# Properties

 Label 475.2.a.e.1.1 Level $475$ Weight $2$ Character 475.1 Self dual yes Analytic conductor $3.793$ Analytic rank $1$ 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] = [475,2,Mod(1,475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(475, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("475.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$475 = 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 475.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: $$3.79289409601$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.169.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 4x - 1$$ x^3 - x^2 - 4*x - 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.37720$$ of defining polynomial Character $$\chi$$ $$=$$ 475.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.37720 q^{2} -1.27389 q^{3} +3.65109 q^{4} +3.02830 q^{6} -0.726109 q^{7} -3.92498 q^{8} -1.37720 q^{9} +O(q^{10})$$ $$q-2.37720 q^{2} -1.27389 q^{3} +3.65109 q^{4} +3.02830 q^{6} -0.726109 q^{7} -3.92498 q^{8} -1.37720 q^{9} -0.273891 q^{11} -4.65109 q^{12} +5.95328 q^{13} +1.72611 q^{14} +2.02830 q^{16} -5.27389 q^{17} +3.27389 q^{18} +1.00000 q^{19} +0.924984 q^{21} +0.651093 q^{22} +3.67939 q^{23} +5.00000 q^{24} -14.1522 q^{26} +5.57608 q^{27} -2.65109 q^{28} -2.27389 q^{29} +3.19887 q^{31} +3.02830 q^{32} +0.348907 q^{33} +12.5371 q^{34} -5.02830 q^{36} -8.12386 q^{37} -2.37720 q^{38} -7.58383 q^{39} -9.43380 q^{41} -2.19887 q^{42} -9.81100 q^{43} -1.00000 q^{44} -8.74666 q^{46} -12.1599 q^{47} -2.58383 q^{48} -6.47277 q^{49} +6.71836 q^{51} +21.7360 q^{52} -5.69781 q^{53} -13.2555 q^{54} +2.84997 q^{56} -1.27389 q^{57} +5.40550 q^{58} -4.20662 q^{59} -0.103312 q^{61} -7.60437 q^{62} +1.00000 q^{63} -11.2555 q^{64} -0.829422 q^{66} +11.7827 q^{67} -19.2555 q^{68} -4.68714 q^{69} +5.75441 q^{71} +5.40550 q^{72} -6.67939 q^{73} +19.3121 q^{74} +3.65109 q^{76} +0.198875 q^{77} +18.0283 q^{78} +3.87826 q^{79} -2.97170 q^{81} +22.4260 q^{82} +0.488265 q^{83} +3.37720 q^{84} +23.3227 q^{86} +2.89669 q^{87} +1.07502 q^{88} -16.4338 q^{89} -4.32273 q^{91} +13.4338 q^{92} -4.07502 q^{93} +28.9066 q^{94} -3.85772 q^{96} +4.44447 q^{97} +15.3871 q^{98} +0.377203 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 2 q^{2} - 2 q^{3} + 4 q^{4} - 3 q^{6} - 4 q^{7} - 3 q^{8} + q^{9}+O(q^{10})$$ 3 * q - 2 * q^2 - 2 * q^3 + 4 * q^4 - 3 * q^6 - 4 * q^7 - 3 * q^8 + q^9 $$3 q - 2 q^{2} - 2 q^{3} + 4 q^{4} - 3 q^{6} - 4 q^{7} - 3 q^{8} + q^{9} + q^{11} - 7 q^{12} - 3 q^{13} + 7 q^{14} - 6 q^{16} - 14 q^{17} + 8 q^{18} + 3 q^{19} - 6 q^{21} - 5 q^{22} - 8 q^{23} + 15 q^{24} - 11 q^{26} + q^{27} - q^{28} - 5 q^{29} - q^{31} - 3 q^{32} + 8 q^{33} + 5 q^{34} - 3 q^{36} - 5 q^{37} - 2 q^{38} - 11 q^{39} + q^{41} + 4 q^{42} + 5 q^{43} - 3 q^{44} - 12 q^{46} - 9 q^{47} + 4 q^{48} - 7 q^{49} + 18 q^{51} + 22 q^{52} - 31 q^{53} - 5 q^{54} - 9 q^{56} - 2 q^{57} - q^{58} - 6 q^{59} + 3 q^{61} + 5 q^{62} + 3 q^{63} + q^{64} - q^{66} + 13 q^{67} - 23 q^{68} + q^{69} + 7 q^{71} - q^{72} - q^{73} - q^{74} + 4 q^{76} - 10 q^{77} + 42 q^{78} - 18 q^{79} - 21 q^{81} + 34 q^{82} - 3 q^{83} + 5 q^{84} + 40 q^{86} + 12 q^{87} + 12 q^{88} - 20 q^{89} + 17 q^{91} + 11 q^{92} - 21 q^{93} + 45 q^{94} + 2 q^{96} + 13 q^{97} - 4 q^{98} - 4 q^{99}+O(q^{100})$$ 3 * q - 2 * q^2 - 2 * q^3 + 4 * q^4 - 3 * q^6 - 4 * q^7 - 3 * q^8 + q^9 + q^11 - 7 * q^12 - 3 * q^13 + 7 * q^14 - 6 * q^16 - 14 * q^17 + 8 * q^18 + 3 * q^19 - 6 * q^21 - 5 * q^22 - 8 * q^23 + 15 * q^24 - 11 * q^26 + q^27 - q^28 - 5 * q^29 - q^31 - 3 * q^32 + 8 * q^33 + 5 * q^34 - 3 * q^36 - 5 * q^37 - 2 * q^38 - 11 * q^39 + q^41 + 4 * q^42 + 5 * q^43 - 3 * q^44 - 12 * q^46 - 9 * q^47 + 4 * q^48 - 7 * q^49 + 18 * q^51 + 22 * q^52 - 31 * q^53 - 5 * q^54 - 9 * q^56 - 2 * q^57 - q^58 - 6 * q^59 + 3 * q^61 + 5 * q^62 + 3 * q^63 + q^64 - q^66 + 13 * q^67 - 23 * q^68 + q^69 + 7 * q^71 - q^72 - q^73 - q^74 + 4 * q^76 - 10 * q^77 + 42 * q^78 - 18 * q^79 - 21 * q^81 + 34 * q^82 - 3 * q^83 + 5 * q^84 + 40 * q^86 + 12 * q^87 + 12 * q^88 - 20 * q^89 + 17 * q^91 + 11 * q^92 - 21 * q^93 + 45 * q^94 + 2 * q^96 + 13 * q^97 - 4 * q^98 - 4 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.37720 −1.68094 −0.840468 0.541861i $$-0.817720\pi$$
−0.840468 + 0.541861i $$0.817720\pi$$
$$3$$ −1.27389 −0.735481 −0.367741 0.929928i $$-0.619869\pi$$
−0.367741 + 0.929928i $$0.619869\pi$$
$$4$$ 3.65109 1.82555
$$5$$ 0 0
$$6$$ 3.02830 1.23630
$$7$$ −0.726109 −0.274444 −0.137222 0.990540i $$-0.543817\pi$$
−0.137222 + 0.990540i $$0.543817\pi$$
$$8$$ −3.92498 −1.38769
$$9$$ −1.37720 −0.459068
$$10$$ 0 0
$$11$$ −0.273891 −0.0825811 −0.0412906 0.999147i $$-0.513147\pi$$
−0.0412906 + 0.999147i $$0.513147\pi$$
$$12$$ −4.65109 −1.34266
$$13$$ 5.95328 1.65114 0.825571 0.564298i $$-0.190853\pi$$
0.825571 + 0.564298i $$0.190853\pi$$
$$14$$ 1.72611 0.461322
$$15$$ 0 0
$$16$$ 2.02830 0.507074
$$17$$ −5.27389 −1.27911 −0.639553 0.768747i $$-0.720881\pi$$
−0.639553 + 0.768747i $$0.720881\pi$$
$$18$$ 3.27389 0.771663
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ 0.924984 0.201848
