# Properties

 Label 105.4.a.f.1.1 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: $$\Q(\sqrt{65})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 16$$ x^2 - x - 16 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-3.53113$$ 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)

 $$f(q)$$ $$=$$ $$q-3.53113 q^{2} +3.00000 q^{3} +4.46887 q^{4} +5.00000 q^{5} -10.5934 q^{6} -7.00000 q^{7} +12.4689 q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q-3.53113 q^{2} +3.00000 q^{3} +4.46887 q^{4} +5.00000 q^{5} -10.5934 q^{6} -7.00000 q^{7} +12.4689 q^{8} +9.00000 q^{9} -17.6556 q^{10} -2.93774 q^{11} +13.4066 q^{12} -19.0623 q^{13} +24.7179 q^{14} +15.0000 q^{15} -79.7802 q^{16} +122.498 q^{17} -31.7802 q^{18} +107.436 q^{19} +22.3444 q^{20} -21.0000 q^{21} +10.3735 q^{22} +210.623 q^{23} +37.4066 q^{24} +25.0000 q^{25} +67.3113 q^{26} +27.0000 q^{27} -31.2821 q^{28} +95.4942 q^{29} -52.9669 q^{30} -94.3074 q^{31} +181.963 q^{32} -8.81323 q^{33} -432.556 q^{34} -35.0000 q^{35} +40.2198 q^{36} +97.1206 q^{37} -379.370 q^{38} -57.1868 q^{39} +62.3444 q^{40} -491.113 q^{41} +74.1537 q^{42} -43.0039 q^{43} -13.1284 q^{44} +45.0000 q^{45} -743.735 q^{46} +473.494 q^{47} -239.340 q^{48} +49.0000 q^{49} -88.2782 q^{50} +367.494 q^{51} -85.1868 q^{52} -183.677 q^{53} -95.3405 q^{54} -14.6887 q^{55} -87.2821 q^{56} +322.307 q^{57} -337.202 q^{58} -760.615 q^{59} +67.0331 q^{60} -198.747 q^{61} +333.012 q^{62} -63.0000 q^{63} -4.29373 q^{64} -95.3113 q^{65} +31.1206 q^{66} -309.992 q^{67} +547.428 q^{68} +631.868 q^{69} +123.590 q^{70} +665.693 q^{71} +112.220 q^{72} +621.288 q^{73} -342.945 q^{74} +75.0000 q^{75} +480.117 q^{76} +20.5642 q^{77} +201.934 q^{78} -24.7626 q^{79} -398.901 q^{80} +81.0000 q^{81} +1734.18 q^{82} -406.724 q^{83} -93.8463 q^{84} +612.490 q^{85} +151.852 q^{86} +286.483 q^{87} -36.6303 q^{88} +261.751 q^{89} -158.901 q^{90} +133.436 q^{91} +941.245 q^{92} -282.922 q^{93} -1671.97 q^{94} +537.179 q^{95} +545.889 q^{96} -1004.77 q^{97} -173.025 q^{98} -26.4397 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} + 6 q^{3} + 17 q^{4} + 10 q^{5} + 3 q^{6} - 14 q^{7} + 33 q^{8} + 18 q^{9}+O(q^{10})$$ 2 * q + q^2 + 6 * q^3 + 17 * q^4 + 10 * q^5 + 3 * q^6 - 14 * q^7 + 33 * q^8 + 18 * q^9 $$2 q + q^{2} + 6 q^{3} + 17 q^{4} + 10 q^{5} + 3 q^{6} - 14 q^{7} + 33 q^{8} + 18 q^{9} + 5 q^{10} - 22 q^{11} + 51 q^{12} - 22 q^{13} - 7 q^{14} + 30 q^{15} - 87 q^{16} + 116 q^{17} + 9 q^{18} + 102 q^{19} + 85 q^{20} - 42 q^{21} - 76 q^{22} + 260 q^{23} + 99 q^{24} + 50 q^{25} + 54 q^{26} + 54 q^{27} - 119 q^{28} - 196 q^{29} + 15 q^{30} + 150 q^{31} - 15 q^{32} - 66 q^{33} - 462 q^{34} - 70 q^{35} + 153 q^{36} - 96 q^{37} - 404 q^{38} - 66 q^{39} + 165 q^{40} - 176 q^{41} - 21 q^{42} - 344 q^{43} - 252 q^{44} + 90 q^{45} - 520 q^{46} + 560 q^{47} - 261 q^{48} + 98 q^{49} + 25 q^{50} + 348 q^{51} - 122 q^{52} + 326 q^{53} + 27 q^{54} - 110 q^{55} - 231 q^{56} + 306 q^{57} - 1658 q^{58} - 844 q^{59} + 255 q^{60} - 204 q^{61} + 1440 q^{62} - 126 q^{63} - 839 q^{64} - 110 q^{65} - 228 q^{66} - 104 q^{67} + 466 q^{68} + 780 q^{69} - 35 q^{70} + 1670 q^{71} + 297 q^{72} - 386 q^{73} - 1218 q^{74} + 150 q^{75} + 412 q^{76} + 154 q^{77} + 162 q^{78} - 888 q^{79} - 435 q^{80} + 162 q^{81} + 3162 q^{82} + 928 q^{83} - 357 q^{84} + 580 q^{85} - 1212 q^{86} - 588 q^{87} - 428 q^{88} + 588 q^{89} + 45 q^{90} + 154 q^{91} + 1560 q^{92} + 450 q^{93} - 1280 q^{94} + 510 q^{95} - 45 q^{96} + 522 q^{97} + 49 q^{98} - 198 q^{99}+O(q^{100})$$ 2 * q + q^2 + 6 * q^3 + 17 * q^4 + 10 * q^5 + 3 * q^6 - 14 * q^7 + 33 * q^8 + 18 * q^9 + 5 * q^10 - 22 * q^11 + 51 * q^12 - 22 * q^13 - 7 * q^14 + 30 * q^15 - 87 * q^16 + 116 * q^17 + 9 * q^18 + 102 * q^19 + 85 * q^20 - 42 * q^21 - 76 * q^22 + 260 * q^23 + 99 * q^24 + 50 * q^25 + 54 * q^26 + 54 * q^27 - 119 * q^28 - 196 * q^29 + 15 * q^30 + 150 * q^31 - 15 * q^32 - 66 * q^33 - 462 * q^34 - 70 * q^35 + 153 * q^36 - 96 * q^37 - 404 * q^38 - 66 * q^39 + 165 * q^40 - 176 * q^41 - 21 * q^42 - 344 * q^43 - 252 * q^44 + 90 * q^45 - 520 * q^46 + 560 * q^47 - 261 * q^48 + 98 * q^49 + 25 * q^50 + 348 * q^51 - 122 * q^52 + 326 * q^53 + 27 * q^54 - 110 * q^55 - 231 * q^56 + 306 * q^57 - 1658 * q^58 - 844 * q^59 + 255 * q^60 - 204 * q^61 + 1440 * q^62 - 126 * q^63 - 839 * q^64 - 110 * q^65 - 228 * q^66 - 104 * q^67 + 466 * q^68 + 780 * q^69 - 35 * q^70 + 1670 * q^71 + 297 * q^72 - 386 * q^73 - 1218 * q^74 + 150 * q^75 + 412 * q^76 + 154 * q^77 + 162 * q^78 - 888 * q^79 - 435 * q^80 + 162 * q^81 + 3162 * q^82 + 928 * q^83 - 357 * q^84 + 580 * q^85 - 1212 * q^86 - 588 * q^87 - 428 * q^88 + 588 * q^89 + 45 * q^90 + 154 * q^91 + 1560 * q^92 + 450 * q^93 - 1280 * q^94 + 510 * q^95 - 45 * q^96 + 522 * q^97 + 49 * q^98 - 198 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −3.53113 −1.24844 −0.624221 0.781248i $$-0.714584\pi$$
−0.624221 + 0.781248i $$0.714584\pi$$
$$3$$ 3.00000 0.577350
$$4$$ 4.46887 0.558609
$$5$$ 5.00000 0.447214
$$6$$ −10.5934 −0.720789
$$7$$ −7.00000 −0.377964
$$8$$ 12.4689 0.551051
$$9$$ 9.00000 0.333333
$$10$$ −17.6556 −0.558320
$$11$$ −2.93774 −0.0805239 −0.0402619 0.999189i $$-0.512819\pi$$
−0.0402619 + 0.999189i $$0.512819\pi$$
$$12$$ 13.4066 0.322513
$$13$$ −19.0623 −0.406686 −0.203343 0.979108i $$-0.565181\pi$$
