# Properties

 Label 475.2.a.d.1.2 Level $475$ Weight $2$ Character 475.1 Self dual yes Analytic conductor $3.793$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Learn more

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: $$\Q(\zeta_{14})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 2x + 1$$ x^3 - x^2 - 2*x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$0.445042$$ 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)

 $$f(q)$$ $$=$$ $$q-1.44504 q^{2} -2.24698 q^{3} +0.0881460 q^{4} +3.24698 q^{6} -1.35690 q^{7} +2.76271 q^{8} +2.04892 q^{9} +O(q^{10})$$ $$q-1.44504 q^{2} -2.24698 q^{3} +0.0881460 q^{4} +3.24698 q^{6} -1.35690 q^{7} +2.76271 q^{8} +2.04892 q^{9} +4.85086 q^{11} -0.198062 q^{12} -0.198062 q^{13} +1.96077 q^{14} -4.16852 q^{16} +1.13706 q^{17} -2.96077 q^{18} -1.00000 q^{19} +3.04892 q^{21} -7.00969 q^{22} -2.55496 q^{23} -6.20775 q^{24} +0.286208 q^{26} +2.13706 q^{27} -0.119605 q^{28} -10.2349 q^{29} +2.51573 q^{31} +0.498271 q^{32} -10.8998 q^{33} -1.64310 q^{34} +0.180604 q^{36} +0.137063 q^{37} +1.44504 q^{38} +0.445042 q^{39} -11.7506 q^{41} -4.40581 q^{42} +7.59179 q^{43} +0.427583 q^{44} +3.69202 q^{46} -2.69202 q^{47} +9.36658 q^{48} -5.15883 q^{49} -2.55496 q^{51} -0.0174584 q^{52} -12.8780 q^{53} -3.08815 q^{54} -3.74871 q^{56} +2.24698 q^{57} +14.7899 q^{58} +5.82371 q^{59} -7.58211 q^{61} -3.63533 q^{62} -2.78017 q^{63} +7.61702 q^{64} +15.7506 q^{66} -8.01507 q^{67} +0.100228 q^{68} +5.74094 q^{69} -8.82371 q^{71} +5.66056 q^{72} +11.9705 q^{73} -0.198062 q^{74} -0.0881460 q^{76} -6.58211 q^{77} -0.643104 q^{78} +10.7409 q^{79} -10.9487 q^{81} +16.9801 q^{82} -3.77479 q^{83} +0.268750 q^{84} -10.9705 q^{86} +22.9976 q^{87} +13.4015 q^{88} +9.36658 q^{89} +0.268750 q^{91} -0.225209 q^{92} -5.65279 q^{93} +3.89008 q^{94} -1.11960 q^{96} -0.198062 q^{97} +7.45473 q^{98} +9.93900 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 4 q^{2} - 2 q^{3} + 4 q^{4} + 5 q^{6} - 9 q^{8} - 3 q^{9}+O(q^{10})$$ 3 * q - 4 * q^2 - 2 * q^3 + 4 * q^4 + 5 * q^6 - 9 * q^8 - 3 * q^9 $$3 q - 4 q^{2} - 2 q^{3} + 4 q^{4} + 5 q^{6} - 9 q^{8} - 3 q^{9} + q^{11} - 5 q^{12} - 5 q^{13} - 7 q^{14} + 18 q^{16} - 2 q^{17} + 4 q^{18} - 3 q^{19} + q^{22} - 8 q^{23} - q^{24} + 9 q^{26} + q^{27} + 21 q^{28} - 7 q^{29} - 5 q^{31} - 27 q^{32} - 10 q^{33} - 9 q^{34} - 11 q^{36} - 5 q^{37} + 4 q^{38} + q^{39} + q^{41} - 5 q^{43} - 15 q^{44} + 6 q^{46} - 3 q^{47} + 2 q^{48} - 7 q^{49} - 8 q^{51} - 16 q^{52} - 19 q^{53} - 13 q^{54} - 35 q^{56} + 2 q^{57} + 21 q^{58} + 10 q^{59} - 17 q^{61} + 23 q^{62} - 7 q^{63} + 49 q^{64} + 11 q^{66} + q^{67} + 23 q^{68} + 3 q^{69} - 19 q^{71} + 37 q^{72} + q^{73} - 5 q^{74} - 4 q^{76} - 14 q^{77} - 6 q^{78} + 18 q^{79} - q^{81} - 6 q^{82} - 13 q^{83} - 7 q^{84} + 2 q^{86} + 28 q^{87} + 46 q^{88} + 2 q^{89} - 7 q^{91} + q^{92} + q^{93} + 11 q^{94} + 18 q^{96} - 5 q^{97} + 20 q^{99}+O(q^{100})$$ 3 * q - 4 * q^2 - 2 * q^3 + 4 * q^4 + 5 * q^6 - 9 * q^8 - 3 * q^9 + q^11 - 5 * q^12 - 5 * q^13 - 7 * q^14 + 18 * q^16 - 2 * q^17 + 4 * q^18 - 3 * q^19 + q^22 - 8 * q^23 - q^24 + 9 * q^26 + q^27 + 21 * q^28 - 7 * q^29 - 5 * q^31 - 27 * q^32 - 10 * q^33 - 9 * q^34 - 11 * q^36 - 5 * q^37 + 4 * q^38 + q^39 + q^41 - 5 * q^43 - 15 * q^44 + 6 * q^46 - 3 * q^47 + 2 * q^48 - 7 * q^49 - 8 * q^51 - 16 * q^52 - 19 * q^53 - 13 * q^54 - 35 * q^56 + 2 * q^57 + 21 * q^58 + 10 * q^59 - 17 * q^61 + 23 * q^62 - 7 * q^63 + 49 * q^64 + 11 * q^66 + q^67 + 23 * q^68 + 3 * q^69 - 19 * q^71 + 37 * q^72 + q^73 - 5 * q^74 - 4 * q^76 - 14 * q^77 - 6 * q^78 + 18 * q^79 - q^81 - 6 * q^82 - 13 * q^83 - 7 * q^84 + 2 * q^86 + 28 * q^87 + 46 * q^88 + 2 * q^89 - 7 * q^91 + q^92 + q^93 + 11 * q^94 + 18 * q^96 - 5 * q^97 + 20 * q^99

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.44504 −1.02180 −0.510899 0.859640i $$-0.670688\pi$$
−0.510899 + 0.859640i $$0.670688\pi$$
$$3$$ −2.24698 −1.29729 −0.648647 0.761089i $$-0.724665\pi$$
−0.648647 + 0.761089i $$0.724665\pi$$
$$4$$ 0.0881460 0.0440730
$$5$$ 0 0
$$6$$ 3.24698 1.32557
$$7$$ −1.35690 −0.512858 −0.256429 0.966563i $$-0.582546\pi$$
−0.256429 + 0.966563i $$0.582546\pi$$
$$8$$ 2.76271 0.976765
$$9$$ 2.04892 0.682972
$$10$$ 0 0
$$11$$ 4.85086 1.46259 0.731294 0.682062i $$-0.238917\pi$$
0.731294 + 0.682062i $$0.238917\pi$$
$$12$$ −0.198062 −0.0571757
$$13$$ −0.198062 −0.0549326 −0.0274663 0.999623i $$-0.508744\pi$$
−0.0274663 + 0.999623i $$0.508744\pi$$
$$14$$ 1.96077 0.524038
$$15$$ 0 0
$$16$$ −4.16852 −1.04213
$$17$$ 1.13706 0.275778 0.137889 0.990448i $$-0.455968\pi$$
0.137889 + 0.990448i $$0.455968\pi$$
$$18$$ −2.96077 −0.697860
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ 3.04892 0.665328
$$22$$ −7.00969 −1.49447
$$23$$ −2.55496 −0.532746 −0.266373 0.963870i $$-0.585825\pi$$
−0.266373 + 0.963870i $$0.585825\pi$$
$$24$$ −6.20775 −1.26715
$$25$$ 0 0
$$26$$ 0.286208 0.0561301
$$27$$ 2.13706 0.411278
$$28$$ −0.119605 −0.0226032
$$29$$ −10.2349 −1.90057 −0.950286 0.311377i $$-0.899210\pi$$
−0.950286 + 0.311377i $$0.899210\pi$$
$$30$$ 0 0
