# Properties

 Label 105.4.a.g.1.2 Level $105$ Weight $4$ Character 105.1 Self dual yes Analytic conductor $6.195$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [105,4,Mod(1,105)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(105, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("105.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$105 = 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 105.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: $$6.19520055060$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{41})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 10$$ x^2 - x - 10 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 $$3.70156$$ of defining polynomial Character $$\chi$$ $$=$$ 105.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+4.70156 q^{2} +3.00000 q^{3} +14.1047 q^{4} -5.00000 q^{5} +14.1047 q^{6} +7.00000 q^{7} +28.7016 q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q+4.70156 q^{2} +3.00000 q^{3} +14.1047 q^{4} -5.00000 q^{5} +14.1047 q^{6} +7.00000 q^{7} +28.7016 q^{8} +9.00000 q^{9} -23.5078 q^{10} +24.5969 q^{11} +42.3141 q^{12} -35.0156 q^{13} +32.9109 q^{14} -15.0000 q^{15} +22.1047 q^{16} -18.4187 q^{17} +42.3141 q^{18} -67.4031 q^{19} -70.5234 q^{20} +21.0000 q^{21} +115.644 q^{22} -145.675 q^{23} +86.1047 q^{24} +25.0000 q^{25} -164.628 q^{26} +27.0000 q^{27} +98.7328 q^{28} +214.419 q^{29} -70.5234 q^{30} -88.6594 q^{31} -125.686 q^{32} +73.7906 q^{33} -86.5969 q^{34} -35.0000 q^{35} +126.942 q^{36} +162.125 q^{37} -316.900 q^{38} -105.047 q^{39} -143.508 q^{40} -337.769 q^{41} +98.7328 q^{42} +122.156 q^{43} +346.931 q^{44} -45.0000 q^{45} -684.900 q^{46} +354.219 q^{47} +66.3141 q^{48} +49.0000 q^{49} +117.539 q^{50} -55.2562 q^{51} -493.884 q^{52} +676.691 q^{53} +126.942 q^{54} -122.984 q^{55} +200.911 q^{56} -202.209 q^{57} +1008.10 q^{58} +501.319 q^{59} -211.570 q^{60} -708.931 q^{61} -416.837 q^{62} +63.0000 q^{63} -767.758 q^{64} +175.078 q^{65} +346.931 q^{66} -907.956 q^{67} -259.791 q^{68} -437.025 q^{69} -164.555 q^{70} +430.334 q^{71} +258.314 q^{72} +41.3406 q^{73} +762.241 q^{74} +75.0000 q^{75} -950.700 q^{76} +172.178 q^{77} -493.884 q^{78} +890.388 q^{79} -110.523 q^{80} +81.0000 q^{81} -1588.04 q^{82} -1057.15 q^{83} +296.198 q^{84} +92.0937 q^{85} +574.325 q^{86} +643.256 q^{87} +705.969 q^{88} +1473.72 q^{89} -211.570 q^{90} -245.109 q^{91} -2054.70 q^{92} -265.978 q^{93} +1665.38 q^{94} +337.016 q^{95} -377.058 q^{96} +555.034 q^{97} +230.377 q^{98} +221.372 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{2} + 6 q^{3} + 9 q^{4} - 10 q^{5} + 9 q^{6} + 14 q^{7} + 51 q^{8} + 18 q^{9}+O(q^{10})$$ 2 * q + 3 * q^2 + 6 * q^3 + 9 * q^4 - 10 * q^5 + 9 * q^6 + 14 * q^7 + 51 * q^8 + 18 * q^9 $$2 q + 3 q^{2} + 6 q^{3} + 9 q^{4} - 10 q^{5} + 9 q^{6} + 14 q^{7} + 51 q^{8} + 18 q^{9} - 15 q^{10} + 62 q^{11} + 27 q^{12} - 6 q^{13} + 21 q^{14} - 30 q^{15} + 25 q^{16} + 40 q^{17} + 27 q^{18} - 122 q^{19} - 45 q^{20} + 42 q^{21} + 52 q^{22} + 16 q^{23} + 153 q^{24} + 50 q^{25} - 214 q^{26} + 54 q^{27} + 63 q^{28} + 352 q^{29} - 45 q^{30} + 66 q^{31} - 309 q^{32} + 186 q^{33} - 186 q^{34} - 70 q^{35} + 81 q^{36} - 188 q^{37} - 224 q^{38} - 18 q^{39} - 255 q^{40} + 16 q^{41} + 63 q^{42} - 396 q^{43} + 156 q^{44} - 90 q^{45} - 960 q^{46} - 188 q^{47} + 75 q^{48} + 98 q^{49} + 75 q^{50} + 120 q^{51} - 642 q^{52} + 982 q^{53} + 81 q^{54} - 310 q^{55} + 357 q^{56} - 366 q^{57} + 774 q^{58} + 516 q^{59} - 135 q^{60} - 880 q^{61} - 680 q^{62} + 126 q^{63} - 479 q^{64} + 30 q^{65} + 156 q^{66} - 356 q^{67} - 558 q^{68} + 48 q^{69} - 105 q^{70} + 310 q^{71} + 459 q^{72} + 326 q^{73} + 1358 q^{74} + 150 q^{75} - 672 q^{76} + 434 q^{77} - 642 q^{78} + 1832 q^{79} - 125 q^{80} + 162 q^{81} - 2190 q^{82} - 680 q^{83} + 189 q^{84} - 200 q^{85} + 1456 q^{86} + 1056 q^{87} + 1540 q^{88} + 796 q^{89} - 135 q^{90} - 42 q^{91} - 2880 q^{92} + 198 q^{93} + 2588 q^{94} + 610 q^{95} - 927 q^{96} - 670 q^{97} + 147 q^{98} + 558 q^{99}+O(q^{100})$$ 2 * q + 3 * q^2 + 6 * q^3 + 9 * q^4 - 10 * q^5 + 9 * q^6 + 14 * q^7 + 51 * q^8 + 18 * q^9 - 15 * q^10 + 62 * q^11 + 27 * q^12 - 6 * q^13 + 21 * q^14 - 30 * q^15 + 25 * q^16 + 40 * q^17 + 27 * q^18 - 122 * q^19 - 45 * q^20 + 42 * q^21 + 52 * q^22 + 16 * q^23 + 153 * q^24 + 50 * q^25 - 214 * q^26 + 54 * q^27 + 63 * q^28 + 352 * q^29 - 45 * q^30 + 66 * q^31 - 309 * q^32 + 186 * q^33 - 186 * q^34 - 70 * q^35 + 81 * q^36 - 188 * q^37 - 224 * q^38 - 18 * q^39 - 255 * q^40 + 16 * q^41 + 63 * q^42 - 396 * q^43 + 156 * q^44 - 90 * q^45 - 960 * q^46 - 188 * q^47 + 75 * q^48 + 98 * q^49 + 75 * q^50 + 120 * q^51 - 642 * q^52 + 982 * q^53 + 81 * q^54 - 310 * q^55 + 357 * q^56 - 366 * q^57 + 774 * q^58 + 516 * q^59 - 135 * q^60 - 880 * q^61 - 680 * q^62 + 126 * q^63 - 479 * q^64 + 30 * q^65 + 156 * q^66 - 356 * q^67 - 558 * q^68 + 48 * q^69 - 105 * q^70 + 310 * q^71 + 459 * q^72 + 326 * q^73 + 1358 * q^74 + 150 * q^75 - 672 * q^76 + 434 * q^77 - 642 * q^78 + 1832 * q^79 - 125 * q^80 + 162 * q^81 - 2190 * q^82 - 680 * q^83 + 189 * q^84 - 200 * q^85 + 1456 * q^86 + 1056 * q^87 + 1540 * q^88 + 796 * q^89 - 135 * q^90 - 42 * q^91 - 2880 * q^92 + 198 * q^93 + 2588 * q^94 + 610 * q^95 - 927 * q^96 - 670 * q^97 + 147 * q^98 + 558 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 4.70156 1.66225 0.831127 0.556083i $$-0.187696\pi$$
0.831127 + 0.556083i $$0.187696\pi$$
$$3$$ 3.00000 0.577350
$$4$$ 14.1047 1.76309
$$5$$ −5.00000 −0.447214
$$6$$ 14.1047 0.959702
$$7$$ 7.00000 0.377964
