# Properties

 Label 5225.2.a.ba.1.9 Level $5225$ Weight $2$ Character 5225.1 Self dual yes Analytic conductor $41.722$ Analytic rank $1$ Dimension $20$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5225,2,Mod(1,5225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5225, 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("5225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$5225 = 5^{2} \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5225.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: $$41.7218350561$$ Analytic rank: $$1$$ Dimension: $$20$$ Coefficient field: $$\mathbb{Q}[x]/(x^{20} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{20} - 26 x^{18} + 281 x^{16} - 1640 x^{14} + 5623 x^{12} - 11551 x^{10} + 13894 x^{8} - 9095 x^{6} + \cdots + 4$$ x^20 - 26*x^18 + 281*x^16 - 1640*x^14 + 5623*x^12 - 11551*x^10 + 13894*x^8 - 9095*x^6 + 2753*x^4 - 276*x^2 + 4 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{8}$$ Twist minimal: no (minimal twist has level 1045) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.9 Root $$-0.386431$$ of defining polynomial Character $$\chi$$ $$=$$ 5225.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-0.386431 q^{2} -0.261531 q^{3} -1.85067 q^{4} +0.101064 q^{6} -4.13791 q^{7} +1.48802 q^{8} -2.93160 q^{9} +O(q^{10})$$ $$q-0.386431 q^{2} -0.261531 q^{3} -1.85067 q^{4} +0.101064 q^{6} -4.13791 q^{7} +1.48802 q^{8} -2.93160 q^{9} +1.00000 q^{11} +0.484009 q^{12} -0.244900 q^{13} +1.59902 q^{14} +3.12633 q^{16} +4.03954 q^{17} +1.13286 q^{18} -1.00000 q^{19} +1.08219 q^{21} -0.386431 q^{22} -0.483419 q^{23} -0.389163 q^{24} +0.0946368 q^{26} +1.55130 q^{27} +7.65791 q^{28} +5.24273 q^{29} -9.76294 q^{31} -4.18414 q^{32} -0.261531 q^{33} -1.56100 q^{34} +5.42543 q^{36} +1.52969 q^{37} +0.386431 q^{38} +0.0640490 q^{39} +6.06547 q^{41} -0.418193 q^{42} +7.60200 q^{43} -1.85067 q^{44} +0.186808 q^{46} +9.01914 q^{47} -0.817633 q^{48} +10.1223 q^{49} -1.05647 q^{51} +0.453229 q^{52} -5.86595 q^{53} -0.599470 q^{54} -6.15728 q^{56} +0.261531 q^{57} -2.02595 q^{58} -12.7234 q^{59} +7.58454 q^{61} +3.77270 q^{62} +12.1307 q^{63} -4.63577 q^{64} +0.101064 q^{66} -2.72263 q^{67} -7.47585 q^{68} +0.126429 q^{69} +2.89186 q^{71} -4.36227 q^{72} +7.24330 q^{73} -0.591117 q^{74} +1.85067 q^{76} -4.13791 q^{77} -0.0247505 q^{78} +5.91489 q^{79} +8.38909 q^{81} -2.34388 q^{82} +4.94874 q^{83} -2.00278 q^{84} -2.93765 q^{86} -1.37114 q^{87} +1.48802 q^{88} -3.41496 q^{89} +1.01337 q^{91} +0.894649 q^{92} +2.55331 q^{93} -3.48527 q^{94} +1.09428 q^{96} +12.5567 q^{97} -3.91157 q^{98} -2.93160 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20 q + 12 q^{4} - 8 q^{6} + 10 q^{9}+O(q^{10})$$ 20 * q + 12 * q^4 - 8 * q^6 + 10 * q^9 $$20 q + 12 q^{4} - 8 q^{6} + 10 q^{9} + 20 q^{11} - 24 q^{14} - 4 q^{16} - 20 q^{19} - 30 q^{21} - 38 q^{24} + 8 q^{26} - 50 q^{29} - 50 q^{31} - 28 q^{34} - 12 q^{36} - 48 q^{39} - 34 q^{41} + 12 q^{44} - 36 q^{46} + 6 q^{49} - 40 q^{51} + 6 q^{54} - 40 q^{56} - 30 q^{59} - 14 q^{61} - 36 q^{64} - 8 q^{66} + 12 q^{69} - 40 q^{71} - 50 q^{74} - 12 q^{76} - 106 q^{79} + 30 q^{84} + 56 q^{86} - 36 q^{89} - 56 q^{91} - 28 q^{94} + 66 q^{96} + 10 q^{99}+O(q^{100})$$ 20 * q + 12 * q^4 - 8 * q^6 + 10 * q^9 + 20 * q^11 - 24 * q^14 - 4 * q^16 - 20 * q^19 - 30 * q^21 - 38 * q^24 + 8 * q^26 - 50 * q^29 - 50 * q^31 - 28 * q^34 - 12 * q^36 - 48 * q^39 - 34 * q^41 + 12 * q^44 - 36 * q^46 + 6 * q^49 - 40 * q^51 + 6 * q^54 - 40 * q^56 - 30 * q^59 - 14 * q^61 - 36 * q^64 - 8 * q^66 + 12 * q^69 - 40 * q^71 - 50 * q^74 - 12 * q^76 - 106 * q^79 + 30 * q^84 + 56 * q^86 - 36 * q^89 - 56 * q^91 - 28 * q^94 + 66 * q^96 + 10 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −0.386431 −0.273248 −0.136624 0.990623i $$-0.543625\pi$$
−0.136624 + 0.990623i $$0.543625\pi$$
$$3$$ −0.261531 −0.150995 −0.0754976 0.997146i $$-0.524055\pi$$
−0.0754976 + 0.997146i $$0.524055\pi$$
$$4$$ −1.85067 −0.925336
$$5$$ 0 0
$$6$$ 0.101064 0.0412591
$$7$$ −4.13791 −1.56398 −0.781992 0.623289i $$-0.785796\pi$$
−0.781992 + 0.623289i $$0.785796\pi$$
$$8$$ 1.48802 0.526094
$$9$$ −2.93160 −0.977200
$$10$$ 0 0
$$11$$ 1.00000 0.301511
$$12$$ 0.484009 0.139721
$$13$$ −0.244900 −0.0679230 −0.0339615 0.999423i $$-0.510812\pi$$
−0.0339615 + 0.999423i $$0.510812\pi$$
$$14$$ 1.59902 0.427355
$$15$$ 0 0
$$16$$ 3.12633 0.781582
$$17$$ 4.03954 0.979732 0.489866 0.871798i $$-0.337046\pi$$
0.489866 + 0.871798i $$0.337046\pi$$
$$18$$ 1.13286 0.267018
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ 1.08219 0.236154
$$22$$ −0.386431 −0.0823873
$$23$$ −0.483419 −0.100800 −0.0503999 0.998729i $$-0.516050\pi$$
−0.0503999 + 0.998729i $$0.516050\pi$$
$$24$$ −0.389163 −0.0794376
$$25$$ 0 0
$$26$$ 0.0946368 0.0185598
$$27$$ 1.55130 0.298548
$$28$$ 7.65791 1.44721
$$29$$ 5.24273 0.973550 0.486775 0.873527i $$-0.338173\pi$$
0.486775 + 0.873527i $$0.338173\pi$$
$$30$$ 0 0
$$31$$ −9.76294 −1.75348 −0.876738 0.480969i $$-0.840285\pi$$
−0.876738 + 0.480969i $$0.840285\pi$$