−0.203343 + 0.979108i $$0.565181\pi$$
$$14$$ 24.7179 0.471867
$$15$$ 15.0000 0.258199
$$16$$ −79.7802 −1.24656
$$17$$ 122.498 1.74766 0.873828 0.486236i $$-0.161630\pi$$
0.873828 + 0.486236i $$0.161630\pi$$
$$18$$ −31.7802 −0.416148
$$19$$ 107.436 1.29723 0.648617 0.761115i $$-0.275348\pi$$
0.648617 + 0.761115i $$0.275348\pi$$
$$20$$ 22.3444 0.249817
$$21$$ −21.0000 −0.218218
$$22$$ 10.3735 0.100529
$$23$$ 210.623 1.90947 0.954736 0.297455i $$-0.0961379\pi$$
0.954736 + 0.297455i $$0.0961379\pi$$
$$24$$ 37.4066 0.318150
$$25$$ 25.0000 0.200000
$$26$$ 67.3113 0.507724
$$27$$ 27.0000 0.192450
$$28$$ −31.2821 −0.211134
$$29$$ 95.4942 0.611477 0.305738 0.952116i $$-0.401097\pi$$
0.305738 + 0.952116i $$0.401097\pi$$
$$30$$ −52.9669 −0.322346
$$31$$ −94.3074 −0.546391 −0.273195 0.961959i $$-0.588081\pi$$
−0.273195 + 0.961959i $$0.588081\pi$$
$$32$$ 181.963 1.00521
$$33$$ −8.81323 −0.0464905
$$34$$ −432.556 −2.18185
$$35$$ −35.0000 −0.169031
$$36$$ 40.2198 0.186203
$$37$$ 97.1206 0.431528 0.215764 0.976446i $$-0.430776\pi$$
0.215764 + 0.976446i $$0.430776\pi$$
$$38$$ −379.370 −1.61952
$$39$$ −57.1868 −0.234800
$$40$$ 62.3444 0.246438
$$41$$ −491.113 −1.87071 −0.935353 0.353716i $$-0.884918\pi$$
−0.935353 + 0.353716i $$0.884918\pi$$
$$42$$ 74.1537 0.272433
$$43$$ −43.0039 −0.152512 −0.0762562 0.997088i $$-0.524297\pi$$
−0.0762562 + 0.997088i $$0.524297\pi$$
$$44$$ −13.1284 −0.0449814
$$45$$ 45.0000 0.149071
$$46$$ −743.735 −2.38387
$$47$$ 473.494 1.46949 0.734747 0.678341i $$-0.237301\pi$$
0.734747 + 0.678341i $$0.237301\pi$$
$$48$$ −239.340 −0.719705
$$49$$ 49.0000 0.142857
$$50$$ −88.2782 −0.249689
$$51$$ 367.494 1.00901
$$52$$ −85.1868 −0.227178
$$53$$ −183.677 −0.476038 −0.238019 0.971261i $$-0.576498\pi$$
−0.238019 + 0.971261i $$0.576498\pi$$
$$54$$ −95.3405 −0.240263
$$55$$ −14.6887 −0.0360114
$$56$$ −87.2821 −0.208278
$$57$$ 322.307 0.748959
$$58$$ −337.202 −0.763394
$$59$$ −760.615 −1.67837 −0.839183 0.543849i $$-0.816966\pi$$
−0.839183 + 0.543849i $$0.816966\pi$$
$$60$$ 67.0331 0.144232
$$61$$ −198.747 −0.417163 −0.208582 0.978005i $$-0.566885\pi$$
−0.208582 + 0.978005i $$0.566885\pi$$
$$62$$ 333.012 0.682137
$$63$$ −63.0000 −0.125988
$$64$$ −4.29373 −0.00838618
$$65$$ −95.3113 −0.181876
$$66$$ 31.1206 0.0580407
$$67$$ −309.992 −0.565247 −0.282624 0.959231i $$-0.591205\pi$$
−0.282624 + 0.959231i $$0.591205\pi$$
$$68$$ 547.428 0.976256
$$69$$ 631.868 1.10243
$$70$$ 123.590 0.211025
$$71$$ 665.693 1.11272 0.556360 0.830941i $$-0.312197\pi$$
0.556360 + 0.830941i $$0.312197\pi$$
$$72$$ 112.220 0.183684
$$73$$ 621.288 0.996113 0.498057 0.867145i $$-0.334047\pi$$
0.498057 + 0.867145i $$0.334047\pi$$
$$74$$ −342.945 −0.538738
$$75$$ 75.0000 0.115470
$$76$$ 480.117 0.724647
$$77$$ 20.5642 0.0304352
$$78$$ 201.934 0.293135
$$79$$ −24.7626 −0.0352659 −0.0176330 0.999845i $$-0.505613\pi$$
−0.0176330 + 0.999845i $$0.505613\pi$$
$$80$$ −398.901 −0.557481
$$81$$ 81.0000 0.111111
$$82$$ 1734.18 2.33547
$$83$$ −406.724 −0.537876 −0.268938 0.963157i $$-0.586673\pi$$
−0.268938 + 0.963157i $$0.586673\pi$$
$$84$$ −93.8463 −0.121898
$$85$$ 612.490 0.781575
$$86$$ 151.852 0.190403
$$87$$ 286.483 0.353036
$$88$$ −36.6303 −0.0443728
$$89$$ 261.751 0.311748 0.155874 0.987777i $$-0.450181\pi$$
0.155874 + 0.987777i $$0.450181\pi$$
$$90$$ −158.901 −0.186107
$$91$$ 133.436 0.153713
$$92$$ 941.245 1.06665
$$93$$ −282.922 −0.315459
$$94$$ −1671.97 −1.83458
$$95$$ 537.179 0.580141
$$96$$ 545.889 0.580360
$$97$$ −1004.77 −1.05175 −0.525873 0.850563i $$-0.676261\pi$$
−0.525873 + 0.850563i $$0.676261\pi$$
$$98$$ −173.025 −0.178349
$$99$$ −26.4397 −0.0268413
$$100$$ 111.722 0.111722
$$101$$ −128.872 −0.126962 −0.0634812 0.997983i $$-0.520220\pi$$
−0.0634812 + 0.997983i $$0.520220\pi$$
$$102$$ −1297.67 −1.25969
$$103$$ 806.008 0.771051 0.385526 0.922697i $$-0.374020\pi$$
0.385526 + 0.922697i $$0.374020\pi$$
$$104$$ −237.685 −0.224105
$$105$$ −105.000 −0.0975900
$$106$$ 648.587 0.594305
$$107$$ −769.712 −0.695429 −0.347714 0.937600i $$-0.613042\pi$$
−0.347714 + 0.937600i $$0.613042\pi$$
$$108$$ 120.660 0.107504
$$109$$ −780.856 −0.686169 −0.343085 0.939304i $$-0.611472\pi$$
−0.343085 + 0.939304i $$0.611472\pi$$
$$110$$ 51.8677 0.0449581
$$111$$ 291.362 0.249143
$$112$$ 558.461 0.471157
$$113$$ −1115.65 −0.928771 −0.464386 0.885633i $$-0.653725\pi$$
−0.464386 + 0.885633i $$0.653725\pi$$
$$114$$ −1138.11 −0.935032
$$115$$ 1053.11 0.853942
$$116$$ 426.751 0.341576
$$117$$ −171.560 −0.135562
$$118$$ 2685.83 2.09534
$$119$$ −857.486 −0.660552
$$120$$ 187.033 0.142281
$$121$$ −1322.37 −0.993516
$$122$$ 701.802 0.520804
$$123$$ −1473.34 −1.08005
$$124$$ −421.448 −0.305219
$$125$$ 125.000 0.0894427
$$126$$ 222.461 0.157289
$$127$$ −1875.98 −1.31076 −0.655381 0.755299i $$-0.727492\pi$$
−0.655381 + 0.755299i $$0.727492\pi$$
$$128$$ −1440.54 −0.994744
$$129$$ −129.012 −0.0880530
$$130$$ 336.556 0.227061
$$131$$ 364.203 0.242905 0.121452 0.992597i $$-0.461245\pi$$
0.121452 + 0.992597i $$0.461245\pi$$
$$132$$ −39.3852 −0.0259700
$$133$$ −752.051 −0.490309
$$134$$ 1094.62 0.705679
$$135$$ 135.000 0.0860663
$$136$$ 1527.41 0.963048
$$137$$ 1603.13 0.999743 0.499872 0.866099i $$-0.333380\pi$$
0.499872 + 0.866099i $$0.333380\pi$$
$$138$$ −2231.21 −1.37633
$$139$$ 2431.12 1.48349 0.741746 0.670681i $$-0.233998\pi$$