$$8$$ 28.7016 1.26844
$$9$$ 9.00000 0.333333
$$10$$ −23.5078 −0.743382
$$11$$ 24.5969 0.674203 0.337102 0.941468i $$-0.390553\pi$$
0.337102 + 0.941468i $$0.390553\pi$$
$$12$$ 42.3141 1.01792
$$13$$ −35.0156 −0.747045 −0.373523 0.927621i $$-0.621850\pi$$
−0.373523 + 0.927621i $$0.621850\pi$$
$$14$$ 32.9109 0.628273
$$15$$ −15.0000 −0.258199
$$16$$ 22.1047 0.345386
$$17$$ −18.4187 −0.262777 −0.131388 0.991331i $$-0.541943\pi$$
−0.131388 + 0.991331i $$0.541943\pi$$
$$18$$ 42.3141 0.554084
$$19$$ −67.4031 −0.813860 −0.406930 0.913459i $$-0.633401\pi$$
−0.406930 + 0.913459i $$0.633401\pi$$
$$20$$ −70.5234 −0.788476
$$21$$ 21.0000 0.218218
$$22$$ 115.644 1.12070
$$23$$ −145.675 −1.32067 −0.660333 0.750973i $$-0.729585\pi$$
−0.660333 + 0.750973i $$0.729585\pi$$
$$24$$ 86.1047 0.732335
$$25$$ 25.0000 0.200000
$$26$$ −164.628 −1.24178
$$27$$ 27.0000 0.192450
$$28$$ 98.7328 0.666384
$$29$$ 214.419 1.37298 0.686492 0.727137i $$-0.259149\pi$$
0.686492 + 0.727137i $$0.259149\pi$$
$$30$$ −70.5234 −0.429192
$$31$$ −88.6594 −0.513667 −0.256834 0.966456i $$-0.582679\pi$$
−0.256834 + 0.966456i $$0.582679\pi$$
$$32$$ −125.686 −0.694323
$$33$$ 73.7906 0.389251
$$34$$ −86.5969 −0.436801
$$35$$ −35.0000 −0.169031
$$36$$ 126.942 0.587695
$$37$$ 162.125 0.720356 0.360178 0.932884i $$-0.382716\pi$$
0.360178 + 0.932884i $$0.382716\pi$$
$$38$$ −316.900 −1.35284
$$39$$ −105.047 −0.431307
$$40$$ −143.508 −0.567264
$$41$$ −337.769 −1.28660 −0.643300 0.765614i $$-0.722435\pi$$
−0.643300 + 0.765614i $$0.722435\pi$$
$$42$$ 98.7328 0.362733
$$43$$ 122.156 0.433224 0.216612 0.976258i $$-0.430499\pi$$
0.216612 + 0.976258i $$0.430499\pi$$
$$44$$ 346.931 1.18868
$$45$$ −45.0000 −0.149071
$$46$$ −684.900 −2.19528
$$47$$ 354.219 1.09932 0.549661 0.835388i $$-0.314757\pi$$
0.549661 + 0.835388i $$0.314757\pi$$
$$48$$ 66.3141 0.199409
$$49$$ 49.0000 0.142857
$$50$$ 117.539 0.332451
$$51$$ −55.2562 −0.151714
$$52$$ −493.884 −1.31710
$$53$$ 676.691 1.75378 0.876892 0.480687i $$-0.159613\pi$$
0.876892 + 0.480687i $$0.159613\pi$$
$$54$$ 126.942 0.319901
$$55$$ −122.984 −0.301513
$$56$$ 200.911 0.479426
$$57$$ −202.209 −0.469882
$$58$$ 1008.10 2.28225
$$59$$ 501.319 1.10621 0.553103 0.833113i $$-0.313444\pi$$
0.553103 + 0.833113i $$0.313444\pi$$
$$60$$ −211.570 −0.455227
$$61$$ −708.931 −1.48802 −0.744011 0.668167i $$-0.767079\pi$$
−0.744011 + 0.668167i $$0.767079\pi$$
$$62$$ −416.837 −0.853845
$$63$$ 63.0000 0.125988
$$64$$ −767.758 −1.49953
$$65$$ 175.078 0.334089
$$66$$ 346.931 0.647035
$$67$$ −907.956 −1.65559 −0.827795 0.561031i $$-0.810405\pi$$
−0.827795 + 0.561031i $$0.810405\pi$$
$$68$$ −259.791 −0.463298
$$69$$ −437.025 −0.762487
$$70$$ −164.555 −0.280972
$$71$$ 430.334 0.719314 0.359657 0.933085i $$-0.382894\pi$$
0.359657 + 0.933085i $$0.382894\pi$$
$$72$$ 258.314 0.422814
$$73$$ 41.3406 0.0662816 0.0331408 0.999451i $$-0.489449\pi$$
0.0331408 + 0.999451i $$0.489449\pi$$
$$74$$ 762.241 1.19741
$$75$$ 75.0000 0.115470
$$76$$ −950.700 −1.43490
$$77$$ 172.178 0.254825
$$78$$ −493.884 −0.716941
$$79$$ 890.388 1.26806 0.634028 0.773310i $$-0.281400\pi$$
0.634028 + 0.773310i $$0.281400\pi$$
$$80$$ −110.523 −0.154461
$$81$$ 81.0000 0.111111
$$82$$ −1588.04 −2.13866
$$83$$ −1057.15 −1.39804 −0.699020 0.715102i $$-0.746380\pi$$
−0.699020 + 0.715102i $$0.746380\pi$$
$$84$$ 296.198 0.384737
$$85$$ 92.0937 0.117517
$$86$$ 574.325 0.720129
$$87$$ 643.256 0.792693
$$88$$ 705.969 0.855188
$$89$$ 1473.72 1.75522 0.877610 0.479376i $$-0.159137\pi$$
0.877610 + 0.479376i $$0.159137\pi$$
$$90$$ −211.570 −0.247794
$$91$$ −245.109 −0.282356
$$92$$ −2054.70 −2.32845
$$93$$ −265.978 −0.296566
$$94$$ 1665.38 1.82735
$$95$$ 337.016 0.363969
$$96$$ −377.058 −0.400868
$$97$$ 555.034 0.580981 0.290491 0.956878i $$-0.406181\pi$$
0.290491 + 0.956878i $$0.406181\pi$$
$$98$$ 230.377 0.237465
$$99$$ 221.372 0.224734
$$100$$ 352.617 0.352617
$$101$$ 1890.14 1.86214 0.931071 0.364838i $$-0.118876\pi$$
0.931071 + 0.364838i $$0.118876\pi$$
$$102$$ −259.791 −0.252187
$$103$$ 662.700 0.633959 0.316979 0.948432i $$-0.397331\pi$$
0.316979 + 0.948432i $$0.397331\pi$$
$$104$$ −1005.00 −0.947583
$$105$$ −105.000 −0.0975900
$$106$$ 3181.50 2.91523
$$107$$ 1614.53 1.45872 0.729358 0.684132i $$-0.239819\pi$$
0.729358 + 0.684132i $$0.239819\pi$$
$$108$$ 380.827 0.339306
$$109$$ 217.206 0.190868 0.0954339 0.995436i $$-0.469576\pi$$
0.0954339 + 0.995436i $$0.469576\pi$$
$$110$$ −578.219 −0.501191
$$111$$ 486.375 0.415898
$$112$$ 154.733 0.130544
$$113$$ −1658.20 −1.38044 −0.690221 0.723598i $$-0.742487\pi$$
−0.690221 + 0.723598i $$0.742487\pi$$
$$114$$ −950.700 −0.781063
$$115$$ 728.375 0.590620
$$116$$ 3024.31 2.42069
$$117$$ −315.141 −0.249015
$$118$$ 2356.98 1.83879
$$119$$ −128.931 −0.0993202
$$120$$ −430.523 −0.327510
$$121$$ −725.994 −0.545450
$$122$$ −3333.08 −2.47347
$$123$$ −1013.31 −0.742819
$$124$$ −1250.51 −0.905640
$$125$$ −125.000 −0.0894427
$$126$$ 296.198 0.209424
$$127$$ −1108.81 −0.774734 −0.387367 0.921926i $$-0.626615\pi$$
−0.387367 + 0.921926i $$0.626615\pi$$
$$128$$ −2604.17 −1.79827
$$129$$ 366.469 0.250122
$$130$$ 823.141 0.555340
$$131$$ 185.488 0.123711 0.0618554 0.998085i $$-0.480298\pi$$
0.0618554 + 0.998085i $$0.480298\pi$$
$$132$$ 1040.79 0.686284
$$133$$ −471.822 −0.307610
$$134$$ −4268.81 −2.75201
$$135$$ −135.000 −0.0860663
$$136$$ −528.647 −0.333317
$$137$$ −37.9907 −0.0236917 −0.0118458 0.999930i $$-0.503771\pi$$
−0.0118458 + 0.999930i $$0.503771\pi$$
$$138$$ −2054.70 −1.26745
$$139$$ 183.609 0.112040 0.0560199 0.998430i $$-0.482159\pi$$