$$32$$ −4.18414 −0.739659
$$33$$ −0.261531 −0.0455268
$$34$$ −1.56100 −0.267709
$$35$$ 0 0
$$36$$ 5.42543 0.904238
$$37$$ 1.52969 0.251479 0.125739 0.992063i $$-0.459870\pi$$
0.125739 + 0.992063i $$0.459870\pi$$
$$38$$ 0.386431 0.0626873
$$39$$ 0.0640490 0.0102560
$$40$$ 0 0
$$41$$ 6.06547 0.947268 0.473634 0.880722i $$-0.342942\pi$$
0.473634 + 0.880722i $$0.342942\pi$$
$$42$$ −0.418193 −0.0645285
$$43$$ 7.60200 1.15929 0.579647 0.814868i $$-0.303190\pi$$
0.579647 + 0.814868i $$0.303190\pi$$
$$44$$ −1.85067 −0.278999
$$45$$ 0 0
$$46$$ 0.186808 0.0275433
$$47$$ 9.01914 1.31558 0.657788 0.753203i $$-0.271492\pi$$
0.657788 + 0.753203i $$0.271492\pi$$
$$48$$ −0.817633 −0.118015
$$49$$ 10.1223 1.44604
$$50$$ 0 0
$$51$$ −1.05647 −0.147935
$$52$$ 0.453229 0.0628516
$$53$$ −5.86595 −0.805751 −0.402875 0.915255i $$-0.631989\pi$$
−0.402875 + 0.915255i $$0.631989\pi$$
$$54$$ −0.599470 −0.0815775
$$55$$ 0 0
$$56$$ −6.15728 −0.822802
$$57$$ 0.261531 0.0346407
$$58$$ −2.02595 −0.266020
$$59$$ −12.7234 −1.65644 −0.828220 0.560403i $$-0.810646\pi$$
−0.828220 + 0.560403i $$0.810646\pi$$
$$60$$ 0 0
$$61$$ 7.58454 0.971101 0.485551 0.874209i $$-0.338619\pi$$
0.485551 + 0.874209i $$0.338619\pi$$
$$62$$ 3.77270 0.479133
$$63$$ 12.1307 1.52833
$$64$$ −4.63577 −0.579472
$$65$$ 0 0
$$66$$ 0.101064 0.0124401
$$67$$ −2.72263 −0.332623 −0.166311 0.986073i $$-0.553186\pi$$
−0.166311 + 0.986073i $$0.553186\pi$$
$$68$$ −7.47585 −0.906581
$$69$$ 0.126429 0.0152203
$$70$$ 0 0
$$71$$ 2.89186 0.343201 0.171600 0.985167i $$-0.445106\pi$$
0.171600 + 0.985167i $$0.445106\pi$$
$$72$$ −4.36227 −0.514099
$$73$$ 7.24330 0.847764 0.423882 0.905717i $$-0.360667\pi$$
0.423882 + 0.905717i $$0.360667\pi$$
$$74$$ −0.591117 −0.0687160
$$75$$ 0 0
$$76$$ 1.85067 0.212287
$$77$$ −4.13791 −0.471559
$$78$$ −0.0247505 −0.00280244
$$79$$ 5.91489 0.665477 0.332739 0.943019i $$-0.392027\pi$$
0.332739 + 0.943019i $$0.392027\pi$$
$$80$$ 0 0
$$81$$ 8.38909 0.932121
$$82$$ −2.34388 −0.258839
$$83$$ 4.94874 0.543195 0.271598 0.962411i $$-0.412448\pi$$
0.271598 + 0.962411i $$0.412448\pi$$
$$84$$ −2.00278 −0.218522
$$85$$ 0 0
$$86$$ −2.93765 −0.316774
$$87$$ −1.37114 −0.147001
$$88$$ 1.48802 0.158623
$$89$$ −3.41496 −0.361985 −0.180993 0.983484i $$-0.557931\pi$$
−0.180993 + 0.983484i $$0.557931\pi$$
$$90$$ 0 0
$$91$$ 1.01337 0.106230
$$92$$ 0.894649 0.0932736
$$93$$ 2.55331 0.264766
$$94$$ −3.48527 −0.359478
$$95$$ 0 0
$$96$$ 1.09428 0.111685
$$97$$ 12.5567 1.27494 0.637468 0.770477i $$-0.279982\pi$$
0.637468 + 0.770477i $$0.279982\pi$$
$$98$$ −3.91157 −0.395128
$$99$$ −2.93160 −0.294637
$$100$$ 0 0
$$101$$ −11.5009 −1.14438 −0.572192 0.820120i $$-0.693907\pi$$
−0.572192 + 0.820120i $$0.693907\pi$$
$$102$$ 0.408251 0.0404228
$$103$$ −13.0487 −1.28573 −0.642864 0.765980i $$-0.722254\pi$$
−0.642864 + 0.765980i $$0.722254\pi$$
$$104$$ −0.364415 −0.0357339
$$105$$ 0 0
$$106$$ 2.26678 0.220169
$$107$$ 15.0538 1.45531 0.727654 0.685945i $$-0.240611\pi$$
0.727654 + 0.685945i $$0.240611\pi$$
$$108$$ −2.87095 −0.276257
$$109$$ −3.97702 −0.380929 −0.190465 0.981694i $$-0.560999\pi$$
−0.190465 + 0.981694i $$0.560999\pi$$
$$110$$ 0 0
$$111$$ −0.400061 −0.0379721
$$112$$ −12.9365 −1.22238
$$113$$ −0.880484 −0.0828290 −0.0414145 0.999142i $$-0.513186\pi$$
−0.0414145 + 0.999142i $$0.513186\pi$$
$$114$$ −0.101064 −0.00946548
$$115$$ 0 0
$$116$$ −9.70257 −0.900861
$$117$$ 0.717949 0.0663744
$$118$$ 4.91670 0.452619
$$119$$ −16.7152 −1.53228
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ −2.93090 −0.265351
$$123$$ −1.58631 −0.143033
$$124$$ 18.0680 1.62255
$$125$$ 0 0
$$126$$ −4.68768 −0.417611
$$127$$ −14.6685 −1.30162 −0.650809 0.759241i $$-0.725570\pi$$
−0.650809 + 0.759241i $$0.725570\pi$$
$$128$$ 10.1597 0.897998
$$129$$ −1.98816 −0.175048
$$130$$ 0 0
$$131$$ 1.37384 0.120033 0.0600165 0.998197i $$-0.480885\pi$$
0.0600165 + 0.998197i $$0.480885\pi$$
$$132$$ 0.484009 0.0421275
$$133$$ 4.13791 0.358802
$$134$$ 1.05211 0.0908884
$$135$$ 0 0
$$136$$ 6.01090 0.515430
$$137$$ −3.57696 −0.305600 −0.152800 0.988257i $$-0.548829\pi$$
−0.152800 + 0.988257i $$0.548829\pi$$
$$138$$ −0.0488561 −0.00415891
$$139$$ 7.87917 0.668303 0.334151 0.942519i $$-0.391550\pi$$
0.334151 + 0.942519i $$0.391550\pi$$
$$140$$ 0 0
$$141$$ −2.35879 −0.198646
$$142$$ −1.11750 −0.0937788
$$143$$ −0.244900 −0.0204796
$$144$$ −9.16515 −0.763762
$$145$$ 0 0
$$146$$ −2.79903 −0.231650
$$147$$ −2.64730 −0.218346
$$148$$ −2.83094 −0.232702
$$149$$ 0.608700 0.0498666 0.0249333 0.999689i $$-0.492063\pi$$
0.0249333 + 0.999689i $$0.492063\pi$$
$$150$$ 0 0
$$151$$ −22.8471 −1.85927 −0.929636 0.368479i $$-0.879879\pi$$
−0.929636 + 0.368479i $$0.879879\pi$$
$$152$$ −1.48802 −0.120694
$$153$$ −11.8423 −0.957394
$$154$$ 1.59902 0.128852
$$155$$ 0 0
$$156$$ −0.118534 −0.00949029
$$157$$ 9.47273 0.756006 0.378003 0.925804i $$-0.376611\pi$$
0.378003 + 0.925804i $$0.376611\pi$$
$$158$$ −2.28569 −0.181840
$$159$$ 1.53413 0.121664
$$160$$ 0 0
$$161$$ 2.00034 0.157649
