Properties

 Label 475.2.a.g.1.2 Level $475$ Weight $2$ Character 475.1 Self dual yes Analytic conductor $3.793$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [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: $$0$$ 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 1.2 Root $$-0.273891$$ 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)

 $$f(q)$$ $$=$$ $$q+1.27389 q^{2} -1.65109 q^{3} -0.377203 q^{4} -2.10331 q^{6} +3.65109 q^{7} -3.02830 q^{8} -0.273891 q^{9} +O(q^{10})$$ $$q+1.27389 q^{2} -1.65109 q^{3} -0.377203 q^{4} -2.10331 q^{6} +3.65109 q^{7} -3.02830 q^{8} -0.273891 q^{9} +2.65109 q^{11} +0.622797 q^{12} +6.13161 q^{13} +4.65109 q^{14} -3.10331 q^{16} +2.34891 q^{17} -0.348907 q^{18} +1.00000 q^{19} -6.02830 q^{21} +3.37720 q^{22} +5.48052 q^{23} +5.00000 q^{24} +7.81100 q^{26} +5.40550 q^{27} -1.37720 q^{28} +0.651093 q^{29} -6.67939 q^{31} +2.10331 q^{32} -4.37720 q^{33} +2.99225 q^{34} +0.103312 q^{36} -8.70769 q^{37} +1.27389 q^{38} -10.1239 q^{39} +1.93273 q^{41} -7.67939 q^{42} -2.65884 q^{43} -1.00000 q^{44} +6.98158 q^{46} +3.71836 q^{47} +5.12386 q^{48} +6.33048 q^{49} -3.87826 q^{51} -2.31286 q^{52} +13.7544 q^{53} +6.88601 q^{54} -11.0566 q^{56} -1.65109 q^{57} +0.829422 q^{58} -7.84997 q^{59} -1.92498 q^{61} -8.50881 q^{62} -1.00000 q^{63} +8.88601 q^{64} -5.57608 q^{66} -4.44447 q^{67} -0.886014 q^{68} -9.04884 q^{69} +3.54778 q^{71} +0.829422 q^{72} -2.48052 q^{73} -11.0926 q^{74} -0.377203 q^{76} +9.67939 q^{77} -12.8967 q^{78} -15.1599 q^{79} -8.10331 q^{81} +2.46209 q^{82} -14.7282 q^{83} +2.27389 q^{84} -3.38708 q^{86} -1.07502 q^{87} -8.02830 q^{88} -5.06727 q^{89} +22.3871 q^{91} -2.06727 q^{92} +11.0283 q^{93} +4.73678 q^{94} -3.47277 q^{96} +3.22717 q^{97} +8.06434 q^{98} -0.726109 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 2 q^{2} + 2 q^{3} + 4 q^{4} - 3 q^{6} + 4 q^{7} + 3 q^{8} + q^{9}+O(q^{10})$$ 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.27389 0.900777 0.450388 0.892833i $$-0.351286\pi$$
0.450388 + 0.892833i $$0.351286\pi$$
$$3$$ −1.65109 −0.953259 −0.476630 0.879104i $$-0.658142\pi$$
−0.476630 + 0.879104i $$0.658142\pi$$
$$4$$ −0.377203 −0.188601
$$5$$ 0 0
$$6$$ −2.10331 −0.858674
$$7$$ 3.65109 1.37998 0.689992 0.723817i $$-0.257614\pi$$
0.689992 + 0.723817i $$0.257614\pi$$
$$8$$ −3.02830 −1.07066
$$9$$ −0.273891 −0.0912969
$$10$$ 0 0
$$11$$ 2.65109 0.799335 0.399667 0.916660i $$-0.369126\pi$$
0.399667 + 0.916660i $$0.369126\pi$$
$$12$$ 0.622797 0.179786
$$13$$ 6.13161 1.70060 0.850301 0.526297i $$-0.176420\pi$$
0.850301 + 0.526297i $$0.176420\pi$$
$$14$$ 4.65109 1.24306
$$15$$ 0 0
$$16$$ −3.10331 −0.775828
$$17$$ 2.34891 0.569694 0.284847 0.958573i $$-0.408057\pi$$
0.284847 + 0.958573i $$0.408057\pi$$
$$18$$ −0.348907 −0.0822381
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ −6.02830 −1.31548
$$22$$ 3.37720 0.720022
$$23$$ 5.48052 1.14277 0.571383 0.820683i $$-0.306407\pi$$
0.571383 + 0.820683i $$0.306407\pi$$
$$24$$ 5.00000 1.02062
$$25$$ 0 0
$$26$$ 7.81100 1.53186
$$27$$ 5.40550 1.04029
$$28$$ −1.37720 −0.260267
$$29$$ 0.651093 0.120905 0.0604525 0.998171i $$-0.480746\pi$$
0.0604525 + 0.998171i $$0.480746\pi$$
$$30$$ 0 0
$$31$$ −6.67939 −1.19965 −0.599827 0.800130i $$-0.704764\pi$$
−0.599827 + 0.800130i $$0.704764\pi$$
$$32$$ 2.10331 0.371817
$$33$$ −4.37720 −0.761973
$$34$$ 2.99225 0.513167
$$35$$ 0 0
$$36$$ 0.103312 0.0172187
$$37$$ −8.70769 −1.43153 −0.715767 0.698339i $$-0.753923\pi$$
−0.715767 + 0.698339i $$0.753923\pi$$
$$38$$ 1.27389 0.206652
$$39$$ −10.1239 −1.62111
$$40$$ 0 0
$$41$$ 1.93273 0.301842 0.150921 0.988546i $$-0.451776\pi$$
0.150921 + 0.988546i $$0.451776\pi$$
$$42$$ −7.67939 −1.18496
$$43$$ −2.65884 −0.405470 −0.202735 0.979234i $$-0.564983\pi$$
−0.202735 + 0.979234i $$0.564983\pi$$
$$44$$ −1.00000 −0.150756
$$45$$ 0 0
$$46$$ 6.98158 1.02938
$$47$$ 3.71836 0.542378 0.271189 0.962526i $$-0.412583\pi$$
0.271189 + 0.962526i $$0.412583\pi$$
$$48$$ 5.12386 0.739565
$$49$$ 6.33048 0.904355
$$50$$ 0 0
$$51$$ −3.87826 −0.543066
$$52$$ −2.31286 −0.320736
$$53$$ 13.7544 1.88931 0.944656 0.328061i $$-0.106395\pi$$
0.944656 + 0.328061i $$0.106395\pi$$
$$54$$ 6.88601 0.937068
$$55$$ 0 0
$$56$$ −11.0566 −1.47750
$$57$$ −1.65109 −0.218693
$$58$$ 0.829422 0.108908
$$59$$ −7.84997 −1.02198 −0.510989 0.859587i $$-0.670721\pi$$
−0.510989 + 0.859587i $$0.670721\pi$$
$$60$$ 0 0
$$61$$ −1.92498 −0.246469 −0.123234 0.992378i $$-0.539327\pi$$
−0.123234 + 0.992378i $$0.539327\pi$$
$$62$$ −8.50881 −1.08062
$$63$$ −1.00000 −0.125988
$$64$$ 8.88601 1.11075
$$65$$ 0 0
$$66$$ −5.57608 −0.686368
$$67$$ −4.44447 −0.542978 −0.271489 0.962442i $$-0.587516\pi$$
−0.271489 + 0.962442i $$0.587516\pi$$
$$68$$ −0.886014 −0.107445
$$69$$ −9.04884 −1.08935
$$70$$ 0 0