$$31$$ 2.51573 0.451838 0.225919 0.974146i $$-0.427461\pi$$
0.225919 + 0.974146i $$0.427461\pi$$
$$32$$ 0.498271 0.0880827
$$33$$ −10.8998 −1.89741
$$34$$ −1.64310 −0.281790
$$35$$ 0 0
$$36$$ 0.180604 0.0301006
$$37$$ 0.137063 0.0225331 0.0112665 0.999937i $$-0.496414\pi$$
0.0112665 + 0.999937i $$0.496414\pi$$
$$38$$ 1.44504 0.234417
$$39$$ 0.445042 0.0712637
$$40$$ 0 0
$$41$$ −11.7506 −1.83514 −0.917570 0.397575i $$-0.869852\pi$$
−0.917570 + 0.397575i $$0.869852\pi$$
$$42$$ −4.40581 −0.679832
$$43$$ 7.59179 1.15774 0.578869 0.815421i $$-0.303494\pi$$
0.578869 + 0.815421i $$0.303494\pi$$
$$44$$ 0.427583 0.0644606
$$45$$ 0 0
$$46$$ 3.69202 0.544359
$$47$$ −2.69202 −0.392672 −0.196336 0.980537i $$-0.562904\pi$$
−0.196336 + 0.980537i $$0.562904\pi$$
$$48$$ 9.36658 1.35195
$$49$$ −5.15883 −0.736976
$$50$$ 0 0
$$51$$ −2.55496 −0.357766
$$52$$ −0.0174584 −0.00242104
$$53$$ −12.8780 −1.76893 −0.884465 0.466607i $$-0.845476\pi$$
−0.884465 + 0.466607i $$0.845476\pi$$
$$54$$ −3.08815 −0.420243
$$55$$ 0 0
$$56$$ −3.74871 −0.500942
$$57$$ 2.24698 0.297620
$$58$$ 14.7899 1.94200
$$59$$ 5.82371 0.758182 0.379091 0.925359i $$-0.376237\pi$$
0.379091 + 0.925359i $$0.376237\pi$$
$$60$$ 0 0
$$61$$ −7.58211 −0.970789 −0.485395 0.874295i $$-0.661324\pi$$
−0.485395 + 0.874295i $$0.661324\pi$$
$$62$$ −3.63533 −0.461688
$$63$$ −2.78017 −0.350268
$$64$$ 7.61702 0.952128
$$65$$ 0 0
$$66$$ 15.7506 1.93877
$$67$$ −8.01507 −0.979196 −0.489598 0.871948i $$-0.662856\pi$$
−0.489598 + 0.871948i $$0.662856\pi$$
$$68$$ 0.100228 0.0121544
$$69$$ 5.74094 0.691128
$$70$$ 0 0
$$71$$ −8.82371 −1.04718 −0.523591 0.851970i $$-0.675408\pi$$
−0.523591 + 0.851970i $$0.675408\pi$$
$$72$$ 5.66056 0.667104
$$73$$ 11.9705 1.40104 0.700518 0.713635i $$-0.252952\pi$$
0.700518 + 0.713635i $$0.252952\pi$$
$$74$$ −0.198062 −0.0230243
$$75$$ 0 0
$$76$$ −0.0881460 −0.0101110
$$77$$ −6.58211 −0.750101
$$78$$ −0.643104 −0.0728172
$$79$$ 10.7409 1.20845 0.604225 0.796814i $$-0.293483\pi$$
0.604225 + 0.796814i $$0.293483\pi$$
$$80$$ 0 0
$$81$$ −10.9487 −1.21652
$$82$$ 16.9801 1.87514
$$83$$ −3.77479 −0.414337 −0.207169 0.978305i $$-0.566425\pi$$
−0.207169 + 0.978305i $$0.566425\pi$$
$$84$$ 0.268750 0.0293230
$$85$$ 0 0
$$86$$ −10.9705 −1.18298
$$87$$ 22.9976 2.46560
$$88$$ 13.4015 1.42860
$$89$$ 9.36658 0.992856 0.496428 0.868078i $$-0.334645\pi$$
0.496428 + 0.868078i $$0.334645\pi$$
$$90$$ 0 0
$$91$$ 0.268750 0.0281726
$$92$$ −0.225209 −0.0234797
$$93$$ −5.65279 −0.586167
$$94$$ 3.89008 0.401232
$$95$$ 0 0
$$96$$ −1.11960 −0.114269
$$97$$ −0.198062 −0.0201102 −0.0100551 0.999949i $$-0.503201\pi$$
−0.0100551 + 0.999949i $$0.503201\pi$$
$$98$$ 7.45473 0.753042
$$99$$ 9.93900 0.998907
$$100$$ 0 0
$$101$$ 11.5090 1.14519 0.572595 0.819838i $$-0.305937\pi$$
0.572595 + 0.819838i $$0.305937\pi$$
$$102$$ 3.69202 0.365565
$$103$$ −15.1564 −1.49341 −0.746704 0.665156i $$-0.768365\pi$$
−0.746704 + 0.665156i $$0.768365\pi$$
$$104$$ −0.547188 −0.0536562
$$105$$ 0 0
$$106$$ 18.6093 1.80749
$$107$$ −2.65279 −0.256455 −0.128228 0.991745i $$-0.540929\pi$$
−0.128228 + 0.991745i $$0.540929\pi$$
$$108$$ 0.188374 0.0181263
$$109$$ −2.49934 −0.239393 −0.119696 0.992811i $$-0.538192\pi$$
−0.119696 + 0.992811i $$0.538192\pi$$
$$110$$ 0 0
$$111$$ −0.307979 −0.0292320
$$112$$ 5.65625 0.534465
$$113$$ −8.52781 −0.802229 −0.401114 0.916028i $$-0.631377\pi$$
−0.401114 + 0.916028i $$0.631377\pi$$
$$114$$ −3.24698 −0.304108
$$115$$ 0 0
$$116$$ −0.902165 −0.0837639
$$117$$ −0.405813 −0.0375174
$$118$$ −8.41550 −0.774710
$$119$$ −1.54288 −0.141435
$$120$$ 0 0
$$121$$ 12.5308 1.13916
$$122$$ 10.9565 0.991951
$$123$$ 26.4034 2.38072
$$124$$ 0.221751 0.0199139
$$125$$ 0 0
$$126$$ 4.01746 0.357904
$$127$$ −20.4088 −1.81099 −0.905494 0.424359i $$-0.860499\pi$$
−0.905494 + 0.424359i $$0.860499\pi$$
$$128$$ −12.0035 −1.06097
$$129$$ −17.0586 −1.50193
$$130$$ 0 0
$$131$$ −13.2131 −1.15444 −0.577218 0.816590i $$-0.695862\pi$$
−0.577218 + 0.816590i $$0.695862\pi$$
$$132$$ −0.960771 −0.0836244
$$133$$ 1.35690 0.117658
$$134$$ 11.5821 1.00054
$$135$$ 0 0
$$136$$ 3.14138 0.269371
$$137$$ 6.86054 0.586136 0.293068 0.956092i $$-0.405324\pi$$
0.293068 + 0.956092i $$0.405324\pi$$
$$138$$ −8.29590 −0.706194
$$139$$ 4.28621 0.363551 0.181776 0.983340i $$-0.441816\pi$$
0.181776 + 0.983340i $$0.441816\pi$$
$$140$$ 0 0
$$141$$ 6.04892 0.509411
$$142$$ 12.7506 1.07001
$$143$$ −0.960771 −0.0803437
$$144$$ −8.54096 −0.711746
$$145$$ 0 0
$$146$$ −17.2978 −1.43158
$$147$$ 11.5918 0.956075
$$148$$ 0.0120816 0.000993100 0
$$149$$ −15.3545 −1.25789 −0.628945 0.777450i $$-0.716513\pi$$
−0.628945 + 0.777450i $$0.716513\pi$$
$$150$$ 0 0
$$151$$ −10.2295 −0.832467 −0.416233 0.909258i $$-0.636650\pi$$
−0.416233 + 0.909258i $$0.636650\pi$$
$$152$$ −2.76271 −0.224085
$$153$$ 2.32975 0.188349
$$154$$ 9.51142 0.766452
$$155$$ 0 0
$$156$$ 0.0392287 0.00314081
$$157$$ −3.17092 −0.253067 −0.126533 0.991962i $$-0.540385\pi$$
−0.126533 + 0.991962i $$0.540385\pi$$
$$158$$ −15.5211 −1.23479
$$159$$ 28.9366 2.29482
$$160$$ 0 0