$$162$$ −3.24180 −0.254700
$$163$$ 23.5045 1.84101 0.920507 0.390727i $$-0.127776\pi$$
0.920507 + 0.390727i $$0.127776\pi$$
$$164$$ −11.2252 −0.876540
$$165$$ 0 0
$$166$$ −1.91235 −0.148427
$$167$$ −18.0231 −1.39467 −0.697334 0.716746i $$-0.745631\pi$$
−0.697334 + 0.716746i $$0.745631\pi$$
$$168$$ 1.61032 0.124239
$$169$$ −12.9400 −0.995386
$$170$$ 0 0
$$171$$ 2.93160 0.224185
$$172$$ −14.0688 −1.07274
$$173$$ −18.1676 −1.38126 −0.690628 0.723210i $$-0.742666\pi$$
−0.690628 + 0.723210i $$0.742666\pi$$
$$174$$ 0.529850 0.0401678
$$175$$ 0 0
$$176$$ 3.12633 0.235656
$$177$$ 3.32756 0.250115
$$178$$ 1.31965 0.0989117
$$179$$ −21.8821 −1.63555 −0.817774 0.575540i $$-0.804792\pi$$
−0.817774 + 0.575540i $$0.804792\pi$$
$$180$$ 0 0
$$181$$ −17.5731 −1.30620 −0.653100 0.757271i $$-0.726532\pi$$
−0.653100 + 0.757271i $$0.726532\pi$$
$$182$$ −0.391599 −0.0290272
$$183$$ −1.98360 −0.146632
$$184$$ −0.719335 −0.0530301
$$185$$ 0 0
$$186$$ −0.986679 −0.0723468
$$187$$ 4.03954 0.295400
$$188$$ −16.6915 −1.21735
$$189$$ −6.41914 −0.466924
$$190$$ 0 0
$$191$$ −20.5263 −1.48523 −0.742615 0.669719i $$-0.766415\pi$$
−0.742615 + 0.669719i $$0.766415\pi$$
$$192$$ 1.21240 0.0874974
$$193$$ 14.2453 1.02540 0.512699 0.858569i $$-0.328646\pi$$
0.512699 + 0.858569i $$0.328646\pi$$
$$194$$ −4.85228 −0.348373
$$195$$ 0 0
$$196$$ −18.7331 −1.33808
$$197$$ −3.26945 −0.232939 −0.116469 0.993194i $$-0.537158\pi$$
−0.116469 + 0.993194i $$0.537158\pi$$
$$198$$ 1.13286 0.0805089
$$199$$ −14.9387 −1.05897 −0.529487 0.848318i $$-0.677616\pi$$
−0.529487 + 0.848318i $$0.677616\pi$$
$$200$$ 0 0
$$201$$ 0.712054 0.0502244
$$202$$ 4.44431 0.312700
$$203$$ −21.6940 −1.52262
$$204$$ 1.95517 0.136889
$$205$$ 0 0
$$206$$ 5.04243 0.351322
$$207$$ 1.41719 0.0985016
$$208$$ −0.765637 −0.0530874
$$209$$ −1.00000 −0.0691714
$$210$$ 0 0
$$211$$ 7.07216 0.486867 0.243434 0.969918i $$-0.421726\pi$$
0.243434 + 0.969918i $$0.421726\pi$$
$$212$$ 10.8560 0.745590
$$213$$ −0.756312 −0.0518217
$$214$$ −5.81725 −0.397659
$$215$$ 0 0
$$216$$ 2.30836 0.157064
$$217$$ 40.3982 2.74241
$$218$$ 1.53684 0.104088
$$219$$ −1.89435 −0.128008
$$220$$ 0 0
$$221$$ −0.989282 −0.0665463
$$222$$ 0.154596 0.0103758
$$223$$ −6.07334 −0.406701 −0.203351 0.979106i $$-0.565183\pi$$
−0.203351 + 0.979106i $$0.565183\pi$$
$$224$$ 17.3136 1.15681
$$225$$ 0 0
$$226$$ 0.340246 0.0226328
$$227$$ −1.60412 −0.106469 −0.0532345 0.998582i $$-0.516953\pi$$
−0.0532345 + 0.998582i $$0.516953\pi$$
$$228$$ −0.484009 −0.0320542
$$229$$ 5.47550 0.361831 0.180916 0.983499i $$-0.442094\pi$$
0.180916 + 0.983499i $$0.442094\pi$$
$$230$$ 0 0
$$231$$ 1.08219 0.0712031
$$232$$ 7.80127 0.512179
$$233$$ −15.8903 −1.04101 −0.520505 0.853859i $$-0.674256\pi$$
−0.520505 + 0.853859i $$0.674256\pi$$
$$234$$ −0.277437 −0.0181366
$$235$$ 0 0
$$236$$ 23.5468 1.53276
$$237$$ −1.54693 −0.100484
$$238$$ 6.45928 0.418693
$$239$$ −29.9052 −1.93440 −0.967202 0.254008i $$-0.918251\pi$$
−0.967202 + 0.254008i $$0.918251\pi$$
$$240$$ 0 0
$$241$$ −7.02083 −0.452251 −0.226126 0.974098i $$-0.572606\pi$$
−0.226126 + 0.974098i $$0.572606\pi$$
$$242$$ −0.386431 −0.0248407
$$243$$ −6.84791 −0.439294
$$244$$ −14.0365 −0.898595
$$245$$ 0 0
$$246$$ 0.612999 0.0390834
$$247$$ 0.244900 0.0155826
$$248$$ −14.5274 −0.922492
$$249$$ −1.29425 −0.0820198
$$250$$ 0 0
$$251$$ 11.8694 0.749191 0.374596 0.927188i $$-0.377782\pi$$
0.374596 + 0.927188i $$0.377782\pi$$
$$252$$ −22.4500 −1.41421
$$253$$ −0.483419 −0.0303923
$$254$$ 5.66835 0.355664
$$255$$ 0 0
$$256$$ 5.34553 0.334096
$$257$$ 24.0606 1.50086 0.750430 0.660950i $$-0.229846\pi$$
0.750430 + 0.660950i $$0.229846\pi$$
$$258$$ 0.768286 0.0478314
$$259$$ −6.32970 −0.393309
$$260$$ 0 0
$$261$$ −15.3696 −0.951354
$$262$$ −0.530894 −0.0327987
$$263$$ 8.07318 0.497814 0.248907 0.968527i $$-0.419929\pi$$
0.248907 + 0.968527i $$0.419929\pi$$
$$264$$ −0.389163 −0.0239513
$$265$$ 0 0
$$266$$ −1.59902 −0.0980419
$$267$$ 0.893120 0.0546581
$$268$$ 5.03870 0.307788
$$269$$ 9.12119 0.556129 0.278064 0.960562i $$-0.410307\pi$$
0.278064 + 0.960562i $$0.410307\pi$$
$$270$$ 0 0
$$271$$ 17.0227 1.03406 0.517028 0.855969i $$-0.327038\pi$$
0.517028 + 0.855969i $$0.327038\pi$$
$$272$$ 12.6289 0.765740
$$273$$ −0.265029 −0.0160403
$$274$$ 1.38225 0.0835045
$$275$$ 0 0
$$276$$ −0.233979 −0.0140839
$$277$$ 26.3075 1.58066 0.790331 0.612680i $$-0.209908\pi$$
0.790331 + 0.612680i $$0.209908\pi$$
$$278$$ −3.04475 −0.182612
$$279$$ 28.6210 1.71350
$$280$$ 0 0
$$281$$ −0.467398 −0.0278826 −0.0139413 0.999903i $$-0.504438\pi$$
−0.0139413 + 0.999903i $$0.504438\pi$$
$$282$$ 0.911508 0.0542795
$$283$$ −20.3573 −1.21011 −0.605057 0.796182i $$-0.706850\pi$$
−0.605057 + 0.796182i $$0.706850\pi$$
$$284$$ −5.35188 −0.317576
$$285$$ 0 0
$$286$$ 0.0946368 0.00559599
$$287$$ −25.0984 −1.48151
$$288$$ 12.2662 0.722795
$$289$$ −0.682144 −0.0401261
$$290$$ 0 0
$$291$$ −3.28396 −0.192509
$$292$$ −13.4050 −0.784467
$$293$$ −29.4685 −1.72157 −0.860784 0.508970i $$-0.830027\pi$$
−0.860784 + 0.508970i $$0.830027\pi$$
