# Properties

 Label 6525.2.a.cc.1.7 Level $6525$ Weight $2$ Character 6525.1 Self dual yes Analytic conductor $52.102$ Analytic rank $0$ Dimension $9$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6525,2,Mod(1,6525)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6525, 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("6525.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$6525 = 3^{2} \cdot 5^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6525.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: $$52.1023873189$$ Analytic rank: $$0$$ Dimension: $$9$$ Coefficient field: $$\mathbb{Q}[x]/(x^{9} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{9} - 2x^{8} - 12x^{7} + 21x^{6} + 48x^{5} - 68x^{4} - 73x^{3} + 66x^{2} + 40x - 10$$ x^9 - 2*x^8 - 12*x^7 + 21*x^6 + 48*x^5 - 68*x^4 - 73*x^3 + 66*x^2 + 40*x - 10 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.7 Root $$1.77820$$ of defining polynomial Character $$\chi$$ $$=$$ 6525.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.77820 q^{2} +1.16198 q^{4} +2.84862 q^{7} -1.49016 q^{8} +O(q^{10})$$ $$q+1.77820 q^{2} +1.16198 q^{4} +2.84862 q^{7} -1.49016 q^{8} +5.35505 q^{11} +2.50643 q^{13} +5.06541 q^{14} -4.97376 q^{16} +6.50704 q^{17} +2.25433 q^{19} +9.52233 q^{22} -0.355939 q^{23} +4.45692 q^{26} +3.31005 q^{28} -1.00000 q^{29} +5.45828 q^{31} -5.86400 q^{32} +11.5708 q^{34} +2.00281 q^{37} +4.00863 q^{38} -5.37258 q^{41} -12.8773 q^{43} +6.22247 q^{44} -0.632929 q^{46} -4.01725 q^{47} +1.11466 q^{49} +2.91242 q^{52} -3.42758 q^{53} -4.24492 q^{56} -1.77820 q^{58} -4.76617 q^{59} -4.29494 q^{61} +9.70588 q^{62} -0.479810 q^{64} +1.09723 q^{67} +7.56105 q^{68} +5.21539 q^{71} +6.78643 q^{73} +3.56138 q^{74} +2.61948 q^{76} +15.2545 q^{77} -1.84943 q^{79} -9.55349 q^{82} +13.7379 q^{83} -22.8984 q^{86} -7.97990 q^{88} -2.56752 q^{89} +7.13987 q^{91} -0.413594 q^{92} -7.14346 q^{94} -2.73855 q^{97} +1.98209 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$9 q + 2 q^{2} + 10 q^{4} - q^{7} + 9 q^{8}+O(q^{10})$$ 9 * q + 2 * q^2 + 10 * q^4 - q^7 + 9 * q^8 $$9 q + 2 q^{2} + 10 q^{4} - q^{7} + 9 q^{8} - 2 q^{11} - q^{13} + 3 q^{14} + 4 q^{16} + 12 q^{17} - q^{19} - 3 q^{22} + 16 q^{23} - 6 q^{26} + 4 q^{28} - 9 q^{29} + 5 q^{31} + 20 q^{32} + 3 q^{34} + 30 q^{38} + 10 q^{41} - 3 q^{43} + 13 q^{44} + 4 q^{46} + 26 q^{47} - 8 q^{49} + 9 q^{52} + 22 q^{53} - 22 q^{56} - 2 q^{58} - 4 q^{59} + 7 q^{61} + 28 q^{62} + 9 q^{64} - 5 q^{67} + 39 q^{68} + 10 q^{73} + 34 q^{74} - 2 q^{76} + 34 q^{77} + 10 q^{79} + 8 q^{82} + 46 q^{83} - 28 q^{86} - 2 q^{88} - 4 q^{89} - 21 q^{91} + 20 q^{92} + 5 q^{94} - 7 q^{97} + 51 q^{98}+O(q^{100})$$ 9 * q + 2 * q^2 + 10 * q^4 - q^7 + 9 * q^8 - 2 * q^11 - q^13 + 3 * q^14 + 4 * q^16 + 12 * q^17 - q^19 - 3 * q^22 + 16 * q^23 - 6 * q^26 + 4 * q^28 - 9 * q^29 + 5 * q^31 + 20 * q^32 + 3 * q^34 + 30 * q^38 + 10 * q^41 - 3 * q^43 + 13 * q^44 + 4 * q^46 + 26 * q^47 - 8 * q^49 + 9 * q^52 + 22 * q^53 - 22 * q^56 - 2 * q^58 - 4 * q^59 + 7 * q^61 + 28 * q^62 + 9 * q^64 - 5 * q^67 + 39 * q^68 + 10 * q^73 + 34 * q^74 - 2 * q^76 + 34 * q^77 + 10 * q^79 + 8 * q^82 + 46 * q^83 - 28 * q^86 - 2 * q^88 - 4 * q^89 - 21 * q^91 + 20 * q^92 + 5 * q^94 - 7 * q^97 + 51 * q^98

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.77820 1.25737 0.628687 0.777658i $$-0.283593\pi$$
0.628687 + 0.777658i $$0.283593\pi$$
$$3$$ 0 0
$$4$$ 1.16198 0.580990
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 2.84862 1.07668 0.538339 0.842728i $$-0.319052\pi$$
0.538339 + 0.842728i $$0.319052\pi$$
$$8$$ −1.49016 −0.526852
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 5.35505 1.61461 0.807305 0.590135i $$-0.200925\pi$$
0.807305 + 0.590135i $$0.200925\pi$$
$$12$$ 0 0
$$13$$ 2.50643 0.695158 0.347579 0.937651i $$-0.387004\pi$$
0.347579 + 0.937651i $$0.387004\pi$$
$$14$$ 5.06541 1.35379
$$15$$ 0 0
$$16$$ −4.97376 −1.24344
$$17$$ 6.50704 1.57819 0.789095 0.614271i $$-0.210550\pi$$
0.789095 + 0.614271i $$0.210550\pi$$
$$18$$ 0 0
$$19$$ 2.25433 0.517178 0.258589 0.965987i $$-0.416742\pi$$
0.258589 + 0.965987i $$0.416742\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 9.52233 2.03017
$$23$$ −0.355939 −0.0742184 −0.0371092 0.999311i $$-0.511815\pi$$
−0.0371092 + 0.999311i $$0.511815\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 4.45692 0.874074
$$27$$ 0 0
$$28$$ 3.31005 0.625540
$$29$$ −1.00000 −0.185695
$$30$$ 0 0
$$31$$ 5.45828 0.980335 0.490168 0.871628i $$-0.336936\pi$$
0.490168 + 0.871628i $$0.336936\pi$$
$$32$$ −5.86400 −1.03662
$$33$$ 0 0
$$34$$ 11.5708 1.98438
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 2.00281 0.329259 0.164630 0.986355i $$-0.447357\pi$$
0.164630 + 0.986355i $$0.447357\pi$$
$$38$$ 4.00863 0.650286