$$161$$ 3.46681 0.273223
$$162$$ 15.8213 1.24304
$$163$$ 4.63773 0.363255 0.181627 0.983367i $$-0.441864\pi$$
0.181627 + 0.983367i $$0.441864\pi$$
$$164$$ −1.03577 −0.0808801
$$165$$ 0 0
$$166$$ 5.45473 0.423369
$$167$$ 19.6286 1.51891 0.759454 0.650560i $$-0.225466\pi$$
0.759454 + 0.650560i $$0.225466\pi$$
$$168$$ 8.42327 0.649870
$$169$$ −12.9608 −0.996982
$$170$$ 0 0
$$171$$ −2.04892 −0.156685
$$172$$ 0.669186 0.0510250
$$173$$ −20.5646 −1.56350 −0.781751 0.623591i $$-0.785673\pi$$
−0.781751 + 0.623591i $$0.785673\pi$$
$$174$$ −33.2325 −2.51935
$$175$$ 0 0
$$176$$ −20.2209 −1.52421
$$177$$ −13.0858 −0.983585
$$178$$ −13.5351 −1.01450
$$179$$ 6.92154 0.517340 0.258670 0.965966i $$-0.416716\pi$$
0.258670 + 0.965966i $$0.416716\pi$$
$$180$$ 0 0
$$181$$ −17.6461 −1.31162 −0.655812 0.754925i $$-0.727673\pi$$
−0.655812 + 0.754925i $$0.727673\pi$$
$$182$$ −0.388355 −0.0287868
$$183$$ 17.0368 1.25940
$$184$$ −7.05861 −0.520367
$$185$$ 0 0
$$186$$ 8.16852 0.598945
$$187$$ 5.51573 0.403350
$$188$$ −0.237291 −0.0173062
$$189$$ −2.89977 −0.210927
$$190$$ 0 0
$$191$$ 6.92931 0.501387 0.250694 0.968066i $$-0.419341\pi$$
0.250694 + 0.968066i $$0.419341\pi$$
$$192$$ −17.1153 −1.23519
$$193$$ 21.0368 1.51426 0.757132 0.653262i $$-0.226600\pi$$
0.757132 + 0.653262i $$0.226600\pi$$
$$194$$ 0.286208 0.0205486
$$195$$ 0 0
$$196$$ −0.454731 −0.0324808
$$197$$ −21.5646 −1.53642 −0.768209 0.640199i $$-0.778852\pi$$
−0.768209 + 0.640199i $$0.778852\pi$$
$$198$$ −14.3623 −1.02068
$$199$$ 21.9909 1.55889 0.779447 0.626468i $$-0.215500\pi$$
0.779447 + 0.626468i $$0.215500\pi$$
$$200$$ 0 0
$$201$$ 18.0097 1.27031
$$202$$ −16.6310 −1.17015
$$203$$ 13.8877 0.974725
$$204$$ −0.225209 −0.0157678
$$205$$ 0 0
$$206$$ 21.9017 1.52596
$$207$$ −5.23490 −0.363851
$$208$$ 0.825627 0.0572469
$$209$$ −4.85086 −0.335541
$$210$$ 0 0
$$211$$ −20.6233 −1.41976 −0.709882 0.704321i $$-0.751252\pi$$
−0.709882 + 0.704321i $$0.751252\pi$$
$$212$$ −1.13514 −0.0779620
$$213$$ 19.8267 1.35850
$$214$$ 3.83340 0.262046
$$215$$ 0 0
$$216$$ 5.90408 0.401722
$$217$$ −3.41358 −0.231729
$$218$$ 3.61165 0.244611
$$219$$ −26.8974 −1.81756
$$220$$ 0 0
$$221$$ −0.225209 −0.0151492
$$222$$ 0.445042 0.0298693
$$223$$ −7.97716 −0.534190 −0.267095 0.963670i $$-0.586064\pi$$
−0.267095 + 0.963670i $$0.586064\pi$$
$$224$$ −0.676102 −0.0451740
$$225$$ 0 0
$$226$$ 12.3230 0.819717
$$227$$ −19.7942 −1.31379 −0.656893 0.753984i $$-0.728129\pi$$
−0.656893 + 0.753984i $$0.728129\pi$$
$$228$$ 0.198062 0.0131170
$$229$$ −4.03385 −0.266564 −0.133282 0.991078i $$-0.542552\pi$$
−0.133282 + 0.991078i $$0.542552\pi$$
$$230$$ 0 0
$$231$$ 14.7899 0.973101
$$232$$ −28.2760 −1.85641
$$233$$ −26.8159 −1.75677 −0.878385 0.477953i $$-0.841379\pi$$
−0.878385 + 0.477953i $$0.841379\pi$$
$$234$$ 0.586417 0.0383353
$$235$$ 0 0
$$236$$ 0.513337 0.0334154
$$237$$ −24.1347 −1.56772
$$238$$ 2.22952 0.144518
$$239$$ 3.36227 0.217487 0.108744 0.994070i $$-0.465317\pi$$
0.108744 + 0.994070i $$0.465317\pi$$
$$240$$ 0 0
$$241$$ 27.7506 1.78758 0.893788 0.448491i $$-0.148038\pi$$
0.893788 + 0.448491i $$0.148038\pi$$
$$242$$ −18.1075 −1.16400
$$243$$ 18.1903 1.16691
$$244$$ −0.668332 −0.0427856
$$245$$ 0 0
$$246$$ −38.1540 −2.43261
$$247$$ 0.198062 0.0126024
$$248$$ 6.95023 0.441340
$$249$$ 8.48188 0.537517
$$250$$ 0 0
$$251$$ −5.59419 −0.353102 −0.176551 0.984291i $$-0.556494\pi$$
−0.176551 + 0.984291i $$0.556494\pi$$
$$252$$ −0.245061 −0.0154374
$$253$$ −12.3937 −0.779187
$$254$$ 29.4916 1.85047
$$255$$ 0 0
$$256$$ 2.11146 0.131966
$$257$$ −10.4668 −0.652902 −0.326451 0.945214i $$-0.605853\pi$$
−0.326451 + 0.945214i $$0.605853\pi$$
$$258$$ 24.6504 1.53467
$$259$$ −0.185981 −0.0115563
$$260$$ 0 0
$$261$$ −20.9705 −1.29804
$$262$$ 19.0935 1.17960
$$263$$ −15.4795 −0.954506 −0.477253 0.878766i $$-0.658367\pi$$
−0.477253 + 0.878766i $$0.658367\pi$$
$$264$$ −30.1129 −1.85332
$$265$$ 0 0
$$266$$ −1.96077 −0.120223
$$267$$ −21.0465 −1.28803
$$268$$ −0.706496 −0.0431561
$$269$$ 9.13036 0.556688 0.278344 0.960481i $$-0.410215\pi$$
0.278344 + 0.960481i $$0.410215\pi$$
$$270$$ 0 0
$$271$$ 7.44265 0.452109 0.226054 0.974115i $$-0.427417\pi$$
0.226054 + 0.974115i $$0.427417\pi$$
$$272$$ −4.73987 −0.287397
$$273$$ −0.603875 −0.0365482
$$274$$ −9.91377 −0.598913
$$275$$ 0 0
$$276$$ 0.506041 0.0304601
$$277$$ 11.4155 0.685891 0.342946 0.939355i $$-0.388575\pi$$
0.342946 + 0.939355i $$0.388575\pi$$
$$278$$ −6.19375 −0.371476
$$279$$ 5.15452 0.308593
$$280$$ 0 0
$$281$$ 21.5060 1.28294 0.641471 0.767147i $$-0.278324\pi$$
0.641471 + 0.767147i $$0.278324\pi$$
$$282$$ −8.74094 −0.520515
$$283$$ 5.45712 0.324392 0.162196 0.986759i $$-0.448142\pi$$
0.162196 + 0.986759i $$0.448142\pi$$
$$284$$ −0.777775 −0.0461524
$$285$$ 0 0
$$286$$ 1.38835 0.0820951
$$287$$ 15.9444 0.941167
$$288$$ 1.02092 0.0601581
$$289$$ −15.7071 −0.923946
$$290$$ 0 0
$$291$$ 0.445042 0.0260888
$$292$$ 1.05515 0.0617479
$$293$$ −7.39075 −0.431772 −0.215886 0.976419i $$-0.569264\pi$$
−0.215886 + 0.976419i $$0.569264\pi$$
$$294$$ −16.7506 −0.976916
$$295$$ 0 0