$$22$$ 0.651093 0.138814
$$23$$ 3.67939 0.767206 0.383603 0.923498i $$-0.374683\pi$$
0.383603 + 0.923498i $$0.374683\pi$$
$$24$$ 5.00000 1.02062
$$25$$ 0 0
$$26$$ −14.1522 −2.77547
$$27$$ 5.57608 1.07312
$$28$$ −2.65109 −0.501010
$$29$$ −2.27389 −0.422251 −0.211125 0.977459i $$-0.567713\pi$$
−0.211125 + 0.977459i $$0.567713\pi$$
$$30$$ 0 0
$$31$$ 3.19887 0.574535 0.287267 0.957850i $$-0.407253\pi$$
0.287267 + 0.957850i $$0.407253\pi$$
$$32$$ 3.02830 0.535332
$$33$$ 0.348907 0.0607368
$$34$$ 12.5371 2.15010
$$35$$ 0 0
$$36$$ −5.02830 −0.838049
$$37$$ −8.12386 −1.33555 −0.667777 0.744361i $$-0.732754\pi$$
−0.667777 + 0.744361i $$0.732754\pi$$
$$38$$ −2.37720 −0.385633
$$39$$ −7.58383 −1.21438
$$40$$ 0 0
$$41$$ −9.43380 −1.47331 −0.736656 0.676268i $$-0.763596\pi$$
−0.736656 + 0.676268i $$0.763596\pi$$
$$42$$ −2.19887 −0.339294
$$43$$ −9.81100 −1.49616 −0.748082 0.663607i $$-0.769025\pi$$
−0.748082 + 0.663607i $$0.769025\pi$$
$$44$$ −1.00000 −0.150756
$$45$$ 0 0
$$46$$ −8.74666 −1.28962
$$47$$ −12.1599 −1.77370 −0.886852 0.462053i $$-0.847113\pi$$
−0.886852 + 0.462053i $$0.847113\pi$$
$$48$$ −2.58383 −0.372943
$$49$$ −6.47277 −0.924681
$$50$$ 0 0
$$51$$ 6.71836 0.940758
$$52$$ 21.7360 3.01424
$$53$$ −5.69781 −0.782655 −0.391327 0.920252i $$-0.627984\pi$$
−0.391327 + 0.920252i $$0.627984\pi$$
$$54$$ −13.2555 −1.80384
$$55$$ 0 0
$$56$$ 2.84997 0.380843
$$57$$ −1.27389 −0.168731
$$58$$ 5.40550 0.709777
$$59$$ −4.20662 −0.547656 −0.273828 0.961779i $$-0.588290\pi$$
−0.273828 + 0.961779i $$0.588290\pi$$
$$60$$ 0 0
$$61$$ −0.103312 −0.0132278 −0.00661389 0.999978i $$-0.502105\pi$$
−0.00661389 + 0.999978i $$0.502105\pi$$
$$62$$ −7.60437 −0.965756
$$63$$ 1.00000 0.125988
$$64$$ −11.2555 −1.40693
$$65$$ 0 0
$$66$$ −0.829422 −0.102095
$$67$$ 11.7827 1.43949 0.719743 0.694241i $$-0.244260\pi$$
0.719743 + 0.694241i $$0.244260\pi$$
$$68$$ −19.2555 −2.33507
$$69$$ −4.68714 −0.564265
$$70$$ 0 0
$$71$$ 5.75441 0.682922 0.341461 0.939896i $$-0.389078\pi$$
0.341461 + 0.939896i $$0.389078\pi$$
$$72$$ 5.40550 0.637044
$$73$$ −6.67939 −0.781763 −0.390882 0.920441i $$-0.627830\pi$$
−0.390882 + 0.920441i $$0.627830\pi$$
$$74$$ 19.3121 2.24498
$$75$$ 0 0
$$76$$ 3.65109 0.418809
$$77$$ 0.198875 0.0226639
$$78$$ 18.0283 2.04130
$$79$$ 3.87826 0.436339 0.218169 0.975911i $$-0.429991\pi$$
0.218169 + 0.975911i $$0.429991\pi$$
$$80$$ 0 0
$$81$$ −2.97170 −0.330189
$$82$$ 22.4260 2.47654
$$83$$ 0.488265 0.0535941 0.0267970 0.999641i $$-0.491469\pi$$
0.0267970 + 0.999641i $$0.491469\pi$$
$$84$$ 3.37720 0.368483
$$85$$ 0 0
$$86$$ 23.3227 2.51495
$$87$$ 2.89669 0.310558
$$88$$ 1.07502 0.114597
$$89$$ −16.4338 −1.74198 −0.870989 0.491302i $$-0.836521\pi$$
−0.870989 + 0.491302i $$0.836521\pi$$
$$90$$ 0 0
$$91$$ −4.32273 −0.453146
$$92$$ 13.4338 1.40057
$$93$$ −4.07502 −0.422559
$$94$$ 28.9066 2.98148
$$95$$ 0 0
$$96$$ −3.85772 −0.393727
$$97$$ 4.44447 0.451267 0.225634 0.974212i $$-0.427555\pi$$
0.225634 + 0.974212i $$0.427555\pi$$
$$98$$ 15.3871 1.55433
$$99$$ 0.377203 0.0379103
$$100$$ 0 0
$$101$$ 4.38495 0.436319 0.218160 0.975913i $$-0.429995\pi$$
0.218160 + 0.975913i $$0.429995\pi$$
$$102$$ −15.9709 −1.58136
$$103$$ −3.33048 −0.328162 −0.164081 0.986447i $$-0.552466\pi$$
−0.164081 + 0.986447i $$0.552466\pi$$
$$104$$ −23.3665 −2.29128
$$105$$ 0 0
$$106$$ 13.5449 1.31559
$$107$$ −16.4904 −1.59419 −0.797093 0.603857i $$-0.793630\pi$$
−0.797093 + 0.603857i $$0.793630\pi$$
$$108$$ 20.3588 1.95902
$$109$$ 7.79045 0.746190 0.373095 0.927793i $$-0.378297\pi$$
0.373095 + 0.927793i $$0.378297\pi$$
$$110$$ 0 0
$$111$$ 10.3489 0.982275
$$112$$ −1.47277 −0.139163
$$113$$ 0.142282 0.0133848 0.00669238 0.999978i $$-0.497870\pi$$
0.00669238 + 0.999978i $$0.497870\pi$$
$$114$$ 3.02830 0.283626
$$115$$ 0 0
$$116$$ −8.30219 −0.770839
$$117$$ −8.19887 −0.757986
$$118$$ 10.0000 0.920575
$$119$$ 3.82942 0.351043
$$120$$ 0 0
$$121$$ −10.9250 −0.993180
$$122$$ 0.245594 0.0222351
$$123$$ 12.0176 1.08359
$$124$$ 11.6794 1.04884
$$125$$ 0 0
$$126$$ −2.37720 −0.211778
$$127$$ 15.1316 1.34271 0.671357 0.741135i $$-0.265712\pi$$
0.671357 + 0.741135i $$0.265712\pi$$
$$128$$ 20.6999 1.82963
$$129$$ 12.4981 1.10040
$$130$$ 0 0
$$131$$ 5.58383 0.487861 0.243931 0.969793i $$-0.421563\pi$$
0.243931 + 0.969793i $$0.421563\pi$$
$$132$$ 1.27389 0.110878
$$133$$ −0.726109 −0.0629617
$$134$$ −28.0099 −2.41968
$$135$$ 0 0
$$136$$ 20.6999 1.77500
$$137$$ −12.8294 −1.09609 −0.548046 0.836448i $$-0.684628\pi$$
−0.548046 + 0.836448i $$0.684628\pi$$
$$138$$ 11.1423 0.948494
$$139$$ −15.2477 −1.29329 −0.646647 0.762789i $$-0.723829\pi$$
−0.646647 + 0.762789i $$0.723829\pi$$
$$140$$ 0 0
$$141$$ 15.4904 1.30453
$$142$$ −13.6794 −1.14795
$$143$$ −1.63055 −0.136353
$$144$$ −2.79338 −0.232781
$$145$$ 0 0
$$146$$ 15.8783 1.31409
$$147$$ 8.24559 0.680085
$$148$$ −29.6610 −2.43812
$$149$$ 13.8315 1.13312 0.566562 0.824019i $$-0.308273\pi$$
0.566562 + 0.824019i $$0.308273\pi$$
$$150$$ 0 0
$$151$$ −11.7077 −0.952758 −0.476379 0.879240i $$-0.658051\pi$$
−0.476379 + 0.879240i $$0.658051\pi$$
