Properties

 Label 2960.2.a.u.1.3 Level $2960$ Weight $2$ Character 2960.1 Self dual yes Analytic conductor $23.636$ 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] = [2960,2,Mod(1,2960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2960, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2960.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2960 = 2^{4} \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2960.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: $$23.6357189983$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.892.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 8x + 10$$ x^3 - x^2 - 8*x + 10 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 370) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.3 Root $$2.59774$$ of defining polynomial Character $$\chi$$ $$=$$ 2960.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.34596 q^{3} -1.00000 q^{5} +2.59774 q^{7} +8.19547 q^{9} +O(q^{10})$$ $$q+3.34596 q^{3} -1.00000 q^{5} +2.59774 q^{7} +8.19547 q^{9} -4.74823 q^{11} +6.69193 q^{13} -3.34596 q^{15} -0.748228 q^{17} +3.34596 q^{19} +8.69193 q^{21} -1.49646 q^{23} +1.00000 q^{25} +17.3839 q^{27} +3.94370 q^{29} -7.79321 q^{31} -15.8874 q^{33} -2.59774 q^{35} -1.00000 q^{37} +22.3909 q^{39} -6.44724 q^{41} -1.94370 q^{43} -8.19547 q^{45} -1.84951 q^{47} -0.251772 q^{49} -2.50354 q^{51} +10.4472 q^{53} +4.74823 q^{55} +11.1955 q^{57} -5.84951 q^{59} +7.94370 q^{61} +21.2897 q^{63} -6.69193 q^{65} -1.84951 q^{67} -5.00709 q^{69} +3.88740 q^{71} -7.49646 q^{73} +3.34596 q^{75} -12.3346 q^{77} +16.5414 q^{79} +33.5793 q^{81} -15.2334 q^{83} +0.748228 q^{85} +13.1955 q^{87} +6.00000 q^{89} +17.3839 q^{91} -26.0758 q^{93} -3.34596 q^{95} -10.4472 q^{97} -38.9140 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{5} + q^{7} + 11 q^{9}+O(q^{10})$$ 3 * q - 3 * q^5 + q^7 + 11 * q^9 $$3 q - 3 q^{5} + q^{7} + 11 q^{9} - 11 q^{11} + q^{17} + 6 q^{21} + 2 q^{23} + 3 q^{25} + 12 q^{27} - 5 q^{29} - 3 q^{31} - 14 q^{33} - q^{35} - 3 q^{37} + 40 q^{39} - 9 q^{41} + 11 q^{43} - 11 q^{45} - 2 q^{47} - 4 q^{49} - 14 q^{51} + 21 q^{53} + 11 q^{55} + 20 q^{57} - 14 q^{59} + 7 q^{61} + 37 q^{63} - 2 q^{67} - 28 q^{69} - 22 q^{71} - 16 q^{73} + 7 q^{77} + 26 q^{79} + 47 q^{81} - 2 q^{83} - q^{85} + 26 q^{87} + 18 q^{89} + 12 q^{91} - 18 q^{93} - 21 q^{97} - 19 q^{99}+O(q^{100})$$ 3 * q - 3 * q^5 + q^7 + 11 * q^9 - 11 * q^11 + q^17 + 6 * q^21 + 2 * q^23 + 3 * q^25 + 12 * q^27 - 5 * q^29 - 3 * q^31 - 14 * q^33 - q^35 - 3 * q^37 + 40 * q^39 - 9 * q^41 + 11 * q^43 - 11 * q^45 - 2 * q^47 - 4 * q^49 - 14 * q^51 + 21 * q^53 + 11 * q^55 + 20 * q^57 - 14 * q^59 + 7 * q^61 + 37 * q^63 - 2 * q^67 - 28 * q^69 - 22 * q^71 - 16 * q^73 + 7 * q^77 + 26 * q^79 + 47 * q^81 - 2 * q^83 - q^85 + 26 * q^87 + 18 * q^89 + 12 * q^91 - 18 * q^93 - 21 * q^97 - 19 * q^99

Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 3.34596 1.93179 0.965896 0.258929i $$-0.0833695\pi$$
0.965896 + 0.258929i $$0.0833695\pi$$
$$4$$ 0 0
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ 2.59774 0.981852 0.490926 0.871201i $$-0.336659\pi$$
0.490926 + 0.871201i $$0.336659\pi$$
$$8$$ 0 0
$$9$$ 8.19547 2.73182
$$10$$ 0 0
$$11$$ −4.74823 −1.43164 −0.715822 0.698282i $$-0.753948\pi$$
−0.715822 + 0.698282i $$0.753948\pi$$
$$12$$ 0 0
$$13$$ 6.69193 1.85601 0.928003 0.372572i $$-0.121524\pi$$
0.928003 + 0.372572i $$0.121524\pi$$
$$14$$ 0 0
$$15$$ −3.34596 −0.863924
$$16$$ 0 0
$$17$$ −0.748228 −0.181472 −0.0907360 0.995875i $$-0.528922\pi$$
−0.0907360 + 0.995875i $$0.528922\pi$$
$$18$$ 0 0
$$19$$ 3.34596 0.767617 0.383808 0.923413i $$-0.374612\pi$$
0.383808 + 0.923413i $$0.374612\pi$$
$$20$$ 0 0
$$21$$ 8.69193 1.89673
$$22$$ 0 0
$$23$$ −1.49646 −0.312033 −0.156016 0.987754i $$-0.549865\pi$$
−0.156016 + 0.987754i $$0.549865\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 17.3839 3.34552
$$28$$ 0 0
$$29$$ 3.94370 0.732326 0.366163 0.930551i $$-0.380671\pi$$
0.366163 + 0.930551i $$0.380671\pi$$
$$30$$ 0 0
$$31$$ −7.79321 −1.39970 −0.699851 0.714289i $$-0.746750\pi$$
−0.699851 + 0.714289i $$0.746750\pi$$
$$32$$ 0 0
$$33$$ −15.8874 −2.76564
$$34$$ 0 0
$$35$$ −2.59774 −0.439097
$$36$$ 0 0
$$37$$ −1.00000 −0.164399
$$38$$ 0 0
$$39$$ 22.3909 3.58542
$$40$$ 0 0
$$41$$ −6.44724 −1.00689 −0.503445 0.864027i $$-0.667934\pi$$