$$71$$ 3.54778 0.421044 0.210522 0.977589i $$-0.432484\pi$$
0.210522 + 0.977589i $$0.432484\pi$$
$$72$$ 0.829422 0.0977483
$$73$$ −2.48052 −0.290322 −0.145161 0.989408i $$-0.546370\pi$$
−0.145161 + 0.989408i $$0.546370\pi$$
$$74$$ −11.0926 −1.28949
$$75$$ 0 0
$$76$$ −0.377203 −0.0432681
$$77$$ 9.67939 1.10307
$$78$$ −12.8967 −1.46026
$$79$$ −15.1599 −1.70562 −0.852811 0.522219i $$-0.825104\pi$$
−0.852811 + 0.522219i $$0.825104\pi$$
$$80$$ 0 0
$$81$$ −8.10331 −0.900368
$$82$$ 2.46209 0.271893
$$83$$ −14.7282 −1.61663 −0.808317 0.588748i $$-0.799621\pi$$
−0.808317 + 0.588748i $$0.799621\pi$$
$$84$$ 2.27389 0.248102
$$85$$ 0 0
$$86$$ −3.38708 −0.365238
$$87$$ −1.07502 −0.115254
$$88$$ −8.02830 −0.855819
$$89$$ −5.06727 −0.537129 −0.268565 0.963262i $$-0.586549\pi$$
−0.268565 + 0.963262i $$0.586549\pi$$
$$90$$ 0 0
$$91$$ 22.3871 2.34680
$$92$$ −2.06727 −0.215527
$$93$$ 11.0283 1.14358
$$94$$ 4.73678 0.488562
$$95$$ 0 0
$$96$$ −3.47277 −0.354438
$$97$$ 3.22717 0.327670 0.163835 0.986488i $$-0.447614\pi$$
0.163835 + 0.986488i $$0.447614\pi$$
$$98$$ 8.06434 0.814622
$$99$$ −0.726109 −0.0729767
$$100$$ 0 0
$$101$$ 16.8032 1.67199 0.835993 0.548740i $$-0.184892\pi$$
0.835993 + 0.548740i $$0.184892\pi$$
$$102$$ −4.94048 −0.489181
$$103$$ −9.85772 −0.971310 −0.485655 0.874151i $$-0.661419\pi$$
−0.485655 + 0.874151i $$0.661419\pi$$
$$104$$ −18.5683 −1.82077
$$105$$ 0 0
$$106$$ 17.5216 1.70185
$$107$$ −5.13936 −0.496841 −0.248420 0.968652i $$-0.579911\pi$$
−0.248420 + 0.968652i $$0.579911\pi$$
$$108$$ −2.03897 −0.196200
$$109$$ 13.9738 1.33845 0.669225 0.743060i $$-0.266626\pi$$
0.669225 + 0.743060i $$0.266626\pi$$
$$110$$ 0 0
$$111$$ 14.3772 1.36462
$$112$$ −11.3305 −1.07063
$$113$$ −0.527235 −0.0495981 −0.0247990 0.999692i $$-0.507895\pi$$
−0.0247990 + 0.999692i $$0.507895\pi$$
$$114$$ −2.10331 −0.196993
$$115$$ 0 0
$$116$$ −0.245594 −0.0228029
$$117$$ −1.67939 −0.155260
$$118$$ −10.0000 −0.920575
$$119$$ 8.57608 0.786168
$$120$$ 0 0
$$121$$ −3.97170 −0.361064
$$122$$ −2.45222 −0.222013
$$123$$ −3.19112 −0.287734
$$124$$ 2.51948 0.226256
$$125$$ 0 0
$$126$$ −1.27389 −0.113487
$$127$$ −11.8217 −1.04900 −0.524502 0.851409i $$-0.675748\pi$$
−0.524502 + 0.851409i $$0.675748\pi$$
$$128$$ 7.11319 0.628723
$$129$$ 4.39000 0.386518
$$130$$ 0 0
$$131$$ 8.12386 0.709785 0.354892 0.934907i $$-0.384517\pi$$
0.354892 + 0.934907i $$0.384517\pi$$
$$132$$ 1.65109 0.143709
$$133$$ 3.65109 0.316590
$$134$$ −5.66177 −0.489102
$$135$$ 0 0
$$136$$ −7.11319 −0.609951
$$137$$ 17.5761 1.50163 0.750813 0.660515i $$-0.229662\pi$$
0.750813 + 0.660515i $$0.229662\pi$$
$$138$$ −11.5272 −0.981263
$$139$$ 18.4154 1.56197 0.780986 0.624549i $$-0.214717\pi$$
0.780986 + 0.624549i $$0.214717\pi$$
$$140$$ 0 0
$$141$$ −6.13936 −0.517027
$$142$$ 4.51948 0.379267
$$143$$ 16.2555 1.35935
$$144$$ 0.849968 0.0708307
$$145$$ 0 0
$$146$$ −3.15990 −0.261516
$$147$$ −10.4522 −0.862084
$$148$$ 3.28456 0.269989
$$149$$ −17.2915 −1.41658 −0.708288 0.705924i $$-0.750532\pi$$
−0.708288 + 0.705924i $$0.750532\pi$$
$$150$$ 0 0
$$151$$ 2.58383 0.210269 0.105134 0.994458i $$-0.466473\pi$$
0.105134 + 0.994458i $$0.466473\pi$$
$$152$$ −3.02830 −0.245627
$$153$$ −0.643343 −0.0520112
$$154$$ 12.3305 0.993619
$$155$$ 0 0
$$156$$ 3.81875 0.305745
$$157$$ −10.9738 −0.875807 −0.437903 0.899022i $$-0.644279\pi$$
−0.437903 + 0.899022i $$0.644279\pi$$
$$158$$ −19.3121 −1.53638
$$159$$ −22.7098 −1.80100
$$160$$ 0 0
$$161$$ 20.0099 1.57700
$$162$$ −10.3227 −0.811031
$$163$$ 22.6794 1.77639 0.888193 0.459470i $$-0.151961\pi$$
0.888193 + 0.459470i $$0.151961\pi$$
$$164$$ −0.729033 −0.0569279
$$165$$ 0 0
$$166$$ −18.7622 −1.45623
$$167$$ −5.16283 −0.399512 −0.199756 0.979846i $$-0.564015\pi$$
−0.199756 + 0.979846i $$0.564015\pi$$
$$168$$ 18.2555 1.40844
$$169$$ 24.5966 1.89205
$$170$$ 0 0
$$171$$ −0.273891 −0.0209449
$$172$$ 1.00292 0.0764722
$$173$$ −17.2165 −1.30895 −0.654473 0.756085i $$-0.727109\pi$$
−0.654473 + 0.756085i $$0.727109\pi$$
$$174$$ −1.36945 −0.103818
$$175$$ 0 0
$$176$$ −8.22717 −0.620146
$$177$$ 12.9610 0.974211
$$178$$ −6.45514 −0.483833
$$179$$ 10.6999 0.799751 0.399875 0.916570i $$-0.369053\pi$$
0.399875 + 0.916570i $$0.369053\pi$$
$$180$$ 0 0
$$181$$ −16.7720 −1.24666 −0.623328 0.781961i $$-0.714220\pi$$
−0.623328 + 0.781961i $$0.714220\pi$$
$$182$$ 28.5187 2.11395
$$183$$ 3.17833 0.234949
$$184$$ −16.5966 −1.22352
$$185$$ 0 0
$$186$$ 14.0488 1.03011
$$187$$ 6.22717 0.455376
$$188$$ −1.40258 −0.102293
$$189$$ 19.7360 1.43558
$$190$$ 0 0
$$191$$ −13.8598 −1.00286 −0.501431 0.865197i $$-0.667193\pi$$
−0.501431 + 0.865197i $$0.667193\pi$$
$$192$$ −14.6716 −1.05883
$$193$$ 21.0694 1.51661 0.758304 0.651901i $$-0.226028\pi$$
0.758304 + 0.651901i $$0.226028\pi$$
$$194$$ 4.11106 0.295157
$$195$$ 0 0
$$196$$ −2.38788 −0.170563