$$296$$ 0.378666 0.0220095
$$297$$ 10.3666 0.601530
$$298$$ 22.1879 1.28531
$$299$$ 0.506041 0.0292651
$$300$$ 0 0
$$301$$ −10.3013 −0.593756
$$302$$ 14.7821 0.850613
$$303$$ −25.8605 −1.48565
$$304$$ 4.16852 0.239081
$$305$$ 0 0
$$306$$ −3.36658 −0.192455
$$307$$ 32.2131 1.83850 0.919250 0.393674i $$-0.128796\pi$$
0.919250 + 0.393674i $$0.128796\pi$$
$$308$$ −0.580186 −0.0330592
$$309$$ 34.0562 1.93739
$$310$$ 0 0
$$311$$ 14.8442 0.841735 0.420867 0.907122i $$-0.361726\pi$$
0.420867 + 0.907122i $$0.361726\pi$$
$$312$$ 1.22952 0.0696079
$$313$$ 13.1491 0.743234 0.371617 0.928386i $$-0.378804\pi$$
0.371617 + 0.928386i $$0.378804\pi$$
$$314$$ 4.58211 0.258583
$$315$$ 0 0
$$316$$ 0.946771 0.0532600
$$317$$ 5.13467 0.288392 0.144196 0.989549i $$-0.453940\pi$$
0.144196 + 0.989549i $$0.453940\pi$$
$$318$$ −41.8146 −2.34485
$$319$$ −49.6480 −2.77975
$$320$$ 0 0
$$321$$ 5.96077 0.332698
$$322$$ −5.00969 −0.279179
$$323$$ −1.13706 −0.0632679
$$324$$ −0.965083 −0.0536157
$$325$$ 0 0
$$326$$ −6.70171 −0.371173
$$327$$ 5.61596 0.310563
$$328$$ −32.4636 −1.79250
$$329$$ 3.65279 0.201385
$$330$$ 0 0
$$331$$ 2.00969 0.110462 0.0552312 0.998474i $$-0.482410\pi$$
0.0552312 + 0.998474i $$0.482410\pi$$
$$332$$ −0.332733 −0.0182611
$$333$$ 0.280831 0.0153895
$$334$$ −28.3642 −1.55202
$$335$$ 0 0
$$336$$ −12.7095 −0.693359
$$337$$ 1.31873 0.0718359 0.0359180 0.999355i $$-0.488564\pi$$
0.0359180 + 0.999355i $$0.488564\pi$$
$$338$$ 18.7289 1.01872
$$339$$ 19.1618 1.04073
$$340$$ 0 0
$$341$$ 12.2034 0.660853
$$342$$ 2.96077 0.160100
$$343$$ 16.4983 0.890823
$$344$$ 20.9739 1.13084
$$345$$ 0 0
$$346$$ 29.7168 1.59758
$$347$$ −12.0151 −0.645003 −0.322501 0.946569i $$-0.604524\pi$$
−0.322501 + 0.946569i $$0.604524\pi$$
$$348$$ 2.02715 0.108666
$$349$$ −10.5579 −0.565154 −0.282577 0.959245i $$-0.591189\pi$$
−0.282577 + 0.959245i $$0.591189\pi$$
$$350$$ 0 0
$$351$$ −0.423272 −0.0225926
$$352$$ 2.41704 0.128829
$$353$$ 1.01102 0.0538110 0.0269055 0.999638i $$-0.491435\pi$$
0.0269055 + 0.999638i $$0.491435\pi$$
$$354$$ 18.9095 1.00503
$$355$$ 0 0
$$356$$ 0.825627 0.0437581
$$357$$ 3.46681 0.183483
$$358$$ −10.0019 −0.528618
$$359$$ −5.14244 −0.271408 −0.135704 0.990749i $$-0.543330\pi$$
−0.135704 + 0.990749i $$0.543330\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 25.4993 1.34022
$$363$$ −28.1564 −1.47783
$$364$$ 0.0236892 0.00124165
$$365$$ 0 0
$$366$$ −24.6189 −1.28685
$$367$$ 29.5308 1.54149 0.770747 0.637141i $$-0.219883\pi$$
0.770747 + 0.637141i $$0.219883\pi$$
$$368$$ 10.6504 0.555190
$$369$$ −24.0761 −1.25335
$$370$$ 0 0
$$371$$ 17.4741 0.907210
$$372$$ −0.498271 −0.0258342
$$373$$ −8.78986 −0.455121 −0.227561 0.973764i $$-0.573075\pi$$
−0.227561 + 0.973764i $$0.573075\pi$$
$$374$$ −7.97046 −0.412143
$$375$$ 0 0
$$376$$ −7.43727 −0.383548
$$377$$ 2.02715 0.104403
$$378$$ 4.19029 0.215525
$$379$$ 19.1511 0.983724 0.491862 0.870673i $$-0.336316\pi$$
0.491862 + 0.870673i $$0.336316\pi$$
$$380$$ 0 0
$$381$$ 45.8582 2.34938
$$382$$ −10.0131 −0.512317
$$383$$ 6.99894 0.357629 0.178814 0.983883i $$-0.442774\pi$$
0.178814 + 0.983883i $$0.442774\pi$$
$$384$$ 26.9715 1.37638
$$385$$ 0 0
$$386$$ −30.3991 −1.54727
$$387$$ 15.5550 0.790703
$$388$$ −0.0174584 −0.000886316 0
$$389$$ 8.08575 0.409964 0.204982 0.978766i $$-0.434286\pi$$
0.204982 + 0.978766i $$0.434286\pi$$
$$390$$ 0 0
$$391$$ −2.90515 −0.146920
$$392$$ −14.2524 −0.719853
$$393$$ 29.6896 1.49764
$$394$$ 31.1618 1.56991
$$395$$ 0 0
$$396$$ 0.876083 0.0440248
$$397$$ 17.7006 0.888370 0.444185 0.895935i $$-0.353493\pi$$
0.444185 + 0.895935i $$0.353493\pi$$
$$398$$ −31.7778 −1.59288
$$399$$ −3.04892 −0.152637
$$400$$ 0 0
$$401$$ −15.5418 −0.776121 −0.388061 0.921634i $$-0.626855\pi$$
−0.388061 + 0.921634i $$0.626855\pi$$
$$402$$ −26.0248 −1.29800
$$403$$ −0.498271 −0.0248207
$$404$$ 1.01447 0.0504720
$$405$$ 0 0
$$406$$ −20.0683 −0.995973
$$407$$ 0.664874 0.0329566
$$408$$ −7.05861 −0.349453
$$409$$ 13.1661 0.651023 0.325512 0.945538i $$-0.394463\pi$$
0.325512 + 0.945538i $$0.394463\pi$$
$$410$$ 0 0
$$411$$ −15.4155 −0.760391
$$412$$ −1.33598 −0.0658190
$$413$$ −7.90217 −0.388840
$$414$$ 7.56465 0.371782
$$415$$ 0 0
$$416$$ −0.0986887 −0.00483861
$$417$$ −9.63102 −0.471633
$$418$$ 7.00969 0.342855
$$419$$ −25.1739 −1.22983 −0.614913 0.788595i $$-0.710809\pi$$
−0.614913 + 0.788595i $$0.710809\pi$$
$$420$$ 0 0
$$421$$ 26.9420 1.31307 0.656536 0.754295i $$-0.272021\pi$$
0.656536 + 0.754295i $$0.272021\pi$$
$$422$$ 29.8015 1.45071
$$423$$ −5.51573 −0.268184
$$424$$ −35.5782 −1.72783
$$425$$ 0 0
$$426$$ −28.6504 −1.38812
$$427$$ 10.2881 0.497877
$$428$$ −0.233833 −0.0113027
$$429$$ 2.15883 0.104229
$$430$$ 0 0
$$431$$ 18.9487 0.912726 0.456363 0.889794i $$-0.349152\pi$$
0.456363 + 0.889794i $$0.349152\pi$$
$$432$$ −8.90840 −0.428605
$$433$$ 3.68904 0.177284 0.0886419 0.996064i $$-0.471747\pi$$
0.0886419 + 0.996064i $$0.471747\pi$$
$$434$$ 4.93277 0.236781
$$435$$ 0 0
$$436$$ −0.220306 −0.0105508
$$437$$ 2.55496 0.122220
$$438$$ 38.8678 1.85718
$$439$$ −25.6926 −1.22624 −0.613121 0.789989i $$-0.710086\pi$$