$$294$$ 1.02300 0.0596625
$$295$$ 0 0
$$296$$ 2.27620 0.132301
$$297$$ 1.55130 0.0900155
$$298$$ −0.235220 −0.0136259
$$299$$ 0.118389 0.00684662
$$300$$ 0 0
$$301$$ −31.4564 −1.81312
$$302$$ 8.82883 0.508042
$$303$$ 3.00785 0.172797
$$304$$ −3.12633 −0.179307
$$305$$ 0 0
$$306$$ 4.57623 0.261606
$$307$$ 13.9125 0.794028 0.397014 0.917813i $$-0.370046\pi$$
0.397014 + 0.917813i $$0.370046\pi$$
$$308$$ 7.65791 0.436350
$$309$$ 3.41265 0.194139
$$310$$ 0 0
$$311$$ 8.17133 0.463354 0.231677 0.972793i $$-0.425579\pi$$
0.231677 + 0.972793i $$0.425579\pi$$
$$312$$ 0.0953060 0.00539564
$$313$$ 24.9259 1.40889 0.704447 0.709757i $$-0.251195\pi$$
0.704447 + 0.709757i $$0.251195\pi$$
$$314$$ −3.66055 −0.206577
$$315$$ 0 0
$$316$$ −10.9465 −0.615790
$$317$$ 22.6159 1.27023 0.635117 0.772416i $$-0.280952\pi$$
0.635117 + 0.772416i $$0.280952\pi$$
$$318$$ −0.592835 −0.0332445
$$319$$ 5.24273 0.293537
$$320$$ 0 0
$$321$$ −3.93704 −0.219744
$$322$$ −0.772994 −0.0430773
$$323$$ −4.03954 −0.224766
$$324$$ −15.5255 −0.862525
$$325$$ 0 0
$$326$$ −9.08285 −0.503053
$$327$$ 1.04012 0.0575185
$$328$$ 9.02553 0.498351
$$329$$ −37.3204 −2.05754
$$330$$ 0 0
$$331$$ −19.8439 −1.09072 −0.545358 0.838203i $$-0.683606\pi$$
−0.545358 + 0.838203i $$0.683606\pi$$
$$332$$ −9.15850 −0.502638
$$333$$ −4.48443 −0.245745
$$334$$ 6.96467 0.381090
$$335$$ 0 0
$$336$$ 3.38329 0.184574
$$337$$ −17.4056 −0.948144 −0.474072 0.880486i $$-0.657216\pi$$
−0.474072 + 0.880486i $$0.657216\pi$$
$$338$$ 5.00042 0.271987
$$339$$ 0.230274 0.0125068
$$340$$ 0 0
$$341$$ −9.76294 −0.528693
$$342$$ −1.13286 −0.0612581
$$343$$ −12.9199 −0.697607
$$344$$ 11.3119 0.609897
$$345$$ 0 0
$$346$$ 7.02051 0.377425
$$347$$ −3.85633 −0.207019 −0.103509 0.994628i $$-0.533007\pi$$
−0.103509 + 0.994628i $$0.533007\pi$$
$$348$$ 2.53753 0.136026
$$349$$ −5.00844 −0.268096 −0.134048 0.990975i $$-0.542798\pi$$
−0.134048 + 0.990975i $$0.542798\pi$$
$$350$$ 0 0
$$351$$ −0.379913 −0.0202783
$$352$$ −4.18414 −0.223016
$$353$$ −16.8399 −0.896296 −0.448148 0.893959i $$-0.647916\pi$$
−0.448148 + 0.893959i $$0.647916\pi$$
$$354$$ −1.28587 −0.0683432
$$355$$ 0 0
$$356$$ 6.31998 0.334958
$$357$$ 4.37156 0.231368
$$358$$ 8.45593 0.446910
$$359$$ −1.05288 −0.0555691 −0.0277846 0.999614i $$-0.508845\pi$$
−0.0277846 + 0.999614i $$0.508845\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 6.79080 0.356916
$$363$$ −0.261531 −0.0137268
$$364$$ −1.87542 −0.0982988
$$365$$ 0 0
$$366$$ 0.766522 0.0400667
$$367$$ −5.43166 −0.283531 −0.141765 0.989900i $$-0.545278\pi$$
−0.141765 + 0.989900i $$0.545278\pi$$
$$368$$ −1.51133 −0.0787833
$$369$$ −17.7815 −0.925670
$$370$$ 0 0
$$371$$ 24.2728 1.26018
$$372$$ −4.72535 −0.244998
$$373$$ 23.1593 1.19914 0.599572 0.800321i $$-0.295338\pi$$
0.599572 + 0.800321i $$0.295338\pi$$
$$374$$ −1.56100 −0.0807174
$$375$$ 0 0
$$376$$ 13.4206 0.692116
$$377$$ −1.28394 −0.0661265
$$378$$ 2.48055 0.127586
$$379$$ −20.8260 −1.06976 −0.534879 0.844929i $$-0.679643\pi$$
−0.534879 + 0.844929i $$0.679643\pi$$
$$380$$ 0 0
$$381$$ 3.83627 0.196538
$$382$$ 7.93198 0.405835
$$383$$ 23.5549 1.20360 0.601800 0.798647i $$-0.294451\pi$$
0.601800 + 0.798647i $$0.294451\pi$$
$$384$$ −2.65708 −0.135593
$$385$$ 0 0
$$386$$ −5.50481 −0.280187
$$387$$ −22.2860 −1.13286
$$388$$ −23.2382 −1.17974
$$389$$ 16.9626 0.860037 0.430019 0.902820i $$-0.358507\pi$$
0.430019 + 0.902820i $$0.358507\pi$$
$$390$$ 0 0
$$391$$ −1.95279 −0.0987567
$$392$$ 15.0622 0.760755
$$393$$ −0.359302 −0.0181244
$$394$$ 1.26342 0.0636500
$$395$$ 0 0
$$396$$ 5.42543 0.272638
$$397$$ −11.2020 −0.562210 −0.281105 0.959677i $$-0.590701\pi$$
−0.281105 + 0.959677i $$0.590701\pi$$
$$398$$ 5.77276 0.289362
$$399$$ −1.08219 −0.0541774
$$400$$ 0 0
$$401$$ −7.62060 −0.380555 −0.190277 0.981730i $$-0.560939\pi$$
−0.190277 + 0.981730i $$0.560939\pi$$
$$402$$ −0.275159 −0.0137237
$$403$$ 2.39094 0.119101
$$404$$ 21.2844 1.05894
$$405$$ 0 0
$$406$$ 8.38321 0.416052
$$407$$ 1.52969 0.0758237
$$408$$ −1.57204 −0.0778275
$$409$$ −30.7305 −1.51953 −0.759763 0.650200i $$-0.774685\pi$$
−0.759763 + 0.650200i $$0.774685\pi$$
$$410$$ 0 0
$$411$$ 0.935486 0.0461441
$$412$$ 24.1489 1.18973
$$413$$ 52.6482 2.59065
$$414$$ −0.547646 −0.0269153
$$415$$ 0 0
$$416$$ 1.02470 0.0502399
$$417$$ −2.06065 −0.100910
$$418$$ 0.386431 0.0189009
$$419$$ −23.9749 −1.17125 −0.585625 0.810582i $$-0.699151\pi$$
−0.585625 + 0.810582i $$0.699151\pi$$
$$420$$ 0 0
$$421$$ −9.89352 −0.482181 −0.241090 0.970503i $$-0.577505\pi$$
−0.241090 + 0.970503i $$0.577505\pi$$
$$422$$ −2.73290 −0.133035
$$423$$ −26.4405 −1.28558
$$424$$ −8.72864 −0.423900
$$425$$ 0 0
$$426$$ 0.292262 0.0141601
$$427$$ −31.3842 −1.51879
$$428$$ −27.8597 −1.34665
$$429$$ 0.0640490 0.00309231
$$430$$ 0 0
$$431$$ 20.9302 1.00817 0.504086 0.863653i $$-0.331829\pi$$
0.504086 + 0.863653i $$0.331829\pi$$
$$432$$ 4.84987 0.233340
$$433$$ 4.10135 0.197098 0.0985490 0.995132i $$-0.468580\pi$$
0.0985490 + 0.995132i $$0.468580\pi$$
$$434$$ −15.6111 −0.749356
