −3.49472 −0.131062
$$712$$ −0.382833 −0.0143473
$$713$$ 9.01056 0.337448
$$714$$ −6.16724 −0.230803
$$715$$ 0 0
$$716$$ 36.3910 1.36000
$$717$$ 8.91638 0.332988
$$718$$ 5.90225 0.220270
$$719$$ −4.33447 −0.161649 −0.0808243 0.996728i $$-0.525755\pi$$
−0.0808243 + 0.996728i $$0.525755\pi$$
$$720$$ 3.50528 0.130634
$$721$$ 71.3311 2.65651
$$722$$ −3.57186 −0.132931
$$723$$ −6.00000 −0.223142
$$724$$ −33.5266 −1.24600
$$725$$ 6.00000 0.222834
$$726$$ −3.80858 −0.141350
$$727$$ 22.1672 0.822137 0.411069 0.911604i $$-0.365156\pi$$
0.411069 + 0.911604i $$0.365156\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0.912811 0.0337846
$$731$$ 5.01770 0.185586
$$732$$ −19.4911 −0.720414
$$733$$ 2.83976 0.104889 0.0524444 0.998624i $$-0.483299\pi$$
0.0524444 + 0.998624i $$0.483299\pi$$
$$734$$ −2.98230 −0.110079
$$735$$ −17.1708 −0.633355
$$736$$ −20.7799 −0.765959
$$737$$ −35.6655 −1.31376
$$738$$ −1.25443 −0.0461761
$$739$$ 43.9305 1.61601 0.808005 0.589176i $$-0.200547\pi$$
0.808005 + 0.589176i $$0.200547\pi$$
$$740$$ 18.1955 0.668880
$$741$$ 0 0
$$742$$ 0.480585 0.0176428
$$743$$ −41.0872 −1.50734 −0.753671 0.657251i $$-0.771719\pi$$
−0.753671 + 0.657251i $$0.771719\pi$$
$$744$$ −1.61003 −0.0590264
$$745$$ −20.4842 −0.750481
$$746$$ 5.40054 0.197728
$$747$$ −11.2544 −0.411778
$$748$$ −40.8716 −1.49441
$$749$$ −20.5053 −0.749247
$$750$$ −0.289169 −0.0105589
$$751$$ 23.6902 0.864468 0.432234 0.901761i $$-0.357725\pi$$
0.432234 + 0.901761i $$0.357725\pi$$
$$752$$ 19.0045 0.693021
$$753$$ 6.31335 0.230071
$$754$$ 0 0
$$755$$ 16.4111 0.597261
$$756$$ −9.42166 −0.342663
$$757$$ 9.32391 0.338883 0.169442 0.985540i $$-0.445804\pi$$
0.169442 + 0.985540i $$0.445804\pi$$
$$758$$ 8.31438 0.301992
$$759$$ −31.1602 −1.13105
$$760$$ 2.91995 0.105918
$$761$$ 42.8222 1.55230 0.776152 0.630546i $$-0.217169\pi$$
0.776152 + 0.630546i $$0.217169\pi$$
$$762$$ −0.529977 −0.0191991
$$763$$ −18.8433 −0.682174
$$764$$ 43.1355 1.56059
$$765$$ −4.33804 −0.156842
$$766$$ 4.11888 0.148821
$$767$$ 0 0
$$768$$ −9.72191 −0.350809
$$769$$ 17.3239 0.624716 0.312358 0.949964i $$-0.398881\pi$$
0.312358 + 0.949964i $$0.398881\pi$$
$$770$$ 6.98944 0.251882
$$771$$ 11.1567 0.401798
$$772$$ 5.08108 0.182872
$$773$$ 11.6373 0.418563 0.209282 0.977855i $$-0.432887\pi$$
0.209282 + 0.977855i $$0.432887\pi$$
$$774$$ −0.334474 −0.0120224
$$775$$ −1.42166 −0.0510676
$$776$$ −13.9728 −0.501593
$$777$$ −46.6797 −1.67462
$$778$$ 10.0272 0.359494
$$779$$ 11.1849 0.400742
$$780$$ 0 0
$$781$$ 4.50528 0.161212
$$782$$ 7.95061 0.284313
$$783$$ −6.00000 −0.214423
$$784$$ 60.1885 2.14959
$$785$$ −21.6655 −0.773276
$$786$$ −1.68665 −0.0601609
$$787$$ −20.9411 −0.746469 −0.373234 0.927737i $$-0.621751\pi$$
−0.373234 + 0.927737i $$0.621751\pi$$
$$788$$ −24.8927 −0.886766
$$789$$ −8.00000 −0.284808