−0.503445 + 0.864027i $$0.667934\pi$$
$$42$$ 0 0
$$43$$ −1.94370 −0.296411 −0.148206 0.988957i $$-0.547350\pi$$
−0.148206 + 0.988957i $$0.547350\pi$$
$$44$$ 0 0
$$45$$ −8.19547 −1.22171
$$46$$ 0 0
$$47$$ −1.84951 −0.269778 −0.134889 0.990861i $$-0.543068\pi$$
−0.134889 + 0.990861i $$0.543068\pi$$
$$48$$ 0 0
$$49$$ −0.251772 −0.0359674
$$50$$ 0 0
$$51$$ −2.50354 −0.350566
$$52$$ 0 0
$$53$$ 10.4472 1.43504 0.717520 0.696538i $$-0.245277\pi$$
0.717520 + 0.696538i $$0.245277\pi$$
$$54$$ 0 0
$$55$$ 4.74823 0.640251
$$56$$ 0 0
$$57$$ 11.1955 1.48288
$$58$$ 0 0
$$59$$ −5.84951 −0.761541 −0.380770 0.924670i $$-0.624341\pi$$
−0.380770 + 0.924670i $$0.624341\pi$$
$$60$$ 0 0
$$61$$ 7.94370 1.01709 0.508543 0.861036i $$-0.330184\pi$$
0.508543 + 0.861036i $$0.330184\pi$$
$$62$$ 0 0
$$63$$ 21.2897 2.68225
$$64$$ 0 0
$$65$$ −6.69193 −0.830031
$$66$$ 0 0
$$67$$ −1.84951 −0.225953 −0.112977 0.993598i $$-0.536039\pi$$
−0.112977 + 0.993598i $$0.536039\pi$$
$$68$$ 0 0
$$69$$ −5.00709 −0.602783
$$70$$ 0 0
$$71$$ 3.88740 0.461349 0.230675 0.973031i $$-0.425907\pi$$
0.230675 + 0.973031i $$0.425907\pi$$
$$72$$ 0 0
$$73$$ −7.49646 −0.877394 −0.438697 0.898635i $$-0.644560\pi$$
−0.438697 + 0.898635i $$0.644560\pi$$
$$74$$ 0 0
$$75$$ 3.34596 0.386359
$$76$$ 0 0
$$77$$ −12.3346 −1.40566
$$78$$ 0 0
$$79$$ 16.5414 1.86106 0.930528 0.366220i $$-0.119348\pi$$
0.930528 + 0.366220i $$0.119348\pi$$
$$80$$ 0 0
$$81$$ 33.5793 3.73104
$$82$$ 0 0
$$83$$ −15.2334 −1.67208 −0.836039 0.548670i $$-0.815134\pi$$
−0.836039 + 0.548670i $$0.815134\pi$$
$$84$$ 0 0
$$85$$ 0.748228 0.0811567
$$86$$ 0 0
$$87$$ 13.1955 1.41470
$$88$$ 0 0
$$89$$ 6.00000 0.635999 0.317999 0.948091i $$-0.396989\pi$$
0.317999 + 0.948091i $$0.396989\pi$$
$$90$$ 0 0
$$91$$ 17.3839 1.82232
$$92$$ 0 0
$$93$$ −26.0758 −2.70393
$$94$$ 0 0
$$95$$ −3.34596 −0.343289
$$96$$ 0 0
$$97$$ −10.4472 −1.06076 −0.530378 0.847761i $$-0.677950\pi$$
−0.530378 + 0.847761i $$0.677950\pi$$
$$98$$ 0 0
$$99$$ −38.9140 −3.91100
$$100$$ 0 0
$$101$$ −12.1884 −1.21279 −0.606395 0.795164i $$-0.707385\pi$$
−0.606395 + 0.795164i $$0.707385\pi$$
$$102$$ 0 0
$$103$$ 1.30807 0.128888 0.0644442 0.997921i $$-0.479473\pi$$
0.0644442 + 0.997921i $$0.479473\pi$$
$$104$$ 0 0
$$105$$ −8.69193 −0.848245
$$106$$ 0 0
$$107$$ 3.04498 0.294369 0.147185 0.989109i $$-0.452979\pi$$
0.147185 + 0.989109i $$0.452979\pi$$
$$108$$ 0 0
$$109$$ −1.44015 −0.137942 −0.0689709 0.997619i $$-0.521972\pi$$
−0.0689709 + 0.997619i $$0.521972\pi$$
$$110$$ 0 0
$$111$$ −3.34596 −0.317585
$$112$$ 0 0
$$113$$ 11.1392 1.04788 0.523942 0.851754i $$-0.324461\pi$$
0.523942 + 0.851754i $$0.324461\pi$$
$$114$$ 0 0
$$115$$ 1.49646 0.139545
$$116$$ 0 0
$$117$$ 54.8435 5.07028
$$118$$ 0 0
$$119$$ −1.94370 −0.178179
$$120$$ 0 0
$$121$$ 11.5457 1.04961
$$122$$ 0 0
$$123$$ −21.5722 −1.94510
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ 8.84242 0.784638 0.392319 0.919829i $$-0.371673\pi$$
0.392319 + 0.919829i $$0.371673\pi$$
$$128$$ 0 0
$$129$$ −6.50354 −0.572605
$$130$$ 0 0
$$131$$ −6.15049 −0.537371 −0.268686 0.963228i $$-0.586589\pi$$
−0.268686 + 0.963228i $$0.586589\pi$$
$$132$$ 0 0
$$133$$ 8.69193 0.753686
$$134$$ 0 0
$$135$$ −17.3839 −1.49616
$$136$$ 0 0
$$137$$ 10.8945 0.930779 0.465389 0.885106i $$-0.345914\pi$$
0.465389 + 0.885106i $$0.345914\pi$$
$$138$$ 0 0
$$139$$ −1.04921 −0.0889932 −0.0444966 0.999010i $$-0.514168\pi$$
−0.0444966 + 0.999010i $$0.514168\pi$$
$$140$$ 0 0
$$141$$ −6.18838 −0.521156
$$142$$ 0 0
$$143$$ −31.7748 −2.65714
$$144$$ 0 0
$$145$$ −3.94370 −0.327506
$$146$$ 0 0
$$147$$ −0.842420 −0.0694816
$$148$$ 0 0
$$149$$ −4.18838 −0.343126 −0.171563 0.985173i $$-0.554882\pi$$
−0.171563 + 0.985173i $$0.554882\pi$$
$$150$$ 0 0
$$151$$ 6.69193 0.544581 0.272291 0.962215i $$-0.412219\pi$$
0.272291 + 0.962215i $$0.412219\pi$$
$$152$$ 0 0
$$153$$ −6.13208 −0.495749
$$154$$ 0 0
$$155$$ 7.79321 0.625965
$$156$$ 0 0
$$157$$ −3.94370 −0.314741 −0.157371 0.987540i $$-0.550302\pi$$
−0.157371 + 0.987540i $$0.550302\pi$$
$$158$$ 0 0
$$159$$ 34.9561 2.77220
$$160$$ 0 0
$$161$$ −3.88740 −0.306370
$$162$$ 0 0
$$163$$ 12.1463 0.951368 0.475684 0.879616i $$-0.342201\pi$$
0.475684 + 0.879616i $$0.342201\pi$$
$$164$$ 0 0
$$165$$ 15.8874 1.23683
$$166$$ 0 0
$$167$$ −17.0829 −1.32191 −0.660956 0.750425i $$-0.729849\pi$$
−0.660956 + 0.750425i $$0.729849\pi$$
$$168$$ 0 0