0.0560199 + 0.998430i $$0.482159\pi$$
$$140$$ −493.664 −0.298016
$$141$$ 1062.66 0.634694
$$142$$ 2023.24 1.19568
$$143$$ −861.275 −0.503660
$$144$$ 198.942 0.115129
$$145$$ −1072.09 −0.614018
$$146$$ 194.366 0.110177
$$147$$ 147.000 0.0824786
$$148$$ 2286.72 1.27005
$$149$$ −1383.34 −0.760587 −0.380293 0.924866i $$-0.624177\pi$$
−0.380293 + 0.924866i $$0.624177\pi$$
$$150$$ 352.617 0.191940
$$151$$ 765.256 0.412422 0.206211 0.978508i $$-0.433887\pi$$
0.206211 + 0.978508i $$0.433887\pi$$
$$152$$ −1934.57 −1.03233
$$153$$ −165.769 −0.0875922
$$154$$ 809.506 0.423584
$$155$$ 443.297 0.229719
$$156$$ −1481.65 −0.760431
$$157$$ −2366.76 −1.20311 −0.601554 0.798832i $$-0.705452\pi$$
−0.601554 + 0.798832i $$0.705452\pi$$
$$158$$ 4186.21 2.10783
$$159$$ 2030.07 1.01255
$$160$$ 628.430 0.310511
$$161$$ −1019.72 −0.499165
$$162$$ 380.827 0.184695
$$163$$ −3137.69 −1.50775 −0.753875 0.657018i $$-0.771817\pi$$
−0.753875 + 0.657018i $$0.771817\pi$$
$$164$$ −4764.12 −2.26839
$$165$$ −368.953 −0.174079
$$166$$ −4970.26 −2.32390
$$167$$ 146.469 0.0678688 0.0339344 0.999424i $$-0.489196\pi$$
0.0339344 + 0.999424i $$0.489196\pi$$
$$168$$ 602.733 0.276797
$$169$$ −970.906 −0.441924
$$170$$ 432.984 0.195343
$$171$$ −606.628 −0.271287
$$172$$ 1722.98 0.763812
$$173$$ −1424.12 −0.625860 −0.312930 0.949776i $$-0.601311\pi$$
−0.312930 + 0.949776i $$0.601311\pi$$
$$174$$ 3024.31 1.31766
$$175$$ 175.000 0.0755929
$$176$$ 543.706 0.232860
$$177$$ 1503.96 0.638668
$$178$$ 6928.81 2.91762
$$179$$ 1244.70 0.519737 0.259869 0.965644i $$-0.416321\pi$$
0.259869 + 0.965644i $$0.416321\pi$$
$$180$$ −634.711 −0.262825
$$181$$ −3879.09 −1.59299 −0.796493 0.604648i $$-0.793314\pi$$
−0.796493 + 0.604648i $$0.793314\pi$$
$$182$$ −1152.40 −0.469348
$$183$$ −2126.79 −0.859110
$$184$$ −4181.10 −1.67519
$$185$$ −810.625 −0.322153
$$186$$ −1250.51 −0.492968
$$187$$ −453.044 −0.177165
$$188$$ 4996.14 1.93820
$$189$$ 189.000 0.0727393
$$190$$ 1584.50 0.605009
$$191$$ 1574.90 0.596628 0.298314 0.954468i $$-0.403576\pi$$
0.298314 + 0.954468i $$0.403576\pi$$
$$192$$ −2303.27 −0.865752
$$193$$ −4775.67 −1.78114 −0.890572 0.454843i $$-0.849695\pi$$
−0.890572 + 0.454843i $$0.849695\pi$$
$$194$$ 2609.53 0.965738
$$195$$ 525.234 0.192886
$$196$$ 691.130 0.251869
$$197$$ −2803.58 −1.01394 −0.506971 0.861963i $$-0.669235\pi$$
−0.506971 + 0.861963i $$0.669235\pi$$
$$198$$ 1040.79 0.373566
$$199$$ 4102.92 1.46155 0.730774 0.682620i $$-0.239159\pi$$
0.730774 + 0.682620i $$0.239159\pi$$
$$200$$ 717.539 0.253688
$$201$$ −2723.87 −0.955855
$$202$$ 8886.63 3.09535
$$203$$ 1500.93 0.518940
$$204$$ −779.372 −0.267485
$$205$$ 1688.84 0.575385
$$206$$ 3115.72 1.05380
$$207$$ −1311.07 −0.440222
$$208$$ −774.009 −0.258019
$$209$$ −1657.91 −0.548707
$$210$$ −493.664 −0.162219
$$211$$ −823.512 −0.268687 −0.134343 0.990935i $$-0.542893\pi$$
−0.134343 + 0.990935i $$0.542893\pi$$
$$212$$ 9544.51 3.09207
$$213$$ 1291.00 0.415296
$$214$$ 7590.82 2.42476
$$215$$ −610.781 −0.193744
$$216$$ 774.942 0.244112
$$217$$ −620.616 −0.194148
$$218$$ 1021.21 0.317271
$$219$$ 124.022 0.0382677
$$220$$ −1734.66 −0.531593
$$221$$ 644.944 0.196306
$$222$$ 2286.72 0.691328
$$223$$ 817.194 0.245396 0.122698 0.992444i $$-0.460845\pi$$
0.122698 + 0.992444i $$0.460845\pi$$
$$224$$ −879.802 −0.262430
$$225$$ 225.000 0.0666667
$$226$$ −7796.12 −2.29465
$$227$$ 3655.85 1.06893 0.534465 0.845190i $$-0.320513\pi$$
0.534465 + 0.845190i $$0.320513\pi$$
$$228$$ −2852.10 −0.828443
$$229$$ 939.393 0.271078 0.135539 0.990772i $$-0.456723\pi$$
0.135539 + 0.990772i $$0.456723\pi$$
$$230$$ 3424.50 0.981760
$$231$$ 516.534 0.147123
$$232$$ 6154.15 1.74155
$$233$$ −7.64701 −0.00215010 −0.00107505 0.999999i $$-0.500342\pi$$
−0.00107505 + 0.999999i $$0.500342\pi$$
$$234$$ −1481.65 −0.413926
$$235$$ −1771.09 −0.491631
$$236$$ 7070.94 1.95034
$$237$$ 2671.16 0.732112
$$238$$ −606.178 −0.165095
$$239$$ −889.115 −0.240636 −0.120318 0.992735i $$-0.538391\pi$$
−0.120318 + 0.992735i $$0.538391\pi$$
$$240$$ −331.570 −0.0891782
$$241$$ 2140.23 0.572051 0.286026 0.958222i $$-0.407666\pi$$
0.286026 + 0.958222i $$0.407666\pi$$
$$242$$ −3413.30 −0.906676
$$243$$ 243.000 0.0641500
$$244$$ −9999.25 −2.62351
$$245$$ −245.000 −0.0638877
$$246$$ −4764.12 −1.23475
$$247$$ 2360.16 0.607990
$$248$$ −2544.66 −0.651557
$$249$$ −3171.45 −0.807158
$$250$$ −587.695 −0.148676
$$251$$ −6749.81 −1.69739 −0.848693 0.528886i $$-0.822610\pi$$
−0.848693 + 0.528886i $$0.822610\pi$$
$$252$$ 888.595 0.222128
$$253$$ −3583.15 −0.890398
$$254$$ −5213.15 −1.28780
$$255$$ 276.281 0.0678486
$$256$$ −6101.62 −1.48965
$$257$$ 3068.64 0.744811 0.372405 0.928070i $$-0.378533\pi$$
0.372405 + 0.928070i $$0.378533\pi$$
$$258$$ 1722.98 0.415766
$$259$$ 1134.87 0.272269
$$260$$ 2469.42 0.589027
$$261$$ 1929.77 0.457662
$$262$$ 872.081 0.205639
$$263$$ −4674.12 −1.09589 −0.547944 0.836515i $$-0.684589\pi$$
−0.547944 + 0.836515i $$0.684589\pi$$
$$264$$ 2117.91 0.493743
$$265$$ −3383.45 −0.784316
$$266$$ −2218.30 −0.511326
$$267$$ 4421.17 1.01338
$$268$$ −12806.4 −2.91895
$$269$$ 2417.38 0.547919 0.273960 0.961741i $$-0.411667\pi$$
0.273960 + 0.961741i $$0.411667\pi$$
$$270$$ −634.711 −0.143064
$$271$$ 7724.30 1.73143 0.865715 0.500537i $$-0.166864\pi$$
0.865715 + 0.500537i $$0.166864\pi$$
$$272$$ −407.141 −0.0907593
$$273$$ −735.328 −0.163019
$$274$$ −178.616 −0.0393816
$$275$$ 614.922 0.134841
$$276$$ −6164.10 −1.34433
$$277$$ −4576.17 −0.992620 −0.496310 0.868145i $$-0.665312\pi$$
−0.496310 + 0.868145i $$0.665312\pi$$