0.741746 + 0.670681i $$0.233998\pi$$
$$140$$ −156.410 −0.0944221
$$141$$ 1420.48 0.848413
$$142$$ −2350.65 −1.38917
$$143$$ 56.0000 0.0327479
$$144$$ −718.021 −0.415522
$$145$$ 477.471 0.273461
$$146$$ −2193.85 −1.24359
$$147$$ 147.000 0.0824786
$$148$$ 434.020 0.241055
$$149$$ 2341.57 1.28744 0.643722 0.765260i $$-0.277389\pi$$
0.643722 + 0.765260i $$0.277389\pi$$
$$150$$ −264.835 −0.144158
$$151$$ −2104.07 −1.13395 −0.566976 0.823734i $$-0.691887\pi$$
−0.566976 + 0.823734i $$0.691887\pi$$
$$152$$ 1339.60 0.714843
$$153$$ 1102.48 0.582552
$$154$$ −72.6148 −0.0379966
$$155$$ −471.537 −0.244353
$$156$$ −255.560 −0.131162
$$157$$ −593.467 −0.301680 −0.150840 0.988558i $$-0.548198\pi$$
−0.150840 + 0.988558i $$0.548198\pi$$
$$158$$ 87.4399 0.0440275
$$159$$ −551.031 −0.274840
$$160$$ 909.815 0.449545
$$161$$ −1474.36 −0.721712
$$162$$ −286.021 −0.138716
$$163$$ 2178.71 1.04693 0.523465 0.852047i $$-0.324639\pi$$
0.523465 + 0.852047i $$0.324639\pi$$
$$164$$ −2194.72 −1.04499
$$165$$ −44.0661 −0.0207912
$$166$$ 1436.19 0.671508
$$167$$ 799.502 0.370463 0.185231 0.982695i $$-0.440697\pi$$
0.185231 + 0.982695i $$0.440697\pi$$
$$168$$ −261.846 −0.120249
$$169$$ −1833.63 −0.834606
$$170$$ −2162.78 −0.975752
$$171$$ 966.922 0.432412
$$172$$ −192.179 −0.0851947
$$173$$ −1444.36 −0.634754 −0.317377 0.948299i $$-0.602802\pi$$
−0.317377 + 0.948299i $$0.602802\pi$$
$$174$$ −1011.61 −0.440745
$$175$$ −175.000 −0.0755929
$$176$$ 234.374 0.100378
$$177$$ −2281.84 −0.969005
$$178$$ −924.276 −0.389199
$$179$$ 3343.49 1.39611 0.698056 0.716043i $$-0.254048\pi$$
0.698056 + 0.716043i $$0.254048\pi$$
$$180$$ 201.099 0.0832725
$$181$$ 2251.81 0.924729 0.462365 0.886690i $$-0.347001\pi$$
0.462365 + 0.886690i $$0.347001\pi$$
$$182$$ −471.179 −0.191902
$$183$$ −596.241 −0.240849
$$184$$ 2626.23 1.05222
$$185$$ 485.603 0.192985
$$186$$ 999.035 0.393832
$$187$$ −359.868 −0.140728
$$188$$ 2115.98 0.820873
$$189$$ −189.000 −0.0727393
$$190$$ −1896.85 −0.724273
$$191$$ −1001.93 −0.379565 −0.189782 0.981826i $$-0.560778\pi$$
−0.189782 + 0.981826i $$0.560778\pi$$
$$192$$ −12.8812 −0.00484177
$$193$$ −4054.97 −1.51235 −0.756173 0.654372i $$-0.772933\pi$$
−0.756173 + 0.654372i $$0.772933\pi$$
$$194$$ 3547.99 1.31304
$$195$$ −285.934 −0.105006
$$196$$ 218.975 0.0798013
$$197$$ −5140.23 −1.85902 −0.929508 0.368802i $$-0.879768\pi$$
−0.929508 + 0.368802i $$0.879768\pi$$
$$198$$ 93.3619 0.0335098
$$199$$ 585.631 0.208614 0.104307 0.994545i $$-0.466737\pi$$
0.104307 + 0.994545i $$0.466737\pi$$
$$200$$ 311.722 0.110210
$$201$$ −929.977 −0.326346
$$202$$ 455.062 0.158505
$$203$$ −668.459 −0.231116
$$204$$ 1642.28 0.563642
$$205$$ −2455.56 −0.836605
$$206$$ −2846.12 −0.962614
$$207$$ 1895.60 0.636490
$$208$$ 1520.79 0.506961
$$209$$ −315.619 −0.104458
$$210$$ 370.769 0.121836
$$211$$ −1055.16 −0.344266 −0.172133 0.985074i $$-0.555066\pi$$
−0.172133 + 0.985074i $$0.555066\pi$$
$$212$$ −820.829 −0.265919
$$213$$ 1997.08 0.642430
$$214$$ 2717.95 0.868203
$$215$$ −215.019 −0.0682056
$$216$$ 336.660 0.106050
$$217$$ 660.152 0.206516
$$218$$ 2757.30 0.856643
$$219$$ 1863.86 0.575106
$$220$$ −65.6420 −0.0201163
$$221$$ −2335.09 −0.710747
$$222$$ −1028.84 −0.311040
$$223$$ 4675.85 1.40412 0.702059 0.712119i $$-0.252264\pi$$
0.702059 + 0.712119i $$0.252264\pi$$
$$224$$ −1273.74 −0.379935
$$225$$ 225.000 0.0666667
$$226$$ 3939.49 1.15952
$$227$$ 5443.11 1.59151 0.795754 0.605621i $$-0.207075\pi$$
0.795754 + 0.605621i $$0.207075\pi$$
$$228$$ 1440.35 0.418375
$$229$$ −536.303 −0.154759 −0.0773797 0.997002i $$-0.524655\pi$$
−0.0773797 + 0.997002i $$0.524655\pi$$
$$230$$ −3718.68 −1.06610
$$231$$ 61.6926 0.0175717
$$232$$ 1190.70 0.336955
$$233$$ −183.490 −0.0515916 −0.0257958 0.999667i $$-0.508212\pi$$
−0.0257958 + 0.999667i $$0.508212\pi$$
$$234$$ 605.802 0.169241
$$235$$ 2367.47 0.657178
$$236$$ −3399.09 −0.937550
$$237$$ −74.2878 −0.0203608
$$238$$ 3027.90 0.824661
$$239$$ 643.218 0.174085 0.0870425 0.996205i $$-0.472258\pi$$
0.0870425 + 0.996205i $$0.472258\pi$$
$$240$$ −1196.70 −0.321862
$$241$$ −5755.61 −1.53839 −0.769194 0.639015i $$-0.779342\pi$$
−0.769194 + 0.639015i $$0.779342\pi$$
$$242$$ 4669.46 1.24035
$$243$$ 243.000 0.0641500
$$244$$ −888.175 −0.233031
$$245$$ 245.000 0.0638877
$$246$$ 5202.55 1.34838
$$247$$ −2047.97 −0.527567
$$248$$ −1175.91 −0.301089
$$249$$ −1220.17 −0.310543
$$250$$ −441.391 −0.111664
$$251$$ −5132.27 −1.29062 −0.645311 0.763920i $$-0.723272\pi$$
−0.645311 + 0.763920i $$0.723272\pi$$
$$252$$ −281.539 −0.0703781
$$253$$ −618.755 −0.153758
$$254$$ 6624.34 1.63641
$$255$$ 1837.47 0.451243
$$256$$ 5121.09 1.25027
$$257$$ 5041.74 1.22372 0.611859 0.790967i $$-0.290422\pi$$
0.611859 + 0.790967i $$0.290422\pi$$
$$258$$ 455.557 0.109929
$$259$$ −679.844 −0.163102
$$260$$ −425.934 −0.101597
$$261$$ 859.448 0.203826
$$262$$ −1286.05 −0.303253
$$263$$ 7577.00 1.77649 0.888246 0.459367i $$-0.151924\pi$$
0.888246 + 0.459367i $$0.151924\pi$$
$$264$$ −109.891 −0.0256186
$$265$$ −918.385 −0.212890
$$266$$ 2655.59 0.612122
$$267$$ 785.253 0.179988
$$268$$ −1385.32 −0.315752
$$269$$ 1023.10 0.231893 0.115947 0.993255i $$-0.463010\pi$$
0.115947 + 0.993255i $$0.463010\pi$$
$$270$$ −476.702 −0.107449
$$271$$ −2251.98 −0.504790 −0.252395 0.967624i $$-0.581218\pi$$
−0.252395 + 0.967624i $$0.581218\pi$$