−0.613121 + 0.789989i $$0.710086\pi$$
$$440$$ 0 0
$$441$$ −10.5700 −0.503334
$$442$$ 0.325437 0.0154795
$$443$$ 27.3653 1.30016 0.650081 0.759865i $$-0.274735\pi$$
0.650081 + 0.759865i $$0.274735\pi$$
$$444$$ −0.0271471 −0.00128834
$$445$$ 0 0
$$446$$ 11.5273 0.545835
$$447$$ 34.5013 1.63185
$$448$$ −10.3355 −0.488307
$$449$$ 7.55794 0.356681 0.178341 0.983969i $$-0.442927\pi$$
0.178341 + 0.983969i $$0.442927\pi$$
$$450$$ 0 0
$$451$$ −57.0006 −2.68405
$$452$$ −0.751692 −0.0353566
$$453$$ 22.9855 1.07995
$$454$$ 28.6034 1.34242
$$455$$ 0 0
$$456$$ 6.20775 0.290705
$$457$$ 7.85623 0.367499 0.183750 0.982973i $$-0.441176\pi$$
0.183750 + 0.982973i $$0.441176\pi$$
$$458$$ 5.82908 0.272375
$$459$$ 2.42998 0.113422
$$460$$ 0 0
$$461$$ 3.87907 0.180666 0.0903331 0.995912i $$-0.471207\pi$$
0.0903331 + 0.995912i $$0.471207\pi$$
$$462$$ −21.3720 −0.994314
$$463$$ 13.0954 0.608597 0.304298 0.952577i $$-0.401578\pi$$
0.304298 + 0.952577i $$0.401578\pi$$
$$464$$ 42.6644 1.98065
$$465$$ 0 0
$$466$$ 38.7502 1.79507
$$467$$ −6.44026 −0.298020 −0.149010 0.988836i $$-0.547609\pi$$
−0.149010 + 0.988836i $$0.547609\pi$$
$$468$$ −0.0357708 −0.00165351
$$469$$ 10.8756 0.502189
$$470$$ 0 0
$$471$$ 7.12498 0.328302
$$472$$ 16.0892 0.740566
$$473$$ 36.8267 1.69329
$$474$$ 34.8756 1.60189
$$475$$ 0 0
$$476$$ −0.135998 −0.00623348
$$477$$ −26.3860 −1.20813
$$478$$ −4.85862 −0.222228
$$479$$ 5.69096 0.260026 0.130013 0.991512i $$-0.458498\pi$$
0.130013 + 0.991512i $$0.458498\pi$$
$$480$$ 0 0
$$481$$ −0.0271471 −0.00123780
$$482$$ −40.1008 −1.82654
$$483$$ −7.78986 −0.354451
$$484$$ 1.10454 0.0502063
$$485$$ 0 0
$$486$$ −26.2857 −1.19235
$$487$$ 12.9632 0.587417 0.293709 0.955895i $$-0.405110\pi$$
0.293709 + 0.955895i $$0.405110\pi$$
$$488$$ −20.9472 −0.948233
$$489$$ −10.4209 −0.471248
$$490$$ 0 0
$$491$$ −19.6045 −0.884737 −0.442369 0.896833i $$-0.645862\pi$$
−0.442369 + 0.896833i $$0.645862\pi$$
$$492$$ 2.32736 0.104925
$$493$$ −11.6377 −0.524137
$$494$$ −0.286208 −0.0128771
$$495$$ 0 0
$$496$$ −10.4869 −0.470875
$$497$$ 11.9729 0.537056
$$498$$ −12.2567 −0.549234
$$499$$ 5.49827 0.246136 0.123068 0.992398i $$-0.460727\pi$$
0.123068 + 0.992398i $$0.460727\pi$$
$$500$$ 0 0
$$501$$ −44.1051 −1.97047
$$502$$ 8.08383 0.360799
$$503$$ 35.6939 1.59151 0.795757 0.605616i $$-0.207073\pi$$
0.795757 + 0.605616i $$0.207073\pi$$
$$504$$ −7.68079 −0.342130
$$505$$ 0 0
$$506$$ 17.9095 0.796173
$$507$$ 29.1226 1.29338
$$508$$ −1.79895 −0.0798157
$$509$$ −16.4450 −0.728914 −0.364457 0.931220i $$-0.618745\pi$$
−0.364457 + 0.931220i $$0.618745\pi$$
$$510$$ 0 0
$$511$$ −16.2427 −0.718533
$$512$$ 20.9558 0.926123
$$513$$ −2.13706 −0.0943537
$$514$$ 15.1250 0.667134
$$515$$ 0 0
$$516$$ −1.50365 −0.0661944
$$517$$ −13.0586 −0.574317
$$518$$ 0.268750 0.0118082
$$519$$ 46.2083 2.02832
$$520$$ 0 0
$$521$$ −26.5435 −1.16289 −0.581445 0.813586i $$-0.697513\pi$$
−0.581445 + 0.813586i $$0.697513\pi$$
$$522$$ 30.3032 1.32633
$$523$$ 24.1685 1.05682 0.528408 0.848991i $$-0.322789\pi$$
0.528408 + 0.848991i $$0.322789\pi$$
$$524$$ −1.16468 −0.0508795
$$525$$ 0 0
$$526$$ 22.3685 0.975313
$$527$$ 2.86054 0.124607
$$528$$ 45.4359 1.97735
$$529$$ −16.4722 −0.716182
$$530$$ 0 0
$$531$$ 11.9323 0.517818
$$532$$ 0.119605 0.00518553
$$533$$ 2.32736 0.100809
$$534$$ 30.4131 1.31610
$$535$$ 0 0
$$536$$ −22.1433 −0.956445
$$537$$ −15.5526 −0.671143
$$538$$ −13.1938 −0.568823
$$539$$ −25.0248 −1.07789
$$540$$ 0 0
$$541$$ 9.80386 0.421501 0.210750 0.977540i $$-0.432409\pi$$
0.210750 + 0.977540i $$0.432409\pi$$
$$542$$ −10.7549 −0.461964
$$543$$ 39.6504 1.70156
$$544$$ 0.566566 0.0242913
$$545$$ 0 0
$$546$$ 0.872625 0.0373449
$$547$$ −21.1739 −0.905331 −0.452665 0.891681i $$-0.649527\pi$$
−0.452665 + 0.891681i $$0.649527\pi$$
$$548$$ 0.604729 0.0258328
$$549$$ −15.5351 −0.663022
$$550$$ 0 0
$$551$$ 10.2349 0.436021
$$552$$ 15.8605 0.675070
$$553$$ −14.5743 −0.619764
$$554$$ −16.4959 −0.700843
$$555$$ 0 0
$$556$$ 0.377812 0.0160228
$$557$$ −24.4077 −1.03419 −0.517094 0.855928i $$-0.672986\pi$$
−0.517094 + 0.855928i $$0.672986\pi$$
$$558$$ −7.44850 −0.315320
$$559$$ −1.50365 −0.0635975
$$560$$ 0 0
$$561$$ −12.3937 −0.523264
$$562$$ −31.0771 −1.31091
$$563$$ −14.4849 −0.610464 −0.305232 0.952278i $$-0.598734\pi$$
−0.305232 + 0.952278i $$0.598734\pi$$
$$564$$ 0.533188 0.0224513
$$565$$ 0 0
$$566$$ −7.88577 −0.331464
$$567$$ 14.8562 0.623903
$$568$$ −24.3773 −1.02285
$$569$$ 0.811626 0.0340251 0.0170126 0.999855i $$-0.494584\pi$$
0.0170126 + 0.999855i $$0.494584\pi$$
$$570$$ 0 0
$$571$$ −37.6588 −1.57597 −0.787985 0.615694i $$-0.788876\pi$$
−0.787985 + 0.615694i $$0.788876\pi$$
$$572$$ −0.0846882 −0.00354099
$$573$$ −15.5700 −0.650447
$$574$$ −23.0403 −0.961683
$$575$$ 0 0
$$576$$ 15.6066 0.650277
$$577$$ −8.89307 −0.370223 −0.185112 0.982718i $$-0.559265\pi$$
−0.185112 + 0.982718i $$0.559265\pi$$
$$578$$ 22.6974 0.944087
$$579$$ −47.2693 −1.96445
$$580$$ 0 0
$$581$$ 5.12200 0.212496
$$582$$ −0.643104 −0.0266575
$$583$$ −62.4693 −2.58721
$$584$$ 33.0709 1.36848