$$278$$ 863.250 0.186239
$$279$$ −797.934 −0.171222
$$280$$ −1004.55 −0.214406
$$281$$ −1358.56 −0.288415 −0.144208 0.989547i $$-0.546063\pi$$
−0.144208 + 0.989547i $$0.546063\pi$$
$$282$$ 4996.14 1.05502
$$283$$ 3885.04 0.816048 0.408024 0.912971i $$-0.366218\pi$$
0.408024 + 0.912971i $$0.366218\pi$$
$$284$$ 6069.73 1.26821
$$285$$ 1011.05 0.210138
$$286$$ −4049.34 −0.837211
$$287$$ −2364.38 −0.486289
$$288$$ −1131.17 −0.231441
$$289$$ −4573.75 −0.930948
$$290$$ −5040.52 −1.02065
$$291$$ 1665.10 0.335430
$$292$$ 583.097 0.116860
$$293$$ −4033.91 −0.804312 −0.402156 0.915571i $$-0.631739\pi$$
−0.402156 + 0.915571i $$0.631739\pi$$
$$294$$ 691.130 0.137100
$$295$$ −2506.59 −0.494710
$$296$$ 4653.24 0.913730
$$297$$ 664.116 0.129750
$$298$$ −6503.85 −1.26429
$$299$$ 5100.90 0.986598
$$300$$ 1057.85 0.203584
$$301$$ 855.093 0.163743
$$302$$ 3597.90 0.685549
$$303$$ 5670.43 1.07511
$$304$$ −1489.92 −0.281096
$$305$$ 3544.66 0.665464
$$306$$ −779.372 −0.145600
$$307$$ −4620.36 −0.858950 −0.429475 0.903079i $$-0.641301\pi$$
−0.429475 + 0.903079i $$0.641301\pi$$
$$308$$ 2428.52 0.449278
$$309$$ 1988.10 0.366016
$$310$$ 2084.19 0.381851
$$311$$ 6675.89 1.21722 0.608609 0.793470i $$-0.291728\pi$$
0.608609 + 0.793470i $$0.291728\pi$$
$$312$$ −3015.01 −0.547087
$$313$$ 2836.78 0.512283 0.256141 0.966639i $$-0.417549\pi$$
0.256141 + 0.966639i $$0.417549\pi$$
$$314$$ −11127.5 −1.99987
$$315$$ −315.000 −0.0563436
$$316$$ 12558.6 2.23569
$$317$$ 4010.63 0.710597 0.355299 0.934753i $$-0.384379\pi$$
0.355299 + 0.934753i $$0.384379\pi$$
$$318$$ 9544.51 1.68311
$$319$$ 5274.03 0.925671
$$320$$ 3838.79 0.670609
$$321$$ 4843.59 0.842190
$$322$$ −4794.30 −0.829739
$$323$$ 1241.48 0.213863
$$324$$ 1142.48 0.195898
$$325$$ −875.391 −0.149409
$$326$$ −14752.1 −2.50626
$$327$$ 651.619 0.110198
$$328$$ −9694.49 −1.63198
$$329$$ 2479.53 0.415504
$$330$$ −1734.66 −0.289363
$$331$$ 11087.5 1.84117 0.920583 0.390546i $$-0.127714\pi$$
0.920583 + 0.390546i $$0.127714\pi$$
$$332$$ −14910.8 −2.46486
$$333$$ 1459.12 0.240119
$$334$$ 688.631 0.112815
$$335$$ 4539.78 0.740402
$$336$$ 464.198 0.0753693
$$337$$ 12118.7 1.95890 0.979450 0.201689i $$-0.0646431\pi$$
0.979450 + 0.201689i $$0.0646431\pi$$
$$338$$ −4564.78 −0.734589
$$339$$ −4974.59 −0.796999
$$340$$ 1298.95 0.207193
$$341$$ −2180.74 −0.346316
$$342$$ −2852.10 −0.450947
$$343$$ 343.000 0.0539949
$$344$$ 3506.07 0.549520
$$345$$ 2185.12 0.340995
$$346$$ −6695.58 −1.04034
$$347$$ −6361.22 −0.984116 −0.492058 0.870562i $$-0.663755\pi$$
−0.492058 + 0.870562i $$0.663755\pi$$
$$348$$ 9072.93 1.39759
$$349$$ −3115.18 −0.477799 −0.238899 0.971044i $$-0.576787\pi$$
−0.238899 + 0.971044i $$0.576787\pi$$
$$350$$ 822.773 0.125655
$$351$$ −945.422 −0.143769
$$352$$ −3091.48 −0.468115
$$353$$ −11927.4 −1.79839 −0.899194 0.437550i $$-0.855846\pi$$
−0.899194 + 0.437550i $$0.855846\pi$$
$$354$$ 7070.94 1.06163
$$355$$ −2151.67 −0.321687
$$356$$ 20786.4 3.09460
$$357$$ −386.794 −0.0573426
$$358$$ 5852.02 0.863935
$$359$$ −6143.95 −0.903245 −0.451623 0.892209i $$-0.649155\pi$$
−0.451623 + 0.892209i $$0.649155\pi$$
$$360$$ −1291.57 −0.189088
$$361$$ −2315.82 −0.337632
$$362$$ −18237.8 −2.64794
$$363$$ −2177.98 −0.314916
$$364$$ −3457.19 −0.497819
$$365$$ −206.703 −0.0296420
$$366$$ −9999.25 −1.42806
$$367$$ −1927.67 −0.274178 −0.137089 0.990559i $$-0.543775\pi$$
−0.137089 + 0.990559i $$0.543775\pi$$
$$368$$ −3220.10 −0.456139
$$369$$ −3039.92 −0.428867
$$370$$ −3811.20 −0.535500
$$371$$ 4736.83 0.662868
$$372$$ −3751.54 −0.522871
$$373$$ 10452.0 1.45090 0.725449 0.688276i $$-0.241632\pi$$
0.725449 + 0.688276i $$0.241632\pi$$
$$374$$ −2130.01 −0.294493
$$375$$ −375.000 −0.0516398
$$376$$ 10166.6 1.39443
$$377$$ −7508.01 −1.02568
$$378$$ 888.595 0.120911
$$379$$ 7066.43 0.957726 0.478863 0.877890i $$-0.341049\pi$$
0.478863 + 0.877890i $$0.341049\pi$$
$$380$$ 4753.50 0.641709
$$381$$ −3326.44 −0.447293
$$382$$ 7404.51 0.991747
$$383$$ 7168.04 0.956318 0.478159 0.878273i $$-0.341304\pi$$
0.478159 + 0.878273i $$0.341304\pi$$
$$384$$ −7812.52 −1.03823
$$385$$ −860.891 −0.113961
$$386$$ −22453.1 −2.96071
$$387$$ 1099.41 0.144408
$$388$$ 7828.58 1.02432
$$389$$ −7414.06 −0.966344 −0.483172 0.875525i $$-0.660515\pi$$
−0.483172 + 0.875525i $$0.660515\pi$$
$$390$$ 2469.42 0.320626
$$391$$ 2683.15 0.347040
$$392$$ 1406.38 0.181206
$$393$$ 556.463 0.0714245
$$394$$ −13181.2 −1.68543
$$395$$ −4451.94 −0.567092
$$396$$ 3122.38 0.396226
$$397$$ −8936.01 −1.12969 −0.564843 0.825198i $$-0.691063\pi$$
−0.564843 + 0.825198i $$0.691063\pi$$
$$398$$ 19290.1 2.42946
$$399$$ −1415.47 −0.177599
$$400$$ 552.617 0.0690771
$$401$$ 1782.91 0.222031 0.111015 0.993819i $$-0.464590\pi$$
0.111015 + 0.993819i $$0.464590\pi$$
$$402$$ −12806.4 −1.58887
$$403$$ 3104.46 0.383733
$$404$$ 26659.9 3.28312
$$405$$ −405.000 −0.0496904
$$406$$ 7056.72 0.862609
$$407$$ 3987.77 0.485667
$$408$$ −1585.94 −0.192441
$$409$$ −8759.92 −1.05905 −0.529524 0.848295i $$-0.677629\pi$$
−0.529524 + 0.848295i $$0.677629\pi$$
$$410$$ 7940.20 0.956436
$$411$$ −113.972 −0.0136784
$$412$$ 9347.17 1.11772
$$413$$ 3509.23 0.418106
$$414$$ −6164.10 −0.731761
$$415$$ 5285.75 0.625222
$$416$$ 4400.97 0.518691
$$417$$ 550.828 0.0646862
$$418$$ −7794.75 −0.912090
$$419$$ −3212.74 −0.374588 −0.187294 0.982304i $$-0.559972\pi$$
−0.187294 + 0.982304i $$0.559972\pi$$
$$420$$ −1480.99 −0.172060
$$421$$ 15757.8 1.82420 0.912101 0.409965i $$-0.134459\pi$$
0.912101 + 0.409965i $$0.134459\pi$$
$$422$$ −3871.79 −0.446626
$$423$$ 3187.97 0.366440