$$152$$ −3.92498 −0.318358
$$153$$ 7.26322 0.587196
$$154$$ −0.472765 −0.0380965
$$155$$ 0 0
$$156$$ −27.6893 −2.21692
$$157$$ 4.79045 0.382320 0.191160 0.981559i $$-0.438775\pi$$
0.191160 + 0.981559i $$0.438775\pi$$
$$158$$ −9.21942 −0.733458
$$159$$ 7.25839 0.575628
$$160$$ 0 0
$$161$$ −2.67164 −0.210555
$$162$$ 7.06434 0.555027
$$163$$ −12.8011 −1.00266 −0.501331 0.865256i $$-0.667156\pi$$
−0.501331 + 0.865256i $$0.667156\pi$$
$$164$$ −34.4437 −2.68960
$$165$$ 0 0
$$166$$ −1.16071 −0.0900882
$$167$$ 20.9426 1.62059 0.810294 0.586024i $$-0.199308\pi$$
0.810294 + 0.586024i $$0.199308\pi$$
$$168$$ −3.63055 −0.280103
$$169$$ 22.4415 1.72627
$$170$$ 0 0
$$171$$ −1.37720 −0.105317
$$172$$ −35.8209 −2.73132
$$173$$ −15.7282 −1.19580 −0.597898 0.801572i $$-0.703997\pi$$
−0.597898 + 0.801572i $$0.703997\pi$$
$$174$$ −6.88601 −0.522027
$$175$$ 0 0
$$176$$ −0.555531 −0.0418747
$$177$$ 5.35878 0.402791
$$178$$ 39.0665 2.92816
$$179$$ 3.41325 0.255118 0.127559 0.991831i $$-0.459286\pi$$
0.127559 + 0.991831i $$0.459286\pi$$
$$180$$ 0 0
$$181$$ 23.5109 1.74755 0.873777 0.486327i $$-0.161664\pi$$
0.873777 + 0.486327i $$0.161664\pi$$
$$182$$ 10.2760 0.761709
$$183$$ 0.131609 0.00972878
$$184$$ −14.4415 −1.06464
$$185$$ 0 0
$$186$$ 9.68714 0.710296
$$187$$ 1.44447 0.105630
$$188$$ −44.3969 −3.23798
$$189$$ −4.04884 −0.294510
$$190$$ 0 0
$$191$$ 12.4650 0.901937 0.450968 0.892540i $$-0.351079\pi$$
0.450968 + 0.892540i $$0.351079\pi$$
$$192$$ 14.3382 1.03477
$$193$$ −19.2993 −1.38919 −0.694596 0.719400i $$-0.744417\pi$$
−0.694596 + 0.719400i $$0.744417\pi$$
$$194$$ −10.5654 −0.758552
$$195$$ 0 0
$$196$$ −23.6327 −1.68805
$$197$$ −6.63055 −0.472407 −0.236203 0.971704i $$-0.575903\pi$$
−0.236203 + 0.971704i $$0.575903\pi$$
$$198$$ −0.896688 −0.0637248
$$199$$ −23.0849 −1.63644 −0.818222 0.574902i $$-0.805040\pi$$
−0.818222 + 0.574902i $$0.805040\pi$$
$$200$$ 0 0
$$201$$ −15.0099 −1.05871
$$202$$ −10.4239 −0.733425
$$203$$ 1.65109 0.115884
$$204$$ 24.5294 1.71740
$$205$$ 0 0
$$206$$ 7.91723 0.551620
$$207$$ −5.06727 −0.352199
$$208$$ 12.0750 0.837252
$$209$$ −0.273891 −0.0189454
$$210$$ 0 0
$$211$$ −7.54778 −0.519611 −0.259805 0.965661i $$-0.583658\pi$$
−0.259805 + 0.965661i $$0.583658\pi$$
$$212$$ −20.8032 −1.42877
$$213$$ −7.33048 −0.502276
$$214$$ 39.2010 2.67973
$$215$$ 0 0
$$216$$ −21.8860 −1.48915
$$217$$ −2.32273 −0.157677
$$218$$ −18.5195 −1.25430
$$219$$ 8.50881 0.574972
$$220$$ 0 0
$$221$$ −31.3969 −2.11199
$$222$$ −24.6015 −1.65114
$$223$$ 1.09344 0.0732221 0.0366111 0.999330i $$-0.488344\pi$$
0.0366111 + 0.999330i $$0.488344\pi$$
$$224$$ −2.19887 −0.146918
$$225$$ 0 0
$$226$$ −0.338233 −0.0224989
$$227$$ 20.1316 1.33618 0.668091 0.744080i $$-0.267112\pi$$
0.668091 + 0.744080i $$0.267112\pi$$
$$228$$ −4.65109 −0.308026
$$229$$ −5.51656 −0.364545 −0.182272 0.983248i $$-0.558345\pi$$
−0.182272 + 0.983248i $$0.558345\pi$$
$$230$$ 0 0
$$231$$ −0.253344 −0.0166688
$$232$$ 8.92498 0.585954
$$233$$ 18.1805 1.19104 0.595520 0.803340i $$-0.296946\pi$$
0.595520 + 0.803340i $$0.296946\pi$$
$$234$$ 19.4904 1.27413
$$235$$ 0 0
$$236$$ −15.3588 −0.999771
$$237$$ −4.94048 −0.320919
$$238$$ −9.10331 −0.590080
$$239$$ 21.9164 1.41766 0.708828 0.705381i $$-0.249224\pi$$
0.708828 + 0.705381i $$0.249224\pi$$
$$240$$ 0 0
$$241$$ −28.1882 −1.81576 −0.907881 0.419228i $$-0.862301\pi$$
−0.907881 + 0.419228i $$0.862301\pi$$
$$242$$ 25.9709 1.66947
$$243$$ −12.9426 −0.830269
$$244$$ −0.377203 −0.0241479
$$245$$ 0 0
$$246$$ −28.5683 −1.82145
$$247$$ 5.95328 0.378798
$$248$$ −12.5555 −0.797277
$$249$$ −0.621996 −0.0394174
$$250$$ 0 0
$$251$$ 9.00987 0.568698 0.284349 0.958721i $$-0.408223\pi$$
0.284349 + 0.958721i $$0.408223\pi$$
$$252$$ 3.65109 0.229997
$$253$$ −1.00775 −0.0633567
$$254$$ −35.9709 −2.25702
$$255$$ 0 0
$$256$$ −26.6970 −1.66856
$$257$$ −6.86064 −0.427955 −0.213978 0.976839i $$-0.568642\pi$$
−0.213978 + 0.976839i $$0.568642\pi$$
$$258$$ −29.7106 −1.84970
$$259$$ 5.89881 0.366534
$$260$$ 0 0
$$261$$ 3.13161 0.193842
$$262$$ −13.2739 −0.820064
$$263$$ 9.25547 0.570717 0.285358 0.958421i $$-0.407887\pi$$
0.285358 + 0.958421i $$0.407887\pi$$
$$264$$ −1.36945 −0.0842840
$$265$$ 0 0
$$266$$ 1.72611 0.105835
$$267$$ 20.9349 1.28119
$$268$$ 43.0197 2.62785
$$269$$ 0.498939 0.0304208 0.0152104 0.999884i $$-0.495158\pi$$
0.0152104 + 0.999884i $$0.495158\pi$$
$$270$$ 0 0
$$271$$ 3.71061 0.225403 0.112702 0.993629i $$-0.464050\pi$$
0.112702 + 0.993629i $$0.464050\pi$$
$$272$$ −10.6970 −0.648602
$$273$$ 5.50669 0.333280
$$274$$ 30.4981 1.84246
$$275$$ 0 0
$$276$$ −17.1132 −1.03009
$$277$$ −4.58675 −0.275591 −0.137796 0.990461i $$-0.544002\pi$$
−0.137796 + 0.990461i $$0.544002\pi$$
$$278$$ 36.2469 2.17395
$$279$$ −4.40550 −0.263750
$$280$$ 0 0
$$281$$ 27.2653 1.62651 0.813257 0.581905i $$-0.197692\pi$$
0.813257 + 0.581905i $$0.197692\pi$$
$$282$$ −36.8238 −2.19283
$$283$$ 10.2661 0.610259 0.305129 0.952311i $$-0.401300\pi$$
0.305129 + 0.952311i $$0.401300\pi$$
$$284$$ 21.0099 1.24671
$$285$$ 0 0
$$286$$ 3.87614 0.229201
$$287$$ 6.84997 0.404341