$$790$$ −1.01056 −0.0359542
$$791$$ 4.14611 0.147419
$$792$$ 5.56777 0.197842
$$793$$ 0 0
$$794$$ −2.07663 −0.0736967
$$795$$ 0.338044 0.0119892
$$796$$ 5.44890 0.193131
$$797$$ 20.3380 0.720410 0.360205 0.932873i $$-0.382707\pi$$
0.360205 + 0.932873i $$0.382707\pi$$
$$798$$ −3.66553 −0.129758
$$799$$ −23.5194 −0.832057
$$800$$ 3.27861 0.115916
$$801$$ 0.338044 0.0119442
$$802$$ −10.9482 −0.386595
$$803$$ −15.5194 −0.547668
$$804$$ 13.9022 0.490294
$$805$$ 31.1602 1.09825
$$806$$ 0 0
$$807$$ −18.6761 −0.657429
$$808$$ −12.0906 −0.425346
$$809$$ 7.68665 0.270248 0.135124 0.990829i $$-0.456857\pi$$
0.135124 + 0.990829i $$0.456857\pi$$
$$810$$ 0.289169 0.0101603
$$811$$ 44.4111 1.55948 0.779742 0.626101i $$-0.215350\pi$$
0.779742 + 0.626101i $$0.215350\pi$$
$$812$$ 56.5300 1.98381
$$813$$ 6.57834 0.230712
$$814$$ 13.4983 0.473115
$$815$$ −6.07306 −0.212730
$$816$$ 15.2061 0.532319
$$817$$ 2.98230 0.104337
$$818$$ 4.04836 0.141548
$$819$$ 0 0
$$820$$ 8.31335 0.290315
$$821$$ 46.4630 1.62157 0.810785 0.585343i $$-0.199040\pi$$
0.810785 + 0.585343i $$0.199040\pi$$
$$822$$ −4.77384 −0.166507
$$823$$ 46.5089 1.62120 0.810598 0.585603i $$-0.199142\pi$$
0.810598 + 0.585603i $$0.199142\pi$$
$$824$$ 16.4312 0.572408
$$825$$ 4.91638 0.171166
$$826$$ −16.0000 −0.556711
$$827$$ −39.4005 −1.37009 −0.685045 0.728500i $$-0.740218\pi$$
−0.685045 + 0.728500i $$0.740218\pi$$
$$828$$ 12.1461 0.422107
$$829$$ 47.6444 1.65476 0.827379 0.561644i $$-0.189831\pi$$
0.827379 + 0.561644i $$0.189831\pi$$
$$830$$ −3.25443 −0.112963
$$831$$ −25.6655 −0.890327
$$832$$ 0 0
$$833$$ −74.4877 −2.58085
$$834$$ −2.16724 −0.0750453
$$835$$ 0.745574 0.0258017
$$836$$ −24.2922 −0.840164
$$837$$ 1.42166 0.0491399
$$838$$ −9.68665 −0.334620
$$839$$ 39.9058 1.37770 0.688851 0.724903i $$-0.258115\pi$$
0.688851 + 0.724903i $$0.258115\pi$$
$$840$$ −5.56777 −0.192106
$$841$$ 7.00000 0.241379
$$842$$ −3.90939 −0.134726
$$843$$ 3.15667 0.108722
$$844$$ −12.0988 −0.416457
$$845$$ 0 0
$$846$$ 1.56777 0.0539012
$$847$$ −64.7527 −2.22493
$$848$$ −1.18494 −0.0406910
$$849$$ 3.47002 0.119091
$$850$$ −1.25443 −0.0430265
$$851$$ 60.1779 2.06287
$$852$$ −1.75614 −0.0601643
$$853$$ −29.5019 −1.01012 −0.505062 0.863083i $$-0.668530\pi$$
−0.505062 + 0.863083i $$0.668530\pi$$
$$854$$ 14.4595 0.494793
$$855$$ −2.57834 −0.0881773
$$856$$ −4.72342 −0.161443
$$857$$ 8.33804 0.284822 0.142411 0.989808i $$-0.454515\pi$$
0.142411 + 0.989808i $$0.454515\pi$$
$$858$$ 0 0
$$859$$ −4.17081 −0.142306 −0.0711531 0.997465i $$-0.522668\pi$$
−0.0711531 + 0.997465i $$0.522668\pi$$
$$860$$ 2.21663 0.0755864
$$861$$ −21.3275 −0.726839
$$862$$ −3.58890 −0.122238
$$863$$ 3.93051 0.133796 0.0668981 0.997760i $$-0.478690\pi$$
0.0668981 + 0.997760i $$0.478690\pi$$
$$864$$ −3.27861 −0.111540
$$865$$ 0.843326 0.0286739