$$169$$ 31.7819 2.44476
$$170$$ 0 0
$$171$$ 27.4217 2.09699
$$172$$ 0 0
$$173$$ −15.3417 −1.16641 −0.583205 0.812325i $$-0.698202\pi$$
−0.583205 + 0.812325i $$0.698202\pi$$
$$174$$ 0 0
$$175$$ 2.59774 0.196370
$$176$$ 0 0
$$177$$ −19.5722 −1.47114
$$178$$ 0 0
$$179$$ −2.45148 −0.183232 −0.0916161 0.995794i $$-0.529203\pi$$
−0.0916161 + 0.995794i $$0.529203\pi$$
$$180$$ 0 0
$$181$$ 13.1955 0.980812 0.490406 0.871494i $$-0.336849\pi$$
0.490406 + 0.871494i $$0.336849\pi$$
$$182$$ 0 0
$$183$$ 26.5793 1.96480
$$184$$ 0 0
$$185$$ 1.00000 0.0735215
$$186$$ 0 0
$$187$$ 3.55276 0.259803
$$188$$ 0 0
$$189$$ 45.1586 3.28481
$$190$$ 0 0
$$191$$ 21.7790 1.57588 0.787938 0.615755i $$-0.211149\pi$$
0.787938 + 0.615755i $$0.211149\pi$$
$$192$$ 0 0
$$193$$ −12.9929 −0.935250 −0.467625 0.883927i $$-0.654890\pi$$
−0.467625 + 0.883927i $$0.654890\pi$$
$$194$$ 0 0
$$195$$ −22.3909 −1.60345
$$196$$ 0 0
$$197$$ 0.616147 0.0438986 0.0219493 0.999759i $$-0.493013\pi$$
0.0219493 + 0.999759i $$0.493013\pi$$
$$198$$ 0 0
$$199$$ 1.54852 0.109772 0.0548859 0.998493i $$-0.482520\pi$$
0.0548859 + 0.998493i $$0.482520\pi$$
$$200$$ 0 0
$$201$$ −6.18838 −0.436495
$$202$$ 0 0
$$203$$ 10.2447 0.719036
$$204$$ 0 0
$$205$$ 6.44724 0.450295
$$206$$ 0 0
$$207$$ −12.2642 −0.852418
$$208$$ 0 0
$$209$$ −15.8874 −1.09895
$$210$$ 0 0
$$211$$ −20.9366 −1.44134 −0.720668 0.693280i $$-0.756165\pi$$
−0.720668 + 0.693280i $$0.756165\pi$$
$$212$$ 0 0
$$213$$ 13.0071 0.891231
$$214$$ 0 0
$$215$$ 1.94370 0.132559
$$216$$ 0 0
$$217$$ −20.2447 −1.37430
$$218$$ 0 0
$$219$$ −25.0829 −1.69494
$$220$$ 0 0
$$221$$ −5.00709 −0.336813
$$222$$ 0 0
$$223$$ 2.71034 0.181498 0.0907488 0.995874i $$-0.471074\pi$$
0.0907488 + 0.995874i $$0.471074\pi$$
$$224$$ 0 0
$$225$$ 8.19547 0.546365
$$226$$ 0 0
$$227$$ −26.1321 −1.73445 −0.867224 0.497919i $$-0.834098\pi$$
−0.867224 + 0.497919i $$0.834098\pi$$
$$228$$ 0 0
$$229$$ −24.7677 −1.63670 −0.818348 0.574723i $$-0.805110\pi$$
−0.818348 + 0.574723i $$0.805110\pi$$
$$230$$ 0 0
$$231$$ −41.2713 −2.71545
$$232$$ 0 0
$$233$$ −12.2783 −0.804381 −0.402190 0.915556i $$-0.631751\pi$$
−0.402190 + 0.915556i $$0.631751\pi$$
$$234$$ 0 0
$$235$$ 1.84951 0.120649
$$236$$ 0 0
$$237$$ 55.3470 3.59518
$$238$$ 0 0
$$239$$ 17.1771 1.11109 0.555546 0.831486i $$-0.312509\pi$$
0.555546 + 0.831486i $$0.312509\pi$$
$$240$$ 0 0
$$241$$ −15.2713 −0.983708 −0.491854 0.870678i $$-0.663681\pi$$
−0.491854 + 0.870678i $$0.663681\pi$$
$$242$$ 0 0
$$243$$ 60.2036 3.86206
$$244$$ 0 0
$$245$$ 0.251772 0.0160851
$$246$$ 0 0
$$247$$ 22.3909 1.42470
$$248$$ 0 0
$$249$$ −50.9703 −3.23011
$$250$$ 0 0
$$251$$ −10.0379 −0.633586 −0.316793 0.948495i $$-0.602606\pi$$
−0.316793 + 0.948495i $$0.602606\pi$$
$$252$$ 0 0
$$253$$ 7.10552 0.446720
$$254$$ 0 0
$$255$$ 2.50354 0.156778
$$256$$ 0 0
$$257$$ −20.9703 −1.30809 −0.654045 0.756456i $$-0.726929\pi$$
−0.654045 + 0.756456i $$0.726929\pi$$
$$258$$ 0 0
$$259$$ −2.59774 −0.161415
$$260$$ 0 0
$$261$$ 32.3205 2.00059
$$262$$ 0 0
$$263$$ −22.3725 −1.37955 −0.689775 0.724024i $$-0.742290\pi$$
−0.689775 + 0.724024i $$0.742290\pi$$
$$264$$ 0 0
$$265$$ −10.4472 −0.641769
$$266$$ 0 0
$$267$$ 20.0758 1.22862
$$268$$ 0 0
$$269$$ −4.18838 −0.255370 −0.127685 0.991815i $$-0.540755\pi$$
−0.127685 + 0.991815i $$0.540755\pi$$
$$270$$ 0 0
$$271$$ 12.1126 0.735788 0.367894 0.929868i $$-0.380079\pi$$
0.367894 + 0.929868i $$0.380079\pi$$
$$272$$ 0 0
$$273$$ 58.1657 3.52035
$$274$$ 0 0
$$275$$ −4.74823 −0.286329
$$276$$ 0 0
$$277$$ −9.30807 −0.559268 −0.279634 0.960107i $$-0.590213\pi$$
−0.279634 + 0.960107i $$0.590213\pi$$
$$278$$ 0 0
$$279$$ −63.8690 −3.82374
$$280$$ 0 0
$$281$$ 19.2713 1.14963 0.574813 0.818285i $$-0.305075\pi$$
0.574813 + 0.818285i $$0.305075\pi$$
$$282$$ 0 0
$$283$$ 13.6848 0.813479 0.406740 0.913544i $$-0.366666\pi$$
0.406740 + 0.913544i $$0.366666\pi$$
$$284$$ 0 0
$$285$$ −11.1955 −0.663162
$$286$$ 0 0
$$287$$ −16.7482 −0.988617
$$288$$ 0 0
$$289$$ −16.4402 −0.967068
$$290$$ 0 0
$$291$$ −34.9561 −2.04916
$$292$$ 0 0
$$293$$ −10.4472 −0.610334 −0.305167 0.952299i $$-0.598712\pi$$
−0.305167 + 0.952299i $$0.598712\pi$$
$$294$$ 0 0
$$295$$ 5.84951 0.340571
$$296$$ 0 0
$$297$$ −82.5425 −4.78960
$$298$$ 0 0
$$299$$ −10.0142 −0.579135
$$300$$ 0 0
$$301$$ −5.04921 −0.291032
$$302$$ 0 0
$$303$$ −40.7819 −2.34286
$$304$$ 0 0