$$288$$ −4.17058 −0.245754
$$289$$ 10.8139 0.636113
$$290$$ 0 0
$$291$$ −5.66177 −0.331899
$$292$$ −24.3871 −1.42715
$$293$$ 1.87051 0.109277 0.0546383 0.998506i $$-0.482599\pi$$
0.0546383 + 0.998506i $$0.482599\pi$$
$$294$$ −19.6015 −1.14318
$$295$$ 0 0
$$296$$ 31.8860 1.85334
$$297$$ −1.52723 −0.0886192
$$298$$ −32.8804 −1.90471
$$299$$ 21.9044 1.26677
$$300$$ 0 0
$$301$$ 7.12386 0.410612
$$302$$ 27.8315 1.60153
$$303$$ −5.58595 −0.320904
$$304$$ 2.02830 0.116331
$$305$$ 0 0
$$306$$ −17.2661 −0.987040
$$307$$ −0.227171 −0.0129653 −0.00648266 0.999979i $$-0.502064\pi$$
−0.00648266 + 0.999979i $$0.502064\pi$$
$$308$$ 0.726109 0.0413739
$$309$$ 4.24267 0.241357
$$310$$ 0 0
$$311$$ 20.9554 1.18827 0.594136 0.804365i $$-0.297494\pi$$
0.594136 + 0.804365i $$0.297494\pi$$
$$312$$ 29.7664 1.68519
$$313$$ 11.2349 0.635035 0.317518 0.948252i $$-0.397151\pi$$
0.317518 + 0.948252i $$0.397151\pi$$
$$314$$ −11.3879 −0.642655
$$315$$ 0 0
$$316$$ 14.1599 0.796557
$$317$$ −18.6228 −1.04596 −0.522980 0.852345i $$-0.675180\pi$$
−0.522980 + 0.852345i $$0.675180\pi$$
$$318$$ −17.2547 −0.967594
$$319$$ 0.622797 0.0348699
$$320$$ 0 0
$$321$$ 21.0069 1.17249
$$322$$ 6.35103 0.353929
$$323$$ −5.27389 −0.293447
$$324$$ −10.8500 −0.602776
$$325$$ 0 0
$$326$$ 30.4309 1.68541
$$327$$ −9.92418 −0.548809
$$328$$ 37.0275 2.04450
$$329$$ 8.82942 0.486782
$$330$$ 0 0
$$331$$ −14.1054 −0.775305 −0.387652 0.921806i $$-0.626714\pi$$
−0.387652 + 0.921806i $$0.626714\pi$$
$$332$$ 1.78270 0.0978385
$$333$$ 11.1882 0.613110
$$334$$ −49.7848 −2.72410
$$335$$ 0 0
$$336$$ 1.87614 0.102352
$$337$$ −22.9709 −1.25130 −0.625652 0.780102i $$-0.715167\pi$$
−0.625652 + 0.780102i $$0.715167\pi$$
$$338$$ −53.3481 −2.90175
$$339$$ −0.181252 −0.00984424
$$340$$ 0 0
$$341$$ −0.876142 −0.0474457
$$342$$ 3.27389 0.177032
$$343$$ 9.78270 0.528216
$$344$$ 38.5080 2.07621
$$345$$ 0 0
$$346$$ 37.3892 2.01006
$$347$$ 3.93273 0.211120 0.105560 0.994413i $$-0.466336\pi$$
0.105560 + 0.994413i $$0.466336\pi$$
$$348$$ 10.5761 0.566937
$$349$$ −34.4252 −1.84274 −0.921371 0.388685i $$-0.872929\pi$$
−0.921371 + 0.388685i $$0.872929\pi$$
$$350$$ 0 0
$$351$$ 33.1960 1.77187
$$352$$ −0.829422 −0.0442083
$$353$$ 4.25547 0.226496 0.113248 0.993567i $$-0.463875\pi$$
0.113248 + 0.993567i $$0.463875\pi$$
$$354$$ −12.7389 −0.677065
$$355$$ 0 0
$$356$$ −60.0013 −3.18006
$$357$$ −4.87826 −0.258185
$$358$$ −8.11399 −0.428837
$$359$$ −20.2944 −1.07110 −0.535550 0.844504i $$-0.679896\pi$$
−0.535550 + 0.844504i $$0.679896\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ −55.8903 −2.93753
$$363$$ 13.9172 0.730465
$$364$$ −15.7827 −0.827238
$$365$$ 0 0
$$366$$ −0.312860 −0.0163535
$$367$$ −3.85289 −0.201119 −0.100560 0.994931i $$-0.532063\pi$$
−0.100560 + 0.994931i $$0.532063\pi$$
$$368$$ 7.46289 0.389030
$$369$$ 12.9922 0.676350
$$370$$ 0 0
$$371$$ 4.13724 0.214795
$$372$$ −14.8783 −0.771402
$$373$$ −14.6356 −0.757802 −0.378901 0.925437i $$-0.623698\pi$$
−0.378901 + 0.925437i $$0.623698\pi$$
$$374$$ −3.43380 −0.177557
$$375$$ 0 0
$$376$$ 47.7274 2.46135
$$377$$ −13.5371 −0.697197
$$378$$ 9.62492 0.495052
$$379$$ −22.0099 −1.13057 −0.565286 0.824895i $$-0.691234\pi$$
−0.565286 + 0.824895i $$0.691234\pi$$
$$380$$ 0 0
$$381$$ −19.2760 −0.987540
$$382$$ −29.6319 −1.51610
$$383$$ 3.08569 0.157671 0.0788357 0.996888i $$-0.474880\pi$$
0.0788357 + 0.996888i $$0.474880\pi$$
$$384$$ −26.3695 −1.34566
$$385$$ 0 0
$$386$$ 45.8783 2.33514
$$387$$ 13.5117 0.686840
$$388$$ 16.2272 0.823810
$$389$$ 8.77203 0.444760 0.222380 0.974960i $$-0.428618\pi$$
0.222380 + 0.974960i $$0.428618\pi$$
$$390$$ 0 0
$$391$$ −19.4047 −0.981338
$$392$$ 25.4055 1.28317
$$393$$ −7.11319 −0.358813
$$394$$ 15.7622 0.794086
$$395$$ 0 0
$$396$$ 1.37720 0.0692070
$$397$$ 1.59450 0.0800257 0.0400129 0.999199i $$-0.487260\pi$$
0.0400129 + 0.999199i $$0.487260\pi$$
$$398$$ 54.8775 2.75076
$$399$$ 0.924984 0.0463071
$$400$$ 0 0
$$401$$ −17.5526 −0.876535 −0.438268 0.898844i $$-0.644408\pi$$
−0.438268 + 0.898844i $$0.644408\pi$$
$$402$$ 35.6815 1.77963
$$403$$ 19.0438 0.948639
$$404$$ 16.0099 0.796521
$$405$$ 0 0
$$406$$ −3.92498 −0.194794
$$407$$ 2.22505 0.110292
$$408$$ −26.3695 −1.30548
$$409$$ 36.6815 1.81378 0.906892 0.421363i $$-0.138448\pi$$
0.906892 + 0.421363i $$0.138448\pi$$
$$410$$ 0 0
$$411$$ 16.3433 0.806155
$$412$$ −12.1599 −0.599076
$$413$$ 3.05447 0.150301
$$414$$ 12.0459 0.592025
$$415$$ 0 0
$$416$$ 18.0283 0.883910
$$417$$ 19.4239 0.951194
$$418$$ 0.651093 0.0318460
$$419$$ −18.8187 −0.919356 −0.459678 0.888086i $$-0.652035\pi$$
−0.459678 + 0.888086i $$0.652035\pi$$
$$420$$ 0 0
$$421$$ −33.7819 −1.64643 −0.823215 0.567730i $$-0.807822\pi$$
−0.823215 + 0.567730i $$0.807822\pi$$
$$422$$ 17.9426 0.873432
$$423$$ 16.7467 0.814250
$$424$$ 22.3638 1.08608
$$425$$ 0 0
$$426$$ 17.4260 0.844295
$$427$$ 0.0750160 0.00363028
$$428$$ −60.2079 −2.91026
$$429$$ 2.07714 0.100285
$$430$$ 0 0
$$431$$ −12.7651 −0.614872 −0.307436 0.951569i $$-0.599471\pi$$
−0.307436 + 0.951569i $$0.599471\pi$$
$$432$$ 11.3099 0.544150
$$433$$ 16.0771 0.772618 0.386309 0.922369i $$-0.373750\pi$$