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −5.37258 −0.839055 −0.419528 0.907743i $$-0.637804\pi$$
−0.419528 + 0.907743i $$0.637804\pi$$
$$42$$ 0 0
$$43$$ −12.8773 −1.96377 −0.981886 0.189472i $$-0.939322\pi$$
−0.981886 + 0.189472i $$0.939322\pi$$
$$44$$ 6.22247 0.938072
$$45$$ 0 0
$$46$$ −0.632929 −0.0933203
$$47$$ −4.01725 −0.585976 −0.292988 0.956116i $$-0.594650\pi$$
−0.292988 + 0.956116i $$0.594650\pi$$
$$48$$ 0 0
$$49$$ 1.11466 0.159238
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 2.91242 0.403880
$$53$$ −3.42758 −0.470814 −0.235407 0.971897i $$-0.575642\pi$$
−0.235407 + 0.971897i $$0.575642\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −4.24492 −0.567251
$$57$$ 0 0
$$58$$ −1.77820 −0.233489
$$59$$ −4.76617 −0.620503 −0.310251 0.950655i $$-0.600413\pi$$
−0.310251 + 0.950655i $$0.600413\pi$$
$$60$$ 0 0
$$61$$ −4.29494 −0.549911 −0.274955 0.961457i $$-0.588663\pi$$
−0.274955 + 0.961457i $$0.588663\pi$$
$$62$$ 9.70588 1.23265
$$63$$ 0 0
$$64$$ −0.479810 −0.0599762
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 1.09723 0.134048 0.0670239 0.997751i $$-0.478650\pi$$
0.0670239 + 0.997751i $$0.478650\pi$$
$$68$$ 7.56105 0.916913
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 5.21539 0.618953 0.309476 0.950907i $$-0.399846\pi$$
0.309476 + 0.950907i $$0.399846\pi$$
$$72$$ 0 0
$$73$$ 6.78643 0.794291 0.397146 0.917756i $$-0.370001\pi$$
0.397146 + 0.917756i $$0.370001\pi$$
$$74$$ 3.56138 0.414002
$$75$$ 0 0
$$76$$ 2.61948 0.300475
$$77$$ 15.2545 1.73842
$$78$$ 0 0
$$79$$ −1.84943 −0.208077 −0.104038 0.994573i $$-0.533176\pi$$
−0.104038 + 0.994573i $$0.533176\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ −9.55349 −1.05501
$$83$$ 13.7379 1.50793 0.753963 0.656917i $$-0.228140\pi$$
0.753963 + 0.656917i $$0.228140\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −22.8984 −2.46920
$$87$$ 0 0
$$88$$ −7.97990 −0.850660
$$89$$ −2.56752 −0.272157 −0.136078 0.990698i $$-0.543450\pi$$
−0.136078 + 0.990698i $$0.543450\pi$$
$$90$$ 0 0
$$91$$ 7.13987 0.748462
$$92$$ −0.413594 −0.0431202
$$93$$ 0 0
$$94$$ −7.14346 −0.736792
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −2.73855 −0.278057 −0.139029 0.990288i $$-0.544398\pi$$
−0.139029 + 0.990288i $$0.544398\pi$$
$$98$$ 1.98209 0.200221
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −5.73985 −0.571137 −0.285568 0.958358i $$-0.592182\pi$$
−0.285568 + 0.958358i $$0.592182\pi$$
$$102$$ 0 0
$$103$$ −15.7896 −1.55579 −0.777895 0.628394i $$-0.783713\pi$$
−0.777895 + 0.628394i $$0.783713\pi$$
$$104$$ −3.73499 −0.366246
$$105$$ 0 0
$$106$$ −6.09491 −0.591990
$$107$$ 16.5642 1.60132 0.800661 0.599118i $$-0.204482\pi$$
0.800661 + 0.599118i $$0.204482\pi$$
$$108$$ 0 0
$$109$$ 9.32556 0.893226 0.446613 0.894727i $$-0.352630\pi$$
0.446613 + 0.894727i $$0.352630\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −14.1684 −1.33879
$$113$$ 14.1979 1.33563 0.667815 0.744327i $$-0.267230\pi$$
0.667815 + 0.744327i $$0.267230\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −1.16198 −0.107887
$$117$$ 0 0
$$118$$ −8.47518 −0.780204
$$119$$ 18.5361 1.69920
$$120$$ 0 0
$$121$$ 17.6766 1.60696
$$122$$ −7.63724 −0.691444
$$123$$ 0 0
$$124$$ 6.34241 0.569565
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 5.36218 0.475816 0.237908 0.971288i $$-0.423538\pi$$
0.237908 + 0.971288i $$0.423538\pi$$
$$128$$ 10.8748 0.961205
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 10.8431 0.947370 0.473685 0.880694i $$-0.342924\pi$$
0.473685 + 0.880694i $$0.342924\pi$$
$$132$$ 0 0
$$133$$ 6.42173 0.556835
$$134$$ 1.95109 0.168548
$$135$$ 0 0
$$136$$ −9.69656 −0.831473
$$137$$ 7.41637 0.633623 0.316812 0.948489i $$-0.397388\pi$$
0.316812 + 0.948489i $$0.397388\pi$$
$$138$$ 0 0
$$139$$ 19.6639 1.66787 0.833937 0.551859i $$-0.186082\pi$$
0.833937 + 0.551859i $$0.186082\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 9.27398 0.778255
$$143$$ 13.4221 1.12241
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 12.0676 0.998722
$$147$$ 0 0
$$148$$ 2.32722 0.191296
$$149$$ 14.4485 1.18367 0.591834 0.806060i $$-0.298404\pi$$
0.591834 + 0.806060i $$0.298404\pi$$
$$150$$ 0 0
$$151$$ 6.69220 0.544604 0.272302 0.962212i $$-0.412215\pi$$
0.272302 + 0.962212i $$0.412215\pi$$
$$152$$ −3.35931 −0.272476
$$153$$ 0 0
$$154$$ 27.1256 2.18584
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 17.5789 1.40295 0.701473 0.712696i $$-0.252526\pi$$
0.701473 + 0.712696i $$0.252526\pi$$
$$158$$ −3.28864 −0.261630
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −1.01394 −0.0799094
$$162$$ 0 0
$$163$$ −19.6576 −1.53971 −0.769853 0.638222i $$-0.779670\pi$$
−0.769853 + 0.638222i $$0.779670\pi$$
$$164$$ −6.24283 −0.487483
$$165$$ 0 0
$$166$$ 24.4286 1.89603