$$272$$ −9772.91 −2.17857
$$273$$ 400.307 0.0887462
$$274$$ −5660.87 −1.24812
$$275$$ −73.4436 −0.0161048
$$276$$ 2823.74 0.615829
$$277$$ −8630.72 −1.87209 −0.936047 0.351875i $$-0.885544\pi$$
−0.936047 + 0.351875i $$0.885544\pi$$
$$278$$ −8584.61 −1.85205
$$279$$ −848.767 −0.182130
$$280$$ −436.410 −0.0931447
$$281$$ −7521.62 −1.59680 −0.798402 0.602124i $$-0.794321\pi$$
−0.798402 + 0.602124i $$0.794321\pi$$
$$282$$ −5015.91 −1.05919
$$283$$ 14.8169 0.00311226 0.00155613 0.999999i $$-0.499505\pi$$
0.00155613 + 0.999999i $$0.499505\pi$$
$$284$$ 2974.89 0.621576
$$285$$ 1611.54 0.334945
$$286$$ −197.743 −0.0408839
$$287$$ 3437.79 0.707060
$$288$$ 1637.67 0.335071
$$289$$ 10092.8 2.05430
$$290$$ −1686.01 −0.341400
$$291$$ −3014.32 −0.607226
$$292$$ 2776.46 0.556438
$$293$$ 6913.39 1.37844 0.689222 0.724550i $$-0.257952\pi$$
0.689222 + 0.724550i $$0.257952\pi$$
$$294$$ −519.076 −0.102970
$$295$$ −3803.07 −0.750588
$$296$$ 1210.98 0.237794
$$297$$ −79.3190 −0.0154968
$$298$$ −8268.39 −1.60730
$$299$$ −4014.94 −0.776555
$$300$$ 335.165 0.0645026
$$301$$ 301.027 0.0576442
$$302$$ 7429.74 1.41567
$$303$$ −386.615 −0.0733018
$$304$$ −8571.25 −1.61709
$$305$$ −993.735 −0.186561
$$306$$ −3893.01 −0.727283
$$307$$ −7644.12 −1.42108 −0.710542 0.703655i $$-0.751550\pi$$
−0.710542 + 0.703655i $$0.751550\pi$$
$$308$$ 91.8987 0.0170014
$$309$$ 2418.02 0.445167
$$310$$ 1665.06 0.305061
$$311$$ 7593.99 1.38462 0.692308 0.721602i $$-0.256594\pi$$
0.692308 + 0.721602i $$0.256594\pi$$
$$312$$ −713.055 −0.129387
$$313$$ −9127.84 −1.64836 −0.824179 0.566329i $$-0.808363\pi$$
−0.824179 + 0.566329i $$0.808363\pi$$
$$314$$ 2095.61 0.376631
$$315$$ −315.000 −0.0563436
$$316$$ −110.661 −0.0196999
$$317$$ −4929.81 −0.873456 −0.436728 0.899593i $$-0.643863\pi$$
−0.436728 + 0.899593i $$0.643863\pi$$
$$318$$ 1945.76 0.343122
$$319$$ −280.537 −0.0492385
$$320$$ −21.4686 −0.00375042
$$321$$ −2309.14 −0.401506
$$322$$ 5206.15 0.901016
$$323$$ 13160.7 2.26712
$$324$$ 361.979 0.0620677
$$325$$ −476.556 −0.0813372
$$326$$ −7693.30 −1.30703
$$327$$ −2342.57 −0.396160
$$328$$ −6123.62 −1.03086
$$329$$ −3314.46 −0.555417
$$330$$ 155.603 0.0259566
$$331$$ 1221.67 0.202867 0.101433 0.994842i $$-0.467657\pi$$
0.101433 + 0.994842i $$0.467657\pi$$
$$332$$ −1817.60 −0.300463
$$333$$ 874.086 0.143843
$$334$$ −2823.14 −0.462502
$$335$$ −1549.96 −0.252786
$$336$$ 1675.38 0.272023
$$337$$ −8744.83 −1.41354 −0.706768 0.707446i $$-0.749847\pi$$
−0.706768 + 0.707446i $$0.749847\pi$$
$$338$$ 6474.79 1.04196
$$339$$ −3346.94 −0.536226
$$340$$ 2737.14 0.436595
$$341$$ 277.051 0.0439975
$$342$$ −3414.33 −0.539841
$$343$$ −343.000 −0.0539949
$$344$$ −536.210 −0.0840421
$$345$$ 3159.34 0.493023
$$346$$ 5100.21 0.792454
$$347$$ 4589.56 0.710031 0.355015 0.934860i $$-0.384476\pi$$
0.355015 + 0.934860i $$0.384476\pi$$
$$348$$ 1280.25 0.197209
$$349$$ −3989.89 −0.611960 −0.305980 0.952038i $$-0.598984\pi$$
−0.305980 + 0.952038i $$0.598984\pi$$
$$350$$ 617.948 0.0943734
$$351$$ −514.681 −0.0782668
$$352$$ −534.561 −0.0809437
$$353$$ 2416.35 0.364333 0.182166 0.983268i $$-0.441689\pi$$
0.182166 + 0.983268i $$0.441689\pi$$
$$354$$ 8057.49 1.20975
$$355$$ 3328.46 0.497624
$$356$$ 1169.73 0.174145
$$357$$ −2572.46 −0.381370
$$358$$ −11806.3 −1.74297
$$359$$ −2756.24 −0.405206 −0.202603 0.979261i $$-0.564940\pi$$
−0.202603 + 0.979261i $$0.564940\pi$$
$$360$$ 561.099 0.0821459
$$361$$ 4683.45 0.682818
$$362$$ −7951.44 −1.15447
$$363$$ −3967.11 −0.573607
$$364$$ 596.307 0.0858654
$$365$$ 3106.44 0.445475
$$366$$ 2105.40 0.300687
$$367$$ 11112.8 1.58061 0.790307 0.612711i $$-0.209921\pi$$
0.790307 + 0.612711i $$0.209921\pi$$
$$368$$ −16803.5 −2.38028
$$369$$ −4420.02 −0.623569
$$370$$ −1714.73 −0.240931
$$371$$ 1285.74 0.179925
$$372$$ −1264.34 −0.176218
$$373$$ 6091.09 0.845535 0.422768 0.906238i $$-0.361059\pi$$
0.422768 + 0.906238i $$0.361059\pi$$
$$374$$ 1270.74 0.175691
$$375$$ 375.000 0.0516398
$$376$$ 5903.94 0.809767
$$377$$ −1820.33 −0.248679
$$378$$ 667.383 0.0908108
$$379$$ 3984.29 0.539998 0.269999 0.962861i $$-0.412977\pi$$
0.269999 + 0.962861i $$0.412977\pi$$
$$380$$ 2400.58 0.324072
$$381$$ −5627.95 −0.756768
$$382$$ 3537.93 0.473865
$$383$$ 318.475 0.0424890 0.0212445 0.999774i $$-0.493237\pi$$
0.0212445 + 0.999774i $$0.493237\pi$$
$$384$$ −4321.63 −0.574316
$$385$$ 102.821 0.0136110
$$386$$ 14318.6 1.88808
$$387$$ −387.035 −0.0508374
$$388$$ −4490.21 −0.587515
$$389$$ 3885.46 0.506429 0.253214 0.967410i $$-0.418512\pi$$
0.253214 + 0.967410i $$0.418512\pi$$
$$390$$ 1009.67 0.131094
$$391$$ 25800.9 3.33710
$$392$$ 610.975 0.0787216
$$393$$ 1092.61 0.140241
$$394$$ 18150.8 2.32088
$$395$$ −123.813 −0.0157714
$$396$$ −118.156 −0.0149938
$$397$$ 4806.04 0.607578 0.303789 0.952739i $$-0.401748\pi$$
0.303789 + 0.952739i $$0.401748\pi$$
$$398$$ −2067.94 −0.260443
$$399$$ −2256.15 −0.283080
$$400$$ −1994.50 −0.249313
$$401$$ 3618.59 0.450633 0.225316 0.974286i $$-0.427658\pi$$
0.225316 + 0.974286i $$0.427658\pi$$
$$402$$ 3283.87 0.407424
$$403$$ 1797.71 0.222209
$$404$$ −575.911 −0.0709223
$$405$$ 405.000 0.0496904
$$406$$ 2360.42 0.288536
$$407$$ −285.315 −0.0347483
$$408$$ 4582.24 0.556016
$$409$$ −2109.05 −0.254978 −0.127489 0.991840i $$-0.540692\pi$$
−0.127489 + 0.991840i $$0.540692\pi$$
$$410$$ 8670.91 1.04445