0.386309 + 0.922369i $$0.373750\pi$$
$$434$$ 5.52161 0.265046
$$435$$ 0 0
$$436$$ 28.4437 1.36220
$$437$$ 3.67939 0.176009
$$438$$ −20.2272 −0.966492
$$439$$ 1.36945 0.0653604 0.0326802 0.999466i $$-0.489596\pi$$
0.0326802 + 0.999466i $$0.489596\pi$$
$$440$$ 0 0
$$441$$ 8.91431 0.424491
$$442$$ 74.6369 3.55012
$$443$$ −4.62280 −0.219636 −0.109818 0.993952i $$-0.535027\pi$$
−0.109818 + 0.993952i $$0.535027\pi$$
$$444$$ 37.7848 1.79319
$$445$$ 0 0
$$446$$ −2.59933 −0.123082
$$447$$ −17.6199 −0.833391
$$448$$ 8.17270 0.386124
$$449$$ 23.2555 1.09749 0.548747 0.835989i $$-0.315105\pi$$
0.548747 + 0.835989i $$0.315105\pi$$
$$450$$ 0 0
$$451$$ 2.58383 0.121668
$$452$$ 0.519485 0.0244345
$$453$$ 14.9143 0.700735
$$454$$ −47.8569 −2.24604
$$455$$ 0 0
$$456$$ 5.00000 0.234146
$$457$$ 35.8443 1.67673 0.838364 0.545111i $$-0.183513\pi$$
0.838364 + 0.545111i $$0.183513\pi$$
$$458$$ 13.1140 0.612776
$$459$$ −29.4076 −1.37263
$$460$$ 0 0
$$461$$ −14.8812 −0.693086 −0.346543 0.938034i $$-0.612645\pi$$
−0.346543 + 0.938034i $$0.612645\pi$$
$$462$$ 0.602251 0.0280193
$$463$$ 29.9554 1.39215 0.696073 0.717971i $$-0.254929\pi$$
0.696073 + 0.717971i $$0.254929\pi$$
$$464$$ −4.61212 −0.214112
$$465$$ 0 0
$$466$$ −43.2186 −2.00206
$$467$$ 6.73598 0.311704 0.155852 0.987780i $$-0.450188\pi$$
0.155852 + 0.987780i $$0.450188\pi$$
$$468$$ −29.9349 −1.38374
$$469$$ −8.55553 −0.395058
$$470$$ 0 0
$$471$$ −6.10251 −0.281189
$$472$$ 16.5109 0.759977
$$473$$ 2.68714 0.123555
$$474$$ 11.7445 0.539444
$$475$$ 0 0
$$476$$ 13.9816 0.640845
$$477$$ 7.84704 0.359291
$$478$$ −52.0998 −2.38299
$$479$$ 16.6978 0.762943 0.381471 0.924381i $$-0.375418\pi$$
0.381471 + 0.924381i $$0.375418\pi$$
$$480$$ 0 0
$$481$$ −48.3636 −2.20519
$$482$$ 67.0091 3.05218
$$483$$ 3.40338 0.154859
$$484$$ −39.8881 −1.81310
$$485$$ 0 0
$$486$$ 30.7672 1.39563
$$487$$ −3.64042 −0.164963 −0.0824816 0.996593i $$-0.526285\pi$$
−0.0824816 + 0.996593i $$0.526285\pi$$
$$488$$ 0.405499 0.0183561
$$489$$ 16.3072 0.737439
$$490$$ 0 0
$$491$$ −33.3249 −1.50393 −0.751965 0.659203i $$-0.770894\pi$$
−0.751965 + 0.659203i $$0.770894\pi$$
$$492$$ 43.8775 1.97815
$$493$$ 11.9922 0.540104
$$494$$ −14.1522 −0.636736
$$495$$ 0 0
$$496$$ 6.48827 0.291332
$$497$$ −4.17833 −0.187424
$$498$$ 1.47861 0.0662582
$$499$$ −37.9914 −1.70073 −0.850365 0.526193i $$-0.823619\pi$$
−0.850365 + 0.526193i $$0.823619\pi$$
$$500$$ 0 0
$$501$$ −26.6786 −1.19191
$$502$$ −21.4183 −0.955945
$$503$$ 42.1826 1.88083 0.940414 0.340032i $$-0.110438\pi$$
0.940414 + 0.340032i $$0.110438\pi$$
$$504$$ −3.92498 −0.174833
$$505$$ 0 0
$$506$$ 2.39563 0.106499
$$507$$ −28.5881 −1.26964
$$508$$ 55.2469 2.45119
$$509$$ −21.9971 −0.975003 −0.487502 0.873122i $$-0.662092\pi$$
−0.487502 + 0.873122i $$0.662092\pi$$
$$510$$ 0 0
$$511$$ 4.84997 0.214550
$$512$$ 22.0643 0.975115
$$513$$ 5.57608 0.246190
$$514$$ 16.3091 0.719365
$$515$$ 0 0
$$516$$ 45.6319 2.00883
$$517$$ 3.33048 0.146474
$$518$$ −14.0227 −0.616121
$$519$$ 20.0360 0.879485
$$520$$ 0 0
$$521$$ 20.0977 0.880496 0.440248 0.897876i $$-0.354891\pi$$
0.440248 + 0.897876i $$0.354891\pi$$
$$522$$ −7.44447 −0.325836
$$523$$ 4.64817 0.203250 0.101625 0.994823i $$-0.467596\pi$$
0.101625 + 0.994823i $$0.467596\pi$$
$$524$$ 20.3871 0.890614
$$525$$ 0 0
$$526$$ −22.0021 −0.959338
$$527$$ −16.8705 −0.734891
$$528$$ 0.707686 0.0307981
$$529$$ −9.46209 −0.411395
$$530$$ 0 0
$$531$$ 5.79338 0.251411
$$532$$ −2.65109 −0.114939
$$533$$ −56.1620 −2.43265
$$534$$ −49.7664 −2.15360
$$535$$ 0 0
$$536$$ −46.2469 −1.99756
$$537$$ −4.34811 −0.187635
$$538$$ −1.18608 −0.0511355
$$539$$ 1.77283 0.0763612
$$540$$ 0 0
$$541$$ 20.0673 0.862759 0.431380 0.902171i $$-0.358027\pi$$
0.431380 + 0.902171i $$0.358027\pi$$
$$542$$ −8.82087 −0.378889
$$543$$ −29.9504 −1.28529
$$544$$ −15.9709 −0.684747
$$545$$ 0 0
$$546$$ −13.0905 −0.560222
$$547$$ 37.2010 1.59060 0.795300 0.606216i $$-0.207313\pi$$
0.795300 + 0.606216i $$0.207313\pi$$
$$548$$ −46.8414 −2.00097
$$549$$ 0.142282 0.00607245
$$550$$ 0 0
$$551$$ −2.27389 −0.0968710
$$552$$ 18.3969 0.783026
$$553$$ −2.81604 −0.119750
$$554$$ 10.9036 0.463251
$$555$$ 0 0
$$556$$ −55.6708 −2.36097
$$557$$ −44.8393 −1.89990 −0.949951 0.312399i $$-0.898867\pi$$
−0.949951 + 0.312399i $$0.898867\pi$$
$$558$$ 10.4728 0.443347
$$559$$ −58.4076 −2.47038
$$560$$ 0 0
$$561$$ −1.84010 −0.0776889
$$562$$ −64.8152 −2.73407
$$563$$ 21.9172 0.923701 0.461851 0.886958i $$-0.347186\pi$$
0.461851 + 0.886958i $$0.347186\pi$$
$$564$$ 56.5569 2.38147
$$565$$ 0 0
$$566$$ −24.4047 −1.02581
$$567$$ 2.15778 0.0906183
$$568$$ −22.5860 −0.947685
$$569$$ 9.90656 0.415305 0.207652 0.978203i $$-0.433418\pi$$
0.207652 + 0.978203i $$0.433418\pi$$
$$570$$ 0 0
$$571$$ 17.6404 0.738229 0.369114 0.929384i $$-0.379661\pi$$
0.369114 + 0.929384i $$0.379661\pi$$
$$572$$ −5.95328 −0.248919
$$573$$ −15.8791 −0.663357
$$574$$ −16.2838 −0.679671
$$575$$ 0 0
$$576$$ 15.5011 0.645878
$$577$$ 12.7048 0.528906 0.264453 0.964399i $$-0.414809\pi$$
0.264453 + 0.964399i $$0.414809\pi$$
$$578$$ −25.7069 −1.06927