$$167$$ −12.2841 −0.950569 −0.475285 0.879832i $$-0.657655\pi$$
−0.475285 + 0.879832i $$0.657655\pi$$
$$168$$ 0 0
$$169$$ −6.71782 −0.516755
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −14.9632 −1.14093
$$173$$ −12.9440 −0.984118 −0.492059 0.870562i $$-0.663756\pi$$
−0.492059 + 0.870562i $$0.663756\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −26.6348 −2.00767
$$177$$ 0 0
$$178$$ −4.56555 −0.342203
$$179$$ −6.01888 −0.449872 −0.224936 0.974374i $$-0.572217\pi$$
−0.224936 + 0.974374i $$0.572217\pi$$
$$180$$ 0 0
$$181$$ −19.8734 −1.47718 −0.738591 0.674154i $$-0.764508\pi$$
−0.738591 + 0.674154i $$0.764508\pi$$
$$182$$ 12.6961 0.941097
$$183$$ 0 0
$$184$$ 0.530407 0.0391021
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 34.8456 2.54816
$$188$$ −4.66797 −0.340446
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 3.04577 0.220384 0.110192 0.993910i $$-0.464853\pi$$
0.110192 + 0.993910i $$0.464853\pi$$
$$192$$ 0 0
$$193$$ −21.3408 −1.53615 −0.768073 0.640362i $$-0.778784\pi$$
−0.768073 + 0.640362i $$0.778784\pi$$
$$194$$ −4.86967 −0.349622
$$195$$ 0 0
$$196$$ 1.29522 0.0925154
$$197$$ −10.0730 −0.717670 −0.358835 0.933401i $$-0.616826\pi$$
−0.358835 + 0.933401i $$0.616826\pi$$
$$198$$ 0 0
$$199$$ −4.22842 −0.299744 −0.149872 0.988705i $$-0.547886\pi$$
−0.149872 + 0.988705i $$0.547886\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ −10.2066 −0.718133
$$203$$ −2.84862 −0.199934
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −28.0769 −1.95621
$$207$$ 0 0
$$208$$ −12.4664 −0.864388
$$209$$ 12.0720 0.835040
$$210$$ 0 0
$$211$$ 28.2252 1.94310 0.971550 0.236834i $$-0.0761098\pi$$
0.971550 + 0.236834i $$0.0761098\pi$$
$$212$$ −3.98278 −0.273539
$$213$$ 0 0
$$214$$ 29.4544 2.01346
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 15.5486 1.05551
$$218$$ 16.5827 1.12312
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 16.3094 1.09709
$$222$$ 0 0
$$223$$ 8.48452 0.568165 0.284083 0.958800i $$-0.408311\pi$$
0.284083 + 0.958800i $$0.408311\pi$$
$$224$$ −16.7043 −1.11610
$$225$$ 0 0
$$226$$ 25.2467 1.67939
$$227$$ −14.7872 −0.981460 −0.490730 0.871312i $$-0.663270\pi$$
−0.490730 + 0.871312i $$0.663270\pi$$
$$228$$ 0 0
$$229$$ −21.0993 −1.39428 −0.697141 0.716934i $$-0.745545\pi$$
−0.697141 + 0.716934i $$0.745545\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 1.49016 0.0978340
$$233$$ 26.0762 1.70831 0.854155 0.520018i $$-0.174075\pi$$
0.854155 + 0.520018i $$0.174075\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −5.53820 −0.360506
$$237$$ 0 0
$$238$$ 32.9609 2.13653
$$239$$ −8.17070 −0.528519 −0.264259 0.964452i $$-0.585128\pi$$
−0.264259 + 0.964452i $$0.585128\pi$$
$$240$$ 0 0
$$241$$ −10.5741 −0.681135 −0.340568 0.940220i $$-0.610619\pi$$
−0.340568 + 0.940220i $$0.610619\pi$$
$$242$$ 31.4324 2.02055
$$243$$ 0 0
$$244$$ −4.99064 −0.319493
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 5.65031 0.359520
$$248$$ −8.13372 −0.516492
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −21.3791 −1.34944 −0.674720 0.738074i $$-0.735736\pi$$
−0.674720 + 0.738074i $$0.735736\pi$$
$$252$$ 0 0
$$253$$ −1.90607 −0.119834
$$254$$ 9.53500 0.598279
$$255$$ 0 0
$$256$$ 20.2971 1.26857
$$257$$ −6.90806 −0.430913 −0.215456 0.976513i $$-0.569124\pi$$
−0.215456 + 0.976513i $$0.569124\pi$$
$$258$$ 0 0
$$259$$ 5.70524 0.354506
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 19.2812 1.19120
$$263$$ 2.36575 0.145878 0.0729391 0.997336i $$-0.476762\pi$$
0.0729391 + 0.997336i $$0.476762\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 11.4191 0.700150
$$267$$ 0 0
$$268$$ 1.27496 0.0778805
$$269$$ −13.0500 −0.795670 −0.397835 0.917457i $$-0.630238\pi$$
−0.397835 + 0.917457i $$0.630238\pi$$
$$270$$ 0 0
$$271$$ −24.2733 −1.47450 −0.737250 0.675620i $$-0.763876\pi$$
−0.737250 + 0.675620i $$0.763876\pi$$
$$272$$ −32.3645 −1.96239
$$273$$ 0 0
$$274$$ 13.1878 0.796701
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −0.0846766 −0.00508773 −0.00254386 0.999997i $$-0.500810\pi$$
−0.00254386 + 0.999997i $$0.500810\pi$$
$$278$$ 34.9663 2.09714
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −8.18399 −0.488216 −0.244108 0.969748i $$-0.578495\pi$$
−0.244108 + 0.969748i $$0.578495\pi$$
$$282$$ 0 0
$$283$$ −10.6021 −0.630227 −0.315114 0.949054i $$-0.602043\pi$$
−0.315114 + 0.949054i $$0.602043\pi$$
$$284$$ 6.06018 0.359605
$$285$$ 0 0
$$286$$ 23.8670 1.41129
$$287$$ −15.3045 −0.903393
$$288$$ 0 0
$$289$$ 25.3416 1.49068
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 7.88570 0.461475
$$293$$ 26.2617 1.53422 0.767112 0.641513i $$-0.221693\pi$$
0.767112 + 0.641513i $$0.221693\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −2.98451 −0.173471
$$297$$ 0 0
$$298$$ 25.6923 1.48831
$$299$$ −0.892136 −0.0515935