$$305$$ −7.94370 −0.454855
$$306$$ 0 0
$$307$$ 4.95502 0.282798 0.141399 0.989953i $$-0.454840\pi$$
0.141399 + 0.989953i $$0.454840\pi$$
$$308$$ 0 0
$$309$$ 4.37677 0.248985
$$310$$ 0 0
$$311$$ −3.68060 −0.208708 −0.104354 0.994540i $$-0.533277\pi$$
−0.104354 + 0.994540i $$0.533277\pi$$
$$312$$ 0 0
$$313$$ −0.992912 −0.0561227 −0.0280614 0.999606i $$-0.508933\pi$$
−0.0280614 + 0.999606i $$0.508933\pi$$
$$314$$ 0 0
$$315$$ −21.2897 −1.19954
$$316$$ 0 0
$$317$$ −6.55985 −0.368438 −0.184219 0.982885i $$-0.558976\pi$$
−0.184219 + 0.982885i $$0.558976\pi$$
$$318$$ 0 0
$$319$$ −18.7256 −1.04843
$$320$$ 0 0
$$321$$ 10.1884 0.568660
$$322$$ 0 0
$$323$$ −2.50354 −0.139301
$$324$$ 0 0
$$325$$ 6.69193 0.371201
$$326$$ 0 0
$$327$$ −4.81870 −0.266475
$$328$$ 0 0
$$329$$ −4.80453 −0.264882
$$330$$ 0 0
$$331$$ 18.0379 0.991452 0.495726 0.868479i $$-0.334902\pi$$
0.495726 + 0.868479i $$0.334902\pi$$
$$332$$ 0 0
$$333$$ −8.19547 −0.449109
$$334$$ 0 0
$$335$$ 1.84951 0.101049
$$336$$ 0 0
$$337$$ −14.8945 −0.811354 −0.405677 0.914016i $$-0.632964\pi$$
−0.405677 + 0.914016i $$0.632964\pi$$
$$338$$ 0 0
$$339$$ 37.2713 2.02430
$$340$$ 0 0
$$341$$ 37.0039 2.00387
$$342$$ 0 0
$$343$$ −18.8382 −1.01717
$$344$$ 0 0
$$345$$ 5.00709 0.269573
$$346$$ 0 0
$$347$$ 9.38385 0.503752 0.251876 0.967760i $$-0.418953\pi$$
0.251876 + 0.967760i $$0.418953\pi$$
$$348$$ 0 0
$$349$$ −32.9703 −1.76486 −0.882429 0.470446i $$-0.844093\pi$$
−0.882429 + 0.470446i $$0.844093\pi$$
$$350$$ 0 0
$$351$$ 116.331 6.20931
$$352$$ 0 0
$$353$$ 27.4260 1.45974 0.729869 0.683587i $$-0.239581\pi$$
0.729869 + 0.683587i $$0.239581\pi$$
$$354$$ 0 0
$$355$$ −3.88740 −0.206322
$$356$$ 0 0
$$357$$ −6.50354 −0.344204
$$358$$ 0 0
$$359$$ 25.0829 1.32382 0.661912 0.749582i $$-0.269745\pi$$
0.661912 + 0.749582i $$0.269745\pi$$
$$360$$ 0 0
$$361$$ −7.80453 −0.410765
$$362$$ 0 0
$$363$$ 38.6314 2.02762
$$364$$ 0 0
$$365$$ 7.49646 0.392382
$$366$$ 0 0
$$367$$ −2.48513 −0.129723 −0.0648614 0.997894i $$-0.520661\pi$$
−0.0648614 + 0.997894i $$0.520661\pi$$
$$368$$ 0 0
$$369$$ −52.8382 −2.75065
$$370$$ 0 0
$$371$$ 27.1392 1.40900
$$372$$ 0 0
$$373$$ −35.1586 −1.82045 −0.910223 0.414119i $$-0.864090\pi$$
−0.910223 + 0.414119i $$0.864090\pi$$
$$374$$ 0 0
$$375$$ −3.34596 −0.172785
$$376$$ 0 0
$$377$$ 26.3909 1.35920
$$378$$ 0 0
$$379$$ −24.6778 −1.26761 −0.633805 0.773492i $$-0.718508\pi$$
−0.633805 + 0.773492i $$0.718508\pi$$
$$380$$ 0 0
$$381$$ 29.5864 1.51576
$$382$$ 0 0
$$383$$ 13.3839 0.683883 0.341941 0.939721i $$-0.388916\pi$$
0.341941 + 0.939721i $$0.388916\pi$$
$$384$$ 0 0
$$385$$ 12.3346 0.628631
$$386$$ 0 0
$$387$$ −15.9295 −0.809743
$$388$$ 0 0
$$389$$ 18.2220 0.923894 0.461947 0.886908i $$-0.347151\pi$$
0.461947 + 0.886908i $$0.347151\pi$$
$$390$$ 0 0
$$391$$ 1.11969 0.0566252
$$392$$ 0 0
$$393$$ −20.5793 −1.03809
$$394$$ 0 0
$$395$$ −16.5414 −0.832290
$$396$$ 0 0
$$397$$ −2.22521 −0.111680 −0.0558399 0.998440i $$-0.517784\pi$$
−0.0558399 + 0.998440i $$0.517784\pi$$
$$398$$ 0 0
$$399$$ 29.0829 1.45596
$$400$$ 0 0
$$401$$ 12.3909 0.618774 0.309387 0.950936i $$-0.399876\pi$$
0.309387 + 0.950936i $$0.399876\pi$$
$$402$$ 0 0
$$403$$ −52.1516 −2.59785
$$404$$ 0 0
$$405$$ −33.5793 −1.66857
$$406$$ 0 0
$$407$$ 4.74823 0.235361
$$408$$ 0 0
$$409$$ 7.27125 0.359540 0.179770 0.983709i $$-0.442465\pi$$
0.179770 + 0.983709i $$0.442465\pi$$
$$410$$ 0 0
$$411$$ 36.4525 1.79807
$$412$$ 0 0
$$413$$ −15.1955 −0.747720
$$414$$ 0 0
$$415$$ 15.2334 0.747776
$$416$$ 0 0
$$417$$ −3.51063 −0.171916
$$418$$ 0 0
$$419$$ −28.4809 −1.39138 −0.695691 0.718341i $$-0.744902\pi$$
−0.695691 + 0.718341i $$0.744902\pi$$
$$420$$ 0 0
$$421$$ 22.0000 1.07221 0.536107 0.844150i $$-0.319894\pi$$
0.536107 + 0.844150i $$0.319894\pi$$
$$422$$ 0 0
$$423$$ −15.1576 −0.736987
$$424$$ 0 0
$$425$$ −0.748228 −0.0362944
$$426$$ 0 0
$$427$$ 20.6356 0.998628
$$428$$ 0 0
$$429$$ −106.317 −5.13305
$$430$$ 0 0
$$431$$ −1.51487 −0.0729686 −0.0364843 0.999334i $$-0.511616\pi$$
−0.0364843 + 0.999334i $$0.511616\pi$$
$$432$$ 0 0
$$433$$ 24.9929 1.20108 0.600541 0.799594i $$-0.294952\pi$$
0.600541 + 0.799594i $$0.294952\pi$$
$$434$$ 0 0
$$435$$ −13.1955 −0.632674
$$436$$ 0 0
$$437$$ −5.00709 −0.239521
$$438$$ 0 0
$$439$$ 5.10128 0.243471 0.121735 0.992563i $$-0.461154\pi$$
0.121735 + 0.992563i $$0.461154\pi$$
$$440$$ 0 0
$$441$$ −2.06339 −0.0982566