$$866$$ −5.00953 −0.170231
$$867$$ −1.81863 −0.0617639
$$868$$ −13.3944 −0.454637
$$869$$ 17.1814 0.582838
$$870$$ −1.73501 −0.0588224
$$871$$ 0 0
$$872$$ −4.34058 −0.146991
$$873$$ 12.3380 0.417580
$$874$$ 4.72548 0.159842
$$875$$ −4.91638 −0.166204
$$876$$ 6.04939 0.204390
$$877$$ 23.0177 0.777253 0.388626 0.921395i $$-0.372950\pi$$
0.388626 + 0.921395i $$0.372950\pi$$
$$878$$ 0.188362 0.00635692
$$879$$ 28.6550 0.966508
$$880$$ −17.2333 −0.580934
$$881$$ −15.3522 −0.517228 −0.258614 0.965981i $$-0.583266\pi$$
−0.258614 + 0.965981i $$0.583266\pi$$
$$882$$ 4.96526 0.167189
$$883$$ 42.8011 1.44037 0.720185 0.693782i $$-0.244057\pi$$
0.720185 + 0.693782i $$0.244057\pi$$
$$884$$ 0 0
$$885$$ −11.2544 −0.378313
$$886$$ −2.55824 −0.0859459
$$887$$ −53.1885 −1.78590 −0.892948 0.450160i $$-0.851367\pi$$
−0.892948 + 0.450160i $$0.851367\pi$$
$$888$$ −10.7527 −0.360837
$$889$$ −9.01056 −0.302205
$$890$$ 0.0977518 0.00327665
$$891$$ −4.91638 −0.164705
$$892$$ 36.8988 1.23546
$$893$$ −13.9789 −0.467785
$$894$$ 5.92337 0.198107
$$895$$ −18.9894 −0.634747
$$896$$ 40.8566 1.36492
$$897$$ 0 0
$$898$$ −1.25443 −0.0418607
$$899$$ −8.52998 −0.284491
$$900$$ −1.91638 −0.0638794
$$901$$ 1.46645 0.0488546
$$902$$ 6.16724 0.205347
$$903$$ −5.68665 −0.189240
$$904$$ 0.955062 0.0317649
$$905$$ 17.4947 0.581544
$$906$$ −4.74557 −0.157661
$$907$$ 11.8116 0.392199 0.196099 0.980584i $$-0.437172\pi$$
0.196099 + 0.980584i $$0.437172\pi$$
$$908$$ −25.0800 −0.832311
$$909$$ 10.6761 0.354104
$$910$$ 0 0
$$911$$ 44.1955 1.46426 0.732131 0.681164i $$-0.238526\pi$$
0.732131 + 0.681164i $$0.238526\pi$$
$$912$$ 9.03780 0.299271
$$913$$ 55.3311 1.83119
$$914$$ −4.43223 −0.146605
$$915$$ 10.1708 0.336237
$$916$$ −46.9683 −1.55188
$$917$$ −28.6761 −0.946968
$$918$$ 1.25443 0.0414022
$$919$$ 55.2096 1.82120 0.910599 0.413291i $$-0.135621\pi$$
0.910599 + 0.413291i $$0.135621\pi$$
$$920$$ 7.17780 0.236645
$$921$$ −1.92694 −0.0634950
$$922$$ 3.42880 0.112922
$$923$$ 0 0
$$924$$ 46.3205 1.52383
$$925$$ −9.49472 −0.312184
$$926$$ 7.63829 0.251010
$$927$$ −14.5089 −0.476533
$$928$$ 19.6716 0.645753
$$929$$ −22.9930 −0.754376 −0.377188 0.926137i $$-0.623109\pi$$
−0.377188 + 0.926137i $$0.623109\pi$$
$$930$$ 0.411100 0.0134805
$$931$$ −44.2721 −1.45096
$$932$$ −15.9789 −0.523405
$$933$$ 29.9789 0.981464
$$934$$ −9.73604 −0.318573
$$935$$ 21.3275 0.697483
$$936$$ 0 0
$$937$$ 7.97887 0.260658 0.130329 0.991471i $$-0.458397\pi$$
0.130329 + 0.991471i $$0.458397\pi$$
$$938$$ −10.3133 −0.336743
$$939$$ 16.3133 0.532366
$$940$$ −10.3900 −0.338884
$$941$$ −41.5019 −1.35292 −0.676461 0.736478i $$-0.736487\pi$$
−0.676461 + 0.736478i $$0.736487\pi$$
$$942$$ 6.26499 0.204124
$$943$$ 27.4947 0.895351
$$944$$ 39.4499 1.28399
$$945$$ 4.91638 0.159930
$$946$$ 1.64440 0.0534641
$$947$$ −47.4499 −1.54192 −0.770958 0.636886i $$-0.780222\pi$$