$$300$$ 0 0
$$301$$ −36.6826 −2.11435
$$302$$ 11.9000 0.684771
$$303$$ 0 0
$$304$$ −11.2125 −0.643080
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −20.8923 −1.19239 −0.596193 0.802841i $$-0.703321\pi$$
−0.596193 + 0.802841i $$0.703321\pi$$
$$308$$ 17.7255 1.01000
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 5.25221 0.297826 0.148913 0.988850i $$-0.452423\pi$$
0.148913 + 0.988850i $$0.452423\pi$$
$$312$$ 0 0
$$313$$ 32.0146 1.80957 0.904787 0.425865i $$-0.140030\pi$$
0.904787 + 0.425865i $$0.140030\pi$$
$$314$$ 31.2587 1.76403
$$315$$ 0 0
$$316$$ −2.14900 −0.120891
$$317$$ 22.1715 1.24527 0.622637 0.782511i $$-0.286061\pi$$
0.622637 + 0.782511i $$0.286061\pi$$
$$318$$ 0 0
$$319$$ −5.35505 −0.299825
$$320$$ 0 0
$$321$$ 0 0
$$322$$ −1.80298 −0.100476
$$323$$ 14.6690 0.816205
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −34.9551 −1.93599
$$327$$ 0 0
$$328$$ 8.00602 0.442058
$$329$$ −11.4436 −0.630908
$$330$$ 0 0
$$331$$ −31.8652 −1.75147 −0.875734 0.482794i $$-0.839622\pi$$
−0.875734 + 0.482794i $$0.839622\pi$$
$$332$$ 15.9631 0.876090
$$333$$ 0 0
$$334$$ −21.8435 −1.19522
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 4.68105 0.254993 0.127496 0.991839i $$-0.459306\pi$$
0.127496 + 0.991839i $$0.459306\pi$$
$$338$$ −11.9456 −0.649755
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 29.2294 1.58286
$$342$$ 0 0
$$343$$ −16.7651 −0.905231
$$344$$ 19.1893 1.03462
$$345$$ 0 0
$$346$$ −23.0170 −1.23740
$$347$$ −10.0153 −0.537650 −0.268825 0.963189i $$-0.586635\pi$$
−0.268825 + 0.963189i $$0.586635\pi$$
$$348$$ 0 0
$$349$$ 14.9709 0.801376 0.400688 0.916215i $$-0.368771\pi$$
0.400688 + 0.916215i $$0.368771\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −31.4020 −1.67373
$$353$$ −12.0450 −0.641092 −0.320546 0.947233i $$-0.603866\pi$$
−0.320546 + 0.947233i $$0.603866\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −2.98341 −0.158120
$$357$$ 0 0
$$358$$ −10.7027 −0.565658
$$359$$ 4.72615 0.249436 0.124718 0.992192i $$-0.460197\pi$$
0.124718 + 0.992192i $$0.460197\pi$$
$$360$$ 0 0
$$361$$ −13.9180 −0.732527
$$362$$ −35.3389 −1.85737
$$363$$ 0 0
$$364$$ 8.29639 0.434849
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 11.9598 0.624295 0.312148 0.950034i $$-0.398952\pi$$
0.312148 + 0.950034i $$0.398952\pi$$
$$368$$ 1.77036 0.0922862
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −9.76389 −0.506916
$$372$$ 0 0
$$373$$ 8.31719 0.430648 0.215324 0.976543i $$-0.430919\pi$$
0.215324 + 0.976543i $$0.430919\pi$$
$$374$$ 61.9622 3.20399
$$375$$ 0 0
$$376$$ 5.98636 0.308723
$$377$$ −2.50643 −0.129088
$$378$$ 0 0
$$379$$ −24.5715 −1.26215 −0.631075 0.775721i $$-0.717386\pi$$
−0.631075 + 0.775721i $$0.717386\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 5.41598 0.277106
$$383$$ 34.0260 1.73865 0.869325 0.494241i $$-0.164554\pi$$
0.869325 + 0.494241i $$0.164554\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −37.9482 −1.93151
$$387$$ 0 0
$$388$$ −3.18214 −0.161549
$$389$$ 25.3691 1.28626 0.643131 0.765756i $$-0.277635\pi$$
0.643131 + 0.765756i $$0.277635\pi$$
$$390$$ 0 0
$$391$$ −2.31611 −0.117131
$$392$$ −1.66103 −0.0838947
$$393$$ 0 0
$$394$$ −17.9117 −0.902379
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −16.7222 −0.839265 −0.419632 0.907694i $$-0.637841\pi$$
−0.419632 + 0.907694i $$0.637841\pi$$
$$398$$ −7.51895 −0.376891
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 8.01416 0.400208 0.200104 0.979775i $$-0.435872\pi$$
0.200104 + 0.979775i $$0.435872\pi$$
$$402$$ 0 0
$$403$$ 13.6808 0.681488
$$404$$ −6.66960 −0.331825
$$405$$ 0 0
$$406$$ −5.06541 −0.251392
$$407$$ 10.7251 0.531625
$$408$$ 0 0
$$409$$ −19.4335 −0.960924 −0.480462 0.877016i $$-0.659531\pi$$
−0.480462 + 0.877016i $$0.659531\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ −18.3471 −0.903899
$$413$$ −13.5770 −0.668082
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −14.6977 −0.720613
$$417$$ 0 0
$$418$$ 21.4664 1.04996
$$419$$ −6.92370 −0.338245 −0.169122 0.985595i $$-0.554093\pi$$
−0.169122 + 0.985595i $$0.554093\pi$$
$$420$$ 0 0
$$421$$ −12.9872 −0.632956 −0.316478 0.948600i $$-0.602500\pi$$
−0.316478 + 0.948600i $$0.602500\pi$$
$$422$$ 50.1899 2.44320
$$423$$ 0 0
$$424$$ 5.10766 0.248050
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −12.2347 −0.592077
$$428$$ 19.2473 0.930352
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −39.6752 −1.91109 −0.955543 0.294851i $$-0.904730\pi$$
−0.955543 + 0.294851i $$0.904730\pi$$
$$432$$ 0 0
$$433$$ −4.93087 −0.236962 −0.118481 0.992956i $$-0.537803\pi$$
−0.118481 + 0.992956i $$0.537803\pi$$
$$434$$ 27.6484 1.32717
$$435$$ 0 0
$$436$$ 10.8361 0.518956
$$437$$ −0.802403 −0.0383841
$$438$$ 0 0
$$439$$ 22.1458 1.05696 0.528481 0.848945i $$-0.322762\pi$$