$$411$$ 4809.40 0.577202
$$412$$ 3601.94 0.430716
$$413$$ 5324.30 0.634363
$$414$$ −6693.62 −0.794622
$$415$$ −2033.62 −0.240546
$$416$$ −3468.63 −0.408806
$$417$$ 7293.37 0.856494
$$418$$ 1114.49 0.130410
$$419$$ −6905.91 −0.805193 −0.402597 0.915377i $$-0.631892\pi$$
−0.402597 + 0.915377i $$0.631892\pi$$
$$420$$ −469.231 −0.0545146
$$421$$ −9647.54 −1.11685 −0.558423 0.829556i $$-0.688593\pi$$
−0.558423 + 0.829556i $$0.688593\pi$$
$$422$$ 3725.91 0.429797
$$423$$ 4261.45 0.489831
$$424$$ −2290.25 −0.262321
$$425$$ 3062.45 0.349531
$$426$$ −7051.94 −0.802037
$$427$$ 1391.23 0.157673
$$428$$ −3439.74 −0.388473
$$429$$ 168.000 0.0189070
$$430$$ 759.261 0.0851508
$$431$$ −13002.7 −1.45318 −0.726589 0.687073i $$-0.758895\pi$$
−0.726589 + 0.687073i $$0.758895\pi$$
$$432$$ −2154.06 −0.239902
$$433$$ 7356.07 0.816420 0.408210 0.912888i $$-0.366153\pi$$
0.408210 + 0.912888i $$0.366153\pi$$
$$434$$ −2331.08 −0.257824
$$435$$ 1432.41 0.157883
$$436$$ −3489.55 −0.383300
$$437$$ 22628.4 2.47703
$$438$$ −6581.54 −0.717987
$$439$$ 6909.21 0.751159 0.375579 0.926790i $$-0.377444\pi$$
0.375579 + 0.926790i $$0.377444\pi$$
$$440$$ −183.152 −0.0198441
$$441$$ 441.000 0.0476190
$$442$$ 8245.50 0.887327
$$443$$ −14812.6 −1.58864 −0.794318 0.607502i $$-0.792172\pi$$
−0.794318 + 0.607502i $$0.792172\pi$$
$$444$$ 1302.06 0.139173
$$445$$ 1308.75 0.139418
$$446$$ −16511.0 −1.75296
$$447$$ 7024.72 0.743306
$$448$$ 30.0561 0.00316968
$$449$$ −10654.5 −1.11986 −0.559932 0.828538i $$-0.689173\pi$$
−0.559932 + 0.828538i $$0.689173\pi$$
$$450$$ −794.504 −0.0832295
$$451$$ 1442.76 0.150636
$$452$$ −4985.68 −0.518820
$$453$$ −6312.21 −0.654688
$$454$$ −19220.3 −1.98691
$$455$$ 667.179 0.0687425
$$456$$ 4018.81 0.412715
$$457$$ −5855.16 −0.599328 −0.299664 0.954045i $$-0.596875\pi$$
−0.299664 + 0.954045i $$0.596875\pi$$
$$458$$ 1893.76 0.193208
$$459$$ 3307.45 0.336336
$$460$$ 4706.23 0.477019
$$461$$ 3204.74 0.323774 0.161887 0.986809i $$-0.448242\pi$$
0.161887 + 0.986809i $$0.448242\pi$$
$$462$$ −217.844 −0.0219373
$$463$$ 371.658 0.0373054 0.0186527 0.999826i $$-0.494062\pi$$
0.0186527 + 0.999826i $$0.494062\pi$$
$$464$$ −7618.54 −0.762245
$$465$$ −1414.61 −0.141077
$$466$$ 647.927 0.0644091
$$467$$ −19752.3 −1.95723 −0.978614 0.205703i $$-0.934052\pi$$
−0.978614 + 0.205703i $$0.934052\pi$$
$$468$$ −766.681 −0.0757262
$$469$$ 2169.95 0.213643
$$470$$ −8359.84 −0.820449
$$471$$ −1780.40 −0.174175
$$472$$ −9484.01 −0.924866
$$473$$ 126.334 0.0122809
$$474$$ 262.320 0.0254193
$$475$$ 2685.90 0.259447
$$476$$ −3832.00 −0.368990
$$477$$ −1653.09 −0.158679
$$478$$ −2271.28 −0.217335
$$479$$ −20762.0 −1.98046 −0.990232 0.139433i $$-0.955472\pi$$
−0.990232 + 0.139433i $$0.955472\pi$$
$$480$$ 2729.45 0.259545
$$481$$ −1851.34 −0.175496
$$482$$ 20323.8 1.92059
$$483$$ −4423.07 −0.416681
$$484$$ −5909.50 −0.554987
$$485$$ −5023.87 −0.470355
$$486$$ −858.064 −0.0800876
$$487$$ 17647.6 1.64207 0.821035 0.570878i $$-0.193397\pi$$
0.821035 + 0.570878i $$0.193397\pi$$
$$488$$ −2478.15 −0.229878
$$489$$ 6536.12 0.604445
$$490$$ −865.127 −0.0797601
$$491$$ −5637.46 −0.518157 −0.259078 0.965856i $$-0.583419\pi$$
−0.259078 + 0.965856i $$0.583419\pi$$
$$492$$ −6584.16 −0.603327
$$493$$ 11697.9 1.06865
$$494$$ 7231.64 0.658638
$$495$$ −132.198 −0.0120038
$$496$$ 7523.86 0.681112
$$497$$ −4659.85 −0.420569
$$498$$ 4308.58 0.387695
$$499$$ −17474.1 −1.56764 −0.783818 0.620991i $$-0.786730\pi$$
−0.783818 + 0.620991i $$0.786730\pi$$
$$500$$ 558.609 0.0499635
$$501$$ 2398.51 0.213887
$$502$$ 18122.7 1.61127
$$503$$ 7444.81 0.659936 0.329968 0.943992i $$-0.392962\pi$$
0.329968 + 0.943992i $$0.392962\pi$$
$$504$$ −785.539 −0.0694260
$$505$$ −644.358 −0.0567793
$$506$$ 2184.90 0.191958
$$507$$ −5500.89 −0.481860
$$508$$ −8383.53 −0.732203
$$509$$ −3384.48 −0.294724 −0.147362 0.989083i $$-0.547078\pi$$
−0.147362 + 0.989083i $$0.547078\pi$$
$$510$$ −6488.35 −0.563351
$$511$$ −4349.02 −0.376495
$$512$$ −6558.89 −0.566142
$$513$$ 2900.77 0.249653
$$514$$ −17803.0 −1.52774
$$515$$ 4030.04 0.344825
$$516$$ −576.536 −0.0491872
$$517$$ −1391.00 −0.118329
$$518$$ 2400.62 0.203624
$$519$$ −4333.07 −0.366476
$$520$$ −1188.42 −0.100223
$$521$$ 2973.12 0.250009 0.125005 0.992156i $$-0.460105\pi$$
0.125005 + 0.992156i $$0.460105\pi$$
$$522$$ −3034.82 −0.254465
$$523$$ 2689.02 0.224823 0.112412 0.993662i $$-0.464142\pi$$
0.112412 + 0.993662i $$0.464142\pi$$
$$524$$ 1627.57 0.135689
$$525$$ −525.000 −0.0436436
$$526$$ −26755.3 −2.21785
$$527$$ −11552.5 −0.954903
$$528$$ 703.121 0.0579534
$$529$$ 32194.9 2.64608
$$530$$ 3242.94 0.265781
$$531$$ −6845.53 −0.559455
$$532$$ −3360.82 −0.273891
$$533$$ 9361.72 0.760790
$$534$$ −2772.83 −0.224704
$$535$$ −3848.56 −0.311005
$$536$$ −3865.25 −0.311480
$$537$$ 10030.5 0.806046
$$538$$ −3612.69 −0.289506
$$539$$ −143.949 −0.0115034
$$540$$ 603.298 0.0480774
$$541$$ −14429.5 −1.14671 −0.573356 0.819306i $$-0.694359\pi$$
−0.573356 + 0.819306i $$0.694359\pi$$
$$542$$ 7952.03 0.630201
$$543$$ 6755.44 0.533893
$$544$$ 22290.1 1.75677
$$545$$ −3904.28 −0.306864
$$546$$ −1413.54 −0.110795
$$547$$ 13811.2 1.07957 0.539784 0.841804i $$-0.318506\pi$$
0.539784 + 0.841804i $$0.318506\pi$$
$$548$$ 7164.19 0.558466
$$549$$ −1788.72 −0.139054
$$550$$ 259.339 0.0201059
$$551$$ 10259.5 0.793229