−0.770958 + 0.636886i $$0.780222\pi$$
$$948$$ −6.69721 −0.217515
$$949$$ 0 0
$$950$$ −0.745574 −0.0241896
$$951$$ 30.6761 0.994740
$$952$$ −24.1533 −0.782811
$$953$$ 30.3663 0.983661 0.491831 0.870691i $$-0.336328\pi$$
0.491831 + 0.870691i $$0.336328\pi$$
$$954$$ −0.0977518 −0.00316483
$$955$$ −22.5089 −0.728369
$$956$$ 17.0872 0.552639
$$957$$ 29.4983 0.953544
$$958$$ −3.10831 −0.100425
$$959$$ −81.1638 −2.62092
$$960$$ 6.06249 0.195666
$$961$$ −28.9789 −0.934802
$$962$$ 0 0
$$963$$ 4.17081 0.134402
$$964$$ −11.4983 −0.370335
$$965$$ −2.65139 −0.0853514
$$966$$ −9.01056 −0.289910
$$967$$ 41.0943 1.32150 0.660752 0.750604i $$-0.270237\pi$$
0.660752 + 0.750604i $$0.270237\pi$$
$$968$$ −14.9159 −0.479414
$$969$$ −11.1849 −0.359312
$$970$$ 3.56777 0.114554
$$971$$ 38.1744 1.22507 0.612537 0.790442i $$-0.290149\pi$$
0.612537 + 0.790442i $$0.290149\pi$$
$$972$$ 1.91638 0.0614680
$$973$$ −36.8469 −1.18126
$$974$$ −6.57834 −0.210784
$$975$$ 0 0
$$976$$ −35.6515 −1.14118
$$977$$ 10.4806 0.335304 0.167652 0.985846i $$-0.446382\pi$$
0.167652 + 0.985846i $$0.446382\pi$$
$$978$$ 1.75614 0.0561551
$$979$$ −1.66196 −0.0531163
$$980$$ −32.9058 −1.05114
$$981$$ 3.83276 0.122371
$$982$$ −5.11442 −0.163208
$$983$$ 0.0766264 0.00244400 0.00122200 0.999999i $$-0.499611\pi$$
0.00122200 + 0.999999i $$0.499611\pi$$
$$984$$ −4.91281 −0.156615
$$985$$ 12.9894 0.413878
$$986$$ −7.52656 −0.239694
$$987$$ 26.6550 0.848437
$$988$$ 0 0
$$989$$ 7.33105 0.233114
$$990$$ −1.42166 −0.0451834
$$991$$ −13.8575 −0.440197 −0.220098 0.975478i $$-0.570638\pi$$
−0.220098 + 0.975478i $$0.570638\pi$$
$$992$$ −4.66107 −0.147989
$$993$$ −10.0978 −0.320442
$$994$$ 1.30279 0.0413219
$$995$$ −2.84333 −0.0901395
$$996$$ −21.5678 −0.683401
$$997$$ −10.3416 −0.327522 −0.163761 0.986500i $$-0.552363\pi$$
−0.163761 + 0.986500i $$0.552363\pi$$
$$998$$ −5.76328 −0.182433
$$999$$ 9.49472 0.300400
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 2535.2.a.bc.1.2 3
3.2 odd 2 7605.2.a.bx.1.2 3
13.12 even 2 195.2.a.e.1.2 3
39.38 odd 2 585.2.a.n.1.2 3
52.51 odd 2 3120.2.a.bj.1.1 3
65.12 odd 4 975.2.c.i.274.3 6
65.38 odd 4 975.2.c.i.274.4 6
65.64 even 2 975.2.a.o.1.2 3
91.90 odd 2 9555.2.a.bq.1.2 3
156.155 even 2 9360.2.a.dd.1.1 3
195.38 even 4 2925.2.c.w.2224.3 6
195.77 even 4 2925.2.c.w.2224.4 6
195.194 odd 2 2925.2.a.bh.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
195.2.a.e.1.2 3 13.12 even 2
585.2.a.n.1.2 3 39.38 odd 2
975.2.a.o.1.2 3 65.64 even 2
975.2.c.i.274.3 6 65.12 odd 4
975.2.c.i.274.4 6 65.38 odd 4
2535.2.a.bc.1.2 3 1.1 even 1 trivial
2925.2.a.bh.1.2 3 195.194 odd 2
2925.2.c.w.2224.3 6 195.38 even 4
2925.2.c.w.2224.4 6 195.77 even 4
3120.2.a.bj.1.1 3 52.51 odd 2
7605.2.a.bx.1.2 3 3.2 odd 2
9360.2.a.dd.1.1 3 156.155 even 2
9555.2.a.bq.1.2 3 91.90 odd 2