0.528481 + 0.848945i $$0.322762\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 29.0014 1.37945
$$443$$ 1.21156 0.0575631 0.0287815 0.999586i $$-0.490837\pi$$
0.0287815 + 0.999586i $$0.490837\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 15.0871 0.714396
$$447$$ 0 0
$$448$$ −1.36680 −0.0645751
$$449$$ 5.22771 0.246711 0.123355 0.992363i $$-0.460635\pi$$
0.123355 + 0.992363i $$0.460635\pi$$
$$450$$ 0 0
$$451$$ −28.7704 −1.35475
$$452$$ 16.4977 0.775988
$$453$$ 0 0
$$454$$ −26.2945 −1.23406
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 16.9064 0.790849 0.395425 0.918498i $$-0.370597\pi$$
0.395425 + 0.918498i $$0.370597\pi$$
$$458$$ −37.5187 −1.75313
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −7.79634 −0.363112 −0.181556 0.983381i $$-0.558113\pi$$
−0.181556 + 0.983381i $$0.558113\pi$$
$$462$$ 0 0
$$463$$ −22.3232 −1.03745 −0.518724 0.854942i $$-0.673593\pi$$
−0.518724 + 0.854942i $$0.673593\pi$$
$$464$$ 4.97376 0.230901
$$465$$ 0 0
$$466$$ 46.3686 2.14799
$$467$$ 8.19178 0.379070 0.189535 0.981874i $$-0.439302\pi$$
0.189535 + 0.981874i $$0.439302\pi$$
$$468$$ 0 0
$$469$$ 3.12559 0.144327
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 7.10237 0.326913
$$473$$ −68.9587 −3.17072
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 21.5386 0.987220
$$477$$ 0 0
$$478$$ −14.5291 −0.664546
$$479$$ 17.4830 0.798817 0.399409 0.916773i $$-0.369215\pi$$
0.399409 + 0.916773i $$0.369215\pi$$
$$480$$ 0 0
$$481$$ 5.01989 0.228887
$$482$$ −18.8028 −0.856442
$$483$$ 0 0
$$484$$ 20.5398 0.933630
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −23.2620 −1.05410 −0.527051 0.849833i $$-0.676702\pi$$
−0.527051 + 0.849833i $$0.676702\pi$$
$$488$$ 6.40016 0.289722
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 34.4941 1.55670 0.778348 0.627833i $$-0.216058\pi$$
0.778348 + 0.627833i $$0.216058\pi$$
$$492$$ 0 0
$$493$$ −6.50704 −0.293062
$$494$$ 10.0474 0.452052
$$495$$ 0 0
$$496$$ −27.1482 −1.21899
$$497$$ 14.8567 0.666413
$$498$$ 0 0
$$499$$ 12.8752 0.576372 0.288186 0.957574i $$-0.406948\pi$$
0.288186 + 0.957574i $$0.406948\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ −38.0163 −1.69675
$$503$$ 6.19856 0.276380 0.138190 0.990406i $$-0.455872\pi$$
0.138190 + 0.990406i $$0.455872\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −3.38937 −0.150676
$$507$$ 0 0
$$508$$ 6.23074 0.276444
$$509$$ −29.6558 −1.31447 −0.657235 0.753686i $$-0.728274\pi$$
−0.657235 + 0.753686i $$0.728274\pi$$
$$510$$ 0 0
$$511$$ 19.3320 0.855197
$$512$$ 14.3427 0.633863
$$513$$ 0 0
$$514$$ −12.2839 −0.541819
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −21.5126 −0.946123
$$518$$ 10.1450 0.445747
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −31.2676 −1.36986 −0.684929 0.728610i $$-0.740167\pi$$
−0.684929 + 0.728610i $$0.740167\pi$$
$$522$$ 0 0
$$523$$ −18.7657 −0.820567 −0.410284 0.911958i $$-0.634570\pi$$
−0.410284 + 0.911958i $$0.634570\pi$$
$$524$$ 12.5995 0.550413
$$525$$ 0 0
$$526$$ 4.20676 0.183423
$$527$$ 35.5172 1.54716
$$528$$ 0 0
$$529$$ −22.8733 −0.994492
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 7.46192 0.323515
$$533$$ −13.4660 −0.583276
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −1.63505 −0.0706234
$$537$$ 0 0
$$538$$ −23.2054 −1.00045
$$539$$ 5.96908 0.257106
$$540$$ 0 0
$$541$$ −32.7471 −1.40791 −0.703955 0.710245i $$-0.748584\pi$$
−0.703955 + 0.710245i $$0.748584\pi$$
$$542$$ −43.1627 −1.85400
$$543$$ 0 0
$$544$$ −38.1573 −1.63598
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −41.0014 −1.75309 −0.876547 0.481316i $$-0.840159\pi$$
−0.876547 + 0.481316i $$0.840159\pi$$
$$548$$ 8.61767 0.368129
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −2.25433 −0.0960375
$$552$$ 0 0
$$553$$ −5.26832 −0.224032
$$554$$ −0.150572 −0.00639718
$$555$$ 0 0
$$556$$ 22.8491 0.969018
$$557$$ 32.8805 1.39319 0.696596 0.717463i $$-0.254697\pi$$
0.696596 + 0.717463i $$0.254697\pi$$
$$558$$ 0 0
$$559$$ −32.2761 −1.36513
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −14.5527 −0.613870
$$563$$ 19.8934 0.838407 0.419204 0.907892i $$-0.362309\pi$$
0.419204 + 0.907892i $$0.362309\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −18.8525 −0.792432
$$567$$ 0 0
$$568$$ −7.77178 −0.326097
$$569$$ −1.72158 −0.0721724 −0.0360862 0.999349i $$-0.511489\pi$$
−0.0360862 + 0.999349i $$0.511489\pi$$
$$570$$ 0 0
$$571$$ 26.4954 1.10880 0.554399 0.832251i $$-0.312948\pi$$
0.554399 + 0.832251i $$0.312948\pi$$
$$572$$ 15.5962 0.652108
$$573$$ 0 0
$$574$$ −27.2143 −1.13590
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 40.8791 1.70182 0.850909 0.525314i $$-0.176052\pi$$
0.850909 + 0.525314i $$0.176052\pi$$
$$578$$ 45.0623 1.87435
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 39.1340 1.62355