$$585$$ 0 0
$$586$$ 10.6799 0.441184
$$587$$ −20.3327 −0.839222 −0.419611 0.907704i $$-0.637833\pi$$
−0.419611 + 0.907704i $$0.637833\pi$$
$$588$$ 1.02177 0.0421371
$$589$$ −2.51573 −0.103659
$$590$$ 0 0
$$591$$ 48.4553 1.99319
$$592$$ −0.571352 −0.0234824
$$593$$ −17.0049 −0.698308 −0.349154 0.937065i $$-0.613531\pi$$
−0.349154 + 0.937065i $$0.613531\pi$$
$$594$$ −14.9801 −0.614643
$$595$$ 0 0
$$596$$ −1.35344 −0.0554390
$$597$$ −49.4131 −2.02234
$$598$$ −0.731250 −0.0299030
$$599$$ −12.4004 −0.506668 −0.253334 0.967379i $$-0.581527\pi$$
−0.253334 + 0.967379i $$0.581527\pi$$
$$600$$ 0 0
$$601$$ −9.73125 −0.396946 −0.198473 0.980106i $$-0.563598\pi$$
−0.198473 + 0.980106i $$0.563598\pi$$
$$602$$ 14.8858 0.606699
$$603$$ −16.4222 −0.668764
$$604$$ −0.901691 −0.0366893
$$605$$ 0 0
$$606$$ 37.3696 1.51803
$$607$$ 23.4282 0.950920 0.475460 0.879737i $$-0.342282\pi$$
0.475460 + 0.879737i $$0.342282\pi$$
$$608$$ −0.498271 −0.0202076
$$609$$ −31.2054 −1.26450
$$610$$ 0 0
$$611$$ 0.533188 0.0215705
$$612$$ 0.205358 0.00830111
$$613$$ −28.1424 −1.13666 −0.568331 0.822800i $$-0.692411\pi$$
−0.568331 + 0.822800i $$0.692411\pi$$
$$614$$ −46.5493 −1.87858
$$615$$ 0 0
$$616$$ −18.1844 −0.732672
$$617$$ −36.4805 −1.46865 −0.734326 0.678797i $$-0.762502\pi$$
−0.734326 + 0.678797i $$0.762502\pi$$
$$618$$ −49.2127 −1.97962
$$619$$ −38.2097 −1.53578 −0.767888 0.640584i $$-0.778692\pi$$
−0.767888 + 0.640584i $$0.778692\pi$$
$$620$$ 0 0
$$621$$ −5.46011 −0.219107
$$622$$ −21.4504 −0.860083
$$623$$ −12.7095 −0.509195
$$624$$ −1.85517 −0.0742661
$$625$$ 0 0
$$626$$ −19.0011 −0.759435
$$627$$ 10.8998 0.435295
$$628$$ −0.279503 −0.0111534
$$629$$ 0.155850 0.00621413
$$630$$ 0 0
$$631$$ −12.5235 −0.498553 −0.249276 0.968432i $$-0.580193\pi$$
−0.249276 + 0.968432i $$0.580193\pi$$
$$632$$ 29.6741 1.18037
$$633$$ 46.3400 1.84185
$$634$$ −7.41981 −0.294678
$$635$$ 0 0
$$636$$ 2.55065 0.101140
$$637$$ 1.02177 0.0404840
$$638$$ 71.7434 2.84035
$$639$$ −18.0790 −0.715196
$$640$$ 0 0
$$641$$ 37.3564 1.47549 0.737745 0.675080i $$-0.235891\pi$$
0.737745 + 0.675080i $$0.235891\pi$$
$$642$$ −8.61356 −0.339950
$$643$$ 14.5483 0.573727 0.286864 0.957971i $$-0.407387\pi$$
0.286864 + 0.957971i $$0.407387\pi$$
$$644$$ 0.305586 0.0120418
$$645$$ 0 0
$$646$$ 1.64310 0.0646471
$$647$$ 23.4403 0.921532 0.460766 0.887522i $$-0.347575\pi$$
0.460766 + 0.887522i $$0.347575\pi$$
$$648$$ −30.2480 −1.18826
$$649$$ 28.2500 1.10891
$$650$$ 0 0
$$651$$ 7.67025 0.300621
$$652$$ 0.408797 0.0160097
$$653$$ −27.1903 −1.06404 −0.532019 0.846732i $$-0.678567\pi$$
−0.532019 + 0.846732i $$0.678567\pi$$
$$654$$ −8.11529 −0.317333
$$655$$ 0 0
$$656$$ 48.9828 1.91246
$$657$$ 24.5265 0.956869
$$658$$ −5.27844 −0.205775
$$659$$ 2.71486 0.105756 0.0528779 0.998601i $$-0.483161\pi$$
0.0528779 + 0.998601i $$0.483161\pi$$
$$660$$ 0 0
$$661$$ −9.41311 −0.366128 −0.183064 0.983101i $$-0.558601\pi$$
−0.183064 + 0.983101i $$0.558601\pi$$
$$662$$ −2.90408 −0.112870
$$663$$ 0.506041 0.0196530
$$664$$ −10.4286 −0.404710
$$665$$ 0 0
$$666$$ −0.405813 −0.0157249
$$667$$ 26.1497 1.01252
$$668$$ 1.73019 0.0669429
$$669$$ 17.9245 0.693002
$$670$$ 0 0
$$671$$ −36.7797 −1.41986
$$672$$ 1.51919 0.0586039
$$673$$ 44.8001 1.72692 0.863459 0.504419i $$-0.168293\pi$$
0.863459 + 0.504419i $$0.168293\pi$$
$$674$$ −1.90562 −0.0734019
$$675$$ 0 0
$$676$$ −1.14244 −0.0439400
$$677$$ 32.9197 1.26521 0.632604 0.774475i $$-0.281986\pi$$
0.632604 + 0.774475i $$0.281986\pi$$
$$678$$ −27.6896 −1.06341
$$679$$ 0.268750 0.0103137
$$680$$ 0 0
$$681$$ 44.4771 1.70437
$$682$$ −17.6345 −0.675259
$$683$$ −20.3448 −0.778473 −0.389236 0.921138i $$-0.627261\pi$$
−0.389236 + 0.921138i $$0.627261\pi$$
$$684$$ −0.180604 −0.00690556
$$685$$ 0 0
$$686$$ −23.8407 −0.910242
$$687$$ 9.06398 0.345813
$$688$$ −31.6466 −1.20651
$$689$$ 2.55065 0.0971719
$$690$$ 0 0
$$691$$ −46.1473 −1.75553 −0.877764 0.479094i $$-0.840965\pi$$
−0.877764 + 0.479094i $$0.840965\pi$$
$$692$$ −1.81269 −0.0689082
$$693$$ −13.4862 −0.512298
$$694$$ 17.3623 0.659063
$$695$$ 0 0
$$696$$ 63.5357 2.40831
$$697$$ −13.3612 −0.506092
$$698$$ 15.2567 0.577473
$$699$$ 60.2549 2.27905
$$700$$ 0 0
$$701$$ 13.1933 0.498303 0.249152 0.968464i $$-0.419848\pi$$
0.249152 + 0.968464i $$0.419848\pi$$
$$702$$ 0.611645 0.0230851
$$703$$ −0.137063 −0.00516944
$$704$$ 36.9491 1.39257
$$705$$ 0 0
$$706$$ −1.46096 −0.0549840
$$707$$ −15.6165 −0.587321
$$708$$ −1.15346 −0.0433496
$$709$$ 41.1860 1.54677 0.773386 0.633935i $$-0.218562\pi$$
0.773386 + 0.633935i $$0.218562\pi$$
$$710$$ 0 0
$$711$$ 22.0073 0.825338
$$712$$ 25.8771 0.969787
$$713$$ −6.42758 −0.240715
$$714$$ −5.00969 −0.187483
$$715$$ 0 0
$$716$$ 0.610106 0.0228007
$$717$$ −7.55496 −0.282145
$$718$$ 7.43104 0.277324
$$719$$ −35.6256 −1.32861 −0.664306 0.747461i $$-0.731273\pi$$
−0.664306 + 0.747461i $$0.731273\pi$$
$$720$$ 0 0
$$721$$ 20.5657 0.765907
$$722$$ −1.44504 −0.0537789
$$723$$ −62.3551 −2.31901
$$724$$ −1.55543 −0.0578072
$$725$$ 0 0
$$726$$ 40.6872 1.51004
$$727$$ −20.0116 −0.742189 −0.371095 0.928595i $$-0.621017\pi$$