$$552$$ 7878.68 0.607498
$$553$$ 173.338 0.0133293
$$554$$ 30476.2 2.33720
$$555$$ 1456.81 0.111420
$$556$$ 10864.4 0.828692
$$557$$ −6033.26 −0.458954 −0.229477 0.973314i $$-0.573702\pi$$
−0.229477 + 0.973314i $$0.573702\pi$$
$$558$$ 2997.10 0.227379
$$559$$ 819.751 0.0620246
$$560$$ 2792.31 0.210708
$$561$$ −1079.60 −0.0812493
$$562$$ 26559.8 1.99352
$$563$$ −6958.47 −0.520896 −0.260448 0.965488i $$-0.583870\pi$$
−0.260448 + 0.965488i $$0.583870\pi$$
$$564$$ 6347.95 0.473931
$$565$$ −5578.23 −0.415359
$$566$$ −52.3202 −0.00388548
$$567$$ −567.000 −0.0419961
$$568$$ 8300.44 0.613166
$$569$$ −13396.4 −0.987009 −0.493505 0.869743i $$-0.664284\pi$$
−0.493505 + 0.869743i $$0.664284\pi$$
$$570$$ −5690.55 −0.418159
$$571$$ −8055.84 −0.590414 −0.295207 0.955433i $$-0.595389\pi$$
−0.295207 + 0.955433i $$0.595389\pi$$
$$572$$ 250.257 0.0182933
$$573$$ −3005.78 −0.219142
$$574$$ −12139.3 −0.882724
$$575$$ 5265.56 0.381894
$$576$$ −38.6435 −0.00279539
$$577$$ −21456.9 −1.54812 −0.774059 0.633114i $$-0.781777\pi$$
−0.774059 + 0.633114i $$0.781777\pi$$
$$578$$ −35638.9 −2.56468
$$579$$ −12164.9 −0.873153
$$580$$ 2133.76 0.152758
$$581$$ 2847.07 0.203298
$$582$$ 10644.0 0.758087
$$583$$ 539.596 0.0383324
$$584$$ 7746.76 0.548910
$$585$$ −857.802 −0.0606252
$$586$$ −24412.1 −1.72091
$$587$$ 20156.3 1.41728 0.708638 0.705572i $$-0.249310\pi$$
0.708638 + 0.705572i $$0.249310\pi$$
$$588$$ 656.924 0.0460733
$$589$$ −10132.0 −0.708797
$$590$$ 13429.1 0.937066
$$591$$ −15420.7 −1.07330
$$592$$ −7748.30 −0.537928
$$593$$ 599.307 0.0415018 0.0207509 0.999785i $$-0.493394\pi$$
0.0207509 + 0.999785i $$0.493394\pi$$
$$594$$ 280.086 0.0193469
$$595$$ −4287.43 −0.295408
$$596$$ 10464.2 0.719177
$$597$$ 1756.89 0.120444
$$598$$ 14177.3 0.969485
$$599$$ −5493.05 −0.374691 −0.187346 0.982294i $$-0.559988\pi$$
−0.187346 + 0.982294i $$0.559988\pi$$
$$600$$ 935.165 0.0636299
$$601$$ 24292.8 1.64879 0.824396 0.566014i $$-0.191515\pi$$
0.824396 + 0.566014i $$0.191515\pi$$
$$602$$ −1062.97 −0.0719655
$$603$$ −2789.93 −0.188416
$$604$$ −9402.82 −0.633436
$$605$$ −6611.85 −0.444314
$$606$$ 1365.19 0.0915131
$$607$$ 3029.50 0.202576 0.101288 0.994857i $$-0.467704\pi$$
0.101288 + 0.994857i $$0.467704\pi$$
$$608$$ 19549.3 1.30400
$$609$$ −2005.38 −0.133435
$$610$$ 3509.01 0.232911
$$611$$ −9025.87 −0.597623
$$612$$ 4926.85 0.325419
$$613$$ −19339.6 −1.27426 −0.637129 0.770757i $$-0.719878\pi$$
−0.637129 + 0.770757i $$0.719878\pi$$
$$614$$ 26992.4 1.77414
$$615$$ −7366.69 −0.483014
$$616$$ 256.412 0.0167713
$$617$$ −5743.91 −0.374783 −0.187391 0.982285i $$-0.560003\pi$$
−0.187391 + 0.982285i $$0.560003\pi$$
$$618$$ −8538.35 −0.555765
$$619$$ −8243.35 −0.535264 −0.267632 0.963521i $$-0.586241\pi$$
−0.267632 + 0.963521i $$0.586241\pi$$
$$620$$ −2107.24 −0.136498
$$621$$ 5686.81 0.367478
$$622$$ −26815.4 −1.72861
$$623$$ −1832.26 −0.117830
$$624$$ 4562.37 0.292694
$$625$$ 625.000 0.0400000
$$626$$ 32231.6 2.05788
$$627$$ −946.856 −0.0603091
$$628$$ −2652.13 −0.168521
$$629$$ 11897.1 0.754162
$$630$$ 1112.31 0.0703418
$$631$$ −4376.56 −0.276114 −0.138057 0.990424i $$-0.544086\pi$$
−0.138057 + 0.990424i $$0.544086\pi$$
$$632$$ −308.762 −0.0194334
$$633$$ −3165.48 −0.198762
$$634$$ 17407.8 1.09046
$$635$$ −9379.92 −0.586190
$$636$$ −2462.49 −0.153528
$$637$$ −934.051 −0.0580980
$$638$$ 990.613 0.0614714
$$639$$ 5991.23 0.370907
$$640$$ −7202.71 −0.444863
$$641$$ 11836.6 0.729357 0.364678 0.931133i $$-0.381179\pi$$
0.364678 + 0.931133i $$0.381179\pi$$
$$642$$ 8153.86 0.501257
$$643$$ 1448.21 0.0888209 0.0444104 0.999013i $$-0.485859\pi$$
0.0444104 + 0.999013i $$0.485859\pi$$
$$644$$ −6588.72 −0.403155
$$645$$ −645.058 −0.0393785
$$646$$ −46472.0 −2.83037
$$647$$ −8732.95 −0.530646 −0.265323 0.964160i $$-0.585479\pi$$
−0.265323 + 0.964160i $$0.585479\pi$$
$$648$$ 1009.98 0.0612279
$$649$$ 2234.49 0.135149
$$650$$ 1682.78 0.101545
$$651$$ 1980.46 0.119232
$$652$$ 9736.37 0.584824
$$653$$ −21978.4 −1.31712 −0.658562 0.752527i $$-0.728835\pi$$
−0.658562 + 0.752527i $$0.728835\pi$$
$$654$$ 8271.91 0.494583
$$655$$ 1821.01 0.108630
$$656$$ 39181.1 2.33196
$$657$$ 5591.59 0.332038
$$658$$ 11703.8 0.693406
$$659$$ −27761.7 −1.64103 −0.820516 0.571623i $$-0.806314\pi$$
−0.820516 + 0.571623i $$0.806314\pi$$
$$660$$ −196.926 −0.0116141
$$661$$ −8573.72 −0.504507 −0.252254 0.967661i $$-0.581172\pi$$
−0.252254 + 0.967661i $$0.581172\pi$$
$$662$$ −4313.86 −0.253267
$$663$$ −7005.27 −0.410350
$$664$$ −5071.39 −0.296398
$$665$$ −3760.25 −0.219273
$$666$$ −3086.51 −0.179579
$$667$$ 20113.2 1.16760
$$668$$ 3572.87 0.206944
$$669$$ 14027.6 0.810668
$$670$$ 5473.11 0.315589
$$671$$ 583.868 0.0335916
$$672$$ −3821.22 −0.219356
$$673$$ 27159.2 1.55559 0.777795 0.628518i $$-0.216338\pi$$
0.777795 + 0.628518i $$0.216338\pi$$
$$674$$ 30879.1 1.76472
$$675$$ 675.000 0.0384900
$$676$$ −8194.26 −0.466219
$$677$$ −1392.30 −0.0790404 −0.0395202 0.999219i $$-0.512583\pi$$
−0.0395202 + 0.999219i $$0.512583\pi$$
$$678$$ 11818.5 0.669448
$$679$$ 7033.42 0.397523
$$680$$ 7637.06 0.430688
$$681$$ 16329.3 0.918857
$$682$$ −978.302 −0.0549283
$$683$$ −8675.09 −0.486007 −0.243004 0.970025i $$-0.578133\pi$$
−0.243004 + 0.970025i $$0.578133\pi$$
$$684$$ 4321.05 0.241549
$$685$$ 8015.66 0.447099
$$686$$ 1211.18 0.0674096
$$687$$ −1608.91 −0.0893504