$$582$$ 0 0
$$583$$ −18.3549 −0.760181
$$584$$ −10.1129 −0.418474
$$585$$ 0 0
$$586$$ 46.6984 1.92909
$$587$$ 40.0943 1.65487 0.827434 0.561563i $$-0.189800\pi$$
0.827434 + 0.561563i $$0.189800\pi$$
$$588$$ 0 0
$$589$$ 12.3047 0.507008
$$590$$ 0 0
$$591$$ 0 0
$$592$$ −9.96148 −0.409414
$$593$$ 7.94061 0.326082 0.163041 0.986619i $$-0.447870\pi$$
0.163041 + 0.986619i $$0.447870\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 16.7889 0.687700
$$597$$ 0 0
$$598$$ −1.58639 −0.0648724
$$599$$ −37.2143 −1.52054 −0.760268 0.649609i $$-0.774932\pi$$
−0.760268 + 0.649609i $$0.774932\pi$$
$$600$$ 0 0
$$601$$ −9.87927 −0.402984 −0.201492 0.979490i $$-0.564579\pi$$
−0.201492 + 0.979490i $$0.564579\pi$$
$$602$$ −65.2289 −2.65853
$$603$$ 0 0
$$604$$ 7.77621 0.316409
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −3.95850 −0.160670 −0.0803352 0.996768i $$-0.525599\pi$$
−0.0803352 + 0.996768i $$0.525599\pi$$
$$608$$ −13.2194 −0.536116
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −10.0690 −0.407346
$$612$$ 0 0
$$613$$ 19.8179 0.800436 0.400218 0.916420i $$-0.368934\pi$$
0.400218 + 0.916420i $$0.368934\pi$$
$$614$$ −37.1506 −1.49927
$$615$$ 0 0
$$616$$ −22.7317 −0.915888
$$617$$ −8.70648 −0.350510 −0.175255 0.984523i $$-0.556075\pi$$
−0.175255 + 0.984523i $$0.556075\pi$$
$$618$$ 0 0
$$619$$ −18.3012 −0.735589 −0.367794 0.929907i $$-0.619887\pi$$
−0.367794 + 0.929907i $$0.619887\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 9.33946 0.374478
$$623$$ −7.31390 −0.293025
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 56.9283 2.27531
$$627$$ 0 0
$$628$$ 20.4263 0.815098
$$629$$ 13.0323 0.519633
$$630$$ 0 0
$$631$$ −48.9315 −1.94793 −0.973966 0.226693i $$-0.927209\pi$$
−0.973966 + 0.226693i $$0.927209\pi$$
$$632$$ 2.75595 0.109626
$$633$$ 0 0
$$634$$ 39.4252 1.56578
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 2.79382 0.110695
$$638$$ −9.52233 −0.376993
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −5.22083 −0.206210 −0.103105 0.994670i $$-0.532878\pi$$
−0.103105 + 0.994670i $$0.532878\pi$$
$$642$$ 0 0
$$643$$ 25.8165 1.01810 0.509051 0.860736i $$-0.329996\pi$$
0.509051 + 0.860736i $$0.329996\pi$$
$$644$$ −1.17817 −0.0464266
$$645$$ 0 0
$$646$$ 26.0844 1.02628
$$647$$ 1.68358 0.0661885 0.0330943 0.999452i $$-0.489464\pi$$
0.0330943 + 0.999452i $$0.489464\pi$$
$$648$$ 0 0
$$649$$ −25.5231 −1.00187
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −22.8418 −0.894553
$$653$$ 26.5807 1.04018 0.520091 0.854111i $$-0.325898\pi$$
0.520091 + 0.854111i $$0.325898\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 26.7219 1.04332
$$657$$ 0 0
$$658$$ −20.3490 −0.793288
$$659$$ −5.88644 −0.229303 −0.114652 0.993406i $$-0.536575\pi$$
−0.114652 + 0.993406i $$0.536575\pi$$
$$660$$ 0 0
$$661$$ 21.3474 0.830319 0.415160 0.909749i $$-0.363726\pi$$
0.415160 + 0.909749i $$0.363726\pi$$
$$662$$ −56.6625 −2.20225
$$663$$ 0 0
$$664$$ −20.4717 −0.794454
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0.355939 0.0137820
$$668$$ −14.2738 −0.552271
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −22.9996 −0.887891
$$672$$ 0 0
$$673$$ 20.3306 0.783688 0.391844 0.920032i $$-0.371837\pi$$
0.391844 + 0.920032i $$0.371837\pi$$
$$674$$ 8.32382 0.320622
$$675$$ 0 0
$$676$$ −7.80597 −0.300230
$$677$$ 27.0573 1.03990 0.519949 0.854197i $$-0.325951\pi$$
0.519949 + 0.854197i $$0.325951\pi$$
$$678$$ 0 0
$$679$$ −7.80110 −0.299379
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 51.9755 1.99025
$$683$$ 14.8635 0.568737 0.284369 0.958715i $$-0.408216\pi$$
0.284369 + 0.958715i $$0.408216\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −29.8117 −1.13821
$$687$$ 0 0
$$688$$ 64.0487 2.44183
$$689$$ −8.59099 −0.327291
$$690$$ 0 0
$$691$$ −3.66787 −0.139532 −0.0697662 0.997563i $$-0.522225\pi$$
−0.0697662 + 0.997563i $$0.522225\pi$$
$$692$$ −15.0407 −0.571763
$$693$$ 0 0
$$694$$ −17.8092 −0.676028
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −34.9596 −1.32419
$$698$$ 26.6212 1.00763
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 39.0313 1.47419 0.737097 0.675787i $$-0.236196\pi$$
0.737097 + 0.675787i $$0.236196\pi$$
$$702$$ 0 0
$$703$$ 4.51498 0.170286
$$704$$ −2.56941 −0.0968382
$$705$$ 0 0
$$706$$ −21.4184 −0.806092
$$707$$ −16.3507 −0.614931
$$708$$ 0 0
$$709$$ −36.9135 −1.38632 −0.693158 0.720785i $$-0.743781\pi$$
−0.693158 + 0.720785i $$0.743781\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 3.82602 0.143386
$$713$$ −1.94281 −0.0727589
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −6.99382 −0.261371
$$717$$ 0 0
$$718$$ 8.40401 0.313635
$$719$$ −41.9646 −1.56502 −0.782508 0.622640i $$-0.786060\pi$$
−0.782508 + 0.622640i $$0.786060\pi$$
$$720$$ 0 0
$$721$$ −44.9785 −1.67509