$$442$$ 0 0
$$443$$ −25.7369 −1.22280 −0.611399 0.791323i $$-0.709393\pi$$
−0.611399 + 0.791323i $$0.709393\pi$$
$$444$$ 0 0
$$445$$ −6.00000 −0.284427
$$446$$ 0 0
$$447$$ −14.0142 −0.662848
$$448$$ 0 0
$$449$$ −17.7748 −0.838844 −0.419422 0.907791i $$-0.637767\pi$$
−0.419422 + 0.907791i $$0.637767\pi$$
$$450$$ 0 0
$$451$$ 30.6130 1.44151
$$452$$ 0 0
$$453$$ 22.3909 1.05202
$$454$$ 0 0
$$455$$ −17.3839 −0.814968
$$456$$ 0 0
$$457$$ 34.3346 1.60611 0.803053 0.595907i $$-0.203207\pi$$
0.803053 + 0.595907i $$0.203207\pi$$
$$458$$ 0 0
$$459$$ −13.0071 −0.607119
$$460$$ 0 0
$$461$$ −16.9508 −0.789477 −0.394738 0.918794i $$-0.629165\pi$$
−0.394738 + 0.918794i $$0.629165\pi$$
$$462$$ 0 0
$$463$$ −10.9929 −0.510884 −0.255442 0.966824i $$-0.582221\pi$$
−0.255442 + 0.966824i $$0.582221\pi$$
$$464$$ 0 0
$$465$$ 26.0758 1.20924
$$466$$ 0 0
$$467$$ −37.4175 −1.73148 −0.865738 0.500498i $$-0.833150\pi$$
−0.865738 + 0.500498i $$0.833150\pi$$
$$468$$ 0 0
$$469$$ −4.80453 −0.221853
$$470$$ 0 0
$$471$$ −13.1955 −0.608015
$$472$$ 0 0
$$473$$ 9.22912 0.424356
$$474$$ 0 0
$$475$$ 3.34596 0.153523
$$476$$ 0 0
$$477$$ 85.6201 3.92027
$$478$$ 0 0
$$479$$ 0.465654 0.0212763 0.0106381 0.999943i $$-0.496614\pi$$
0.0106381 + 0.999943i $$0.496614\pi$$
$$480$$ 0 0
$$481$$ −6.69193 −0.305126
$$482$$ 0 0
$$483$$ −13.0071 −0.591843
$$484$$ 0 0
$$485$$ 10.4472 0.474385
$$486$$ 0 0
$$487$$ −34.9561 −1.58401 −0.792006 0.610514i $$-0.790963\pi$$
−0.792006 + 0.610514i $$0.790963\pi$$
$$488$$ 0 0
$$489$$ 40.6409 1.83785
$$490$$ 0 0
$$491$$ 24.7819 1.11839 0.559195 0.829036i $$-0.311110\pi$$
0.559195 + 0.829036i $$0.311110\pi$$
$$492$$ 0 0
$$493$$ −2.95079 −0.132897
$$494$$ 0 0
$$495$$ 38.9140 1.74905
$$496$$ 0 0
$$497$$ 10.0984 0.452976
$$498$$ 0 0
$$499$$ 2.33888 0.104702 0.0523512 0.998629i $$-0.483328\pi$$
0.0523512 + 0.998629i $$0.483328\pi$$
$$500$$ 0 0
$$501$$ −57.1586 −2.55366
$$502$$ 0 0
$$503$$ 18.6161 0.830053 0.415026 0.909809i $$-0.363772\pi$$
0.415026 + 0.909809i $$0.363772\pi$$
$$504$$ 0 0
$$505$$ 12.1884 0.542376
$$506$$ 0 0
$$507$$ 106.341 4.72277
$$508$$ 0 0
$$509$$ 6.89448 0.305593 0.152796 0.988258i $$-0.451172\pi$$
0.152796 + 0.988258i $$0.451172\pi$$
$$510$$ 0 0
$$511$$ −19.4738 −0.861471
$$512$$ 0 0
$$513$$ 58.1657 2.56808
$$514$$ 0 0
$$515$$ −1.30807 −0.0576406
$$516$$ 0 0
$$517$$ 8.78188 0.386227
$$518$$ 0 0
$$519$$ −51.3329 −2.25326
$$520$$ 0 0
$$521$$ 4.86083 0.212957 0.106478 0.994315i $$-0.466042\pi$$
0.106478 + 0.994315i $$0.466042\pi$$
$$522$$ 0 0
$$523$$ −22.7677 −0.995562 −0.497781 0.867303i $$-0.665852\pi$$
−0.497781 + 0.867303i $$0.665852\pi$$
$$524$$ 0 0
$$525$$ 8.69193 0.379347
$$526$$ 0 0
$$527$$ 5.83110 0.254007
$$528$$ 0 0
$$529$$ −20.7606 −0.902636
$$530$$ 0 0
$$531$$ −47.9395 −2.08040
$$532$$ 0 0
$$533$$ −43.1445 −1.86879
$$534$$ 0 0
$$535$$ −3.04498 −0.131646
$$536$$ 0 0
$$537$$ −8.20256 −0.353967
$$538$$ 0 0
$$539$$ 1.19547 0.0514926
$$540$$ 0 0
$$541$$ 8.99291 0.386636 0.193318 0.981136i $$-0.438075\pi$$
0.193318 + 0.981136i $$0.438075\pi$$
$$542$$ 0 0
$$543$$ 44.1516 1.89472
$$544$$ 0 0
$$545$$ 1.44015 0.0616894
$$546$$ 0 0
$$547$$ 18.8382 0.805463 0.402731 0.915318i $$-0.368061\pi$$
0.402731 + 0.915318i $$0.368061\pi$$
$$548$$ 0 0
$$549$$ 65.1023 2.77850
$$550$$ 0 0
$$551$$ 13.1955 0.562146
$$552$$ 0 0
$$553$$ 42.9703 1.82728
$$554$$ 0 0
$$555$$ 3.34596 0.142028
$$556$$ 0 0
$$557$$ 3.69901 0.156732 0.0783661 0.996925i $$-0.475030\pi$$
0.0783661 + 0.996925i $$0.475030\pi$$
$$558$$ 0 0
$$559$$ −13.0071 −0.550141
$$560$$ 0 0
$$561$$ 11.8874 0.501886
$$562$$ 0 0
$$563$$ 14.3488 0.604730 0.302365 0.953192i $$-0.402224\pi$$
0.302365 + 0.953192i $$0.402224\pi$$
$$564$$ 0 0
$$565$$ −11.1392 −0.468628
$$566$$ 0 0
$$567$$ 87.2302 3.66332
$$568$$ 0 0
$$569$$ 15.9858 0.670161 0.335080 0.942190i $$-0.391237\pi$$
0.335080 + 0.942190i $$0.391237\pi$$
$$570$$ 0 0
$$571$$ 30.2079 1.26416 0.632080 0.774903i $$-0.282201\pi$$
0.632080 + 0.774903i $$0.282201\pi$$
$$572$$ 0 0
$$573$$ 72.8718 3.04426
$$574$$ 0 0
$$575$$ −1.49646 −0.0624065
$$576$$ 0 0
$$577$$ −19.1813 −0.798528 −0.399264 0.916836i $$-0.630734\pi$$
−0.399264 + 0.916836i $$0.630734\pi$$
$$578$$ 0 0
$$579$$ −43.4738 −1.80671
$$580$$ 0 0
$$581$$ −39.5722 −1.64173
$$582$$ 0 0
$$583$$ −49.6059 −2.05447
$$584$$ 0 0
$$585$$ −54.8435 −2.26750
$$586$$ 0 0