$$197$$ 21.2555 1.51439 0.757195 0.653189i $$-0.226569\pi$$
0.757195 + 0.653189i $$0.226569\pi$$
$$198$$ −0.924984 −0.0657357
$$199$$ −7.69006 −0.545134 −0.272567 0.962137i $$-0.587873\pi$$
−0.272567 + 0.962137i $$0.587873\pi$$
$$200$$ 0 0
$$201$$ 7.33823 0.517599
$$202$$ 21.4055 1.50609
$$203$$ 2.37720 0.166847
$$204$$ 1.46289 0.102423
$$205$$ 0 0
$$206$$ −12.5577 −0.874933
$$207$$ −1.50106 −0.104331
$$208$$ −19.0283 −1.31937
$$209$$ 2.65109 0.183380
$$210$$ 0 0
$$211$$ −1.69781 −0.116882 −0.0584411 0.998291i $$-0.518613\pi$$
−0.0584411 + 0.998291i $$0.518613\pi$$
$$212$$ −5.18820 −0.356327
$$213$$ −5.85772 −0.401364
$$214$$ −6.54698 −0.447542
$$215$$ 0 0
$$216$$ −16.3695 −1.11380
$$217$$ −24.3871 −1.65550
$$218$$ 17.8011 1.20564
$$219$$ 4.09556 0.276752
$$220$$ 0 0
$$221$$ 14.4026 0.968822
$$222$$ 18.3150 1.22922
$$223$$ −25.2632 −1.69175 −0.845875 0.533381i $$-0.820921\pi$$
−0.845875 + 0.533381i $$0.820921\pi$$
$$224$$ 7.67939 0.513101
$$225$$ 0 0
$$226$$ −0.671640 −0.0446768
$$227$$ −16.8217 −1.11649 −0.558247 0.829675i $$-0.688526\pi$$
−0.558247 + 0.829675i $$0.688526\pi$$
$$228$$ 0.622797 0.0412457
$$229$$ −14.6249 −0.966442 −0.483221 0.875498i $$-0.660533\pi$$
−0.483221 + 0.875498i $$0.660533\pi$$
$$230$$ 0 0
$$231$$ −15.9816 −1.05151
$$232$$ −1.97170 −0.129449
$$233$$ 8.91431 0.583996 0.291998 0.956419i $$-0.405680\pi$$
0.291998 + 0.956419i $$0.405680\pi$$
$$234$$ −2.13936 −0.139854
$$235$$ 0 0
$$236$$ 2.96103 0.192747
$$237$$ 25.0304 1.62590
$$238$$ 10.9250 0.708162
$$239$$ −24.6015 −1.59134 −0.795668 0.605733i $$-0.792880\pi$$
−0.795668 + 0.605733i $$0.792880\pi$$
$$240$$ 0 0
$$241$$ −14.6150 −0.941438 −0.470719 0.882283i $$-0.656005\pi$$
−0.470719 + 0.882283i $$0.656005\pi$$
$$242$$ −5.05952 −0.325238
$$243$$ −2.83717 −0.182005
$$244$$ 0.726109 0.0464844
$$245$$ 0 0
$$246$$ −4.06514 −0.259184
$$247$$ 6.13161 0.390145
$$248$$ 20.2272 1.28443
$$249$$ 24.3177 1.54107
$$250$$ 0 0
$$251$$ −13.3382 −0.841902 −0.420951 0.907083i $$-0.638304\pi$$
−0.420951 + 0.907083i $$0.638304\pi$$
$$252$$ 0.377203 0.0237615
$$253$$ 14.5294 0.913453
$$254$$ −15.0595 −0.944918
$$255$$ 0 0
$$256$$ −8.71061 −0.544413
$$257$$ −3.35103 −0.209031 −0.104516 0.994523i $$-0.533329\pi$$
−0.104516 + 0.994523i $$0.533329\pi$$
$$258$$ 5.59238 0.348166
$$259$$ −31.7926 −1.97549
$$260$$ 0 0
$$261$$ −0.178328 −0.0110382
$$262$$ 10.3489 0.639358
$$263$$ 10.8860 0.671260 0.335630 0.941994i $$-0.391051\pi$$
0.335630 + 0.941994i $$0.391051\pi$$
$$264$$ 13.2555 0.815818
$$265$$ 0 0
$$266$$ 4.65109 0.285177
$$267$$ 8.36653 0.512023
$$268$$ 1.67647 0.102406
$$269$$ 18.4338 1.12393 0.561964 0.827162i $$-0.310046\pi$$
0.561964 + 0.827162i $$0.310046\pi$$
$$270$$ 0 0
$$271$$ −20.4076 −1.23967 −0.619837 0.784730i $$-0.712801\pi$$
−0.619837 + 0.784730i $$0.712801\pi$$
$$272$$ −7.28939 −0.441984
$$273$$ −36.9632 −2.23711
$$274$$ 22.3900 1.35263
$$275$$ 0 0
$$276$$ 3.41325 0.205453
$$277$$ −2.69994 −0.162223 −0.0811117 0.996705i $$-0.525847\pi$$
−0.0811117 + 0.996705i $$0.525847\pi$$
$$278$$ 23.4592 1.40699
$$279$$ 1.82942 0.109525
$$280$$ 0 0
$$281$$ −15.2242 −0.908202 −0.454101 0.890950i $$-0.650040\pi$$
−0.454101 + 0.890950i $$0.650040\pi$$
$$282$$ −7.82087 −0.465726
$$283$$ 6.18045 0.367390 0.183695 0.982983i $$-0.441194\pi$$
0.183695 + 0.982983i $$0.441194\pi$$
$$284$$ −1.33823 −0.0794095
$$285$$ 0 0
$$286$$ 20.7077 1.22447
$$287$$ 7.05659 0.416537
$$288$$ −0.576077 −0.0339457
$$289$$ −11.4826 −0.675449
$$290$$ 0 0
$$291$$ −5.32836 −0.312354
$$292$$ 0.935657 0.0547552
$$293$$ 30.6893 1.79289 0.896443 0.443159i $$-0.146142\pi$$
0.896443 + 0.443159i $$0.146142\pi$$
$$294$$ −13.3150 −0.776546
$$295$$ 0 0
$$296$$ 26.3695 1.53269
$$297$$ 14.3305 0.831539
$$298$$ −22.0275 −1.27602
$$299$$ 33.6044 1.94339
$$300$$ 0 0
$$301$$ −9.70769 −0.559542
$$302$$ 3.29151 0.189405
$$303$$ −27.7437 −1.59384
$$304$$ −3.10331 −0.177987
$$305$$ 0 0
$$306$$ −0.819549 −0.0468505
$$307$$ −14.7827 −0.843693 −0.421847 0.906667i $$-0.638618\pi$$
−0.421847 + 0.906667i $$0.638618\pi$$
$$308$$ −3.65109 −0.208040
$$309$$ 16.2760 0.925910
$$310$$ 0 0
$$311$$ −26.9992 −1.53098 −0.765492 0.643445i $$-0.777504\pi$$
−0.765492 + 0.643445i $$0.777504\pi$$
$$312$$ 30.6580 1.73567
$$313$$ −9.74666 −0.550914 −0.275457 0.961313i $$-0.588829\pi$$
−0.275457 + 0.961313i $$0.588829\pi$$
$$314$$ −13.9795 −0.788906
$$315$$ 0 0
$$316$$ 5.71836 0.321683
$$317$$ 19.7261 1.10793 0.553964 0.832540i $$-0.313114\pi$$
0.553964 + 0.832540i $$0.313114\pi$$
$$318$$ −28.9298 −1.62230
$$319$$ 1.72611 0.0966436
$$320$$ 0 0
$$321$$ 8.48556 0.473618
$$322$$ 25.4904 1.42052
$$323$$ 2.34891 0.130697
$$324$$ 3.05659 0.169811
$$325$$ 0 0
$$326$$ 28.8911 1.60013
$$327$$ −23.0721 −1.27589
$$328$$ −5.85289 −0.323172
$$329$$ 13.5761 0.748473
$$330$$ 0 0