$$424$$ 19422.1 2.22457
$$425$$ −460.469 −0.0525553
$$426$$ 6069.73 0.690327
$$427$$ −4962.52 −0.562419
$$428$$ 22772.5 2.57184
$$429$$ −2583.82 −0.290788
$$430$$ −2871.63 −0.322051
$$431$$ −405.917 −0.0453650 −0.0226825 0.999743i $$-0.507221\pi$$
−0.0226825 + 0.999743i $$0.507221\pi$$
$$432$$ 596.827 0.0664695
$$433$$ −7845.25 −0.870713 −0.435357 0.900258i $$-0.643378\pi$$
−0.435357 + 0.900258i $$0.643378\pi$$
$$434$$ −2917.86 −0.322723
$$435$$ −3216.28 −0.354503
$$436$$ 3063.63 0.336516
$$437$$ 9818.95 1.07484
$$438$$ 583.097 0.0636106
$$439$$ 423.029 0.0459911 0.0229955 0.999736i $$-0.492680\pi$$
0.0229955 + 0.999736i $$0.492680\pi$$
$$440$$ −3529.84 −0.382452
$$441$$ 441.000 0.0476190
$$442$$ 3032.24 0.326310
$$443$$ −16058.7 −1.72229 −0.861143 0.508362i $$-0.830251\pi$$
−0.861143 + 0.508362i $$0.830251\pi$$
$$444$$ 6860.17 0.733264
$$445$$ −7368.62 −0.784958
$$446$$ 3842.09 0.407911
$$447$$ −4150.01 −0.439125
$$448$$ −5374.30 −0.566768
$$449$$ 2186.75 0.229842 0.114921 0.993375i $$-0.463338\pi$$
0.114921 + 0.993375i $$0.463338\pi$$
$$450$$ 1057.85 0.110817
$$451$$ −8308.05 −0.867430
$$452$$ −23388.3 −2.43384
$$453$$ 2295.77 0.238112
$$454$$ 17188.2 1.77683
$$455$$ 1225.55 0.126274
$$456$$ −5803.72 −0.596018
$$457$$ −5799.22 −0.593602 −0.296801 0.954939i $$-0.595920\pi$$
−0.296801 + 0.954939i $$0.595920\pi$$
$$458$$ 4416.62 0.450600
$$459$$ −497.306 −0.0505714
$$460$$ 10273.5 1.04131
$$461$$ 9873.35 0.997500 0.498750 0.866746i $$-0.333793\pi$$
0.498750 + 0.866746i $$0.333793\pi$$
$$462$$ 2428.52 0.244556
$$463$$ −6181.84 −0.620506 −0.310253 0.950654i $$-0.600414\pi$$
−0.310253 + 0.950654i $$0.600414\pi$$
$$464$$ 4739.66 0.474209
$$465$$ 1329.89 0.132628
$$466$$ −35.9529 −0.00357400
$$467$$ 6145.50 0.608950 0.304475 0.952520i $$-0.401519\pi$$
0.304475 + 0.952520i $$0.401519\pi$$
$$468$$ −4444.96 −0.439035
$$469$$ −6355.69 −0.625754
$$470$$ −8326.91 −0.817216
$$471$$ −7100.28 −0.694615
$$472$$ 14388.6 1.40316
$$473$$ 3004.66 0.292081
$$474$$ 12558.6 1.21696
$$475$$ −1685.08 −0.162772
$$476$$ −1818.53 −0.175110
$$477$$ 6090.22 0.584595
$$478$$ −4180.23 −0.399999
$$479$$ 10879.4 1.03777 0.518887 0.854843i $$-0.326347\pi$$
0.518887 + 0.854843i $$0.326347\pi$$
$$480$$ 1885.29 0.179274
$$481$$ −5676.91 −0.538139
$$482$$ 10062.4 0.950894
$$483$$ −3059.17 −0.288193
$$484$$ −10239.9 −0.961675
$$485$$ −2775.17 −0.259823
$$486$$ 1142.48 0.106634
$$487$$ 8087.51 0.752526 0.376263 0.926513i $$-0.377209\pi$$
0.376263 + 0.926513i $$0.377209\pi$$
$$488$$ −20347.4 −1.88747
$$489$$ −9413.08 −0.870499
$$490$$ −1151.88 −0.106197
$$491$$ −6959.90 −0.639707 −0.319853 0.947467i $$-0.603634\pi$$
−0.319853 + 0.947467i $$0.603634\pi$$
$$492$$ −14292.4 −1.30965
$$493$$ −3949.32 −0.360788
$$494$$ 11096.4 1.01063
$$495$$ −1106.86 −0.100504
$$496$$ −1959.79 −0.177413
$$497$$ 3012.34 0.271875
$$498$$ −14910.8 −1.34170
$$499$$ 18632.0 1.67151 0.835756 0.549101i $$-0.185030\pi$$
0.835756 + 0.549101i $$0.185030\pi$$
$$500$$ −1763.09 −0.157695
$$501$$ 439.406 0.0391840
$$502$$ −31734.6 −2.82149
$$503$$ 4627.62 0.410209 0.205105 0.978740i $$-0.434247\pi$$
0.205105 + 0.978740i $$0.434247\pi$$
$$504$$ 1808.20 0.159809
$$505$$ −9450.72 −0.832775
$$506$$ −16846.4 −1.48007
$$507$$ −2912.72 −0.255145
$$508$$ −15639.4 −1.36592
$$509$$ −11351.8 −0.988528 −0.494264 0.869312i $$-0.664562\pi$$
−0.494264 + 0.869312i $$0.664562\pi$$
$$510$$ 1298.95 0.112782
$$511$$ 289.384 0.0250521
$$512$$ −7853.76 −0.677911
$$513$$ −1819.88 −0.156627
$$514$$ 14427.4 1.23806
$$515$$ −3313.50 −0.283515
$$516$$ 5168.93 0.440987
$$517$$ 8712.67 0.741166
$$518$$ 5335.68 0.452580
$$519$$ −4272.36 −0.361340
$$520$$ 5025.02 0.423772
$$521$$ 19096.1 1.60579 0.802893 0.596123i $$-0.203293\pi$$
0.802893 + 0.596123i $$0.203293\pi$$
$$522$$ 9072.93 0.760750
$$523$$ −3145.11 −0.262956 −0.131478 0.991319i $$-0.541972\pi$$
−0.131478 + 0.991319i $$0.541972\pi$$
$$524$$ 2616.24 0.218113
$$525$$ 525.000 0.0436436
$$526$$ −21975.7 −1.82164
$$527$$ 1632.99 0.134980
$$528$$ 1631.12 0.134442
$$529$$ 9054.20 0.744160
$$530$$ −15907.5 −1.30373
$$531$$ 4511.87 0.368735
$$532$$ −6654.90 −0.542343
$$533$$ 11827.2 0.961148
$$534$$ 20786.4 1.68449
$$535$$ −8072.66 −0.652358
$$536$$ −26059.8 −2.10002
$$537$$ 3734.09 0.300071
$$538$$ 11365.5 0.910781
$$539$$ 1205.25 0.0963148
$$540$$ −1904.13 −0.151742
$$541$$ 8776.12 0.697440 0.348720 0.937227i $$-0.386616\pi$$
0.348720 + 0.937227i $$0.386616\pi$$
$$542$$ 36316.3 2.87808
$$543$$ −11637.3 −0.919710
$$544$$ 2314.98 0.182452
$$545$$ −1086.03 −0.0853587
$$546$$ −3457.19 −0.270978
$$547$$ −13695.1 −1.07049 −0.535247 0.844696i $$-0.679781\pi$$
−0.535247 + 0.844696i $$0.679781\pi$$
$$548$$ −535.847 −0.0417705
$$549$$ −6380.38 −0.496007
$$550$$ 2891.09 0.224139
$$551$$ −14452.5 −1.11742
$$552$$ −12543.3 −0.967171
$$553$$ 6232.71 0.479280
$$554$$ −21515.2 −1.64999
$$555$$ −2431.87 −0.185995
$$556$$ 2589.75 0.197536
$$557$$ 7850.44 0.597188 0.298594 0.954380i $$-0.403482\pi$$
0.298594 + 0.954380i $$0.403482\pi$$
$$558$$ −3751.54 −0.284615
$$559$$ −4277.38 −0.323638
$$560$$ −773.664 −0.0583808
$$561$$ −1359.13 −0.102286
$$562$$ −6387.33 −0.479419
$$563$$ −4948.81 −0.370457 −0.185229 0.982695i $$-0.559303\pi$$
−0.185229 + 0.982695i $$0.559303\pi$$
$$564$$ 14988.4 1.11902
$$565$$ 8290.98 0.617353
$$566$$ 18265.7 1.35648
$$567$$ 567.000 0.0419961
$$568$$ 12351.3 0.912408
$$569$$ −8115.76 −0.597945 −0.298972 0.954262i $$-0.596644\pi$$
−0.298972 + 0.954262i $$0.596644\pi$$
$$570$$ 4753.50 0.349302