$$587$$ 26.6130 1.09844 0.549218 0.835679i $$-0.314926\pi$$
0.549218 + 0.835679i $$0.314926\pi$$
$$588$$ 0 0
$$589$$ −26.0758 −1.07443
$$590$$ 0 0
$$591$$ 2.06160 0.0848031
$$592$$ 0 0
$$593$$ 26.2642 1.07854 0.539270 0.842133i $$-0.318700\pi$$
0.539270 + 0.842133i $$0.318700\pi$$
$$594$$ 0 0
$$595$$ 1.94370 0.0796839
$$596$$ 0 0
$$597$$ 5.18130 0.212056
$$598$$ 0 0
$$599$$ 38.2415 1.56251 0.781253 0.624215i $$-0.214581\pi$$
0.781253 + 0.624215i $$0.214581\pi$$
$$600$$ 0 0
$$601$$ 30.2447 1.23371 0.616853 0.787078i $$-0.288407\pi$$
0.616853 + 0.787078i $$0.288407\pi$$
$$602$$ 0 0
$$603$$ −15.1576 −0.617264
$$604$$ 0 0
$$605$$ −11.5457 −0.469398
$$606$$ 0 0
$$607$$ 5.90157 0.239537 0.119769 0.992802i $$-0.461785\pi$$
0.119769 + 0.992802i $$0.461785\pi$$
$$608$$ 0 0
$$609$$ 34.2783 1.38903
$$610$$ 0 0
$$611$$ −12.3768 −0.500710
$$612$$ 0 0
$$613$$ 6.15473 0.248587 0.124294 0.992245i $$-0.460334\pi$$
0.124294 + 0.992245i $$0.460334\pi$$
$$614$$ 0 0
$$615$$ 21.5722 0.869877
$$616$$ 0 0
$$617$$ 16.5035 0.664408 0.332204 0.943208i $$-0.392208\pi$$
0.332204 + 0.943208i $$0.392208\pi$$
$$618$$ 0 0
$$619$$ −9.04921 −0.363719 −0.181859 0.983325i $$-0.558212\pi$$
−0.181859 + 0.983325i $$0.558212\pi$$
$$620$$ 0 0
$$621$$ −26.0142 −1.04391
$$622$$ 0 0
$$623$$ 15.5864 0.624456
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ −53.1586 −2.12295
$$628$$ 0 0
$$629$$ 0.748228 0.0298338
$$630$$ 0 0
$$631$$ 0.507780 0.0202144 0.0101072 0.999949i $$-0.496783\pi$$
0.0101072 + 0.999949i $$0.496783\pi$$
$$632$$ 0 0
$$633$$ −70.0531 −2.78436
$$634$$ 0 0
$$635$$ −8.84242 −0.350901
$$636$$ 0 0
$$637$$ −1.68484 −0.0667558
$$638$$ 0 0
$$639$$ 31.8590 1.26032
$$640$$ 0 0
$$641$$ −40.6356 −1.60501 −0.802505 0.596645i $$-0.796500\pi$$
−0.802505 + 0.596645i $$0.796500\pi$$
$$642$$ 0 0
$$643$$ 9.53011 0.375831 0.187915 0.982185i $$-0.439827\pi$$
0.187915 + 0.982185i $$0.439827\pi$$
$$644$$ 0 0
$$645$$ 6.50354 0.256077
$$646$$ 0 0
$$647$$ 21.0071 0.825874 0.412937 0.910760i $$-0.364503\pi$$
0.412937 + 0.910760i $$0.364503\pi$$
$$648$$ 0 0
$$649$$ 27.7748 1.09026
$$650$$ 0 0
$$651$$ −67.7380 −2.65486
$$652$$ 0 0
$$653$$ −45.5496 −1.78249 −0.891247 0.453519i $$-0.850168\pi$$
−0.891247 + 0.453519i $$0.850168\pi$$
$$654$$ 0 0
$$655$$ 6.15049 0.240320
$$656$$ 0 0
$$657$$ −61.4370 −2.39689
$$658$$ 0 0
$$659$$ −9.38385 −0.365543 −0.182772 0.983155i $$-0.558507\pi$$
−0.182772 + 0.983155i $$0.558507\pi$$
$$660$$ 0 0
$$661$$ 4.05630 0.157772 0.0788859 0.996884i $$-0.474864\pi$$
0.0788859 + 0.996884i $$0.474864\pi$$
$$662$$ 0 0
$$663$$ −16.7535 −0.650653
$$664$$ 0 0
$$665$$ −8.69193 −0.337058
$$666$$ 0 0
$$667$$ −5.90157 −0.228510
$$668$$ 0 0
$$669$$ 9.06869 0.350616
$$670$$ 0 0
$$671$$ −37.7185 −1.45611
$$672$$ 0 0
$$673$$ 7.60906 0.293308 0.146654 0.989188i $$-0.453150\pi$$
0.146654 + 0.989188i $$0.453150\pi$$
$$674$$ 0 0
$$675$$ 17.3839 0.669105
$$676$$ 0 0
$$677$$ 28.7677 1.10563 0.552816 0.833303i $$-0.313553\pi$$
0.552816 + 0.833303i $$0.313553\pi$$
$$678$$ 0 0
$$679$$ −27.1392 −1.04151
$$680$$ 0 0
$$681$$ −87.4370 −3.35059
$$682$$ 0 0
$$683$$ −0.0704767 −0.00269671 −0.00134836 0.999999i $$-0.500429\pi$$
−0.00134836 + 0.999999i $$0.500429\pi$$
$$684$$ 0 0
$$685$$ −10.8945 −0.416257
$$686$$ 0 0
$$687$$ −82.8718 −3.16176
$$688$$ 0 0
$$689$$ 69.9122 2.66344
$$690$$ 0 0
$$691$$ −10.6498 −0.405137 −0.202569 0.979268i $$-0.564929\pi$$
−0.202569 + 0.979268i $$0.564929\pi$$
$$692$$ 0 0
$$693$$ −101.088 −3.84002
$$694$$ 0 0
$$695$$ 1.04921 0.0397990
$$696$$ 0 0
$$697$$ 4.82401 0.182722
$$698$$ 0 0
$$699$$ −41.0829 −1.55390
$$700$$ 0 0
$$701$$ −7.98582 −0.301620 −0.150810 0.988563i $$-0.548188\pi$$
−0.150810 + 0.988563i $$0.548188\pi$$
$$702$$ 0 0
$$703$$ −3.34596 −0.126195
$$704$$ 0 0
$$705$$ 6.18838 0.233068
$$706$$ 0 0
$$707$$ −31.6622 −1.19078
$$708$$ 0 0
$$709$$ −37.2149 −1.39764 −0.698818 0.715299i $$-0.746290\pi$$
−0.698818 + 0.715299i $$0.746290\pi$$
$$710$$ 0 0
$$711$$ 135.565 5.08408
$$712$$ 0 0
$$713$$ 11.6622 0.436752
$$714$$ 0 0
$$715$$ 31.7748 1.18831
$$716$$ 0 0
$$717$$ 57.4738 2.14640
$$718$$ 0 0
$$719$$ −30.1742 −1.12531 −0.562654 0.826692i $$-0.690220\pi$$
−0.562654 + 0.826692i $$0.690220\pi$$
$$720$$ 0 0
$$721$$ 3.39803 0.126549
$$722$$ 0 0
$$723$$ −51.0970 −1.90032
$$724$$ 0 0
$$725$$ 3.94370 0.146465
$$726$$ 0 0
$$727$$ 18.7677 0.696056 0.348028 0.937484i $$-0.386851\pi$$