$$331$$ 19.9426 1.09614 0.548072 0.836431i $$-0.315362\pi$$
0.548072 + 0.836431i $$0.315362\pi$$
$$332$$ 5.55553 0.304899
$$333$$ 2.38495 0.130695
$$334$$ −6.57688 −0.359871
$$335$$ 0 0
$$336$$ 18.7077 1.02059
$$337$$ 2.05952 0.112189 0.0560945 0.998425i $$-0.482135\pi$$
0.0560945 + 0.998425i $$0.482135\pi$$
$$338$$ 31.3334 1.70431
$$339$$ 0.870514 0.0472798
$$340$$ 0 0
$$341$$ −17.7077 −0.958925
$$342$$ −0.348907 −0.0188667
$$343$$ −2.44447 −0.131989
$$344$$ 8.05177 0.434122
$$345$$ 0 0
$$346$$ −21.9319 −1.17907
$$347$$ −10.5011 −0.563727 −0.281863 0.959455i $$-0.590952\pi$$
−0.281863 + 0.959455i $$0.590952\pi$$
$$348$$ 0.405499 0.0217370
$$349$$ 16.5059 0.883540 0.441770 0.897128i $$-0.354351\pi$$
0.441770 + 0.897128i $$0.354351\pi$$
$$350$$ 0 0
$$351$$ 33.1444 1.76912
$$352$$ 5.57608 0.297206
$$353$$ 15.8860 0.845527 0.422764 0.906240i $$-0.361060\pi$$
0.422764 + 0.906240i $$0.361060\pi$$
$$354$$ 16.5109 0.877546
$$355$$ 0 0
$$356$$ 1.91139 0.101303
$$357$$ −14.1599 −0.749422
$$358$$ 13.6305 0.720397
$$359$$ 1.28376 0.0677544 0.0338772 0.999426i $$-0.489214\pi$$
0.0338772 + 0.999426i $$0.489214\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ −21.3657 −1.12296
$$363$$ 6.55765 0.344188
$$364$$ −8.44447 −0.442610
$$365$$ 0 0
$$366$$ 4.04884 0.211636
$$367$$ −19.8804 −1.03775 −0.518874 0.854851i $$-0.673649\pi$$
−0.518874 + 0.854851i $$0.673649\pi$$
$$368$$ −17.0078 −0.886590
$$369$$ −0.529358 −0.0275573
$$370$$ 0 0
$$371$$ 50.2186 2.60722
$$372$$ −4.15990 −0.215681
$$373$$ −16.4359 −0.851020 −0.425510 0.904954i $$-0.639905\pi$$
−0.425510 + 0.904954i $$0.639905\pi$$
$$374$$ 7.93273 0.410192
$$375$$ 0 0
$$376$$ −11.2603 −0.580705
$$377$$ 3.99225 0.205611
$$378$$ 25.1415 1.29314
$$379$$ 0.338233 0.0173739 0.00868694 0.999962i $$-0.497235\pi$$
0.00868694 + 0.999962i $$0.497235\pi$$
$$380$$ 0 0
$$381$$ 19.5187 0.999972
$$382$$ −17.6559 −0.903355
$$383$$ −13.7339 −0.701767 −0.350884 0.936419i $$-0.614119\pi$$
−0.350884 + 0.936419i $$0.614119\pi$$
$$384$$ −11.7445 −0.599336
$$385$$ 0 0
$$386$$ 26.8401 1.36612
$$387$$ 0.728232 0.0370181
$$388$$ −1.21730 −0.0617989
$$389$$ −2.26109 −0.114642 −0.0573210 0.998356i $$-0.518256\pi$$
−0.0573210 + 0.998356i $$0.518256\pi$$
$$390$$ 0 0
$$391$$ 12.8732 0.651027
$$392$$ −19.1706 −0.968260
$$393$$ −13.4132 −0.676609
$$394$$ 27.0771 1.36413
$$395$$ 0 0
$$396$$ 0.273891 0.0137635
$$397$$ −7.82942 −0.392947 −0.196474 0.980509i $$-0.562949\pi$$
−0.196474 + 0.980509i $$0.562949\pi$$
$$398$$ −9.79630 −0.491044
$$399$$ −6.02830 −0.301792
$$400$$ 0 0
$$401$$ −35.0510 −1.75036 −0.875181 0.483796i $$-0.839258\pi$$
−0.875181 + 0.483796i $$0.839258\pi$$
$$402$$ 9.34811 0.466241
$$403$$ −40.9554 −2.04013
$$404$$ −6.33823 −0.315339
$$405$$ 0 0
$$406$$ 3.02830 0.150292
$$407$$ −23.0849 −1.14428
$$408$$ 11.7445 0.581441
$$409$$ −8.34811 −0.412787 −0.206394 0.978469i $$-0.566173\pi$$
−0.206394 + 0.978469i $$0.566173\pi$$
$$410$$ 0 0
$$411$$ −29.0197 −1.43144
$$412$$ 3.71836 0.183190
$$413$$ −28.6610 −1.41031
$$414$$ −1.91219 −0.0939789
$$415$$ 0 0
$$416$$ 12.8967 0.632312
$$417$$ −30.4055 −1.48896
$$418$$ 3.37720 0.165184
$$419$$ −19.8705 −0.970738 −0.485369 0.874309i $$-0.661315\pi$$
−0.485369 + 0.874309i $$0.661315\pi$$
$$420$$ 0 0
$$421$$ −0.400672 −0.0195276 −0.00976379 0.999952i $$-0.503108\pi$$
−0.00976379 + 0.999952i $$0.503108\pi$$
$$422$$ −2.16283 −0.105285
$$423$$ −1.01842 −0.0495174
$$424$$ −41.6524 −2.02282
$$425$$ 0 0
$$426$$ −7.46209 −0.361540
$$427$$ −7.02830 −0.340123
$$428$$ 1.93858 0.0937048
$$429$$ −26.8393 −1.29581
$$430$$ 0 0
$$431$$ −14.2533 −0.686559 −0.343280 0.939233i $$-0.611538\pi$$
−0.343280 + 0.939233i $$0.611538\pi$$
$$432$$ −16.7750 −0.807085
$$433$$ 12.8393 0.617017 0.308509 0.951222i $$-0.400170\pi$$
0.308509 + 0.951222i $$0.400170\pi$$
$$434$$ −31.0665 −1.49124
$$435$$ 0 0
$$436$$ −5.27097 −0.252434
$$437$$ 5.48052 0.262169
$$438$$ 5.21730 0.249292
$$439$$ −13.2555 −0.632649 −0.316324 0.948651i $$-0.602449\pi$$
−0.316324 + 0.948651i $$0.602449\pi$$
$$440$$ 0 0
$$441$$ −1.73386 −0.0825647
$$442$$ 18.3473 0.872692
$$443$$ 5.72611 0.272056 0.136028 0.990705i $$-0.456566\pi$$
0.136028 + 0.990705i $$0.456566\pi$$
$$444$$ −5.42312 −0.257370
$$445$$ 0 0
$$446$$ −32.1826 −1.52389
$$447$$ 28.5499 1.35036
$$448$$ 32.4437 1.53282
$$449$$ 3.11399 0.146958 0.0734790 0.997297i $$-0.476590\pi$$
0.0734790 + 0.997297i $$0.476590\pi$$
$$450$$ 0 0
$$451$$ 5.12386 0.241273
$$452$$ 0.198875 0.00935427
$$453$$ −4.26614 −0.200441
$$454$$ −21.4290 −1.00571
$$455$$ 0 0
$$456$$ 5.00000 0.234146
$$457$$ 27.4535 1.28422 0.642111 0.766611i $$-0.278059\pi$$
0.642111 + 0.766611i $$0.278059\pi$$
$$458$$ −18.6305 −0.870548
$$459$$ 12.6970 0.592646
$$460$$ 0 0
$$461$$ 13.9837 0.651286 0.325643 0.945493i $$-0.394419\pi$$
0.325643 + 0.945493i $$0.394419\pi$$