$$571$$ 5656.42 0.414560 0.207280 0.978282i $$-0.433539\pi$$
0.207280 + 0.978282i $$0.433539\pi$$
$$572$$ −12148.0 −0.887996
$$573$$ 4724.71 0.344463
$$574$$ −11116.3 −0.808336
$$575$$ −3641.87 −0.264133
$$576$$ −6909.82 −0.499842
$$577$$ −9536.77 −0.688078 −0.344039 0.938955i $$-0.611795\pi$$
−0.344039 + 0.938955i $$0.611795\pi$$
$$578$$ −21503.8 −1.54747
$$579$$ −14327.0 −1.02834
$$580$$ −15121.5 −1.08257
$$581$$ −7400.05 −0.528409
$$582$$ 7828.58 0.557569
$$583$$ 16644.5 1.18241
$$584$$ 1186.54 0.0840743
$$585$$ 1575.70 0.111363
$$586$$ −18965.7 −1.33697
$$587$$ −13089.6 −0.920383 −0.460191 0.887820i $$-0.652219\pi$$
−0.460191 + 0.887820i $$0.652219\pi$$
$$588$$ 2073.39 0.145417
$$589$$ 5975.92 0.418053
$$590$$ −11784.9 −0.822334
$$591$$ −8410.73 −0.585400
$$592$$ 3583.72 0.248801
$$593$$ 4281.96 0.296524 0.148262 0.988948i $$-0.452632\pi$$
0.148262 + 0.988948i $$0.452632\pi$$
$$594$$ 3122.38 0.215678
$$595$$ 644.656 0.0444173
$$596$$ −19511.5 −1.34098
$$597$$ 12308.7 0.843825
$$598$$ 23982.2 1.63997
$$599$$ 3699.92 0.252378 0.126189 0.992006i $$-0.459725\pi$$
0.126189 + 0.992006i $$0.459725\pi$$
$$600$$ 2152.62 0.146467
$$601$$ −17286.1 −1.17323 −0.586616 0.809865i $$-0.699540\pi$$
−0.586616 + 0.809865i $$0.699540\pi$$
$$602$$ 4020.28 0.272183
$$603$$ −8171.61 −0.551863
$$604$$ 10793.7 0.727135
$$605$$ 3629.97 0.243933
$$606$$ 26659.9 1.78710
$$607$$ 14456.7 0.966689 0.483344 0.875430i $$-0.339422\pi$$
0.483344 + 0.875430i $$0.339422\pi$$
$$608$$ 8471.63 0.565082
$$609$$ 4502.79 0.299610
$$610$$ 16665.4 1.10617
$$611$$ −12403.2 −0.821243
$$612$$ −2338.12 −0.154433
$$613$$ 17981.9 1.18480 0.592400 0.805644i $$-0.298181\pi$$
0.592400 + 0.805644i $$0.298181\pi$$
$$614$$ −21722.9 −1.42779
$$615$$ 5066.53 0.332199
$$616$$ 4941.78 0.323231
$$617$$ 19614.7 1.27983 0.639916 0.768445i $$-0.278969\pi$$
0.639916 + 0.768445i $$0.278969\pi$$
$$618$$ 9347.17 0.608412
$$619$$ −10462.9 −0.679385 −0.339692 0.940537i $$-0.610323\pi$$
−0.339692 + 0.940537i $$0.610323\pi$$
$$620$$ 6252.56 0.405014
$$621$$ −3933.22 −0.254162
$$622$$ 31387.1 2.02332
$$623$$ 10316.1 0.663411
$$624$$ −2322.03 −0.148967
$$625$$ 625.000 0.0400000
$$626$$ 13337.3 0.851544
$$627$$ −4973.72 −0.316796
$$628$$ −33382.4 −2.12118
$$629$$ −2986.14 −0.189293
$$630$$ −1480.99 −0.0936574
$$631$$ 24481.9 1.54454 0.772272 0.635292i $$-0.219120\pi$$
0.772272 + 0.635292i $$0.219120\pi$$
$$632$$ 25555.5 1.60846
$$633$$ −2470.54 −0.155126
$$634$$ 18856.2 1.18119
$$635$$ 5544.06 0.346471
$$636$$ 28633.5 1.78521
$$637$$ −1715.77 −0.106721
$$638$$ 24796.2 1.53870
$$639$$ 3873.01 0.239771
$$640$$ 13020.9 0.804211
$$641$$ −1109.39 −0.0683595 −0.0341797 0.999416i $$-0.510882\pi$$
−0.0341797 + 0.999416i $$0.510882\pi$$
$$642$$ 22772.5 1.39993
$$643$$ 30112.5 1.84684 0.923422 0.383787i $$-0.125380\pi$$
0.923422 + 0.383787i $$0.125380\pi$$
$$644$$ −14382.9 −0.880071
$$645$$ −1832.34 −0.111858
$$646$$ 5836.90 0.355495
$$647$$ −4260.27 −0.258869 −0.129435 0.991588i $$-0.541316\pi$$
−0.129435 + 0.991588i $$0.541316\pi$$
$$648$$ 2324.83 0.140938
$$649$$ 12330.9 0.745808
$$650$$ −4115.70 −0.248356
$$651$$ −1861.85 −0.112091
$$652$$ −44256.2 −2.65829
$$653$$ −10576.8 −0.633844 −0.316922 0.948452i $$-0.602649\pi$$
−0.316922 + 0.948452i $$0.602649\pi$$
$$654$$ 3063.63 0.183176
$$655$$ −927.438 −0.0553252
$$656$$ −7466.27 −0.444373
$$657$$ 372.066 0.0220939
$$658$$ 11657.7 0.690674
$$659$$ 3394.70 0.200666 0.100333 0.994954i $$-0.468009\pi$$
0.100333 + 0.994954i $$0.468009\pi$$
$$660$$ −5203.97 −0.306915
$$661$$ −33174.4 −1.95210 −0.976048 0.217554i $$-0.930192\pi$$
−0.976048 + 0.217554i $$0.930192\pi$$
$$662$$ 52128.7 3.06048
$$663$$ 1934.83 0.113337
$$664$$ −30341.9 −1.77333
$$665$$ 2359.11 0.137567
$$666$$ 6860.17 0.399138
$$667$$ −31235.4 −1.81326
$$668$$ 2065.89 0.119658
$$669$$ 2451.58 0.141680
$$670$$ 21344.1 1.23074
$$671$$ −17437.5 −1.00323
$$672$$ −2639.40 −0.151514
$$673$$ 753.881 0.0431797 0.0215899 0.999767i $$-0.493127\pi$$
0.0215899 + 0.999767i $$0.493127\pi$$
$$674$$ 56976.9 3.25619
$$675$$ 675.000 0.0384900
$$676$$ −13694.3 −0.779149
$$677$$ 15668.8 0.889511 0.444756 0.895652i $$-0.353291\pi$$
0.444756 + 0.895652i $$0.353291\pi$$
$$678$$ −23388.3 −1.32481
$$679$$ 3885.24 0.219590
$$680$$ 2643.23 0.149064
$$681$$ 10967.5 0.617147
$$682$$ −10252.9 −0.575665
$$683$$ −11557.4 −0.647485 −0.323742 0.946145i $$-0.604941\pi$$
−0.323742 + 0.946145i $$0.604941\pi$$
$$684$$ −8556.30 −0.478302
$$685$$ 189.953 0.0105952
$$686$$ 1612.64 0.0897532
$$687$$ 2818.18 0.156507
$$688$$ 2700.22 0.149630
$$689$$ −23694.7 −1.31016
$$690$$ 10273.5 0.566819
$$691$$ −18503.1 −1.01866 −0.509328 0.860572i $$-0.670106\pi$$
−0.509328 + 0.860572i $$0.670106\pi$$
$$692$$ −20086.8 −1.10344
$$693$$ 1549.60 0.0849416
$$694$$ −29907.7 −1.63585
$$695$$ −918.046 −0.0501057
$$696$$ 18462.5 1.00549
$$697$$ 6221.28 0.338088
$$698$$ −14646.2 −0.794223
$$699$$ −22.9410 −0.00124136
$$700$$ 2468.32 0.133277
$$701$$ 22580.4 1.21662 0.608311 0.793699i $$-0.291847\pi$$
0.608311 + 0.793699i $$0.291847\pi$$
$$702$$ −4444.96 −0.238980
$$703$$ −10927.7 −0.586269
$$704$$ −18884.4 −1.01099
$$705$$ −5313.28 −0.283844
$$706$$ −56077.4 −2.98938
$$707$$ 13231.0 0.703823
$$708$$ 21212.8 1.12603
$$709$$ −27426.6 −1.45279 −0.726394 0.687278i $$-0.758805\pi$$
−0.726394 + 0.687278i $$0.758805\pi$$
$$710$$ −10116.2 −0.534725
$$711$$ 8013.49 0.422685
$$712$$ 42298.2 2.22639
$$713$$ 12915.5 0.678383
$$714$$ −1818.53 −0.0953178
$$715$$ 4306.37 0.225244
$$716$$ 17556.1 0.916342
$$717$$ −2667.35 −0.138931