0.348028 + 0.937484i $$0.386851\pi$$
$$728$$ 0 0
$$729$$ 100.701 3.72967
$$730$$ 0 0
$$731$$ 1.45433 0.0537903
$$732$$ 0 0
$$733$$ 26.1094 0.964374 0.482187 0.876068i $$-0.339843\pi$$
0.482187 + 0.876068i $$0.339843\pi$$
$$734$$ 0 0
$$735$$ 0.842420 0.0310731
$$736$$ 0 0
$$737$$ 8.78188 0.323485
$$738$$ 0 0
$$739$$ −42.6130 −1.56754 −0.783772 0.621049i $$-0.786707\pi$$
−0.783772 + 0.621049i $$0.786707\pi$$
$$740$$ 0 0
$$741$$ 74.9193 2.75223
$$742$$ 0 0
$$743$$ 35.4922 1.30208 0.651042 0.759042i $$-0.274332\pi$$
0.651042 + 0.759042i $$0.274332\pi$$
$$744$$ 0 0
$$745$$ 4.18838 0.153450
$$746$$ 0 0
$$747$$ −124.845 −4.56782
$$748$$ 0 0
$$749$$ 7.91005 0.289027
$$750$$ 0 0
$$751$$ −21.5722 −0.787182 −0.393591 0.919286i $$-0.628767\pi$$
−0.393591 + 0.919286i $$0.628767\pi$$
$$752$$ 0 0
$$753$$ −33.5864 −1.22396
$$754$$ 0 0
$$755$$ −6.69193 −0.243544
$$756$$ 0 0
$$757$$ −23.7890 −0.864625 −0.432312 0.901724i $$-0.642302\pi$$
−0.432312 + 0.901724i $$0.642302\pi$$
$$758$$ 0 0
$$759$$ 23.7748 0.862970
$$760$$ 0 0
$$761$$ −18.6724 −0.676876 −0.338438 0.940989i $$-0.609898\pi$$
−0.338438 + 0.940989i $$0.609898\pi$$
$$762$$ 0 0
$$763$$ −3.74114 −0.135438
$$764$$ 0 0
$$765$$ 6.13208 0.221706
$$766$$ 0 0
$$767$$ −39.1445 −1.41342
$$768$$ 0 0
$$769$$ −2.48937 −0.0897689 −0.0448845 0.998992i $$-0.514292\pi$$
−0.0448845 + 0.998992i $$0.514292\pi$$
$$770$$ 0 0
$$771$$ −70.1657 −2.52696
$$772$$ 0 0
$$773$$ 0.950786 0.0341974 0.0170987 0.999854i $$-0.494557\pi$$
0.0170987 + 0.999854i $$0.494557\pi$$
$$774$$ 0 0
$$775$$ −7.79321 −0.279940
$$776$$ 0 0
$$777$$ −8.69193 −0.311821
$$778$$ 0 0
$$779$$ −21.5722 −0.772906
$$780$$ 0 0
$$781$$ −18.4582 −0.660488
$$782$$ 0 0
$$783$$ 68.5567 2.45002
$$784$$ 0 0
$$785$$ 3.94370 0.140757
$$786$$ 0 0
$$787$$ 29.4359 1.04928 0.524639 0.851325i $$-0.324200\pi$$
0.524639 + 0.851325i $$0.324200\pi$$
$$788$$ 0 0
$$789$$ −74.8577 −2.66500
$$790$$ 0 0
$$791$$ 28.9366 1.02887
$$792$$ 0 0
$$793$$ 53.1586 1.88772
$$794$$ 0 0
$$795$$ −34.9561 −1.23976
$$796$$ 0 0
$$797$$ −14.1742 −0.502076 −0.251038 0.967977i $$-0.580772\pi$$
−0.251038 + 0.967977i $$0.580772\pi$$
$$798$$ 0 0
$$799$$ 1.38385 0.0489572
$$800$$ 0 0
$$801$$ 49.1728 1.73744
$$802$$ 0 0
$$803$$ 35.5949 1.25612
$$804$$ 0 0
$$805$$ 3.88740 0.137013
$$806$$ 0 0
$$807$$ −14.0142 −0.493322
$$808$$ 0 0
$$809$$ −32.9929 −1.15997 −0.579985 0.814627i $$-0.696941\pi$$
−0.579985 + 0.814627i $$0.696941\pi$$
$$810$$ 0 0
$$811$$ −40.5567 −1.42414 −0.712069 0.702110i $$-0.752242\pi$$
−0.712069 + 0.702110i $$0.752242\pi$$
$$812$$ 0 0
$$813$$ 40.5283 1.42139
$$814$$ 0 0
$$815$$ −12.1463 −0.425465
$$816$$ 0 0
$$817$$ −6.50354 −0.227530
$$818$$ 0 0
$$819$$ 142.469 4.97826
$$820$$ 0 0
$$821$$ −38.8661 −1.35644 −0.678219 0.734860i $$-0.737248\pi$$
−0.678219 + 0.734860i $$0.737248\pi$$
$$822$$ 0 0
$$823$$ 49.6243 1.72979 0.864897 0.501949i $$-0.167384\pi$$
0.864897 + 0.501949i $$0.167384\pi$$
$$824$$ 0 0
$$825$$ −15.8874 −0.553128
$$826$$ 0 0
$$827$$ 30.0195 1.04388 0.521940 0.852982i $$-0.325209\pi$$
0.521940 + 0.852982i $$0.325209\pi$$
$$828$$ 0 0
$$829$$ 10.5598 0.366759 0.183379 0.983042i $$-0.441296\pi$$
0.183379 + 0.983042i $$0.441296\pi$$
$$830$$ 0 0
$$831$$ −31.1445 −1.08039
$$832$$ 0 0
$$833$$ 0.188383 0.00652708
$$834$$ 0 0
$$835$$ 17.0829 0.591177
$$836$$ 0 0
$$837$$ −135.476 −4.68273
$$838$$ 0 0
$$839$$ −29.3839 −1.01444 −0.507222 0.861816i $$-0.669327\pi$$
−0.507222 + 0.861816i $$0.669327\pi$$
$$840$$ 0 0
$$841$$ −13.4472 −0.463698
$$842$$ 0 0
$$843$$ 64.4809 2.22084
$$844$$ 0 0
$$845$$ −31.7819 −1.09333
$$846$$ 0 0
$$847$$ 29.9926 1.03056
$$848$$ 0 0
$$849$$ 45.7890 1.57147
$$850$$ 0 0
$$851$$ 1.49646 0.0512979
$$852$$ 0 0
$$853$$ 6.22521 0.213147 0.106573 0.994305i $$-0.466012\pi$$
0.106573 + 0.994305i $$0.466012\pi$$
$$854$$ 0 0
$$855$$ −27.4217 −0.937804
$$856$$ 0 0
$$857$$ −3.65119 −0.124722 −0.0623611 0.998054i $$-0.519863\pi$$
−0.0623611 + 0.998054i $$0.519863\pi$$
$$858$$ 0 0
$$859$$ 36.3162 1.23909 0.619547 0.784960i $$-0.287316\pi$$
0.619547 + 0.784960i $$0.287316\pi$$
$$860$$ 0 0
$$861$$ −56.0390 −1.90980
$$862$$ 0 0
$$863$$ −24.5751 −0.836546 −0.418273 0.908321i $$-0.637364\pi$$
−0.418273 + 0.908321i $$0.637364\pi$$
$$864$$ 0 0
$$865$$ 15.3417 0.521634
$$866$$ 0 0
$$867$$ −55.0082 −1.86817
$$868$$ 0 0