$$435$$ 0 0
$$436$$ 7.36016 0.352488
$$437$$ 0.483419 0.0231251
$$438$$ 0.732035 0.0349780
$$439$$ 29.1785 1.39262 0.696308 0.717743i $$-0.254825\pi$$
0.696308 + 0.717743i $$0.254825\pi$$
$$440$$ 0 0
$$441$$ −29.6746 −1.41308
$$442$$ 0.382289 0.0181836
$$443$$ −19.1722 −0.910899 −0.455450 0.890262i $$-0.650522\pi$$
−0.455450 + 0.890262i $$0.650522\pi$$
$$444$$ 0.740381 0.0351369
$$445$$ 0 0
$$446$$ 2.34693 0.111130
$$447$$ −0.159194 −0.00752962
$$448$$ 19.1824 0.906284
$$449$$ 3.01676 0.142370 0.0711849 0.997463i $$-0.477322\pi$$
0.0711849 + 0.997463i $$0.477322\pi$$
$$450$$ 0 0
$$451$$ 6.06547 0.285612
$$452$$ 1.62949 0.0766446
$$453$$ 5.97524 0.280741
$$454$$ 0.619880 0.0290924
$$455$$ 0 0
$$456$$ 0.389163 0.0182242
$$457$$ −6.32717 −0.295973 −0.147986 0.988989i $$-0.547279\pi$$
−0.147986 + 0.988989i $$0.547279\pi$$
$$458$$ −2.11590 −0.0988695
$$459$$ 6.26653 0.292497
$$460$$ 0 0
$$461$$ −26.9740 −1.25631 −0.628153 0.778090i $$-0.716189\pi$$
−0.628153 + 0.778090i $$0.716189\pi$$
$$462$$ −0.418193 −0.0194561
$$463$$ 5.17767 0.240627 0.120313 0.992736i $$-0.461610\pi$$
0.120313 + 0.992736i $$0.461610\pi$$
$$464$$ 16.3905 0.760909
$$465$$ 0 0
$$466$$ 6.14051 0.284454
$$467$$ 12.5932 0.582744 0.291372 0.956610i $$-0.405888\pi$$
0.291372 + 0.956610i $$0.405888\pi$$
$$468$$ −1.32869 −0.0614186
$$469$$ 11.2660 0.520216
$$470$$ 0 0
$$471$$ −2.47742 −0.114153
$$472$$ −18.9326 −0.871443
$$473$$ 7.60200 0.349540
$$474$$ 0.597781 0.0274570
$$475$$ 0 0
$$476$$ 30.9344 1.41788
$$477$$ 17.1966 0.787380
$$478$$ 11.5563 0.528571
$$479$$ 36.1790 1.65306 0.826530 0.562893i $$-0.190312\pi$$
0.826530 + 0.562893i $$0.190312\pi$$
$$480$$ 0 0
$$481$$ −0.374620 −0.0170812
$$482$$ 2.71306 0.123577
$$483$$ −0.523153 −0.0238043
$$484$$ −1.85067 −0.0841214
$$485$$ 0 0
$$486$$ 2.64624 0.120036
$$487$$ 13.3983 0.607133 0.303566 0.952810i $$-0.401823\pi$$
0.303566 + 0.952810i $$0.401823\pi$$
$$488$$ 11.2859 0.510890
$$489$$ −6.14716 −0.277984
$$490$$ 0 0
$$491$$ −26.7013 −1.20501 −0.602506 0.798115i $$-0.705831\pi$$
−0.602506 + 0.798115i $$0.705831\pi$$
$$492$$ 2.93574 0.132353
$$493$$ 21.1782 0.953818
$$494$$ −0.0946368 −0.00425791
$$495$$ 0 0
$$496$$ −30.5221 −1.37048
$$497$$ −11.9663 −0.536760
$$498$$ 0.500138 0.0224117
$$499$$ 4.18714 0.187442 0.0937211 0.995598i $$-0.470124\pi$$
0.0937211 + 0.995598i $$0.470124\pi$$
$$500$$ 0 0
$$501$$ 4.71360 0.210588
$$502$$ −4.58671 −0.204715
$$503$$ 3.27450 0.146003 0.0730014 0.997332i $$-0.476742\pi$$
0.0730014 + 0.997332i $$0.476742\pi$$
$$504$$ 18.0507 0.804042
$$505$$ 0 0
$$506$$ 0.186808 0.00830462
$$507$$ 3.38422 0.150299
$$508$$ 27.1466 1.20443
$$509$$ 4.77423 0.211614 0.105807 0.994387i $$-0.466257\pi$$
0.105807 + 0.994387i $$0.466257\pi$$
$$510$$ 0 0
$$511$$ −29.9721 −1.32589
$$512$$ −22.3851 −0.989289
$$513$$ −1.55130 −0.0684915
$$514$$ −9.29776 −0.410107
$$515$$ 0 0
$$516$$ 3.67943 0.161978
$$517$$ 9.01914 0.396661
$$518$$ 2.44599 0.107471
$$519$$ 4.75139 0.208563
$$520$$ 0 0
$$521$$ −40.1413 −1.75862 −0.879311 0.476247i $$-0.841997\pi$$
−0.879311 + 0.476247i $$0.841997\pi$$
$$522$$ 5.93928 0.259955
$$523$$ −25.7012 −1.12384 −0.561918 0.827193i $$-0.689936\pi$$
−0.561918 + 0.827193i $$0.689936\pi$$
$$524$$ −2.54253 −0.111071
$$525$$ 0 0
$$526$$ −3.11972 −0.136026
$$527$$ −39.4377 −1.71794
$$528$$ −0.817633 −0.0355829
$$529$$ −22.7663 −0.989839
$$530$$ 0 0
$$531$$ 37.2998 1.61867
$$532$$ −7.65791 −0.332013
$$533$$ −1.48543 −0.0643413
$$534$$ −0.345129 −0.0149352
$$535$$ 0 0
$$536$$ −4.05133 −0.174991
$$537$$ 5.72286 0.246960
$$538$$ −3.52471 −0.151961
$$539$$ 10.1223 0.435999
$$540$$ 0 0
$$541$$ 13.5359 0.581952 0.290976 0.956730i $$-0.406020\pi$$
0.290976 + 0.956730i $$0.406020\pi$$
$$542$$ −6.57809 −0.282553
$$543$$ 4.59592 0.197230
$$544$$ −16.9020 −0.724667
$$545$$ 0 0
$$546$$ 0.102415 0.00438297
$$547$$ −16.5470 −0.707498 −0.353749 0.935340i $$-0.615093\pi$$
−0.353749 + 0.935340i $$0.615093\pi$$
$$548$$ 6.61977 0.282783
$$549$$ −22.2349 −0.948961
$$550$$ 0 0
$$551$$ −5.24273 −0.223348
$$552$$ 0.188129 0.00800729
$$553$$ −24.4753 −1.04080
$$554$$ −10.1660 −0.431912
$$555$$ 0 0
$$556$$ −14.5818 −0.618404
$$557$$ −20.7288 −0.878307 −0.439153 0.898412i $$-0.644721\pi$$
−0.439153 + 0.898412i $$0.644721\pi$$
$$558$$ −11.0600 −0.468209
$$559$$ −1.86173 −0.0787427
$$560$$ 0 0
$$561$$ −1.05647 −0.0446040
$$562$$ 0.180617 0.00761886
$$563$$ −4.80258 −0.202405 −0.101202 0.994866i $$-0.532269\pi$$
−0.101202 + 0.994866i $$0.532269\pi$$
$$564$$ 4.36534 0.183814
$$565$$ 0 0
$$566$$ 7.86667 0.330661
$$567$$ −34.7133 −1.45782
$$568$$ 4.30314 0.180556
$$569$$ 18.7387 0.785565 0.392783 0.919631i $$-0.371512\pi$$
0.392783 + 0.919631i $$0.371512\pi$$
$$570$$ 0 0
$$571$$ −34.1368 −1.42858 −0.714291 0.699849i $$-0.753251\pi$$
−0.714291 + 0.699849i $$0.753251\pi$$
$$572$$ 0.453229 0.0189505
$$573$$ 5.36826 0.224262
$$574$$ 9.69878 0.404819
$$575$$ 0 0
$$576$$ 13.5902 0.566260
$$577$$ 43.9533 1.82980 0.914901 0.403679i $$-0.132269\pi$$
0.914901 + 0.403679i $$0.132269\pi$$