$$688$$ 3430.86 0.190117
$$689$$ 3501.30 0.193598
$$690$$ −11156.0 −0.615511
$$691$$ −21426.0 −1.17957 −0.589785 0.807561i $$-0.700787\pi$$
−0.589785 + 0.807561i $$0.700787\pi$$
$$692$$ −6454.65 −0.354579
$$693$$ 185.078 0.0101451
$$694$$ −16206.3 −0.886433
$$695$$ 12155.6 0.663438
$$696$$ 3572.11 0.194541
$$697$$ −60160.4 −3.26935
$$698$$ 14088.8 0.763997
$$699$$ −550.470 −0.0297864
$$700$$ −782.052 −0.0422269
$$701$$ 24840.5 1.33839 0.669197 0.743085i $$-0.266638\pi$$
0.669197 + 0.743085i $$0.266638\pi$$
$$702$$ 1817.40 0.0977116
$$703$$ 10434.2 0.559793
$$704$$ 12.6139 0.000675288 0
$$705$$ 7102.41 0.379422
$$706$$ −8532.45 −0.454848
$$707$$ 902.101 0.0479873
$$708$$ −10197.3 −0.541295
$$709$$ 12525.0 0.663450 0.331725 0.943376i $$-0.392369\pi$$
0.331725 + 0.943376i $$0.392369\pi$$
$$710$$ −11753.2 −0.621255
$$711$$ −222.863 −0.0117553
$$712$$ 3263.74 0.171789
$$713$$ −19863.3 −1.04332
$$714$$ 9083.69 0.476118
$$715$$ 280.000 0.0146453
$$716$$ 14941.6 0.779881
$$717$$ 1929.65 0.100508
$$718$$ 9732.66 0.505877
$$719$$ 28085.0 1.45674 0.728369 0.685185i $$-0.240279\pi$$
0.728369 + 0.685185i $$0.240279\pi$$
$$720$$ −3590.11 −0.185827
$$721$$ −5642.05 −0.291430
$$722$$ −16537.9 −0.852460
$$723$$ −17266.8 −0.888189
$$724$$ 10063.1 0.516562
$$725$$ 2387.35 0.122295
$$726$$ 14008.4 0.716115
$$727$$ −14326.2 −0.730851 −0.365426 0.930841i $$-0.619077\pi$$
−0.365426 + 0.930841i $$0.619077\pi$$
$$728$$ 1663.79 0.0847037
$$729$$ 729.000 0.0370370
$$730$$ −10969.2 −0.556150
$$731$$ −5267.89 −0.266539
$$732$$ −2664.53 −0.134541
$$733$$ 6727.85 0.339016 0.169508 0.985529i $$-0.445782\pi$$
0.169508 + 0.985529i $$0.445782\pi$$
$$734$$ −39240.8 −1.97331
$$735$$ 735.000 0.0368856
$$736$$ 38325.5 1.91943
$$737$$ 910.677 0.0455159
$$738$$ 15607.6 0.778490
$$739$$ −3418.51 −0.170165 −0.0850826 0.996374i $$-0.527115\pi$$
−0.0850826 + 0.996374i $$0.527115\pi$$
$$740$$ 2170.10 0.107803
$$741$$ −6143.91 −0.304591
$$742$$ −4540.11 −0.224626
$$743$$ 8095.50 0.399724 0.199862 0.979824i $$-0.435951\pi$$
0.199862 + 0.979824i $$0.435951\pi$$
$$744$$ −3527.72 −0.173834
$$745$$ 11707.9 0.575762
$$746$$ −21508.4 −1.05560
$$747$$ −3660.51 −0.179292
$$748$$ −1608.20 −0.0786119
$$749$$ 5387.99 0.262847
$$750$$ −1324.17 −0.0644693
$$751$$ 13446.8 0.653371 0.326686 0.945133i $$-0.394068\pi$$
0.326686 + 0.945133i $$0.394068\pi$$
$$752$$ −37775.4 −1.83182
$$753$$ −15396.8 −0.745141
$$754$$ 6427.84 0.310462
$$755$$ −10520.4 −0.507119
$$756$$ −844.617 −0.0406328
$$757$$ −2593.24 −0.124508 −0.0622541 0.998060i $$-0.519829\pi$$
−0.0622541 + 0.998060i $$0.519829\pi$$
$$758$$ −14069.0 −0.674156
$$759$$ −1856.26 −0.0887722
$$760$$ 6698.02 0.319688
$$761$$ 27079.4 1.28992 0.644959 0.764217i $$-0.276875\pi$$
0.644959 + 0.764217i $$0.276875\pi$$
$$762$$ 19873.0 0.944782
$$763$$ 5465.99 0.259348
$$764$$ −4477.48 −0.212028
$$765$$ 5512.41 0.260525
$$766$$ −1124.58 −0.0530451
$$767$$ 14499.0 0.682568
$$768$$ 15363.3 0.721842
$$769$$ 2138.72 0.100292 0.0501458 0.998742i $$-0.484031\pi$$
0.0501458 + 0.998742i $$0.484031\pi$$
$$770$$ −363.074 −0.0169926
$$771$$ 15125.2 0.706513
$$772$$ −18121.1 −0.844810
$$773$$ 25864.0 1.20345 0.601724 0.798704i $$-0.294481\pi$$
0.601724 + 0.798704i $$0.294481\pi$$
$$774$$ 1366.67 0.0634676
$$775$$ −2357.69 −0.109278
$$776$$ −12528.4 −0.579566
$$777$$ −2039.53 −0.0941671
$$778$$ −13720.1 −0.632247
$$779$$ −52763.1 −2.42675
$$780$$ −1277.80 −0.0586572
$$781$$ −1955.63 −0.0896006
$$782$$ −91106.2 −4.16618
$$783$$ 2578.34 0.117679
$$784$$ −3909.23 −0.178081
$$785$$ −2967.33 −0.134916
$$786$$ −3858.14 −0.175083
$$787$$ −32371.3 −1.46621 −0.733107 0.680113i $$-0.761931\pi$$
−0.733107 + 0.680113i $$0.761931\pi$$
$$788$$ −22971.0 −1.03846
$$789$$ 22731.0 1.02566
$$790$$ 437.200 0.0196897
$$791$$ 7809.52 0.351043
$$792$$ −329.673 −0.0147909
$$793$$ 3788.57 0.169654
$$794$$ −16970.8 −0.758526
$$795$$ −2755.16 −0.122912
$$796$$ 2617.11 0.116534
$$797$$ 2024.33 0.0899691 0.0449845 0.998988i $$-0.485676\pi$$
0.0449845 + 0.998988i $$0.485676\pi$$
$$798$$ 7966.76 0.353409
$$799$$ 58002.1 2.56817
$$800$$ 4549.08 0.201043
$$801$$ 2355.76 0.103916
$$802$$ −12777.7 −0.562589
$$803$$ −1825.18 −0.0802109
$$804$$ −4155.95 −0.182300
$$805$$ −7371.79 −0.322760
$$806$$ −6347.95 −0.277416
$$807$$ 3069.29 0.133884
$$808$$ −1606.88 −0.0699628
$$809$$ 12391.7 0.538526 0.269263 0.963067i $$-0.413220\pi$$
0.269263 + 0.963067i $$0.413220\pi$$
$$810$$ −1430.11 −0.0620356
$$811$$ 14654.5 0.634511 0.317256 0.948340i $$-0.397239\pi$$
0.317256 + 0.948340i $$0.397239\pi$$
$$812$$ −2987.26 −0.129104
$$813$$ −6755.94 −0.291441
$$814$$ 1007.49 0.0433813
$$815$$ 10893.5 0.468201
$$816$$ −29318.7 −1.25780
$$817$$ −4620.16 −0.197844
$$818$$ 7447.33 0.318325
$$819$$ 1200.92 0.0512376
$$820$$ −10973.6 −0.467335
$$821$$ 23887.9 1.01546 0.507731 0.861516i $$-0.330485\pi$$
0.507731 + 0.861516i $$0.330485\pi$$
$$822$$ −16982.6 −0.720604
$$823$$ −4008.41 −0.169774 −0.0848871 0.996391i $$-0.527053\pi$$
−0.0848871 + 0.996391i $$0.527053\pi$$
$$824$$ 10050.0 0.424889
$$825$$ −220.331 −0.00929810
$$826$$ −18800.8 −0.791966
$$827$$ −45110.4 −1.89679 −0.948394 0.317096i $$-0.897292\pi$$
−0.948394 + 0.317096i $$0.897292\pi$$
$$828$$ 8471.21 0.355549
$$829$$ 16165.4 0.677260 0.338630 0.940920i $$-0.390036\pi$$
0.338630 + 0.940920i $$0.390036\pi$$