$$869$$ −78.5425 −2.66437
$$870$$ 0 0
$$871$$ −12.3768 −0.419371
$$872$$ 0 0
$$873$$ −85.6201 −2.89780
$$874$$ 0 0
$$875$$ −2.59774 −0.0878195
$$876$$ 0 0
$$877$$ 20.0563 0.677253 0.338627 0.940921i $$-0.390038\pi$$
0.338627 + 0.940921i $$0.390038\pi$$
$$878$$ 0 0
$$879$$ −34.9561 −1.17904
$$880$$ 0 0
$$881$$ −15.0266 −0.506258 −0.253129 0.967433i $$-0.581460\pi$$
−0.253129 + 0.967433i $$0.581460\pi$$
$$882$$ 0 0
$$883$$ 6.62145 0.222830 0.111415 0.993774i $$-0.464462\pi$$
0.111415 + 0.993774i $$0.464462\pi$$
$$884$$ 0 0
$$885$$ 19.5722 0.657914
$$886$$ 0 0
$$887$$ 37.5538 1.26093 0.630467 0.776216i $$-0.282863\pi$$
0.630467 + 0.776216i $$0.282863\pi$$
$$888$$ 0 0
$$889$$ 22.9703 0.770398
$$890$$ 0 0
$$891$$ −159.442 −5.34152
$$892$$ 0 0
$$893$$ −6.18838 −0.207086
$$894$$ 0 0
$$895$$ 2.45148 0.0819439
$$896$$ 0 0
$$897$$ −33.5071 −1.11877
$$898$$ 0 0
$$899$$ −30.7341 −1.02504
$$900$$ 0 0
$$901$$ −7.81692 −0.260419
$$902$$ 0 0
$$903$$ −16.8945 −0.562213
$$904$$ 0 0
$$905$$ −13.1955 −0.438632
$$906$$ 0 0
$$907$$ −50.4667 −1.67572 −0.837860 0.545885i $$-0.816193\pi$$
−0.837860 + 0.545885i $$0.816193\pi$$
$$908$$ 0 0
$$909$$ −99.8895 −3.31313
$$910$$ 0 0
$$911$$ −15.9395 −0.528098 −0.264049 0.964509i $$-0.585058\pi$$
−0.264049 + 0.964509i $$0.585058\pi$$
$$912$$ 0 0
$$913$$ 72.3315 2.39382
$$914$$ 0 0
$$915$$ −26.5793 −0.878685
$$916$$ 0 0
$$917$$ −15.9774 −0.527619
$$918$$ 0 0
$$919$$ −32.5414 −1.07344 −0.536721 0.843760i $$-0.680337\pi$$
−0.536721 + 0.843760i $$0.680337\pi$$
$$920$$ 0 0
$$921$$ 16.5793 0.546307
$$922$$ 0 0
$$923$$ 26.0142 0.856267
$$924$$ 0 0
$$925$$ −1.00000 −0.0328798
$$926$$ 0 0
$$927$$ 10.7203 0.352100
$$928$$ 0 0
$$929$$ −34.1094 −1.11909 −0.559547 0.828799i $$-0.689025\pi$$
−0.559547 + 0.828799i $$0.689025\pi$$
$$930$$ 0 0
$$931$$ −0.842420 −0.0276092
$$932$$ 0 0
$$933$$ −12.3152 −0.403180
$$934$$ 0 0
$$935$$ −3.55276 −0.116188
$$936$$ 0 0
$$937$$ 0.0141751 0.000463082 0 0.000231541 1.00000i $$-0.499926\pi$$
0.000231541 1.00000i $$0.499926\pi$$
$$938$$ 0 0
$$939$$ −3.32225 −0.108417
$$940$$ 0 0
$$941$$ 11.1813 0.364500 0.182250 0.983252i $$-0.441662\pi$$
0.182250 + 0.983252i $$0.441662\pi$$
$$942$$ 0 0
$$943$$ 9.64802 0.314183
$$944$$ 0 0
$$945$$ −45.1586 −1.46901
$$946$$ 0 0
$$947$$ 53.6427 1.74315 0.871577 0.490259i $$-0.163098\pi$$
0.871577 + 0.490259i $$0.163098\pi$$
$$948$$ 0 0
$$949$$ −50.1657 −1.62845
$$950$$ 0 0
$$951$$ −21.9490 −0.711745
$$952$$ 0 0
$$953$$ −9.88740 −0.320284 −0.160142 0.987094i $$-0.551195\pi$$
−0.160142 + 0.987094i $$0.551195\pi$$
$$954$$ 0 0
$$955$$ −21.7790 −0.704753
$$956$$ 0 0
$$957$$ −62.6551 −2.02535
$$958$$ 0 0
$$959$$ 28.3010 0.913886
$$960$$ 0 0
$$961$$ 29.7341 0.959163
$$962$$ 0 0
$$963$$ 24.9550 0.804164
$$964$$ 0 0
$$965$$ 12.9929 0.418257
$$966$$ 0 0
$$967$$ −51.6254 −1.66016 −0.830080 0.557644i $$-0.811705\pi$$
−0.830080 + 0.557644i $$0.811705\pi$$
$$968$$ 0 0
$$969$$ −8.37677 −0.269100
$$970$$ 0 0
$$971$$ −20.5598 −0.659797 −0.329898 0.944016i $$-0.607014\pi$$
−0.329898 + 0.944016i $$0.607014\pi$$
$$972$$ 0 0
$$973$$ −2.72558 −0.0873781
$$974$$ 0 0
$$975$$ 22.3909 0.717084
$$976$$ 0 0
$$977$$ 38.4472 1.23004 0.615018 0.788513i $$-0.289149\pi$$
0.615018 + 0.788513i $$0.289149\pi$$
$$978$$ 0 0
$$979$$ −28.4894 −0.910524
$$980$$ 0 0
$$981$$ −11.8027 −0.376833
$$982$$ 0 0
$$983$$ −11.8973 −0.379466 −0.189733 0.981836i $$-0.560762\pi$$
−0.189733 + 0.981836i $$0.560762\pi$$
$$984$$ 0 0
$$985$$ −0.616147 −0.0196321
$$986$$ 0 0
$$987$$ −16.0758 −0.511698
$$988$$ 0 0
$$989$$ 2.90866 0.0924900
$$990$$ 0 0
$$991$$ 19.7932 0.628752 0.314376 0.949299i $$-0.398205\pi$$
0.314376 + 0.949299i $$0.398205\pi$$
$$992$$ 0 0
$$993$$ 60.3541 1.91528
$$994$$ 0 0
$$995$$ −1.54852 −0.0490914
$$996$$ 0 0
$$997$$ 5.88740 0.186456 0.0932279 0.995645i $$-0.470281\pi$$
0.0932279 + 0.995645i $$0.470281\pi$$
$$998$$ 0 0
$$999$$ −17.3839 −0.550001
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 2960.2.a.u.1.3 3
4.3 odd 2 370.2.a.g.1.1 3
12.11 even 2 3330.2.a.bg.1.1 3
20.3 even 4 1850.2.b.o.149.1 6
20.7 even 4 1850.2.b.o.149.6 6
20.19 odd 2 1850.2.a.z.1.3 3

By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.a.g.1.1 3 4.3 odd 2
1850.2.a.z.1.3 3 20.19 odd 2
1850.2.b.o.149.1 6 20.3 even 4
1850.2.b.o.149.6 6 20.7 even 4
2960.2.a.u.1.3 3 1.1 even 1 trivial
3330.2.a.bg.1.1 3 12.11 even 2