$$578$$ 0.263601 0.0109644
$$579$$ −3.72559 −0.154830
$$580$$ 0 0
$$581$$ −20.4775 −0.849548
$$582$$ 1.26902 0.0526027
$$583$$ −5.86595 −0.242943
$$584$$ 10.7782 0.446003
$$585$$ 0 0
$$586$$ 11.3875 0.470415
$$587$$ −15.3671 −0.634267 −0.317134 0.948381i $$-0.602720\pi$$
−0.317134 + 0.948381i $$0.602720\pi$$
$$588$$ 4.89929 0.202043
$$589$$ 9.76294 0.402275
$$590$$ 0 0
$$591$$ 0.855064 0.0351726
$$592$$ 4.78230 0.196551
$$593$$ −16.1352 −0.662595 −0.331297 0.943526i $$-0.607486\pi$$
−0.331297 + 0.943526i $$0.607486\pi$$
$$594$$ −0.599470 −0.0245965
$$595$$ 0 0
$$596$$ −1.12650 −0.0461434
$$597$$ 3.90693 0.159900
$$598$$ −0.0457492 −0.00187082
$$599$$ −9.65248 −0.394390 −0.197195 0.980364i $$-0.563183\pi$$
−0.197195 + 0.980364i $$0.563183\pi$$
$$600$$ 0 0
$$601$$ 29.8944 1.21942 0.609708 0.792626i $$-0.291287\pi$$
0.609708 + 0.792626i $$0.291287\pi$$
$$602$$ 12.1557 0.495430
$$603$$ 7.98168 0.325039
$$604$$ 42.2825 1.72045
$$605$$ 0 0
$$606$$ −1.16233 −0.0472162
$$607$$ −15.3867 −0.624528 −0.312264 0.949995i $$-0.601087\pi$$
−0.312264 + 0.949995i $$0.601087\pi$$
$$608$$ 4.18414 0.169689
$$609$$ 5.67365 0.229908
$$610$$ 0 0
$$611$$ −2.20879 −0.0893579
$$612$$ 21.9162 0.885911
$$613$$ −43.3990 −1.75287 −0.876435 0.481521i $$-0.840085\pi$$
−0.876435 + 0.481521i $$0.840085\pi$$
$$614$$ −5.37621 −0.216966
$$615$$ 0 0
$$616$$ −6.15728 −0.248084
$$617$$ −32.5803 −1.31164 −0.655818 0.754919i $$-0.727676\pi$$
−0.655818 + 0.754919i $$0.727676\pi$$
$$618$$ −1.31875 −0.0530480
$$619$$ −4.04214 −0.162467 −0.0812336 0.996695i $$-0.525886\pi$$
−0.0812336 + 0.996695i $$0.525886\pi$$
$$620$$ 0 0
$$621$$ −0.749927 −0.0300935
$$622$$ −3.15765 −0.126610
$$623$$ 14.1308 0.566139
$$624$$ 0.200238 0.00801594
$$625$$ 0 0
$$626$$ −9.63212 −0.384977
$$627$$ 0.261531 0.0104446
$$628$$ −17.5309 −0.699560
$$629$$ 6.17922 0.246382
$$630$$ 0 0
$$631$$ 39.0445 1.55434 0.777169 0.629292i $$-0.216655\pi$$
0.777169 + 0.629292i $$0.216655\pi$$
$$632$$ 8.80146 0.350103
$$633$$ −1.84959 −0.0735146
$$634$$ −8.73946 −0.347088
$$635$$ 0 0
$$636$$ −2.83917 −0.112580
$$637$$ −2.47895 −0.0982197
$$638$$ −2.02595 −0.0802082
$$639$$ −8.47778 −0.335376
$$640$$ 0 0
$$641$$ 30.5292 1.20583 0.602916 0.797805i $$-0.294006\pi$$
0.602916 + 0.797805i $$0.294006\pi$$
$$642$$ 1.52139 0.0600446
$$643$$ 27.4261 1.08158 0.540790 0.841158i $$-0.318125\pi$$
0.540790 + 0.841158i $$0.318125\pi$$
$$644$$ −3.70198 −0.145878
$$645$$ 0 0
$$646$$ 1.56100 0.0614167
$$647$$ −42.2102 −1.65945 −0.829726 0.558171i $$-0.811503\pi$$
−0.829726 + 0.558171i $$0.811503\pi$$
$$648$$ 12.4831 0.490383
$$649$$ −12.7234 −0.499436
$$650$$ 0 0
$$651$$ −10.5654 −0.414090
$$652$$ −43.4991 −1.70356
$$653$$ −28.8516 −1.12905 −0.564526 0.825415i $$-0.690941\pi$$
−0.564526 + 0.825415i $$0.690941\pi$$
$$654$$ −0.401932 −0.0157168
$$655$$ 0 0
$$656$$ 18.9627 0.740367
$$657$$ −21.2345 −0.828436
$$658$$ 14.4217 0.562218
$$659$$ −11.5297 −0.449135 −0.224567 0.974459i $$-0.572097\pi$$
−0.224567 + 0.974459i $$0.572097\pi$$
$$660$$ 0 0
$$661$$ 16.8814 0.656610 0.328305 0.944572i $$-0.393523\pi$$
0.328305 + 0.944572i $$0.393523\pi$$
$$662$$ 7.66827 0.298036
$$663$$ 0.258728 0.0100482
$$664$$ 7.36381 0.285771
$$665$$ 0 0
$$666$$ 1.73292 0.0671493
$$667$$ −2.53443 −0.0981337
$$668$$ 33.3548 1.29054
$$669$$ 1.58837 0.0614099
$$670$$ 0 0
$$671$$ 7.58454 0.292798
$$672$$ −4.52805 −0.174673
$$673$$ 7.27270 0.280342 0.140171 0.990127i $$-0.455235\pi$$
0.140171 + 0.990127i $$0.455235\pi$$
$$674$$ 6.72606 0.259078
$$675$$ 0 0
$$676$$ 23.9477 0.921067
$$677$$ −0.515123 −0.0197978 −0.00989888 0.999951i $$-0.503151\pi$$
−0.00989888 + 0.999951i $$0.503151\pi$$
$$678$$ −0.0889850 −0.00341745
$$679$$ −51.9583 −1.99398
$$680$$ 0 0
$$681$$ 0.419527 0.0160763
$$682$$ 3.77270 0.144464
$$683$$ 29.0661 1.11218 0.556091 0.831121i $$-0.312300\pi$$
0.556091 + 0.831121i $$0.312300\pi$$
$$684$$ −5.42543 −0.207447
$$685$$ 0 0
$$686$$ 4.99263 0.190619
$$687$$ −1.43201 −0.0546348
$$688$$ 23.7663 0.906083
$$689$$ 1.43657 0.0547290
$$690$$ 0 0
$$691$$ −1.60956 −0.0612306 −0.0306153 0.999531i $$-0.509747\pi$$
−0.0306153 + 0.999531i $$0.509747\pi$$
$$692$$ 33.6222 1.27813
$$693$$ 12.1307 0.460807
$$694$$ 1.49020 0.0565673
$$695$$ 0 0
$$696$$ −2.04028 −0.0773365
$$697$$ 24.5017 0.928068
$$698$$ 1.93541 0.0732565
$$699$$ 4.15582 0.157187
$$700$$ 0 0
$$701$$ −26.6908 −1.00810 −0.504048 0.863675i $$-0.668157\pi$$
−0.504048 + 0.863675i $$0.668157\pi$$
$$702$$ 0.146810 0.00554099
$$703$$ −1.52969 −0.0576932
$$704$$ −4.63577 −0.174717
$$705$$ 0 0
$$706$$ 6.50744 0.244911
$$707$$ 47.5898 1.78980
$$708$$ −6.15822 −0.231440
$$709$$ 34.7167 1.30381 0.651907 0.758299i $$-0.273969\pi$$
0.651907 + 0.758299i $$0.273969\pi$$
$$710$$ 0 0
$$711$$ −17.3401 −0.650305
$$712$$ −5.08153 −0.190438
$$713$$ 4.71959 0.176750
$$714$$ −1.68930 −0.0632206
$$715$$ 0 0
$$716$$ 40.4966 1.51343
$$717$$ 7.82114 0.292086
$$718$$ 0.406867 0.0151841
$$719$$ 0.454145 0.0169368 0.00846838 0.999964i $$-0.497304\pi$$
0.00846838 + 0.999964i $$0.497304\pi$$