−0.371095 + 0.928595i $$0.621017\pi$$
$$728$$ 0.742478 0.0275181
$$729$$ −8.02715 −0.297302
$$730$$ 0 0
$$731$$ 8.63235 0.319279
$$732$$ 1.50173 0.0555055
$$733$$ 18.9952 0.701604 0.350802 0.936450i $$-0.385909\pi$$
0.350802 + 0.936450i $$0.385909\pi$$
$$734$$ −42.6732 −1.57510
$$735$$ 0 0
$$736$$ −1.27306 −0.0469257
$$737$$ −38.8799 −1.43216
$$738$$ 34.7909 1.28067
$$739$$ 48.1704 1.77198 0.885989 0.463706i $$-0.153481\pi$$
0.885989 + 0.463706i $$0.153481\pi$$
$$740$$ 0 0
$$741$$ −0.445042 −0.0163490
$$742$$ −25.2508 −0.926987
$$743$$ 13.2446 0.485897 0.242948 0.970039i $$-0.421885\pi$$
0.242948 + 0.970039i $$0.421885\pi$$
$$744$$ −15.6170 −0.572548
$$745$$ 0 0
$$746$$ 12.7017 0.465043
$$747$$ −7.73423 −0.282981
$$748$$ 0.486189 0.0177768
$$749$$ 3.59956 0.131525
$$750$$ 0 0
$$751$$ 12.0562 0.439937 0.219969 0.975507i $$-0.429404\pi$$
0.219969 + 0.975507i $$0.429404\pi$$
$$752$$ 11.2218 0.409215
$$753$$ 12.5700 0.458077
$$754$$ −2.92931 −0.106679
$$755$$ 0 0
$$756$$ −0.255603 −0.00929620
$$757$$ −15.0054 −0.545380 −0.272690 0.962102i $$-0.587913\pi$$
−0.272690 + 0.962102i $$0.587913\pi$$
$$758$$ −27.6741 −1.00517
$$759$$ 27.8485 1.01084
$$760$$ 0 0
$$761$$ 44.3967 1.60938 0.804690 0.593695i $$-0.202332\pi$$
0.804690 + 0.593695i $$0.202332\pi$$
$$762$$ −66.2669 −2.40060
$$763$$ 3.39134 0.122775
$$764$$ 0.610791 0.0220976
$$765$$ 0 0
$$766$$ −10.1138 −0.365425
$$767$$ −1.15346 −0.0416489
$$768$$ −4.74440 −0.171199
$$769$$ −39.7211 −1.43238 −0.716190 0.697906i $$-0.754115\pi$$
−0.716190 + 0.697906i $$0.754115\pi$$
$$770$$ 0 0
$$771$$ 23.5187 0.847006
$$772$$ 1.85431 0.0667382
$$773$$ 1.72779 0.0621444 0.0310722 0.999517i $$-0.490108\pi$$
0.0310722 + 0.999517i $$0.490108\pi$$
$$774$$ −22.4776 −0.807939
$$775$$ 0 0
$$776$$ −0.547188 −0.0196429
$$777$$ 0.417895 0.0149919
$$778$$ −11.6843 −0.418901
$$779$$ 11.7506 0.421010
$$780$$ 0 0
$$781$$ −42.8025 −1.53159
$$782$$ 4.19806 0.150122
$$783$$ −21.8726 −0.781664
$$784$$ 21.5047 0.768025
$$785$$ 0 0
$$786$$ −42.9028 −1.53029
$$787$$ −42.6329 −1.51970 −0.759850 0.650098i $$-0.774728\pi$$
−0.759850 + 0.650098i $$0.774728\pi$$
$$788$$ −1.90084 −0.0677145
$$789$$ 34.7821 1.23828
$$790$$ 0 0
$$791$$ 11.5714 0.411430
$$792$$ 27.4586 0.975698
$$793$$ 1.50173 0.0533280
$$794$$ −25.5782 −0.907735
$$795$$ 0 0
$$796$$ 1.93841 0.0687051
$$797$$ −38.5864 −1.36680 −0.683401 0.730044i $$-0.739500\pi$$
−0.683401 + 0.730044i $$0.739500\pi$$
$$798$$ 4.40581 0.155964
$$799$$ −3.06100 −0.108290
$$800$$ 0 0
$$801$$ 19.1914 0.678093
$$802$$ 22.4586 0.793040
$$803$$ 58.0670 2.04914
$$804$$ 1.58748 0.0559862
$$805$$ 0 0
$$806$$ 0.720023 0.0253617
$$807$$ −20.5157 −0.722188
$$808$$ 31.7961 1.11858
$$809$$ −8.38298 −0.294730 −0.147365 0.989082i $$-0.547079\pi$$
−0.147365 + 0.989082i $$0.547079\pi$$
$$810$$ 0 0
$$811$$ −0.340765 −0.0119659 −0.00598295 0.999982i $$-0.501904\pi$$
−0.00598295 + 0.999982i $$0.501904\pi$$
$$812$$ 1.22414 0.0429590
$$813$$ −16.7235 −0.586518
$$814$$ −0.960771 −0.0336750
$$815$$ 0 0
$$816$$ 10.6504 0.372839
$$817$$ −7.59179 −0.265603
$$818$$ −19.0256 −0.665215
$$819$$ 0.550646 0.0192411
$$820$$ 0 0
$$821$$ −33.7506 −1.17791 −0.588953 0.808168i $$-0.700460\pi$$
−0.588953 + 0.808168i $$0.700460\pi$$
$$822$$ 22.2760 0.776966
$$823$$ −37.2669 −1.29904 −0.649522 0.760343i $$-0.725031\pi$$
−0.649522 + 0.760343i $$0.725031\pi$$
$$824$$ −41.8728 −1.45871
$$825$$ 0 0
$$826$$ 11.4190 0.397316
$$827$$ 21.1691 0.736122 0.368061 0.929802i $$-0.380022\pi$$
0.368061 + 0.929802i $$0.380022\pi$$
$$828$$ −0.461435 −0.0160360
$$829$$ −31.0374 −1.07797 −0.538987 0.842314i $$-0.681193\pi$$
−0.538987 + 0.842314i $$0.681193\pi$$
$$830$$ 0 0
$$831$$ −25.6504 −0.889803
$$832$$ −1.50864 −0.0523028
$$833$$ −5.86592 −0.203242
$$834$$ 13.9172 0.481914
$$835$$ 0 0
$$836$$ −0.427583 −0.0147883
$$837$$ 5.37627 0.185831
$$838$$ 36.3773 1.25663
$$839$$ 33.2403 1.14758 0.573791 0.819002i $$-0.305472\pi$$
0.573791 + 0.819002i $$0.305472\pi$$
$$840$$ 0 0
$$841$$ 75.7531 2.61218
$$842$$ −38.9323 −1.34170
$$843$$ −48.3236 −1.66435
$$844$$ −1.81786 −0.0625732
$$845$$ 0 0
$$846$$ 7.97046 0.274030
$$847$$ −17.0030 −0.584229
$$848$$ 53.6822 1.84346
$$849$$ −12.2620 −0.420832
$$850$$ 0 0
$$851$$ −0.350191 −0.0120044
$$852$$ 1.74764 0.0598733
$$853$$ 24.3086 0.832310 0.416155 0.909294i $$-0.363377\pi$$
0.416155 + 0.909294i $$0.363377\pi$$
$$854$$ −14.8668 −0.508731
$$855$$ 0 0
$$856$$ −7.32889 −0.250496
$$857$$ 57.3889 1.96037 0.980185 0.198087i $$-0.0634727\pi$$
0.980185 + 0.198087i $$0.0634727\pi$$
$$858$$ −3.11960 −0.106502
$$859$$ −13.8135 −0.471312 −0.235656 0.971837i $$-0.575724\pi$$
−0.235656 + 0.971837i $$0.575724\pi$$
$$860$$ 0 0
$$861$$ −35.8267 −1.22097
$$862$$ −27.3817 −0.932623
$$863$$ 9.01938 0.307023 0.153512 0.988147i $$-0.450942\pi$$
0.153512 + 0.988147i $$0.450942\pi$$
$$864$$ 1.06484 0.0362265
$$865$$ 0 0
$$866$$ −5.33081 −0.181148
$$867$$ 35.2935 1.19863
$$868$$ −0.300894 −0.0102130
$$869$$ 52.1027 1.76746
$$870$$ 0 0
$$871$$ 1.58748 0.0537898
$$872$$ −6.90494 −0.233831