$$462$$ −20.3588 −0.947176
$$463$$ 17.9992 0.836494 0.418247 0.908333i $$-0.362645\pi$$
0.418247 + 0.908333i $$0.362645\pi$$
$$464$$ −2.02055 −0.0938015
$$465$$ 0 0
$$466$$ 11.3559 0.526050
$$467$$ 12.6871 0.587091 0.293545 0.955945i $$-0.405165\pi$$
0.293545 + 0.955945i $$0.405165\pi$$
$$468$$ 0.633471 0.0292822
$$469$$ −16.2272 −0.749301
$$470$$ 0 0
$$471$$ 18.1188 0.834871
$$472$$ 23.7720 1.09420
$$473$$ −7.04884 −0.324106
$$474$$ 31.8860 1.46457
$$475$$ 0 0
$$476$$ −3.23492 −0.148272
$$477$$ −3.76720 −0.172488
$$478$$ −31.3396 −1.43344
$$479$$ 24.7544 1.13106 0.565529 0.824729i $$-0.308672\pi$$
0.565529 + 0.824729i $$0.308672\pi$$
$$480$$ 0 0
$$481$$ −53.3921 −2.43447
$$482$$ −18.6180 −0.848025
$$483$$ −33.0382 −1.50329
$$484$$ 1.49814 0.0680972
$$485$$ 0 0
$$486$$ −3.61425 −0.163946
$$487$$ −4.08277 −0.185008 −0.0925039 0.995712i $$-0.529487\pi$$
−0.0925039 + 0.995712i $$0.529487\pi$$
$$488$$ 5.82942 0.263886
$$489$$ −37.4458 −1.69336
$$490$$ 0 0
$$491$$ 29.2547 1.32024 0.660122 0.751158i $$-0.270504\pi$$
0.660122 + 0.751158i $$0.270504\pi$$
$$492$$ 1.20370 0.0542670
$$493$$ 1.52936 0.0688788
$$494$$ 7.81100 0.351433
$$495$$ 0 0
$$496$$ 20.7282 0.930725
$$497$$ 12.9533 0.581034
$$498$$ 30.9781 1.38816
$$499$$ 1.57315 0.0704240 0.0352120 0.999380i $$-0.488789\pi$$
0.0352120 + 0.999380i $$0.488789\pi$$
$$500$$ 0 0
$$501$$ 8.52431 0.380838
$$502$$ −16.9914 −0.758365
$$503$$ 20.7819 0.926619 0.463310 0.886196i $$-0.346662\pi$$
0.463310 + 0.886196i $$0.346662\pi$$
$$504$$ 3.02830 0.134891
$$505$$ 0 0
$$506$$ 18.5088 0.822817
$$507$$ −40.6113 −1.80361
$$508$$ 4.45917 0.197844
$$509$$ −31.8238 −1.41056 −0.705282 0.708926i $$-0.749180\pi$$
−0.705282 + 0.708926i $$0.749180\pi$$
$$510$$ 0 0
$$511$$ −9.05659 −0.400640
$$512$$ −25.3227 −1.11912
$$513$$ 5.40550 0.238659
$$514$$ −4.26884 −0.188291
$$515$$ 0 0
$$516$$ −1.65592 −0.0728978
$$517$$ 9.85772 0.433542
$$518$$ −40.5003 −1.77948
$$519$$ 28.4260 1.24776
$$520$$ 0 0
$$521$$ −27.4720 −1.20357 −0.601784 0.798659i $$-0.705543\pi$$
−0.601784 + 0.798659i $$0.705543\pi$$
$$522$$ −0.227171 −0.00994300
$$523$$ −10.4466 −0.456798 −0.228399 0.973568i $$-0.573349\pi$$
−0.228399 + 0.973568i $$0.573349\pi$$
$$524$$ −3.06434 −0.133866
$$525$$ 0 0
$$526$$ 13.8676 0.604656
$$527$$ −15.6893 −0.683435
$$528$$ 13.5838 0.591160
$$529$$ 7.03605 0.305915
$$530$$ 0 0
$$531$$ 2.15003 0.0933034
$$532$$ −1.37720 −0.0597093
$$533$$ 11.8508 0.513314
$$534$$ 10.6580 0.461219
$$535$$ 0 0
$$536$$ 13.4592 0.581348
$$537$$ −17.6666 −0.762370
$$538$$ 23.4826 1.01241
$$539$$ 16.7827 0.722882
$$540$$ 0 0
$$541$$ 13.4989 0.580365 0.290182 0.956971i $$-0.406284\pi$$
0.290182 + 0.956971i $$0.406284\pi$$
$$542$$ −25.9971 −1.11667
$$543$$ 27.6922 1.18839
$$544$$ 4.94048 0.211822
$$545$$ 0 0
$$546$$ −47.0870 −2.01514
$$547$$ 8.54698 0.365442 0.182721 0.983165i $$-0.441509\pi$$
0.182721 + 0.983165i $$0.441509\pi$$
$$548$$ −6.62975 −0.283209
$$549$$ 0.527235 0.0225018
$$550$$ 0 0
$$551$$ 0.651093 0.0277375
$$552$$ 27.4026 1.16633
$$553$$ −55.3502 −2.35373
$$554$$ −3.43942 −0.146127
$$555$$ 0 0
$$556$$ −6.94633 −0.294590
$$557$$ 27.2378 1.15410 0.577052 0.816707i $$-0.304203\pi$$
0.577052 + 0.816707i $$0.304203\pi$$
$$558$$ 2.33048 0.0986572
$$559$$ −16.3030 −0.689543
$$560$$ 0 0
$$561$$ −10.2816 −0.434091
$$562$$ −19.3940 −0.818088
$$563$$ −1.44235 −0.0607876 −0.0303938 0.999538i $$-0.509676\pi$$
−0.0303938 + 0.999538i $$0.509676\pi$$
$$564$$ 2.31578 0.0975121
$$565$$ 0 0
$$566$$ 7.87322 0.330936
$$567$$ −29.5860 −1.24249
$$568$$ −10.7437 −0.450797
$$569$$ −14.2632 −0.597945 −0.298973 0.954262i $$-0.596644\pi$$
−0.298973 + 0.954262i $$0.596644\pi$$
$$570$$ 0 0
$$571$$ 9.91723 0.415023 0.207512 0.978233i $$-0.433464\pi$$
0.207512 + 0.978233i $$0.433464\pi$$
$$572$$ −6.13161 −0.256375
$$573$$ 22.8839 0.955988
$$574$$ 8.98933 0.375207
$$575$$ 0 0
$$576$$ −2.43380 −0.101408
$$577$$ −8.23997 −0.343034 −0.171517 0.985181i $$-0.554867\pi$$
−0.171517 + 0.985181i $$0.554867\pi$$
$$578$$ −14.6276 −0.608429
$$579$$ −34.7875 −1.44572
$$580$$ 0 0
$$581$$ −53.7742 −2.23093
$$582$$ −6.78775 −0.281361
$$583$$ 36.4642 1.51019
$$584$$ 7.51173 0.310838
$$585$$ 0 0
$$586$$ 39.0948 1.61499
$$587$$ 36.9554 1.52531 0.762656 0.646804i $$-0.223895\pi$$
0.762656 + 0.646804i $$0.223895\pi$$
$$588$$ 3.94261 0.162590
$$589$$ −6.67939 −0.275219
$$590$$ 0 0
$$591$$ −35.0948 −1.44361
$$592$$ 27.0227 1.11062
$$593$$ −1.36170 −0.0559184 −0.0279592 0.999609i $$-0.508901\pi$$
−0.0279592 + 0.999609i $$0.508901\pi$$
$$594$$ 18.2555 0.749031
$$595$$ 0 0
$$596$$ 6.52241 0.267168
$$597$$ 12.6970 0.519654
$$598$$ 42.8083 1.75056
$$599$$ 33.5753 1.37185 0.685924 0.727673i $$-0.259398\pi$$
0.685924 + 0.727673i $$0.259398\pi$$
$$600$$ 0 0
$$601$$ 12.6561 0.516255 0.258127 0.966111i $$-0.416895\pi$$