$$720$$ 0 0
$$721$$ 53.9945 2.01086
$$722$$ −0.386431 −0.0143815
$$723$$ 1.83617 0.0682878
$$724$$ 32.5221 1.20867
$$725$$ 0 0
$$726$$ 0.101064 0.00375083
$$727$$ −34.4667 −1.27830 −0.639150 0.769082i $$-0.720714\pi$$
−0.639150 + 0.769082i $$0.720714\pi$$
$$728$$ 1.50792 0.0558872
$$729$$ −23.3763 −0.865790
$$730$$ 0 0
$$731$$ 30.7086 1.13580
$$732$$ 3.67098 0.135683
$$733$$ −33.3530 −1.23192 −0.615960 0.787777i $$-0.711232\pi$$
−0.615960 + 0.787777i $$0.711232\pi$$
$$734$$ 2.09896 0.0774741
$$735$$ 0 0
$$736$$ 2.02269 0.0745574
$$737$$ −2.72263 −0.100290
$$738$$ 6.87133 0.252937
$$739$$ 14.6031 0.537185 0.268592 0.963254i $$-0.413442\pi$$
0.268592 + 0.963254i $$0.413442\pi$$
$$740$$ 0 0
$$741$$ −0.0640490 −0.00235290
$$742$$ −9.37975 −0.344341
$$743$$ 29.2470 1.07297 0.536484 0.843910i $$-0.319752\pi$$
0.536484 + 0.843910i $$0.319752\pi$$
$$744$$ 3.79938 0.139292
$$745$$ 0 0
$$746$$ −8.94946 −0.327663
$$747$$ −14.5077 −0.530810
$$748$$ −7.47585 −0.273344
$$749$$ −62.2914 −2.27608
$$750$$ 0 0
$$751$$ −26.1796 −0.955307 −0.477654 0.878548i $$-0.658513\pi$$
−0.477654 + 0.878548i $$0.658513\pi$$
$$752$$ 28.1968 1.02823
$$753$$ −3.10422 −0.113124
$$754$$ 0.496155 0.0180689
$$755$$ 0 0
$$756$$ 11.8797 0.432061
$$757$$ −35.4067 −1.28688 −0.643439 0.765497i $$-0.722493\pi$$
−0.643439 + 0.765497i $$0.722493\pi$$
$$758$$ 8.04778 0.292309
$$759$$ 0.126429 0.00458909
$$760$$ 0 0
$$761$$ 3.04755 0.110474 0.0552368 0.998473i $$-0.482409\pi$$
0.0552368 + 0.998473i $$0.482409\pi$$
$$762$$ −1.48245 −0.0537036
$$763$$ 16.4566 0.595767
$$764$$ 37.9874 1.37434
$$765$$ 0 0
$$766$$ −9.10233 −0.328881
$$767$$ 3.11595 0.112510
$$768$$ −1.39802 −0.0504469
$$769$$ 48.6008 1.75259 0.876295 0.481776i $$-0.160008\pi$$
0.876295 + 0.481776i $$0.160008\pi$$
$$770$$ 0 0
$$771$$ −6.29261 −0.226623
$$772$$ −26.3633 −0.948837
$$773$$ −51.0426 −1.83587 −0.917937 0.396727i $$-0.870146\pi$$
−0.917937 + 0.396727i $$0.870146\pi$$
$$774$$ 8.61200 0.309552
$$775$$ 0 0
$$776$$ 18.6845 0.670735
$$777$$ 1.65542 0.0593877
$$778$$ −6.55486 −0.235003
$$779$$ −6.06547 −0.217318
$$780$$ 0 0
$$781$$ 2.89186 0.103479
$$782$$ 0.754617 0.0269850
$$783$$ 8.13304 0.290651
$$784$$ 31.6457 1.13020
$$785$$ 0 0
$$786$$ 0.138845 0.00495245
$$787$$ 3.01361 0.107424 0.0537118 0.998556i $$-0.482895\pi$$
0.0537118 + 0.998556i $$0.482895\pi$$
$$788$$ 6.05068 0.215547
$$789$$ −2.11139 −0.0751675
$$790$$ 0 0
$$791$$ 3.64337 0.129543
$$792$$ −4.36227 −0.155007
$$793$$ −1.85745 −0.0659601
$$794$$ 4.32878 0.153623
$$795$$ 0 0
$$796$$ 27.6466 0.979907
$$797$$ 45.9063 1.62608 0.813042 0.582206i $$-0.197810\pi$$
0.813042 + 0.582206i $$0.197810\pi$$
$$798$$ 0.418193 0.0148039
$$799$$ 36.4331 1.28891
$$800$$ 0 0
$$801$$ 10.0113 0.353732
$$802$$ 2.94483 0.103986
$$803$$ 7.24330 0.255611
$$804$$ −1.31778 −0.0464744
$$805$$ 0 0
$$806$$ −0.923933 −0.0325442
$$807$$ −2.38548 −0.0839728
$$808$$ −17.1136 −0.602053
$$809$$ 41.8470 1.47126 0.735631 0.677383i $$-0.236886\pi$$
0.735631 + 0.677383i $$0.236886\pi$$
$$810$$ 0 0
$$811$$ 28.5357 1.00202 0.501011 0.865441i $$-0.332962\pi$$
0.501011 + 0.865441i $$0.332962\pi$$
$$812$$ 40.1484 1.40893
$$813$$ −4.45197 −0.156137
$$814$$ −0.591117 −0.0207186
$$815$$ 0 0
$$816$$ −3.30286 −0.115623
$$817$$ −7.60200 −0.265960
$$818$$ 11.8752 0.415207
$$819$$ −2.97081 −0.103808
$$820$$ 0 0
$$821$$ −18.4483 −0.643849 −0.321924 0.946765i $$-0.604330\pi$$
−0.321924 + 0.946765i $$0.604330\pi$$
$$822$$ −0.361500 −0.0126088
$$823$$ 8.59026 0.299438 0.149719 0.988729i $$-0.452163\pi$$
0.149719 + 0.988729i $$0.452163\pi$$
$$824$$ −19.4167 −0.676414
$$825$$ 0 0
$$826$$ −20.3449 −0.707888
$$827$$ −22.3427 −0.776931 −0.388466 0.921463i $$-0.626995\pi$$
−0.388466 + 0.921463i $$0.626995\pi$$
$$828$$ −2.62275 −0.0911470
$$829$$ −15.8776 −0.551450 −0.275725 0.961237i $$-0.588918\pi$$
−0.275725 + 0.961237i $$0.588918\pi$$
$$830$$ 0 0
$$831$$ −6.88023 −0.238672
$$832$$ 1.13530 0.0393595
$$833$$ 40.8895 1.41674
$$834$$ 0.796298 0.0275736
$$835$$ 0 0
$$836$$ 1.85067 0.0640068
$$837$$ −15.1452 −0.523496
$$838$$ 9.26463 0.320042
$$839$$ 50.1265 1.73056 0.865280 0.501289i $$-0.167141\pi$$
0.865280 + 0.501289i $$0.167141\pi$$
$$840$$ 0 0
$$841$$ −1.51378 −0.0521994
$$842$$ 3.82316 0.131755
$$843$$ 0.122239 0.00421014
$$844$$ −13.0882 −0.450516
$$845$$ 0 0
$$846$$ 10.2174 0.351282
$$847$$ −4.13791 −0.142180
$$848$$ −18.3389 −0.629760
$$849$$ 5.32406 0.182721
$$850$$ 0 0
$$851$$ −0.739478 −0.0253490
$$852$$ 1.39969 0.0479524
$$853$$ −31.6970 −1.08528 −0.542642 0.839964i $$-0.682576\pi$$
−0.542642 + 0.839964i $$0.682576\pi$$
$$854$$ 12.1278 0.415005
$$855$$ 0 0
$$856$$ 22.4003 0.765628
$$857$$ −20.4944 −0.700075 −0.350037 0.936736i $$-0.613831\pi$$
−0.350037 + 0.936736i $$0.613831\pi$$
$$858$$ −0.0247505 −0.000844968 0
$$859$$ 8.63383 0.294583 0.147291 0.989093i $$-0.452945\pi$$
0.147291 + 0.989093i $$0.452945\pi$$
$$860$$ 0 0
$$861$$ 6.56401 0.223701
$$862$$ −8.08807 −0.275481
$$863$$ −49.4127 −1.68203 −0.841014 0.541014i $$-0.818041\pi$$