$$722$$ −24.7489 −0.921060
$$723$$ 0 0
$$724$$ −23.0926 −0.858228
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 37.0774 1.37513 0.687563 0.726125i $$-0.258680\pi$$
0.687563 + 0.726125i $$0.258680\pi$$
$$728$$ −10.6396 −0.394329
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −83.7932 −3.09920
$$732$$ 0 0
$$733$$ −6.29483 −0.232505 −0.116252 0.993220i $$-0.537088\pi$$
−0.116252 + 0.993220i $$0.537088\pi$$
$$734$$ 21.2668 0.784973
$$735$$ 0 0
$$736$$ 2.08723 0.0769361
$$737$$ 5.87572 0.216435
$$738$$ 0 0
$$739$$ −34.6989 −1.27642 −0.638210 0.769862i $$-0.720325\pi$$
−0.638210 + 0.769862i $$0.720325\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −17.3621 −0.637383
$$743$$ 10.8648 0.398590 0.199295 0.979940i $$-0.436135\pi$$
0.199295 + 0.979940i $$0.436135\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 14.7896 0.541485
$$747$$ 0 0
$$748$$ 40.4898 1.48046
$$749$$ 47.1852 1.72411
$$750$$ 0 0
$$751$$ 6.02497 0.219854 0.109927 0.993940i $$-0.464938\pi$$
0.109927 + 0.993940i $$0.464938\pi$$
$$752$$ 19.9809 0.728627
$$753$$ 0 0
$$754$$ −4.45692 −0.162311
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −34.1212 −1.24016 −0.620078 0.784540i $$-0.712899\pi$$
−0.620078 + 0.784540i $$0.712899\pi$$
$$758$$ −43.6929 −1.58700
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −14.0126 −0.507956 −0.253978 0.967210i $$-0.581739\pi$$
−0.253978 + 0.967210i $$0.581739\pi$$
$$762$$ 0 0
$$763$$ 26.5650 0.961718
$$764$$ 3.53913 0.128041
$$765$$ 0 0
$$766$$ 60.5050 2.18613
$$767$$ −11.9461 −0.431347
$$768$$ 0 0
$$769$$ 34.0766 1.22883 0.614417 0.788982i $$-0.289391\pi$$
0.614417 + 0.788982i $$0.289391\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −24.7976 −0.892486
$$773$$ −14.3678 −0.516773 −0.258386 0.966042i $$-0.583191\pi$$
−0.258386 + 0.966042i $$0.583191\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 4.08088 0.146495
$$777$$ 0 0
$$778$$ 45.1112 1.61731
$$779$$ −12.1115 −0.433941
$$780$$ 0 0
$$781$$ 27.9287 0.999367
$$782$$ −4.11850 −0.147277
$$783$$ 0 0
$$784$$ −5.54407 −0.198002
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 1.65976 0.0591642 0.0295821 0.999562i $$-0.490582\pi$$
0.0295821 + 0.999562i $$0.490582\pi$$
$$788$$ −11.7046 −0.416959
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 40.4446 1.43804
$$792$$ 0 0
$$793$$ −10.7650 −0.382275
$$794$$ −29.7354 −1.05527
$$795$$ 0 0
$$796$$ −4.91334 −0.174149
$$797$$ −8.57389 −0.303703 −0.151851 0.988403i $$-0.548523\pi$$
−0.151851 + 0.988403i $$0.548523\pi$$
$$798$$ 0 0
$$799$$ −26.1404 −0.924782
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 14.2507 0.503211
$$803$$ 36.3417 1.28247
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 24.3271 0.856886
$$807$$ 0 0
$$808$$ 8.55332 0.300905
$$809$$ −44.3986 −1.56097 −0.780485 0.625175i $$-0.785028\pi$$
−0.780485 + 0.625175i $$0.785028\pi$$
$$810$$ 0 0
$$811$$ −13.2677 −0.465892 −0.232946 0.972490i $$-0.574837\pi$$
−0.232946 + 0.972490i $$0.574837\pi$$
$$812$$ −3.31005 −0.116160
$$813$$ 0 0
$$814$$ 19.0714 0.668451
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −29.0297 −1.01562
$$818$$ −34.5565 −1.20824
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −18.5405 −0.647069 −0.323534 0.946216i $$-0.604871\pi$$
−0.323534 + 0.946216i $$0.604871\pi$$
$$822$$ 0 0
$$823$$ −4.42801 −0.154351 −0.0771753 0.997018i $$-0.524590\pi$$
−0.0771753 + 0.997018i $$0.524590\pi$$
$$824$$ 23.5290 0.819672
$$825$$ 0 0
$$826$$ −24.1426 −0.840029
$$827$$ 26.1067 0.907819 0.453910 0.891048i $$-0.350029\pi$$
0.453910 + 0.891048i $$0.350029\pi$$
$$828$$ 0 0
$$829$$ −25.1458 −0.873350 −0.436675 0.899619i $$-0.643844\pi$$
−0.436675 + 0.899619i $$0.643844\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ −1.20261 −0.0416930
$$833$$ 7.25316 0.251307
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 14.0275 0.485150
$$837$$ 0 0
$$838$$ −12.3117 −0.425300
$$839$$ −31.3784 −1.08330 −0.541651 0.840603i $$-0.682201\pi$$
−0.541651 + 0.840603i $$0.682201\pi$$
$$840$$ 0 0
$$841$$ 1.00000 0.0344828
$$842$$ −23.0937 −0.795863
$$843$$ 0 0
$$844$$ 32.7971 1.12892
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 50.3540 1.73018
$$848$$ 17.0480 0.585430
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −0.712877 −0.0244371
$$852$$ 0 0
$$853$$ −36.1547 −1.23791 −0.618956 0.785425i $$-0.712444\pi$$
−0.618956 + 0.785425i $$0.712444\pi$$
$$854$$ −21.7556 −0.744463
$$855$$ 0 0
$$856$$ −24.6834 −0.843660
$$857$$ 20.2270 0.690943 0.345471 0.938429i $$-0.387719\pi$$
0.345471 + 0.938429i $$0.387719\pi$$
$$858$$ 0 0
$$859$$ 29.8005 1.01678 0.508390 0.861127i $$-0.330241\pi$$
0.508390 + 0.861127i $$0.330241\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −70.5502 −2.40295