0.258127 + 0.966111i $$0.416895\pi$$
$$602$$ −12.3665 −0.504022
$$603$$ 1.21730 0.0495722
$$604$$ −0.974627 −0.0396570
$$605$$ 0 0
$$606$$ −35.3425 −1.43569
$$607$$ 17.4047 0.706435 0.353217 0.935541i $$-0.385088\pi$$
0.353217 + 0.935541i $$0.385088\pi$$
$$608$$ 2.10331 0.0853006
$$609$$ −3.92498 −0.159048
$$610$$ 0 0
$$611$$ 22.7995 0.922370
$$612$$ 0.242671 0.00980939
$$613$$ −37.2603 −1.50493 −0.752465 0.658633i $$-0.771135\pi$$
−0.752465 + 0.658633i $$0.771135\pi$$
$$614$$ −18.8315 −0.759979
$$615$$ 0 0
$$616$$ −29.3121 −1.18102
$$617$$ −18.3764 −0.739806 −0.369903 0.929070i $$-0.620609\pi$$
−0.369903 + 0.929070i $$0.620609\pi$$
$$618$$ 20.7339 0.834038
$$619$$ 14.5526 0.584919 0.292459 0.956278i $$-0.405526\pi$$
0.292459 + 0.956278i $$0.405526\pi$$
$$620$$ 0 0
$$621$$ 29.6249 1.18881
$$622$$ −34.3940 −1.37907
$$623$$ −18.5011 −0.741229
$$624$$ 31.4175 1.25771
$$625$$ 0 0
$$626$$ −12.4162 −0.496250
$$627$$ −4.37720 −0.174809
$$628$$ 4.13936 0.165178
$$629$$ −20.4535 −0.815536
$$630$$ 0 0
$$631$$ 22.0304 0.877017 0.438509 0.898727i $$-0.355507\pi$$
0.438509 + 0.898727i $$0.355507\pi$$
$$632$$ 45.9087 1.82615
$$633$$ 2.80325 0.111419
$$634$$ 25.1289 0.997996
$$635$$ 0 0
$$636$$ 8.56620 0.339672
$$637$$ 38.8160 1.53795
$$638$$ 2.19887 0.0870543
$$639$$ −0.971704 −0.0384400
$$640$$ 0 0
$$641$$ −43.6765 −1.72512 −0.862558 0.505958i $$-0.831139\pi$$
−0.862558 + 0.505958i $$0.831139\pi$$
$$642$$ 10.8097 0.426624
$$643$$ −25.9263 −1.02243 −0.511217 0.859452i $$-0.670805\pi$$
−0.511217 + 0.859452i $$0.670805\pi$$
$$644$$ −7.54778 −0.297424
$$645$$ 0 0
$$646$$ 2.99225 0.117728
$$647$$ −42.1046 −1.65530 −0.827652 0.561242i $$-0.810324\pi$$
−0.827652 + 0.561242i $$0.810324\pi$$
$$648$$ 24.5392 0.963992
$$649$$ −20.8110 −0.816903
$$650$$ 0 0
$$651$$ 40.2653 1.57812
$$652$$ −8.55473 −0.335029
$$653$$ 40.4671 1.58360 0.791801 0.610779i $$-0.209144\pi$$
0.791801 + 0.610779i $$0.209144\pi$$
$$654$$ −29.3913 −1.14929
$$655$$ 0 0
$$656$$ −5.99788 −0.234178
$$657$$ 0.679390 0.0265055
$$658$$ 17.2944 0.674207
$$659$$ 33.4204 1.30187 0.650937 0.759131i $$-0.274376\pi$$
0.650937 + 0.759131i $$0.274376\pi$$
$$660$$ 0 0
$$661$$ 19.4883 0.758006 0.379003 0.925396i $$-0.376267\pi$$
0.379003 + 0.925396i $$0.376267\pi$$
$$662$$ 25.4047 0.987382
$$663$$ −23.7800 −0.923539
$$664$$ 44.6015 1.73087
$$665$$ 0 0
$$666$$ 3.03817 0.117727
$$667$$ 3.56833 0.138166
$$668$$ 1.94743 0.0753485
$$669$$ 41.7119 1.61268
$$670$$ 0 0
$$671$$ −5.10331 −0.197011
$$672$$ −12.6794 −0.489118
$$673$$ −0.747456 −0.0288123 −0.0144062 0.999896i $$-0.504586\pi$$
−0.0144062 + 0.999896i $$0.504586\pi$$
$$674$$ 2.62360 0.101057
$$675$$ 0 0
$$676$$ −9.27792 −0.356843
$$677$$ 25.0275 0.961885 0.480942 0.876752i $$-0.340295\pi$$
0.480942 + 0.876752i $$0.340295\pi$$
$$678$$ 1.10894 0.0425886
$$679$$ 11.7827 0.452179
$$680$$ 0 0
$$681$$ 27.7742 1.06431
$$682$$ −22.5577 −0.863777
$$683$$ −36.7253 −1.40525 −0.702627 0.711558i $$-0.747990\pi$$
−0.702627 + 0.711558i $$0.747990\pi$$
$$684$$ 0.103312 0.00395024
$$685$$ 0 0
$$686$$ −3.11399 −0.118893
$$687$$ 24.1471 0.921270
$$688$$ 8.25122 0.314575
$$689$$ 84.3366 3.21297
$$690$$ 0 0
$$691$$ −21.3177 −0.810963 −0.405482 0.914103i $$-0.632896\pi$$
−0.405482 + 0.914103i $$0.632896\pi$$
$$692$$ 6.49411 0.246869
$$693$$ −2.65109 −0.100707
$$694$$ −13.3772 −0.507792
$$695$$ 0 0
$$696$$ 3.25547 0.123398
$$697$$ 4.53981 0.171958
$$698$$ 21.0267 0.795872
$$699$$ −14.7184 −0.556699
$$700$$ 0 0
$$701$$ 27.1161 1.02416 0.512081 0.858937i $$-0.328875\pi$$
0.512081 + 0.858937i $$0.328875\pi$$
$$702$$ 42.2223 1.59358
$$703$$ −8.70769 −0.328417
$$704$$ 23.5577 0.887862
$$705$$ 0 0
$$706$$ 20.2370 0.761631
$$707$$ 61.3502 2.30731
$$708$$ −4.88894 −0.183738
$$709$$ −10.1161 −0.379918 −0.189959 0.981792i $$-0.560836\pi$$
−0.189959 + 0.981792i $$0.560836\pi$$
$$710$$ 0 0
$$711$$ 4.15215 0.155718
$$712$$ 15.3452 0.575085
$$713$$ −36.6065 −1.37092
$$714$$ −18.0382 −0.675062
$$715$$ 0 0
$$716$$ −4.03605 −0.150834
$$717$$ 40.6193 1.51696
$$718$$ 1.63537 0.0610316
$$719$$ 24.1471 0.900535 0.450268 0.892894i $$-0.351329\pi$$
0.450268 + 0.892894i $$0.351329\pi$$
$$720$$ 0 0
$$721$$ −35.9914 −1.34039
$$722$$ 1.27389 0.0474093
$$723$$ 24.1308 0.897434
$$724$$ 6.32646 0.235121
$$725$$ 0 0
$$726$$ 8.35373 0.310036
$$727$$ −19.8988 −0.738006 −0.369003 0.929428i $$-0.620301\pi$$
−0.369003 + 0.929428i $$0.620301\pi$$
$$728$$ −67.7947 −2.51264
$$729$$ 28.9944 1.07387
$$730$$ 0 0
$$731$$ −6.24537 −0.230994
$$732$$ −1.19887 −0.0443117
$$733$$ 19.9455 0.736705 0.368352 0.929686i $$-0.379922\pi$$
0.368352 + 0.929686i $$0.379922\pi$$
$$734$$ −25.3254 −0.934779
$$735$$ 0 0
$$736$$ 11.5272 0.424900
$$737$$ −11.7827 −0.434021
$$738$$ −0.674344 −0.0248229
$$739$$ −38.2624 −1.40751 −0.703753 0.710445i $$-0.748494\pi$$