$$718$$ −28886.1 −1.50142
$$719$$ 19383.0 1.00538 0.502688 0.864468i $$-0.332344\pi$$
0.502688 + 0.864468i $$0.332344\pi$$
$$720$$ −994.711 −0.0514871
$$721$$ 4638.90 0.239614
$$722$$ −10888.0 −0.561230
$$723$$ 6420.69 0.330274
$$724$$ −54713.3 −2.80857
$$725$$ 5360.47 0.274597
$$726$$ −10239.9 −0.523469
$$727$$ −12317.3 −0.628368 −0.314184 0.949362i $$-0.601731\pi$$
−0.314184 + 0.949362i $$0.601731\pi$$
$$728$$ −7035.02 −0.358153
$$729$$ 729.000 0.0370370
$$730$$ −971.828 −0.0492726
$$731$$ −2249.96 −0.113841
$$732$$ −29997.8 −1.51468
$$733$$ 1234.02 0.0621822 0.0310911 0.999517i $$-0.490102\pi$$
0.0310911 + 0.999517i $$0.490102\pi$$
$$734$$ −9063.05 −0.455754
$$735$$ −735.000 −0.0368856
$$736$$ 18309.3 0.916970
$$737$$ −22332.9 −1.11620
$$738$$ −14292.4 −0.712885
$$739$$ −15257.3 −0.759473 −0.379736 0.925095i $$-0.623985\pi$$
−0.379736 + 0.925095i $$0.623985\pi$$
$$740$$ −11433.6 −0.567984
$$741$$ 7080.49 0.351023
$$742$$ 22270.5 1.10186
$$743$$ −35565.1 −1.75606 −0.878032 0.478602i $$-0.841144\pi$$
−0.878032 + 0.478602i $$0.841144\pi$$
$$744$$ −7633.99 −0.376177
$$745$$ 6916.69 0.340145
$$746$$ 49140.8 2.41176
$$747$$ −9514.35 −0.466013
$$748$$ −6390.04 −0.312357
$$749$$ 11301.7 0.551343
$$750$$ −1763.09 −0.0858384
$$751$$ 14266.7 0.693209 0.346605 0.938011i $$-0.387335\pi$$
0.346605 + 0.938011i $$0.387335\pi$$
$$752$$ 7829.89 0.379690
$$753$$ −20249.4 −0.979986
$$754$$ −35299.4 −1.70494
$$755$$ −3826.28 −0.184441
$$756$$ 2665.79 0.128246
$$757$$ −15927.9 −0.764744 −0.382372 0.924009i $$-0.624893\pi$$
−0.382372 + 0.924009i $$0.624893\pi$$
$$758$$ 33223.3 1.59198
$$759$$ −10749.4 −0.514071
$$760$$ 9672.87 0.461674
$$761$$ −2566.48 −0.122253 −0.0611266 0.998130i $$-0.519469\pi$$
−0.0611266 + 0.998130i $$0.519469\pi$$
$$762$$ −15639.4 −0.743514
$$763$$ 1520.44 0.0721413
$$764$$ 22213.5 1.05191
$$765$$ 828.844 0.0391724
$$766$$ 33701.0 1.58964
$$767$$ −17554.0 −0.826386
$$768$$ −18304.9 −0.860052
$$769$$ 14433.1 0.676816 0.338408 0.940999i $$-0.390112\pi$$
0.338408 + 0.940999i $$0.390112\pi$$
$$770$$ −4047.53 −0.189432
$$771$$ 9205.91 0.430017
$$772$$ −67359.4 −3.14031
$$773$$ −29443.2 −1.36999 −0.684993 0.728550i $$-0.740195\pi$$
−0.684993 + 0.728550i $$0.740195\pi$$
$$774$$ 5168.93 0.240043
$$775$$ −2216.48 −0.102733
$$776$$ 15930.4 0.736941
$$777$$ 3404.62 0.157195
$$778$$ −34857.7 −1.60631
$$779$$ 22766.7 1.04711
$$780$$ 7408.27 0.340075
$$781$$ 10584.9 0.484964
$$782$$ 12615.0 0.576869
$$783$$ 5789.31 0.264231
$$784$$ 1083.13 0.0493408
$$785$$ 11833.8 0.538046
$$786$$ 2616.24 0.118726
$$787$$ 26390.6 1.19533 0.597664 0.801747i $$-0.296096\pi$$
0.597664 + 0.801747i $$0.296096\pi$$
$$788$$ −39543.6 −1.78767
$$789$$ −14022.4 −0.632711
$$790$$ −20931.1 −0.942650
$$791$$ −11607.4 −0.521758
$$792$$ 6353.72 0.285063
$$793$$ 24823.7 1.11162
$$794$$ −42013.2 −1.87783
$$795$$ −10150.4 −0.452825
$$796$$ 57870.3 2.57683
$$797$$ −3738.33 −0.166146 −0.0830730 0.996543i $$-0.526473\pi$$
−0.0830730 + 0.996543i $$0.526473\pi$$
$$798$$ −6654.90 −0.295214
$$799$$ −6524.26 −0.288876
$$800$$ −3142.15 −0.138865
$$801$$ 13263.5 0.585073
$$802$$ 8382.48 0.369072
$$803$$ 1016.85 0.0446873
$$804$$ −38419.3 −1.68525
$$805$$ 5098.62 0.223233
$$806$$ 14595.8 0.637861
$$807$$ 7252.14 0.316341
$$808$$ 54250.1 2.36202
$$809$$ 43204.1 1.87760 0.938798 0.344468i $$-0.111941\pi$$
0.938798 + 0.344468i $$0.111941\pi$$
$$810$$ −1904.13 −0.0825980
$$811$$ −30192.4 −1.30727 −0.653637 0.756809i $$-0.726758\pi$$
−0.653637 + 0.756809i $$0.726758\pi$$
$$812$$ 21170.2 0.914935
$$813$$ 23172.9 0.999642
$$814$$ 18748.7 0.807301
$$815$$ 15688.5 0.674286
$$816$$ −1221.42 −0.0523999
$$817$$ −8233.71 −0.352584
$$818$$ −41185.3 −1.76041
$$819$$ −2205.98 −0.0941188
$$820$$ 23820.6 1.01445
$$821$$ −40274.7 −1.71206 −0.856028 0.516929i $$-0.827075\pi$$
−0.856028 + 0.516929i $$0.827075\pi$$
$$822$$ −535.847 −0.0227370
$$823$$ 25184.2 1.06667 0.533334 0.845905i $$-0.320939\pi$$
0.533334 + 0.845905i $$0.320939\pi$$
$$824$$ 19020.5 0.804140
$$825$$ 1844.77 0.0778503
$$826$$ 16498.9 0.694999
$$827$$ −38941.7 −1.63741 −0.818703 0.574218i $$-0.805306\pi$$
−0.818703 + 0.574218i $$0.805306\pi$$
$$828$$ −18492.3 −0.776150
$$829$$ −8327.05 −0.348867 −0.174433 0.984669i $$-0.555809\pi$$
−0.174433 + 0.984669i $$0.555809\pi$$
$$830$$ 24851.3 1.03928
$$831$$ −13728.5 −0.573089
$$832$$ 26883.5 1.12021
$$833$$ −902.519 −0.0375395
$$834$$ 2589.75 0.107525
$$835$$ −732.343 −0.0303518
$$836$$ −23384.2 −0.967418
$$837$$ −2393.80 −0.0988553
$$838$$ −15104.9 −0.622660
$$839$$ 8784.41 0.361468 0.180734 0.983532i $$-0.442153\pi$$
0.180734 + 0.983532i $$0.442153\pi$$
$$840$$ −3013.66 −0.123787
$$841$$ 21586.4 0.885087
$$842$$ 74086.4 3.03229
$$843$$ −4075.67 −0.166517
$$844$$ −11615.4 −0.473718
$$845$$ 4854.53 0.197634
$$846$$ 14988.4 0.609117
$$847$$ −5081.96 −0.206161
$$848$$ 14958.0 0.605732
$$849$$ 11655.1 0.471145
$$850$$ −2164.92 −0.0873602
$$851$$ −23617.6 −0.951350
$$852$$ 18209.2 0.732203
$$853$$ −9076.15 −0.364316 −0.182158 0.983269i $$-0.558308\pi$$
−0.182158 + 0.983269i $$0.558308\pi$$
$$854$$ −23331.6 −0.934884
$$855$$ 3033.14 0.121323
$$856$$ 46339.6 1.85030
$$857$$ 36396.7 1.45074 0.725372 0.688357i $$-0.241668\pi$$
0.725372 + 0.688357i $$0.241668\pi$$
$$858$$ −12148.0 −0.483364
$$859$$ 8915.27 0.354115 0.177058 0.984200i $$-0.443342\pi$$
0.177058 + 0.984200i $$0.443342\pi$$
$$860$$ −8614.88 −0.341587
$$861$$ −7093.14 −0.280759
$$862$$ −1908.44 −0.0754081
$$863$$ −6148.26 −0.242514 −0.121257 0.992621i $$-0.538692\pi$$