$$863$$ −35.7369 −1.21650 −0.608248 0.793747i $$-0.708128\pi$$
−0.608248 + 0.793747i $$0.708128\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −8.76805 −0.297951
$$867$$ 0 0
$$868$$ 18.0671 0.613239
$$869$$ −9.90378 −0.335963
$$870$$ 0 0
$$871$$ 2.75013 0.0931845
$$872$$ −13.8966 −0.470598
$$873$$ 0 0
$$874$$ −1.42683 −0.0482632
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −0.626314 −0.0211491 −0.0105746 0.999944i $$-0.503366\pi$$
−0.0105746 + 0.999944i $$0.503366\pi$$
$$878$$ 39.3796 1.32900
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 6.65299 0.224145 0.112072 0.993700i $$-0.464251\pi$$
0.112072 + 0.993700i $$0.464251\pi$$
$$882$$ 0 0
$$883$$ 1.03215 0.0347345 0.0173673 0.999849i $$-0.494472\pi$$
0.0173673 + 0.999849i $$0.494472\pi$$
$$884$$ 18.9512 0.637399
$$885$$ 0 0
$$886$$ 2.15440 0.0723783
$$887$$ 55.8755 1.87611 0.938057 0.346480i $$-0.112623\pi$$
0.938057 + 0.346480i $$0.112623\pi$$
$$888$$ 0 0
$$889$$ 15.2748 0.512301
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 9.85884 0.330098
$$893$$ −9.05620 −0.303054
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 30.9782 1.03491
$$897$$ 0 0
$$898$$ 9.29588 0.310208
$$899$$ −5.45828 −0.182044
$$900$$ 0 0
$$901$$ −22.3034 −0.743035
$$902$$ −51.1595 −1.70342
$$903$$ 0 0
$$904$$ −21.1572 −0.703679
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −23.0687 −0.765982 −0.382991 0.923752i $$-0.625106\pi$$
−0.382991 + 0.923752i $$0.625106\pi$$
$$908$$ −17.1824 −0.570219
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 52.8451 1.75084 0.875418 0.483367i $$-0.160586\pi$$
0.875418 + 0.483367i $$0.160586\pi$$
$$912$$ 0 0
$$913$$ 73.5670 2.43471
$$914$$ 30.0629 0.994393
$$915$$ 0 0
$$916$$ −24.5170 −0.810064
$$917$$ 30.8881 1.02001
$$918$$ 0 0
$$919$$ −49.6417 −1.63753 −0.818765 0.574129i $$-0.805341\pi$$
−0.818765 + 0.574129i $$0.805341\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −13.8634 −0.456567
$$923$$ 13.0720 0.430270
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −39.6951 −1.30446
$$927$$ 0 0
$$928$$ 5.86400 0.192495
$$929$$ 10.7335 0.352153 0.176077 0.984376i $$-0.443659\pi$$
0.176077 + 0.984376i $$0.443659\pi$$
$$930$$ 0 0
$$931$$ 2.51281 0.0823542
$$932$$ 30.3001 0.992512
$$933$$ 0 0
$$934$$ 14.5666 0.476633
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 11.4028 0.372513 0.186256 0.982501i $$-0.440365\pi$$
0.186256 + 0.982501i $$0.440365\pi$$
$$938$$ 5.55792 0.181472
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 26.5670 0.866060 0.433030 0.901380i $$-0.357444\pi$$
0.433030 + 0.901380i $$0.357444\pi$$
$$942$$ 0 0
$$943$$ 1.91231 0.0622734
$$944$$ 23.7058 0.771558
$$945$$ 0 0
$$946$$ −122.622 −3.98679
$$947$$ 15.3658 0.499322 0.249661 0.968333i $$-0.419681\pi$$
0.249661 + 0.968333i $$0.419681\pi$$
$$948$$ 0 0
$$949$$ 17.0097 0.552158
$$950$$ 0 0
$$951$$ 0 0
$$952$$ −27.6218 −0.895229
$$953$$ 12.7062 0.411596 0.205798 0.978595i $$-0.434021\pi$$
0.205798 + 0.978595i $$0.434021\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −9.49419 −0.307064
$$957$$ 0 0
$$958$$ 31.0881 1.00441
$$959$$ 21.1264 0.682209
$$960$$ 0 0
$$961$$ −1.20722 −0.0389426
$$962$$ 8.92634 0.287797
$$963$$ 0 0
$$964$$ −12.2869 −0.395733
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −15.8218 −0.508795 −0.254398 0.967100i $$-0.581877\pi$$
−0.254398 + 0.967100i $$0.581877\pi$$
$$968$$ −26.3410 −0.846632
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 47.8287 1.53490 0.767448 0.641111i $$-0.221526\pi$$
0.767448 + 0.641111i $$0.221526\pi$$
$$972$$ 0 0
$$973$$ 56.0152 1.79577
$$974$$ −41.3644 −1.32540
$$975$$ 0 0
$$976$$ 21.3620 0.683781
$$977$$ 46.2147 1.47854 0.739270 0.673409i $$-0.235171\pi$$
0.739270 + 0.673409i $$0.235171\pi$$
$$978$$ 0 0
$$979$$ −13.7492 −0.439426
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 61.3372 1.95735
$$983$$ 40.2132 1.28260 0.641301 0.767289i $$-0.278395\pi$$
0.641301 + 0.767289i $$0.278395\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −11.5708 −0.368489
$$987$$ 0 0
$$988$$ 6.56555 0.208878
$$989$$ 4.58354 0.145748
$$990$$ 0 0
$$991$$ −8.54051 −0.271298 −0.135649 0.990757i $$-0.543312\pi$$
−0.135649 + 0.990757i $$0.543312\pi$$
$$992$$ −32.0073 −1.01623
$$993$$ 0 0
$$994$$ 26.4181 0.837931
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −26.7017 −0.845651 −0.422825 0.906211i $$-0.638962\pi$$
−0.422825 + 0.906211i $$0.638962\pi$$
$$998$$ 22.8946 0.724715
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 6525.2.a.cc.1.7 yes 9
3.2 odd 2 6525.2.a.ca.1.3 9
5.4 even 2 6525.2.a.cb.1.3 yes 9
15.14 odd 2 6525.2.a.cd.1.7 yes 9

By twisted newform
Twist Min Dim Char Parity Ord Type
6525.2.a.ca.1.3 9 3.2 odd 2
6525.2.a.cb.1.3 yes 9 5.4 even 2
6525.2.a.cc.1.7 yes 9 1.1 even 1 trivial
6525.2.a.cd.1.7 yes 9 15.14 odd 2