$$579$$ 24.5851 1.02172
$$580$$ 0 0
$$581$$ −0.354534 −0.0147085
$$582$$ 13.4592 0.557900
$$583$$ 1.56058 0.0646325
$$584$$ 26.2165 1.08485
$$585$$ 0 0
$$586$$ −4.44659 −0.183687
$$587$$ −15.0438 −0.620924 −0.310462 0.950586i $$-0.600484\pi$$
−0.310462 + 0.950586i $$0.600484\pi$$
$$588$$ 30.1054 1.24153
$$589$$ 3.19887 0.131807
$$590$$ 0 0
$$591$$ 8.44659 0.347446
$$592$$ −16.4776 −0.677225
$$593$$ 16.4231 0.674417 0.337208 0.941430i $$-0.390517\pi$$
0.337208 + 0.941430i $$0.390517\pi$$
$$594$$ 3.63055 0.148963
$$595$$ 0 0
$$596$$ 50.5003 2.06857
$$597$$ 29.4076 1.20357
$$598$$ −52.0713 −2.12935
$$599$$ −19.1260 −0.781466 −0.390733 0.920504i $$-0.627778\pi$$
−0.390733 + 0.920504i $$0.627778\pi$$
$$600$$ 0 0
$$601$$ 31.4124 1.28134 0.640670 0.767816i $$-0.278657\pi$$
0.640670 + 0.767816i $$0.278657\pi$$
$$602$$ −16.9349 −0.690213
$$603$$ −16.2272 −0.660821
$$604$$ −42.7459 −1.73930
$$605$$ 0 0
$$606$$ 13.2789 0.539420
$$607$$ 41.5315 1.68571 0.842855 0.538140i $$-0.180873\pi$$
0.842855 + 0.538140i $$0.180873\pi$$
$$608$$ 3.02830 0.122814
$$609$$ −2.10331 −0.0852305
$$610$$ 0 0
$$611$$ −72.3913 −2.92864
$$612$$ 26.5187 1.07195
$$613$$ −21.7274 −0.877563 −0.438781 0.898594i $$-0.644590\pi$$
−0.438781 + 0.898594i $$0.644590\pi$$
$$614$$ 0.540031 0.0217939
$$615$$ 0 0
$$616$$ −0.780579 −0.0314504
$$617$$ −33.6065 −1.35295 −0.676473 0.736467i $$-0.736493\pi$$
−0.676473 + 0.736467i $$0.736493\pi$$
$$618$$ −10.0857 −0.405706
$$619$$ −27.6036 −1.10948 −0.554741 0.832023i $$-0.687183\pi$$
−0.554741 + 0.832023i $$0.687183\pi$$
$$620$$ 0 0
$$621$$ 20.5166 0.823301
$$622$$ −49.8152 −1.99741
$$623$$ 11.9327 0.478075
$$624$$ −15.3822 −0.615783
$$625$$ 0 0
$$626$$ −26.7077 −1.06745
$$627$$ 0.348907 0.0139340
$$628$$ 17.4904 0.697942
$$629$$ 42.8443 1.70832
$$630$$ 0 0
$$631$$ 1.94048 0.0772495 0.0386247 0.999254i $$-0.487702\pi$$
0.0386247 + 0.999254i $$0.487702\pi$$
$$632$$ −15.2221 −0.605504
$$633$$ 9.61505 0.382164
$$634$$ 44.2702 1.75819
$$635$$ 0 0
$$636$$ 26.5011 1.05084
$$637$$ −38.5342 −1.52678
$$638$$ −1.48052 −0.0586142
$$639$$ −7.92498 −0.313508
$$640$$ 0 0
$$641$$ 1.01975 0.0402775 0.0201388 0.999797i $$-0.493589\pi$$
0.0201388 + 0.999797i $$0.493589\pi$$
$$642$$ −49.9378 −1.97089
$$643$$ −36.9866 −1.45861 −0.729305 0.684189i $$-0.760156\pi$$
−0.729305 + 0.684189i $$0.760156\pi$$
$$644$$ −9.75441 −0.384377
$$645$$ 0 0
$$646$$ 12.5371 0.493266
$$647$$ −24.1182 −0.948186 −0.474093 0.880475i $$-0.657224\pi$$
−0.474093 + 0.880475i $$0.657224\pi$$
$$648$$ 11.6639 0.458201
$$649$$ 1.15215 0.0452260
$$650$$ 0 0
$$651$$ 2.95891 0.115969
$$652$$ −46.7381 −1.83041
$$653$$ −37.2603 −1.45811 −0.729054 0.684456i $$-0.760040\pi$$
−0.729054 + 0.684456i $$0.760040\pi$$
$$654$$ 23.5918 0.922512
$$655$$ 0 0
$$656$$ −19.1345 −0.747078
$$657$$ 9.19887 0.358882
$$658$$ −20.9893 −0.818249
$$659$$ 21.4386 0.835130 0.417565 0.908647i $$-0.362884\pi$$
0.417565 + 0.908647i $$0.362884\pi$$
$$660$$ 0 0
$$661$$ 0.783503 0.0304747 0.0152374 0.999884i $$-0.495150\pi$$
0.0152374 + 0.999884i $$0.495150\pi$$
$$662$$ 33.5315 1.30324
$$663$$ 39.9963 1.55333
$$664$$ −1.91643 −0.0743720
$$665$$ 0 0
$$666$$ −26.5966 −1.03060
$$667$$ −8.36653 −0.323953
$$668$$ 76.4634 2.95846
$$669$$ −1.39292 −0.0538535
$$670$$ 0 0
$$671$$ 0.0282963 0.00109237
$$672$$ 2.80113 0.108056
$$673$$ 50.1903 1.93469 0.967347 0.253454i $$-0.0815667\pi$$
0.967347 + 0.253454i $$0.0815667\pi$$
$$674$$ 54.6065 2.10336
$$675$$ 0 0
$$676$$ 81.9362 3.15139
$$677$$ 29.8804 1.14840 0.574198 0.818716i $$-0.305314\pi$$
0.574198 + 0.818716i $$0.305314\pi$$
$$678$$ 0.430872 0.0165475
$$679$$ −3.22717 −0.123847
$$680$$ 0 0
$$681$$ −25.6455 −0.982736
$$682$$ 2.08277 0.0797532
$$683$$ −12.3326 −0.471894 −0.235947 0.971766i $$-0.575819\pi$$
−0.235947 + 0.971766i $$0.575819\pi$$
$$684$$ −5.02830 −0.192262
$$685$$ 0 0
$$686$$ −23.2555 −0.887898
$$687$$ 7.02750 0.268116
$$688$$ −19.8996 −0.758666
$$689$$ −33.9207 −1.29227
$$690$$ 0 0
$$691$$ 3.62200 0.137787 0.0688936 0.997624i $$-0.478053\pi$$
0.0688936 + 0.997624i $$0.478053\pi$$
$$692$$ −57.4252 −2.18298
$$693$$ −0.273891 −0.0104042
$$694$$ −9.34891 −0.354880
$$695$$ 0 0
$$696$$ −11.3695 −0.430958
$$697$$ 49.7528 1.88452
$$698$$ 81.8358 3.09753
$$699$$ −23.1599 −0.875988
$$700$$ 0 0
$$701$$ 34.1209 1.28873 0.644365 0.764718i $$-0.277122\pi$$
0.644365 + 0.764718i $$0.277122\pi$$
$$702$$ −78.9135 −2.97840
$$703$$ −8.12386 −0.306397
$$704$$ 3.08277 0.116186
$$705$$ 0 0
$$706$$ −10.1161 −0.380725
$$707$$ −3.18396 −0.119745
$$708$$ 19.5654 0.735313
$$709$$ −17.1209 −0.642990 −0.321495 0.946911i $$-0.604185\pi$$
−0.321495 + 0.946911i $$0.604185\pi$$
$$710$$ 0 0
$$711$$ −5.34116 −0.200309
$$712$$ 64.5024 2.41733
$$713$$ 11.7699 0.440786
$$714$$ 11.5966 0.433993
$$715$$ 0 0
$$716$$ 12.4621 0.465730
$$717$$ −27.9191 −1.04266
$$718$$ 48.2440 1.80045
$$719$$ −7.02750 −0.262081 −0.131041 0.991377i $$-0.541832\pi$$
−0.131041 + 0.991377i $$0.541832\pi$$
$$720$$ 0 0
$$721$$ 2.41830 0.0900620
$$722$$ −2.37720 −0.0884703
$$723$$ 35.9087 1.33546
$$724$$ 85.8406 3.19024