−0.703753 + 0.710445i $$0.748494\pi$$
$$740$$ 0 0
$$741$$ −10.1239 −0.371909
$$742$$ 63.9730 2.34852
$$743$$ −12.5547 −0.460588 −0.230294 0.973121i $$-0.573969\pi$$
−0.230294 + 0.973121i $$0.573969\pi$$
$$744$$ −33.3969 −1.22439
$$745$$ 0 0
$$746$$ −20.9376 −0.766579
$$747$$ 4.03392 0.147594
$$748$$ −2.34891 −0.0858845
$$749$$ −18.7643 −0.685632
$$750$$ 0 0
$$751$$ 23.2215 0.847366 0.423683 0.905810i $$-0.360737\pi$$
0.423683 + 0.905810i $$0.360737\pi$$
$$752$$ −11.5392 −0.420792
$$753$$ 22.0227 0.802551
$$754$$ 5.08569 0.185210
$$755$$ 0 0
$$756$$ −7.44447 −0.270753
$$757$$ 18.4105 0.669143 0.334571 0.942370i $$-0.391408\pi$$
0.334571 + 0.942370i $$0.391408\pi$$
$$758$$ 0.430872 0.0156500
$$759$$ −23.9893 −0.870757
$$760$$ 0 0
$$761$$ 11.7982 0.427684 0.213842 0.976868i $$-0.431402\pi$$
0.213842 + 0.976868i $$0.431402\pi$$
$$762$$ 24.8647 0.900752
$$763$$ 51.0197 1.84704
$$764$$ 5.22797 0.189141
$$765$$ 0 0
$$766$$ −17.4954 −0.632136
$$767$$ −48.1329 −1.73798
$$768$$ 14.3820 0.518967
$$769$$ 41.2653 1.48807 0.744033 0.668143i $$-0.232910\pi$$
0.744033 + 0.668143i $$0.232910\pi$$
$$770$$ 0 0
$$771$$ 5.53286 0.199261
$$772$$ −7.94743 −0.286034
$$773$$ 5.51736 0.198446 0.0992229 0.995065i $$-0.468364\pi$$
0.0992229 + 0.995065i $$0.468364\pi$$
$$774$$ 0.927688 0.0333451
$$775$$ 0 0
$$776$$ −9.77283 −0.350824
$$777$$ 52.4925 1.88316
$$778$$ −2.88039 −0.103267
$$779$$ 1.93273 0.0692474
$$780$$ 0 0
$$781$$ 9.40550 0.336555
$$782$$ 16.3991 0.586430
$$783$$ 3.51948 0.125776
$$784$$ −19.6455 −0.701624
$$785$$ 0 0
$$786$$ −17.0870 −0.609474
$$787$$ 16.7771 0.598038 0.299019 0.954247i $$-0.403341\pi$$
0.299019 + 0.954247i $$0.403341\pi$$
$$788$$ −8.01762 −0.285616
$$789$$ −17.9738 −0.639885
$$790$$ 0 0
$$791$$ −1.92498 −0.0684446
$$792$$ 2.19887 0.0781336
$$793$$ −11.8032 −0.419146
$$794$$ −9.97383 −0.353958
$$795$$ 0 0
$$796$$ 2.90071 0.102813
$$797$$ −29.4260 −1.04232 −0.521162 0.853458i $$-0.674501\pi$$
−0.521162 + 0.853458i $$0.674501\pi$$
$$798$$ −7.67939 −0.271847
$$799$$ 8.73408 0.308989
$$800$$ 0 0
$$801$$ 1.38788 0.0490382
$$802$$ −44.6511 −1.57668
$$803$$ −6.57608 −0.232065
$$804$$ −2.76800 −0.0976199
$$805$$ 0 0
$$806$$ −52.1727 −1.83771
$$807$$ −30.4359 −1.07140
$$808$$ −50.8852 −1.79014
$$809$$ 5.48614 0.192883 0.0964413 0.995339i $$-0.469254\pi$$
0.0964413 + 0.995339i $$0.469254\pi$$
$$810$$ 0 0
$$811$$ 16.3927 0.575626 0.287813 0.957687i $$-0.407072\pi$$
0.287813 + 0.957687i $$0.407072\pi$$
$$812$$ −0.896688 −0.0314676
$$813$$ 33.6949 1.18173
$$814$$ −29.4076 −1.03074
$$815$$ 0 0
$$816$$ 12.0355 0.421326
$$817$$ −2.65884 −0.0930212
$$818$$ −10.6346 −0.371829
$$819$$ −6.13161 −0.214256
$$820$$ 0 0
$$821$$ 15.1628 0.529186 0.264593 0.964360i $$-0.414762\pi$$
0.264593 + 0.964360i $$0.414762\pi$$
$$822$$ −36.9680 −1.28941
$$823$$ 16.6094 0.578968 0.289484 0.957183i $$-0.406516\pi$$
0.289484 + 0.957183i $$0.406516\pi$$
$$824$$ 29.8521 1.03995
$$825$$ 0 0
$$826$$ −36.5109 −1.27038
$$827$$ 15.2293 0.529574 0.264787 0.964307i $$-0.414698\pi$$
0.264787 + 0.964307i $$0.414698\pi$$
$$828$$ 0.566205 0.0196770
$$829$$ 47.4565 1.64823 0.824116 0.566422i $$-0.191673\pi$$
0.824116 + 0.566422i $$0.191673\pi$$
$$830$$ 0 0
$$831$$ 4.45785 0.154641
$$832$$ 54.4856 1.88895
$$833$$ 14.8697 0.515205
$$834$$ −38.7333 −1.34122
$$835$$ 0 0
$$836$$ −1.00000 −0.0345857
$$837$$ −36.1054 −1.24799
$$838$$ −25.3129 −0.874418
$$839$$ 2.26614 0.0782359 0.0391179 0.999235i $$-0.487545\pi$$
0.0391179 + 0.999235i $$0.487545\pi$$
$$840$$ 0 0
$$841$$ −28.5761 −0.985382
$$842$$ −0.510413 −0.0175900
$$843$$ 25.1367 0.865752
$$844$$ 0.640420 0.0220442
$$845$$ 0 0
$$846$$ −1.29736 −0.0446042
$$847$$ −14.5011 −0.498262
$$848$$ −42.6842 −1.46578
$$849$$ −10.2045 −0.350218
$$850$$ 0 0
$$851$$ −47.7226 −1.63591
$$852$$ 2.20955 0.0756979
$$853$$ 51.8753 1.77618 0.888089 0.459672i $$-0.152033\pi$$
0.888089 + 0.459672i $$0.152033\pi$$
$$854$$ −8.95328 −0.306375
$$855$$ 0 0
$$856$$ 15.5635 0.531949
$$857$$ −6.17058 −0.210783 −0.105391 0.994431i $$-0.533610\pi$$
−0.105391 + 0.994431i $$0.533610\pi$$
$$858$$ −34.1903 −1.16724
$$859$$ −31.0635 −1.05987 −0.529937 0.848037i $$-0.677785\pi$$
−0.529937 + 0.848037i $$0.677785\pi$$
$$860$$ 0 0
$$861$$ −11.6511 −0.397068
$$862$$ −18.1572 −0.618437
$$863$$ −3.24772 −0.110554 −0.0552768 0.998471i $$-0.517604\pi$$
−0.0552768 + 0.998471i $$0.517604\pi$$
$$864$$ 11.3695 0.386797
$$865$$ 0 0
$$866$$ 16.3559 0.555795
$$867$$ 18.9589 0.643878
$$868$$ 9.19887 0.312230
$$869$$ −40.1903 −1.36336
$$870$$ 0 0
$$871$$ −27.2517 −0.923390
$$872$$ −42.3169 −1.43303
$$873$$ −0.883892 −0.0299152
$$874$$ 6.98158 0.236155
$$875$$ 0 0
$$876$$ −1.54486 −0.0521959
$$877$$ −41.7042 −1.40825 −0.704125 0.710076i $$-0.748661\pi$$
−0.704125 + 0.710076i $$0.748661\pi$$