−0.841014 + 0.541014i $$0.818041\pi$$
$$864$$ −6.49086 −0.220824
$$865$$ 0 0
$$866$$ −1.58489 −0.0538566
$$867$$ 0.178402 0.00605885
$$868$$ −74.7638 −2.53765
$$869$$ 5.91489 0.200649
$$870$$ 0 0
$$871$$ 0.666773 0.0225927
$$872$$ −5.91787 −0.200404
$$873$$ −36.8111 −1.24587
$$874$$ −0.186808 −0.00631887
$$875$$ 0 0
$$876$$ 3.50582 0.118451
$$877$$ −12.9391 −0.436922 −0.218461 0.975846i $$-0.570104\pi$$
−0.218461 + 0.975846i $$0.570104\pi$$
$$878$$ −11.2755 −0.380529
$$879$$ 7.70694 0.259949
$$880$$ 0 0
$$881$$ −18.7876 −0.632970 −0.316485 0.948598i $$-0.602503\pi$$
−0.316485 + 0.948598i $$0.602503\pi$$
$$882$$ 11.4672 0.386120
$$883$$ 49.3212 1.65979 0.829895 0.557919i $$-0.188400\pi$$
0.829895 + 0.557919i $$0.188400\pi$$
$$884$$ 1.83084 0.0615777
$$885$$ 0 0
$$886$$ 7.40873 0.248901
$$887$$ −43.4343 −1.45838 −0.729191 0.684310i $$-0.760103\pi$$
−0.729191 + 0.684310i $$0.760103\pi$$
$$888$$ −0.595297 −0.0199769
$$889$$ 60.6969 2.03571
$$890$$ 0 0
$$891$$ 8.38909 0.281045
$$892$$ 11.2398 0.376335
$$893$$ −9.01914 −0.301814
$$894$$ 0.0615175 0.00205745
$$895$$ 0 0
$$896$$ −42.0399 −1.40445
$$897$$ −0.0309625 −0.00103381
$$898$$ −1.16577 −0.0389022
$$899$$ −51.1844 −1.70710
$$900$$ 0 0
$$901$$ −23.6957 −0.789419
$$902$$ −2.34388 −0.0780428
$$903$$ 8.22684 0.273772
$$904$$ −1.31018 −0.0435758
$$905$$ 0 0
$$906$$ −2.30901 −0.0767119
$$907$$ −39.3136 −1.30539 −0.652694 0.757622i $$-0.726361\pi$$
−0.652694 + 0.757622i $$0.726361\pi$$
$$908$$ 2.96869 0.0985196
$$909$$ 33.7161 1.11829
$$910$$ 0 0
$$911$$ 2.30597 0.0764001 0.0382001 0.999270i $$-0.487838\pi$$
0.0382001 + 0.999270i $$0.487838\pi$$
$$912$$ 0.817633 0.0270745
$$913$$ 4.94874 0.163779
$$914$$ 2.44501 0.0808738
$$915$$ 0 0
$$916$$ −10.1334 −0.334815
$$917$$ −5.68483 −0.187730
$$918$$ −2.42158 −0.0799240
$$919$$ 35.1934 1.16092 0.580462 0.814287i $$-0.302872\pi$$
0.580462 + 0.814287i $$0.302872\pi$$
$$920$$ 0 0
$$921$$ −3.63855 −0.119894
$$922$$ 10.4236 0.343283
$$923$$ −0.708216 −0.0233112
$$924$$ −2.00278 −0.0658868
$$925$$ 0 0
$$926$$ −2.00081 −0.0657507
$$927$$ 38.2537 1.25641
$$928$$ −21.9363 −0.720095
$$929$$ 26.4097 0.866474 0.433237 0.901280i $$-0.357371\pi$$
0.433237 + 0.901280i $$0.357371\pi$$
$$930$$ 0 0
$$931$$ −10.1223 −0.331745
$$932$$ 29.4078 0.963284
$$933$$ −2.13706 −0.0699642
$$934$$ −4.86640 −0.159233
$$935$$ 0 0
$$936$$ 1.06832 0.0349191
$$937$$ −25.8042 −0.842987 −0.421493 0.906831i $$-0.638494\pi$$
−0.421493 + 0.906831i $$0.638494\pi$$
$$938$$ −4.35353 −0.142148
$$939$$ −6.51890 −0.212736
$$940$$ 0 0
$$941$$ 57.3538 1.86968 0.934841 0.355067i $$-0.115542\pi$$
0.934841 + 0.355067i $$0.115542\pi$$
$$942$$ 0.957349 0.0311921
$$943$$ −2.93216 −0.0954843
$$944$$ −39.7774 −1.29464
$$945$$ 0 0
$$946$$ −2.93765 −0.0955111
$$947$$ 59.3155 1.92749 0.963747 0.266818i $$-0.0859723\pi$$
0.963747 + 0.266818i $$0.0859723\pi$$
$$948$$ 2.86286 0.0929813
$$949$$ −1.77388 −0.0575827
$$950$$ 0 0
$$951$$ −5.91476 −0.191799
$$952$$ −24.8726 −0.806125
$$953$$ 1.58112 0.0512174 0.0256087 0.999672i $$-0.491848\pi$$
0.0256087 + 0.999672i $$0.491848\pi$$
$$954$$ −6.64531 −0.215150
$$955$$ 0 0
$$956$$ 55.3446 1.78997
$$957$$ −1.37114 −0.0443226
$$958$$ −13.9807 −0.451695
$$959$$ 14.8011 0.477953
$$960$$ 0 0
$$961$$ 64.3150 2.07468
$$962$$ 0.144765 0.00466740
$$963$$ −44.1318 −1.42213
$$964$$ 12.9932 0.418484
$$965$$ 0 0
$$966$$ 0.202162 0.00650446
$$967$$ 38.7297 1.24546 0.622731 0.782436i $$-0.286023\pi$$
0.622731 + 0.782436i $$0.286023\pi$$
$$968$$ 1.48802 0.0478267
$$969$$ 1.05647 0.0339386
$$970$$ 0 0
$$971$$ −44.2299 −1.41941 −0.709703 0.704501i $$-0.751171\pi$$
−0.709703 + 0.704501i $$0.751171\pi$$
$$972$$ 12.6732 0.406494
$$973$$ −32.6033 −1.04521
$$974$$ −5.17750 −0.165898
$$975$$ 0 0
$$976$$ 23.7118 0.758995
$$977$$ 0.637601 0.0203987 0.0101993 0.999948i $$-0.496753\pi$$
0.0101993 + 0.999948i $$0.496753\pi$$
$$978$$ 2.37545 0.0759585
$$979$$ −3.41496 −0.109143
$$980$$ 0 0
$$981$$ 11.6590 0.372244
$$982$$ 10.3182 0.329267
$$983$$ 27.4345 0.875025 0.437512 0.899212i $$-0.355860\pi$$
0.437512 + 0.899212i $$0.355860\pi$$
$$984$$ −2.36046 −0.0752486
$$985$$ 0 0
$$986$$ −8.18390 −0.260629
$$987$$ 9.76046 0.310679
$$988$$ −0.453229 −0.0144191
$$989$$ −3.67495 −0.116857
$$990$$ 0 0
$$991$$ −49.4562 −1.57103 −0.785514 0.618844i $$-0.787601\pi$$
−0.785514 + 0.618844i $$0.787601\pi$$
$$992$$ 40.8495 1.29697
$$993$$ 5.18979 0.164693
$$994$$ 4.62413 0.146669
$$995$$ 0 0
$$996$$ 2.39523 0.0758959
$$997$$ 23.2499 0.736331 0.368165 0.929760i $$-0.379986\pi$$
0.368165 + 0.929760i $$0.379986\pi$$
$$998$$ −1.61804 −0.0512182
$$999$$ 2.37300 0.0750784
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 5225.2.a.ba.1.9 20
5.2 odd 4 1045.2.b.c.419.9 20
5.3 odd 4 1045.2.b.c.419.12 yes 20
5.4 even 2 inner 5225.2.a.ba.1.12 20

By twisted newform
Twist Min Dim Char Parity Ord Type
1045.2.b.c.419.9 20 5.2 odd 4
1045.2.b.c.419.12 yes 20 5.3 odd 4
5225.2.a.ba.1.9 20 1.1 even 1 trivial
5225.2.a.ba.1.12 20 5.4 even 2 inner