$$725$$ 0 0
$$726$$ −33.0841 −1.22787
$$727$$ 11.8938 0.441115 0.220558 0.975374i $$-0.429212\pi$$
0.220558 + 0.975374i $$0.429212\pi$$
$$728$$ 16.9667 0.628826
$$729$$ 25.4026 0.940836
$$730$$ 0 0
$$731$$ 51.7421 1.91375
$$732$$ 0.480515 0.0177604
$$733$$ −20.7154 −0.765142 −0.382571 0.923926i $$-0.624961\pi$$
−0.382571 + 0.923926i $$0.624961\pi$$
$$734$$ 9.15910 0.338069
$$735$$ 0 0
$$736$$ 11.1423 0.410710
$$737$$ −3.22717 −0.118874
$$738$$ −30.8852 −1.13690
$$739$$ 33.8620 1.24563 0.622816 0.782368i $$-0.285988\pi$$
0.622816 + 0.782368i $$0.285988\pi$$
$$740$$ 0 0
$$741$$ −7.58383 −0.278599
$$742$$ −9.83505 −0.361056
$$743$$ −42.7381 −1.56791 −0.783955 0.620818i $$-0.786800\pi$$
−0.783955 + 0.620818i $$0.786800\pi$$
$$744$$ 15.9944 0.586382
$$745$$ 0 0
$$746$$ 34.7918 1.27382
$$747$$ −0.672440 −0.0246033
$$748$$ 5.27389 0.192833
$$749$$ 11.9738 0.437514
$$750$$ 0 0
$$751$$ 11.9581 0.436358 0.218179 0.975909i $$-0.429988\pi$$
0.218179 + 0.975909i $$0.429988\pi$$
$$752$$ −24.6639 −0.899400
$$753$$ −11.4776 −0.418267
$$754$$ 32.1805 1.17194
$$755$$ 0 0
$$756$$ −14.7827 −0.537642
$$757$$ −29.1103 −1.05803 −0.529015 0.848612i $$-0.677439\pi$$
−0.529015 + 0.848612i $$0.677439\pi$$
$$758$$ 52.3219 1.90042
$$759$$ 1.28376 0.0465977
$$760$$ 0 0
$$761$$ −22.3014 −0.808425 −0.404212 0.914665i $$-0.632454\pi$$
−0.404212 + 0.914665i $$0.632454\pi$$
$$762$$ 45.8230 1.65999
$$763$$ −5.65672 −0.204787
$$764$$ 45.5109 1.64653
$$765$$ 0 0
$$766$$ −7.33531 −0.265036
$$767$$ −25.0432 −0.904258
$$768$$ 34.0091 1.22720
$$769$$ 3.95891 0.142762 0.0713809 0.997449i $$-0.477259\pi$$
0.0713809 + 0.997449i $$0.477259\pi$$
$$770$$ 0 0
$$771$$ 8.73971 0.314753
$$772$$ −70.4634 −2.53603
$$773$$ −27.8139 −1.00040 −0.500199 0.865911i $$-0.666740\pi$$
−0.500199 + 0.865911i $$0.666740\pi$$
$$774$$ −32.1201 −1.15453
$$775$$ 0 0
$$776$$ −17.4445 −0.626220
$$777$$ −7.51444 −0.269579
$$778$$ −20.8529 −0.747612
$$779$$ −9.43380 −0.338001
$$780$$ 0 0
$$781$$ −1.57608 −0.0563965
$$782$$ 46.1289 1.64957
$$783$$ −12.6794 −0.453124
$$784$$ −13.1287 −0.468882
$$785$$ 0 0
$$786$$ 16.9095 0.603141
$$787$$ 1.82460 0.0650398 0.0325199 0.999471i $$-0.489647\pi$$
0.0325199 + 0.999471i $$0.489647\pi$$
$$788$$ −24.2087 −0.862401
$$789$$ −11.7905 −0.419751
$$790$$ 0 0
$$791$$ −0.103312 −0.00367336
$$792$$ −1.48052 −0.0526078
$$793$$ −0.615047 −0.0218410
$$794$$ −3.79045 −0.134518
$$795$$ 0 0
$$796$$ −84.2851 −2.98741
$$797$$ 21.0360 0.745135 0.372567 0.928005i $$-0.378478\pi$$
0.372567 + 0.928005i $$0.378478\pi$$
$$798$$ −2.19887 −0.0778393
$$799$$ 64.1300 2.26876
$$800$$ 0 0
$$801$$ 22.6327 0.799686
$$802$$ 41.7261 1.47340
$$803$$ 1.82942 0.0645589
$$804$$ −54.8024 −1.93273
$$805$$ 0 0
$$806$$ −45.2710 −1.59460
$$807$$ −0.635593 −0.0223739
$$808$$ −17.2109 −0.605476
$$809$$ −0.0819654 −0.00288175 −0.00144088 0.999999i $$-0.500459\pi$$
−0.00144088 + 0.999999i $$0.500459\pi$$
$$810$$ 0 0
$$811$$ −6.72531 −0.236158 −0.118079 0.993004i $$-0.537674\pi$$
−0.118079 + 0.993004i $$0.537674\pi$$
$$812$$ 6.02830 0.211552
$$813$$ −4.72691 −0.165780
$$814$$ −5.28939 −0.185393
$$815$$ 0 0
$$816$$ 13.6268 0.477034
$$817$$ −9.81100 −0.343243
$$818$$ −87.1994 −3.04886
$$819$$ 5.95328 0.208024
$$820$$ 0 0
$$821$$ 30.9426 1.07990 0.539952 0.841696i $$-0.318442\pi$$
0.539952 + 0.841696i $$0.318442\pi$$
$$822$$ −38.8513 −1.35509
$$823$$ −26.5908 −0.926896 −0.463448 0.886124i $$-0.653388\pi$$
−0.463448 + 0.886124i $$0.653388\pi$$
$$824$$ 13.0721 0.455388
$$825$$ 0 0
$$826$$ −7.26109 −0.252646
$$827$$ 5.57900 0.194001 0.0970004 0.995284i $$-0.469075\pi$$
0.0970004 + 0.995284i $$0.469075\pi$$
$$828$$ −18.5011 −0.642956
$$829$$ 18.9765 0.659082 0.329541 0.944141i $$-0.393106\pi$$
0.329541 + 0.944141i $$0.393106\pi$$
$$830$$ 0 0
$$831$$ 5.84302 0.202692
$$832$$ −67.0069 −2.32305
$$833$$ 34.1367 1.18276
$$834$$ −46.1746 −1.59890
$$835$$ 0 0
$$836$$ −1.00000 −0.0345857
$$837$$ 17.8372 0.616543
$$838$$ 44.7360 1.54538
$$839$$ 12.9143 0.445852 0.222926 0.974835i $$-0.428439\pi$$
0.222926 + 0.974835i $$0.428439\pi$$
$$840$$ 0 0
$$841$$ −23.8294 −0.821704
$$842$$ 80.3064 2.76754
$$843$$ −34.7331 −1.19627
$$844$$ −27.5577 −0.948574
$$845$$ 0 0
$$846$$ −39.8102 −1.36870
$$847$$ 7.93273 0.272572
$$848$$ −11.5569 −0.396864
$$849$$ −13.0779 −0.448834
$$850$$ 0 0
$$851$$ −29.8908 −1.02464
$$852$$ −26.7643 −0.916929
$$853$$ −6.46077 −0.221213 −0.110606 0.993864i $$-0.535279\pi$$
−0.110606 + 0.993864i $$0.535279\pi$$
$$854$$ −0.178328 −0.00610227
$$855$$ 0 0
$$856$$ 64.7245 2.21224
$$857$$ 12.4055 0.423764 0.211882 0.977295i $$-0.432041\pi$$
0.211882 + 0.977295i $$0.432041\pi$$
$$858$$ −4.93778 −0.168573
$$859$$ 40.3425 1.37647 0.688234 0.725489i $$-0.258386\pi$$
0.688234 + 0.725489i $$0.258386\pi$$
$$860$$ 0 0
$$861$$ −8.72611 −0.297385
$$862$$ 30.3452 1.03356
$$863$$ −1.83235 −0.0623738 −0.0311869 0.999514i $$-0.509929\pi$$
−0.0311869 + 0.999514i $$0.509929\pi$$
$$864$$ 16.8860 0.574474
$$865$$ 0 0
$$866$$ −38.2186 −1.29872
$$867$$ −13.7758 −0.467849
$$868$$ −8.48052 −0.287847
$$869$$ −1.06222 −0.0360333
$$870$$ 0 0