−0.121257 + 0.992621i $$0.538692\pi$$
$$864$$ −3393.52 −0.133623
$$865$$ 7120.59 0.279893
$$866$$ −36884.9 −1.44735
$$867$$ −13721.2 −0.537483
$$868$$ −8753.59 −0.342300
$$869$$ 21900.8 0.854928
$$870$$ −15121.5 −0.589274
$$871$$ 31792.6 1.23680
$$872$$ 6234.16 0.242105
$$873$$ 4995.31 0.193660
$$874$$ 46164.4 1.78665
$$875$$ −875.000 −0.0338062
$$876$$ 1749.29 0.0674692
$$877$$ −14287.0 −0.550101 −0.275050 0.961430i $$-0.588694\pi$$
−0.275050 + 0.961430i $$0.588694\pi$$
$$878$$ 1988.90 0.0764488
$$879$$ −12101.7 −0.464370
$$880$$ −2718.53 −0.104138
$$881$$ −13315.9 −0.509221 −0.254610 0.967044i $$-0.581947\pi$$
−0.254610 + 0.967044i $$0.581947\pi$$
$$882$$ 2073.39 0.0791549
$$883$$ −5271.78 −0.200917 −0.100458 0.994941i $$-0.532031\pi$$
−0.100458 + 0.994941i $$0.532031\pi$$
$$884$$ 9096.73 0.346104
$$885$$ −7519.78 −0.285621
$$886$$ −75501.1 −2.86288
$$887$$ 2606.07 0.0986507 0.0493253 0.998783i $$-0.484293\pi$$
0.0493253 + 0.998783i $$0.484293\pi$$
$$888$$ 13959.7 0.527542
$$889$$ −7761.69 −0.292822
$$890$$ −34644.0 −1.30480
$$891$$ 1992.35 0.0749115
$$892$$ 11526.3 0.432654
$$893$$ −23875.4 −0.894694
$$894$$ −19511.5 −0.729937
$$895$$ −6223.48 −0.232434
$$896$$ −18229.2 −0.679682
$$897$$ 15302.7 0.569612
$$898$$ 10281.1 0.382056
$$899$$ −19010.2 −0.705258
$$900$$ 3173.55 0.117539
$$901$$ −12463.8 −0.460854
$$902$$ −39060.8 −1.44189
$$903$$ 2565.28 0.0945373
$$904$$ −47592.8 −1.75101
$$905$$ 19395.4 0.712405
$$906$$ 10793.7 0.395802
$$907$$ 18610.6 0.681317 0.340659 0.940187i $$-0.389350\pi$$
0.340659 + 0.940187i $$0.389350\pi$$
$$908$$ 51564.6 1.88462
$$909$$ 17011.3 0.620714
$$910$$ 5761.98 0.209899
$$911$$ 41091.7 1.49443 0.747216 0.664581i $$-0.231390\pi$$
0.747216 + 0.664581i $$0.231390\pi$$
$$912$$ −4469.77 −0.162291
$$913$$ −26002.6 −0.942563
$$914$$ −27265.4 −0.986716
$$915$$ 10634.0 0.384206
$$916$$ 13249.8 0.477934
$$917$$ 1298.41 0.0467583
$$918$$ −2338.12 −0.0840624
$$919$$ 38891.3 1.39598 0.697990 0.716107i $$-0.254078\pi$$
0.697990 + 0.716107i $$0.254078\pi$$
$$920$$ 20905.5 0.749167
$$921$$ −13861.1 −0.495915
$$922$$ 46420.2 1.65810
$$923$$ −15068.4 −0.537360
$$924$$ 7285.56 0.259391
$$925$$ 4053.12 0.144071
$$926$$ −29064.3 −1.03144
$$927$$ 5964.30 0.211320
$$928$$ −26949.4 −0.953295
$$929$$ 18699.4 0.660396 0.330198 0.943912i $$-0.392885\pi$$
0.330198 + 0.943912i $$0.392885\pi$$
$$930$$ 6252.56 0.220462
$$931$$ −3302.75 −0.116266
$$932$$ −107.859 −0.00379080
$$933$$ 20027.7 0.702761
$$934$$ 28893.4 1.01223
$$935$$ 2265.22 0.0792305
$$936$$ −9045.03 −0.315861
$$937$$ 21509.6 0.749933 0.374967 0.927038i $$-0.377654\pi$$
0.374967 + 0.927038i $$0.377654\pi$$
$$938$$ −29881.7 −1.04016
$$939$$ 8510.35 0.295767
$$940$$ −24980.7 −0.866788
$$941$$ −11241.7 −0.389448 −0.194724 0.980858i $$-0.562381\pi$$
−0.194724 + 0.980858i $$0.562381\pi$$
$$942$$ −33382.4 −1.15463
$$943$$ 49204.5 1.69917
$$944$$ 11081.5 0.382068
$$945$$ −945.000 −0.0325300
$$946$$ 14126.6 0.485513
$$947$$ −36556.3 −1.25441 −0.627203 0.778856i $$-0.715800\pi$$
−0.627203 + 0.778856i $$0.715800\pi$$
$$948$$ 37675.9 1.29078
$$949$$ −1447.57 −0.0495153
$$950$$ −7922.50 −0.270568
$$951$$ 12031.9 0.410263
$$952$$ −3700.53 −0.125982
$$953$$ −36633.4 −1.24520 −0.622598 0.782542i $$-0.713923\pi$$
−0.622598 + 0.782542i $$0.713923\pi$$
$$954$$ 28633.5 0.971745
$$955$$ −7874.52 −0.266820
$$956$$ −12540.7 −0.424263
$$957$$ 15822.1 0.534436
$$958$$ 51150.3 1.72504
$$959$$ −265.935 −0.00895462
$$960$$ 11516.4 0.387176
$$961$$ −21930.5 −0.736146
$$962$$ −26690.3 −0.894523
$$963$$ 14530.8 0.486239
$$964$$ 30187.3 1.00858
$$965$$ 23878.4 0.796551
$$966$$ −14382.9 −0.479050
$$967$$ −35515.8 −1.18109 −0.590544 0.807006i $$-0.701087\pi$$
−0.590544 + 0.807006i $$0.701087\pi$$
$$968$$ −20837.2 −0.691871
$$969$$ 3724.44 0.123474
$$970$$ −13047.6 −0.431891
$$971$$ −39661.0 −1.31080 −0.655398 0.755283i $$-0.727499\pi$$
−0.655398 + 0.755283i $$0.727499\pi$$
$$972$$ 3427.44 0.113102
$$973$$ 1285.26 0.0423471
$$974$$ 38023.9 1.25089
$$975$$ −2626.17 −0.0862613
$$976$$ −15670.7 −0.513942
$$977$$ 50325.3 1.64795 0.823977 0.566624i $$-0.191751\pi$$
0.823977 + 0.566624i $$0.191751\pi$$
$$978$$ −44256.2 −1.44699
$$979$$ 36249.0 1.18337
$$980$$ −3455.65 −0.112639
$$981$$ 1954.86 0.0636226
$$982$$ −32722.4 −1.06335
$$983$$ 51189.0 1.66091 0.830456 0.557084i $$-0.188080\pi$$
0.830456 + 0.557084i $$0.188080\pi$$
$$984$$ −29083.5 −0.942223
$$985$$ 14017.9 0.453449
$$986$$ −18568.0 −0.599721
$$987$$ 7438.59 0.239892
$$988$$ 33289.3 1.07194
$$989$$ −17795.1 −0.572145
$$990$$ −5203.97 −0.167064
$$991$$ −55137.3 −1.76740 −0.883700 0.468054i $$-0.844955\pi$$
−0.883700 + 0.468054i $$0.844955\pi$$
$$992$$ 11143.2 0.356651
$$993$$ 33262.6 1.06300
$$994$$ 14162.7 0.451925
$$995$$ −20514.6 −0.653624
$$996$$ −44732.3 −1.42309
$$997$$ 41606.5 1.32166 0.660828 0.750537i $$-0.270205\pi$$
0.660828 + 0.750537i $$0.270205\pi$$
$$998$$ 87599.6 2.77848
$$999$$ 4377.37 0.138633
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 105.4.a.g.1.2 2
3.2 odd 2 315.4.a.g.1.1 2
4.3 odd 2 1680.4.a.y.1.2 2
5.2 odd 4 525.4.d.j.274.4 4
5.3 odd 4 525.4.d.j.274.1 4
5.4 even 2 525.4.a.i.1.1 2
7.6 odd 2 735.4.a.q.1.2 2
15.14 odd 2 1575.4.a.y.1.2 2
21.20 even 2 2205.4.a.v.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
105.4.a.g.1.2 2 1.1 even 1 trivial
315.4.a.g.1.1 2 3.2 odd 2
525.4.a.i.1.1 2 5.4 even 2
525.4.d.j.274.1 4 5.3 odd 4
525.4.d.j.274.4 4 5.2 odd 4
735.4.a.q.1.2 2 7.6 odd 2
1575.4.a.y.1.2 2 15.14 odd 2
1680.4.a.y.1.2 2 4.3 odd 2
2205.4.a.v.1.1 2 21.20 even 2