$$878$$ −16.8860 −0.569875
$$879$$ −50.6708 −1.70908
$$880$$ 0 0
$$881$$ 7.42392 0.250118 0.125059 0.992149i $$-0.460088\pi$$
0.125059 + 0.992149i $$0.460088\pi$$
$$882$$ −2.20875 −0.0743724
$$883$$ 32.0333 1.07801 0.539004 0.842303i $$-0.318801\pi$$
0.539004 + 0.842303i $$0.318801\pi$$
$$884$$ −5.43269 −0.182721
$$885$$ 0 0
$$886$$ 7.29444 0.245061
$$887$$ 20.0275 0.672457 0.336229 0.941780i $$-0.390848\pi$$
0.336229 + 0.941780i $$0.390848\pi$$
$$888$$ −43.5384 −1.46105
$$889$$ −43.1620 −1.44761
$$890$$ 0 0
$$891$$ −21.4826 −0.719695
$$892$$ 9.52936 0.319066
$$893$$ 3.71836 0.124430
$$894$$ 36.3695 1.21638
$$895$$ 0 0
$$896$$ 25.9709 0.867627
$$897$$ −55.4840 −1.85256
$$898$$ 3.96688 0.132376
$$899$$ −4.34891 −0.145044
$$900$$ 0 0
$$901$$ 32.3078 1.07633
$$902$$ 6.52723 0.217333
$$903$$ 16.0283 0.533388
$$904$$ 1.59662 0.0531029
$$905$$ 0 0
$$906$$ −5.43460 −0.180552
$$907$$ 4.94823 0.164303 0.0821517 0.996620i $$-0.473821\pi$$
0.0821517 + 0.996620i $$0.473821\pi$$
$$908$$ 6.34518 0.210572
$$909$$ −4.60225 −0.152647
$$910$$ 0 0
$$911$$ 32.3014 1.07019 0.535096 0.844791i $$-0.320275\pi$$
0.535096 + 0.844791i $$0.320275\pi$$
$$912$$ 5.12386 0.169668
$$913$$ −39.0459 −1.29223
$$914$$ 34.9728 1.15680
$$915$$ 0 0
$$916$$ 5.51656 0.182272
$$917$$ 29.6610 0.979491
$$918$$ 16.1746 0.533841
$$919$$ 21.3072 0.702861 0.351430 0.936214i $$-0.385695\pi$$
0.351430 + 0.936214i $$0.385695\pi$$
$$920$$ 0 0
$$921$$ 24.4076 0.804258
$$922$$ 17.8137 0.586663
$$923$$ 21.7536 0.716029
$$924$$ 6.02830 0.198316
$$925$$ 0 0
$$926$$ 22.9290 0.753494
$$927$$ 2.69994 0.0886775
$$928$$ 1.36945 0.0449545
$$929$$ 24.7848 0.813164 0.406582 0.913614i $$-0.366721\pi$$
0.406582 + 0.913614i $$0.366721\pi$$
$$930$$ 0 0
$$931$$ 6.33048 0.207473
$$932$$ −3.36250 −0.110142
$$933$$ 44.5782 1.45942
$$934$$ 16.1620 0.528838
$$935$$ 0 0
$$936$$ 5.08569 0.166231
$$937$$ −32.3687 −1.05744 −0.528719 0.848797i $$-0.677327\pi$$
−0.528719 + 0.848797i $$0.677327\pi$$
$$938$$ −20.6716 −0.674953
$$939$$ 16.0926 0.525163
$$940$$ 0 0
$$941$$ −1.48264 −0.0483326 −0.0241663 0.999708i $$-0.507693\pi$$
−0.0241663 + 0.999708i $$0.507693\pi$$
$$942$$ 23.0814 0.752032
$$943$$ 10.5924 0.344935
$$944$$ 24.3609 0.792880
$$945$$ 0 0
$$946$$ −8.97945 −0.291947
$$947$$ −8.86064 −0.287932 −0.143966 0.989583i $$-0.545986\pi$$
−0.143966 + 0.989583i $$0.545986\pi$$
$$948$$ −9.44155 −0.306647
$$949$$ −15.2095 −0.493723
$$950$$ 0 0
$$951$$ −32.5696 −1.05614
$$952$$ −25.9709 −0.841722
$$953$$ 10.1239 0.327944 0.163972 0.986465i $$-0.447569\pi$$
0.163972 + 0.986465i $$0.447569\pi$$
$$954$$ −4.79900 −0.155373
$$955$$ 0 0
$$956$$ 9.27974 0.300128
$$957$$ −2.84997 −0.0921264
$$958$$ 31.5344 1.01883
$$959$$ 64.1719 2.07222
$$960$$ 0 0
$$961$$ 13.6142 0.439169
$$962$$ −68.0157 −2.19291
$$963$$ 1.40762 0.0453600
$$964$$ 5.51284 0.177557
$$965$$ 0 0
$$966$$ −42.0870 −1.35413
$$967$$ 41.8139 1.34465 0.672323 0.740258i $$-0.265297\pi$$
0.672323 + 0.740258i $$0.265297\pi$$
$$968$$ 12.0275 0.386578
$$969$$ −3.87826 −0.124588
$$970$$ 0 0
$$971$$ −5.53016 −0.177471 −0.0887356 0.996055i $$-0.528283\pi$$
−0.0887356 + 0.996055i $$0.528283\pi$$
$$972$$ 1.07019 0.0343263
$$973$$ 67.2362 2.15549
$$974$$ −5.20100 −0.166651
$$975$$ 0 0
$$976$$ 5.97383 0.191218
$$977$$ −31.9447 −1.02200 −0.511001 0.859580i $$-0.670725\pi$$
−0.511001 + 0.859580i $$0.670725\pi$$
$$978$$ −47.7018 −1.52534
$$979$$ −13.4338 −0.429346
$$980$$ 0 0
$$981$$ −3.82730 −0.122196
$$982$$ 37.2672 1.18925
$$983$$ 8.82862 0.281589 0.140795 0.990039i $$-0.455034\pi$$
0.140795 + 0.990039i $$0.455034\pi$$
$$984$$ 9.66367 0.308067
$$985$$ 0 0
$$986$$ 1.94823 0.0620444
$$987$$ −22.4154 −0.713489
$$988$$ −2.31286 −0.0735819
$$989$$ −14.5718 −0.463357
$$990$$ 0 0
$$991$$ −43.7304 −1.38914 −0.694570 0.719425i $$-0.744405\pi$$
−0.694570 + 0.719425i $$0.744405\pi$$
$$992$$ −14.0488 −0.446051
$$993$$ −32.9271 −1.04491
$$994$$ 16.5011 0.523382
$$995$$ 0 0
$$996$$ −9.17270 −0.290648
$$997$$ −46.6738 −1.47817 −0.739086 0.673611i $$-0.764743\pi$$
−0.739086 + 0.673611i $$0.764743\pi$$
$$998$$ 2.00403 0.0634363
$$999$$ −47.0694 −1.48921
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.g.1.2 yes 3
3.2 odd 2 4275.2.a.ba.1.2 3
4.3 odd 2 7600.2.a.bh.1.3 3
5.2 odd 4 475.2.b.b.324.4 6
5.3 odd 4 475.2.b.b.324.3 6
5.4 even 2 475.2.a.e.1.2 3
15.14 odd 2 4275.2.a.bm.1.2 3
19.18 odd 2 9025.2.a.y.1.2 3
20.19 odd 2 7600.2.a.cc.1.1 3
95.94 odd 2 9025.2.a.bc.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
475.2.a.e.1.2 3 5.4 even 2
475.2.a.g.1.2 yes 3 1.1 even 1 trivial
475.2.b.b.324.3 6 5.3 odd 4
475.2.b.b.324.4 6 5.2 odd 4
4275.2.a.ba.1.2 3 3.2 odd 2
4275.2.a.bm.1.2 3 15.14 odd 2
7600.2.a.bh.1.3 3 4.3 odd 2
7600.2.a.cc.1.1 3 20.19 odd 2
9025.2.a.y.1.2 3 19.18 odd 2
9025.2.a.bc.1.2 3 95.94 odd 2