$$873$$ −0.405813 −0.0137347
$$874$$ −3.69202 −0.124884
$$875$$ 0 0
$$876$$ −2.37090 −0.0801052
$$877$$ 2.37675 0.0802570 0.0401285 0.999195i $$-0.487223\pi$$
0.0401285 + 0.999195i $$0.487223\pi$$
$$878$$ 37.1269 1.25297
$$879$$ 16.6069 0.560135
$$880$$ 0 0
$$881$$ −7.99330 −0.269301 −0.134650 0.990893i $$-0.542991\pi$$
−0.134650 + 0.990893i $$0.542991\pi$$
$$882$$ 15.2741 0.514307
$$883$$ 1.76377 0.0593557 0.0296779 0.999560i $$-0.490552\pi$$
0.0296779 + 0.999560i $$0.490552\pi$$
$$884$$ −0.0198513 −0.000667672 0
$$885$$ 0 0
$$886$$ −39.5439 −1.32850
$$887$$ −45.9197 −1.54183 −0.770917 0.636936i $$-0.780202\pi$$
−0.770917 + 0.636936i $$0.780202\pi$$
$$888$$ −0.850855 −0.0285528
$$889$$ 27.6926 0.928780
$$890$$ 0 0
$$891$$ −53.1105 −1.77927
$$892$$ −0.703155 −0.0235434
$$893$$ 2.69202 0.0900851
$$894$$ −49.8558 −1.66743
$$895$$ 0 0
$$896$$ 16.2874 0.544125
$$897$$ −1.13706 −0.0379654
$$898$$ −10.9215 −0.364457
$$899$$ −25.7482 −0.858752
$$900$$ 0 0
$$901$$ −14.6431 −0.487833
$$902$$ 82.3682 2.74256
$$903$$ 23.1468 0.770276
$$904$$ −23.5599 −0.783589
$$905$$ 0 0
$$906$$ −33.2150 −1.10350
$$907$$ −55.0549 −1.82807 −0.914034 0.405638i $$-0.867049\pi$$
−0.914034 + 0.405638i $$0.867049\pi$$
$$908$$ −1.74478 −0.0579024
$$909$$ 23.5810 0.782134
$$910$$ 0 0
$$911$$ 51.8998 1.71952 0.859758 0.510702i $$-0.170614\pi$$
0.859758 + 0.510702i $$0.170614\pi$$
$$912$$ −9.36658 −0.310159
$$913$$ −18.3110 −0.606004
$$914$$ −11.3526 −0.375510
$$915$$ 0 0
$$916$$ −0.355568 −0.0117483
$$917$$ 17.9288 0.592062
$$918$$ −3.51142 −0.115894
$$919$$ 11.4614 0.378078 0.189039 0.981970i $$-0.439463\pi$$
0.189039 + 0.981970i $$0.439463\pi$$
$$920$$ 0 0
$$921$$ −72.3822 −2.38508
$$922$$ −5.60541 −0.184604
$$923$$ 1.74764 0.0575244
$$924$$ 1.30367 0.0428875
$$925$$ 0 0
$$926$$ −18.9235 −0.621864
$$927$$ −31.0543 −1.01996
$$928$$ −5.09975 −0.167408
$$929$$ 36.5295 1.19849 0.599246 0.800565i $$-0.295467\pi$$
0.599246 + 0.800565i $$0.295467\pi$$
$$930$$ 0 0
$$931$$ 5.15883 0.169074
$$932$$ −2.36372 −0.0774261
$$933$$ −33.3545 −1.09198
$$934$$ 9.30644 0.304516
$$935$$ 0 0
$$936$$ −1.12114 −0.0366457
$$937$$ 48.7845 1.59372 0.796860 0.604164i $$-0.206493\pi$$
0.796860 + 0.604164i $$0.206493\pi$$
$$938$$ −15.7157 −0.513136
$$939$$ −29.5459 −0.964193
$$940$$ 0 0
$$941$$ −18.1817 −0.592705 −0.296353 0.955079i $$-0.595770\pi$$
−0.296353 + 0.955079i $$0.595770\pi$$
$$942$$ −10.2959 −0.335458
$$943$$ 30.0224 0.977663
$$944$$ −24.2763 −0.790125
$$945$$ 0 0
$$946$$ −53.2161 −1.73021
$$947$$ 7.55150 0.245391 0.122695 0.992444i $$-0.460846\pi$$
0.122695 + 0.992444i $$0.460846\pi$$
$$948$$ −2.12737 −0.0690939
$$949$$ −2.37090 −0.0769626
$$950$$ 0 0
$$951$$ −11.5375 −0.374129
$$952$$ −4.26252 −0.138149
$$953$$ −13.8592 −0.448944 −0.224472 0.974481i $$-0.572066\pi$$
−0.224472 + 0.974481i $$0.572066\pi$$
$$954$$ 38.1288 1.23447
$$955$$ 0 0
$$956$$ 0.296371 0.00958532
$$957$$ 111.558 3.60616
$$958$$ −8.22367 −0.265695
$$959$$ −9.30904 −0.300605
$$960$$ 0 0
$$961$$ −24.6711 −0.795842
$$962$$ 0.0392287 0.00126478
$$963$$ −5.43535 −0.175152
$$964$$ 2.44611 0.0787838
$$965$$ 0 0
$$966$$ 11.2567 0.362177
$$967$$ −4.89977 −0.157566 −0.0787830 0.996892i $$-0.525103\pi$$
−0.0787830 + 0.996892i $$0.525103\pi$$
$$968$$ 34.6189 1.11269
$$969$$ 2.55496 0.0820771
$$970$$ 0 0
$$971$$ 14.5133 0.465755 0.232878 0.972506i $$-0.425186\pi$$
0.232878 + 0.972506i $$0.425186\pi$$
$$972$$ 1.60340 0.0514291
$$973$$ −5.81594 −0.186450
$$974$$ −18.7323 −0.600222
$$975$$ 0 0
$$976$$ 31.6062 1.01169
$$977$$ 19.6644 0.629120 0.314560 0.949238i $$-0.398143\pi$$
0.314560 + 0.949238i $$0.398143\pi$$
$$978$$ 15.0586 0.481521
$$979$$ 45.4359 1.45214
$$980$$ 0 0
$$981$$ −5.12093 −0.163499
$$982$$ 28.3293 0.904023
$$983$$ 15.4397 0.492449 0.246224 0.969213i $$-0.420810\pi$$
0.246224 + 0.969213i $$0.420810\pi$$
$$984$$ 72.9450 2.32540
$$985$$ 0 0
$$986$$ 16.8170 0.535562
$$987$$ −8.20775 −0.261256
$$988$$ 0.0174584 0.000555426 0
$$989$$ −19.3967 −0.616780
$$990$$ 0 0
$$991$$ 11.9377 0.379213 0.189606 0.981860i $$-0.439279\pi$$
0.189606 + 0.981860i $$0.439279\pi$$
$$992$$ 1.25352 0.0397991
$$993$$ −4.51573 −0.143302
$$994$$ −17.3013 −0.548763
$$995$$ 0 0
$$996$$ 0.747644 0.0236900
$$997$$ −17.9390 −0.568134 −0.284067 0.958804i $$-0.591684\pi$$
−0.284067 + 0.958804i $$0.591684\pi$$
$$998$$ −7.94523 −0.251502
$$999$$ 0.292913 0.00926736
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.d.1.2 3
3.2 odd 2 4275.2.a.bn.1.2 3
4.3 odd 2 7600.2.a.bw.1.3 3
5.2 odd 4 475.2.b.c.324.2 6
5.3 odd 4 475.2.b.c.324.5 6
5.4 even 2 475.2.a.h.1.2 yes 3
15.14 odd 2 4275.2.a.z.1.2 3
19.18 odd 2 9025.2.a.be.1.2 3
20.19 odd 2 7600.2.a.bn.1.1 3
95.94 odd 2 9025.2.a.w.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
475.2.a.d.1.2 3 1.1 even 1 trivial
475.2.a.h.1.2 yes 3 5.4 even 2
475.2.b.c.324.2 6 5.2 odd 4
475.2.b.c.324.5 6 5.3 odd 4
4275.2.a.z.1.2 3 15.14 odd 2
4275.2.a.bn.1.2 3 3.2 odd 2
7600.2.a.bn.1.1 3 20.19 odd 2
7600.2.a.bw.1.3 3 4.3 odd 2
9025.2.a.w.1.2 3 95.94 odd 2
9025.2.a.be.1.2 3 19.18 odd 2