$$871$$ 70.1457 2.37680
$$872$$ −30.5774 −1.03548
$$873$$ −6.12094 −0.207162
$$874$$ −8.74666 −0.295860
$$875$$ 0 0
$$876$$ 31.0665 1.04964
$$877$$ 8.14419 0.275010 0.137505 0.990501i $$-0.456092\pi$$
0.137505 + 0.990501i $$0.456092\pi$$
$$878$$ −3.25547 −0.109867
$$879$$ −2.38283 −0.0803709
$$880$$ 0 0
$$881$$ 12.1706 0.410037 0.205019 0.978758i $$-0.434275\pi$$
0.205019 + 0.978758i $$0.434275\pi$$
$$882$$ −21.1911 −0.713542
$$883$$ −46.7614 −1.57364 −0.786822 0.617179i $$-0.788275\pi$$
−0.786822 + 0.617179i $$0.788275\pi$$
$$884$$ −114.633 −3.85553
$$885$$ 0 0
$$886$$ 10.9893 0.369194
$$887$$ 34.8804 1.17117 0.585584 0.810611i $$-0.300865\pi$$
0.585584 + 0.810611i $$0.300865\pi$$
$$888$$ −40.6193 −1.36309
$$889$$ −10.9872 −0.368499
$$890$$ 0 0
$$891$$ 0.813922 0.0272674
$$892$$ 3.99225 0.133670
$$893$$ −12.1599 −0.406916
$$894$$ 41.8860 1.40088
$$895$$ 0 0
$$896$$ −15.0304 −0.502131
$$897$$ −27.9039 −0.931683
$$898$$ −55.2830 −1.84482
$$899$$ −7.27389 −0.242598
$$900$$ 0 0
$$901$$ 30.0496 1.00110
$$902$$ −6.14228 −0.204516
$$903$$ −9.07502 −0.301998
$$904$$ −0.558455 −0.0185739
$$905$$ 0 0
$$906$$ −35.4543 −1.17789
$$907$$ 25.5080 0.846980 0.423490 0.905901i $$-0.360805\pi$$
0.423490 + 0.905901i $$0.360805\pi$$
$$908$$ 73.5024 2.43926
$$909$$ −6.03897 −0.200300
$$910$$ 0 0
$$911$$ −21.5032 −0.712432 −0.356216 0.934404i $$-0.615933\pi$$
−0.356216 + 0.934404i $$0.615933\pi$$
$$912$$ −2.58383 −0.0855591
$$913$$ −0.133731 −0.00442586
$$914$$ −85.2093 −2.81847
$$915$$ 0 0
$$916$$ −20.1415 −0.665493
$$917$$ −4.05447 −0.133890
$$918$$ 69.9079 2.30730
$$919$$ 37.1386 1.22509 0.612544 0.790436i $$-0.290146\pi$$
0.612544 + 0.790436i $$0.290146\pi$$
$$920$$ 0 0
$$921$$ 0.289391 0.00953575
$$922$$ 35.3756 1.16503
$$923$$ 34.2576 1.12760
$$924$$ −0.924984 −0.0304297
$$925$$ 0 0
$$926$$ −71.2101 −2.34011
$$927$$ 4.58675 0.150649
$$928$$ −6.88601 −0.226044
$$929$$ −3.36170 −0.110294 −0.0551469 0.998478i $$-0.517563\pi$$
−0.0551469 + 0.998478i $$0.517563\pi$$
$$930$$ 0 0
$$931$$ −6.47277 −0.212136
$$932$$ 66.3785 2.17430
$$933$$ −26.6949 −0.873951
$$934$$ −16.0128 −0.523955
$$935$$ 0 0
$$936$$ 32.1805 1.05185
$$937$$ −10.0694 −0.328953 −0.164476 0.986381i $$-0.552593\pi$$
−0.164476 + 0.986381i $$0.552593\pi$$
$$938$$ 20.3382 0.664067
$$939$$ −14.3121 −0.467056
$$940$$ 0 0
$$941$$ 20.8139 0.678514 0.339257 0.940694i $$-0.389824\pi$$
0.339257 + 0.940694i $$0.389824\pi$$
$$942$$ 14.5069 0.472661
$$943$$ −34.7106 −1.13033
$$944$$ −8.53228 −0.277702
$$945$$ 0 0
$$946$$ −6.38788 −0.207688
$$947$$ 30.4904 0.990804 0.495402 0.868664i $$-0.335021\pi$$
0.495402 + 0.868664i $$0.335021\pi$$
$$948$$ −18.0382 −0.585852
$$949$$ −39.7643 −1.29080
$$950$$ 0 0
$$951$$ 23.7234 0.769284
$$952$$ −15.0304 −0.487139
$$953$$ −7.58383 −0.245664 −0.122832 0.992427i $$-0.539198\pi$$
−0.122832 + 0.992427i $$0.539198\pi$$
$$954$$ −18.6540 −0.603946
$$955$$ 0 0
$$956$$ 80.0189 2.58800
$$957$$ −0.793375 −0.0256462
$$958$$ −39.6941 −1.28246
$$959$$ 9.31556 0.300815
$$960$$ 0 0
$$961$$ −20.7672 −0.669910
$$962$$ 114.970 3.70678
$$963$$ 22.7106 0.731839
$$964$$ −102.918 −3.31476
$$965$$ 0 0
$$966$$ −8.09052 −0.260308
$$967$$ −54.6687 −1.75803 −0.879014 0.476797i $$-0.841798\pi$$
−0.879014 + 0.476797i $$0.841798\pi$$
$$968$$ 42.8804 1.37823
$$969$$ 6.71836 0.215825
$$970$$ 0 0
$$971$$ −39.9632 −1.28248 −0.641239 0.767341i $$-0.721579\pi$$
−0.641239 + 0.767341i $$0.721579\pi$$
$$972$$ −47.2547 −1.51569
$$973$$ 11.0715 0.354936
$$974$$ 8.65402 0.277293
$$975$$ 0 0
$$976$$ −0.209548 −0.00670747
$$977$$ −15.2400 −0.487570 −0.243785 0.969829i $$-0.578389\pi$$
−0.243785 + 0.969829i $$0.578389\pi$$
$$978$$ −38.7656 −1.23959
$$979$$ 4.50106 0.143855
$$980$$ 0 0
$$981$$ −10.7290 −0.342552
$$982$$ 79.2199 2.52801
$$983$$ 45.3609 1.44679 0.723394 0.690435i $$-0.242581\pi$$
0.723394 + 0.690435i $$0.242581\pi$$
$$984$$ −47.1690 −1.50369
$$985$$ 0 0
$$986$$ −28.5080 −0.907880
$$987$$ −11.2477 −0.358019
$$988$$ 21.7360 0.691514
$$989$$ −36.0985 −1.14787
$$990$$ 0 0
$$991$$ −16.3537 −0.519493 −0.259747 0.965677i $$-0.583639\pi$$
−0.259747 + 0.965677i $$0.583639\pi$$
$$992$$ 9.68714 0.307567
$$993$$ 17.9688 0.570222
$$994$$ 9.93273 0.315047
$$995$$ 0 0
$$996$$ −2.27097 −0.0719583
$$997$$ 33.2037 1.05157 0.525786 0.850617i $$-0.323771\pi$$
0.525786 + 0.850617i $$0.323771\pi$$
$$998$$ 90.3134 2.85882
$$999$$ −45.2993 −1.43321
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 475.2.a.e.1.1 3
3.2 odd 2 4275.2.a.bm.1.3 3
4.3 odd 2 7600.2.a.cc.1.2 3
5.2 odd 4 475.2.b.b.324.1 6
5.3 odd 4 475.2.b.b.324.6 6
5.4 even 2 475.2.a.g.1.3 yes 3
15.14 odd 2 4275.2.a.ba.1.1 3
19.18 odd 2 9025.2.a.bc.1.3 3
20.19 odd 2 7600.2.a.bh.1.2 3
95.94 odd 2 9025.2.a.y.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
475.2.a.e.1.1 3 1.1 even 1 trivial
475.2.a.g.1.3 yes 3 5.4 even 2
475.2.b.b.324.1 6 5.2 odd 4
475.2.b.b.324.6 6 5.3 odd 4
4275.2.a.ba.1.1 3 15.14 odd 2
4275.2.a.bm.1.3 3 3.2 odd 2
7600.2.a.bh.1.2 3 20.19 odd 2
7600.2.a.cc.1.2 3 4.3 odd 2
9025.2.a.y.1.1 3 95.94 odd 2
9025.2.a.bc.1.3 3 19.18 odd 2