$$830$$ 7180.97 0.300307
$$831$$ −25892.2 −1.08085
$$832$$ 81.8481 0.00341054
$$833$$ 6002.41 0.249665
$$834$$ −25753.8 −1.06928
$$835$$ 3997.51 0.165676
$$836$$ −1410.46 −0.0583514
$$837$$ −2546.30 −0.105153
$$838$$ 24385.7 1.00524
$$839$$ −25244.4 −1.03878 −0.519388 0.854538i $$-0.673840\pi$$
−0.519388 + 0.854538i $$0.673840\pi$$
$$840$$ −1309.23 −0.0537771
$$841$$ −15269.9 −0.626096
$$842$$ 34066.7 1.39432
$$843$$ −22564.9 −0.921916
$$844$$ −4715.37 −0.192310
$$845$$ −9168.15 −0.373247
$$846$$ −15047.7 −0.611526
$$847$$ 9256.59 0.375514
$$848$$ 14653.8 0.593412
$$849$$ 44.4506 0.00179687
$$850$$ −10813.9 −0.436370
$$851$$ 20455.8 0.823990
$$852$$ 8924.68 0.358867
$$853$$ −30168.1 −1.21094 −0.605472 0.795867i $$-0.707016\pi$$
−0.605472 + 0.795867i $$0.707016\pi$$
$$854$$ −4912.61 −0.196846
$$855$$ 4834.61 0.193380
$$856$$ −9597.44 −0.383217
$$857$$ −13393.6 −0.533857 −0.266929 0.963716i $$-0.586009\pi$$
−0.266929 + 0.963716i $$0.586009\pi$$
$$858$$ −593.230 −0.0236043
$$859$$ 19060.4 0.757081 0.378541 0.925585i $$-0.376426\pi$$
0.378541 + 0.925585i $$0.376426\pi$$
$$860$$ −960.894 −0.0381002
$$861$$ 10313.4 0.408222
$$862$$ 45914.3 1.81421
$$863$$ −9466.86 −0.373413 −0.186707 0.982416i $$-0.559781\pi$$
−0.186707 + 0.982416i $$0.559781\pi$$
$$864$$ 4913.00 0.193453
$$865$$ −7221.79 −0.283871
$$866$$ −25975.2 −1.01925
$$867$$ 30278.3 1.18605
$$868$$ 2950.13 0.115362
$$869$$ 72.7461 0.00283975
$$870$$ −5058.03 −0.197107
$$871$$ 5909.15 0.229878
$$872$$ −9736.39 −0.378115
$$873$$ −9042.97 −0.350582
$$874$$ −79903.8 −3.09243
$$875$$ −875.000 −0.0338062
$$876$$ 8329.37 0.321259
$$877$$ 37740.6 1.45315 0.726573 0.687090i $$-0.241112\pi$$
0.726573 + 0.687090i $$0.241112\pi$$
$$878$$ −24397.3 −0.937778
$$879$$ 20740.2 0.795845
$$880$$ 1171.87 0.0448905
$$881$$ 25991.5 0.993957 0.496979 0.867763i $$-0.334443\pi$$
0.496979 + 0.867763i $$0.334443\pi$$
$$882$$ −1557.23 −0.0594496
$$883$$ 39420.3 1.50238 0.751189 0.660087i $$-0.229481\pi$$
0.751189 + 0.660087i $$0.229481\pi$$
$$884$$ −10435.2 −0.397030
$$885$$ −11409.2 −0.433352
$$886$$ 52305.0 1.98332
$$887$$ 46005.2 1.74149 0.870745 0.491735i $$-0.163637\pi$$
0.870745 + 0.491735i $$0.163637\pi$$
$$888$$ 3632.95 0.137290
$$889$$ 13131.9 0.495421
$$890$$ −4621.38 −0.174055
$$891$$ −237.957 −0.00894710
$$892$$ 20895.8 0.784353
$$893$$ 50870.2 1.90628
$$894$$ −24805.2 −0.927975
$$895$$ 16717.5 0.624361
$$896$$ 10083.8 0.375978
$$897$$ −12044.8 −0.448345
$$898$$ 37622.6 1.39809
$$899$$ −9005.81 −0.334105
$$900$$ 1005.50 0.0372406
$$901$$ −22500.1 −0.831950
$$902$$ −5094.58 −0.188061
$$903$$ 903.081 0.0332809
$$904$$ −13910.8 −0.511801
$$905$$ 11259.1 0.413551
$$906$$ 22289.2 0.817340
$$907$$ −2838.97 −0.103932 −0.0519661 0.998649i $$-0.516549\pi$$
−0.0519661 + 0.998649i $$0.516549\pi$$
$$908$$ 24324.6 0.889030
$$909$$ −1159.84 −0.0423208
$$910$$ −2355.90 −0.0858211
$$911$$ 39890.9 1.45076 0.725382 0.688347i $$-0.241663\pi$$
0.725382 + 0.688347i $$0.241663\pi$$
$$912$$ −25713.7 −0.933626
$$913$$ 1194.85 0.0433119
$$914$$ 20675.3 0.748226
$$915$$ −2981.21 −0.107711
$$916$$ −2396.67 −0.0864500
$$917$$ −2549.42 −0.0918094
$$918$$ −11679.0 −0.419897
$$919$$ −646.475 −0.0232048 −0.0116024 0.999933i $$-0.503693\pi$$
−0.0116024 + 0.999933i $$0.503693\pi$$
$$920$$ 13131.1 0.470566
$$921$$ −22932.4 −0.820463
$$922$$ −11316.3 −0.404213
$$923$$ −12689.6 −0.452528
$$924$$ 275.696 0.00981574
$$925$$ 2428.02 0.0863056
$$926$$ −1312.37 −0.0465737
$$927$$ 7254.07 0.257017
$$928$$ 17376.4 0.614665
$$929$$ 51188.2 1.80778 0.903892 0.427760i $$-0.140697\pi$$
0.903892 + 0.427760i $$0.140697\pi$$
$$930$$ 4995.17 0.176127
$$931$$ 5264.35 0.185319
$$932$$ −819.993 −0.0288195
$$933$$ 22782.0 0.799409
$$934$$ 69747.8 2.44349
$$935$$ −1799.34 −0.0629355
$$936$$ −2139.16 −0.0747017
$$937$$ 29786.1 1.03849 0.519247 0.854624i $$-0.326212\pi$$
0.519247 + 0.854624i $$0.326212\pi$$
$$938$$ −7662.36 −0.266722
$$939$$ −27383.5 −0.951680
$$940$$ 10579.9 0.367105
$$941$$ −44817.4 −1.55261 −0.776304 0.630358i $$-0.782908\pi$$
−0.776304 + 0.630358i $$0.782908\pi$$
$$942$$ 6286.82 0.217448
$$943$$ −103439. −3.57206
$$944$$ 60682.0 2.09219
$$945$$ −945.000 −0.0325300
$$946$$ −446.103 −0.0153320
$$947$$ 54697.1 1.87689 0.938446 0.345425i $$-0.112265\pi$$
0.938446 + 0.345425i $$0.112265\pi$$
$$948$$ −331.983 −0.0113737
$$949$$ −11843.2 −0.405105
$$950$$ −9484.24 −0.323905
$$951$$ −14789.4 −0.504290
$$952$$ −10691.9 −0.363998
$$953$$ −7577.51 −0.257565 −0.128783 0.991673i $$-0.541107\pi$$
−0.128783 + 0.991673i $$0.541107\pi$$
$$954$$ 5837.29 0.198102
$$955$$ −5009.63 −0.169746
$$956$$ 2874.46 0.0972454
$$957$$ −841.612 −0.0284278
$$958$$ 73313.4 2.47249
$$959$$ −11221.9 −0.377868
$$960$$ −64.4059 −0.00216530
$$961$$ −20897.1 −0.701457
$$962$$ 6537.32 0.219097
$$963$$ −6927.41 −0.231810
$$964$$ −25721.1 −0.859357
$$965$$ −20274.8 −0.676342
$$966$$ 15618.4 0.520202
$$967$$ 50779.0 1.68867 0.844334 0.535817i $$-0.179996\pi$$
0.844334 + 0.535817i $$0.179996\pi$$
$$968$$ −16488.5 −0.547478
$$969$$ 39482.0 1.30892
$$970$$ 17739.9 0.587212
$$971$$ −15313.2 −0.506102 −0.253051 0.967453i $$-0.581434\pi$$
−0.253051 + 0.967453i $$0.581434\pi$$
$$972$$ 1085.94 0.0358348
$$973$$ −17017.9 −0.560707
$$974$$ −62315.9 −2.05003
$$975$$ −1429.67 −0.0469601
$$976$$ 15856.1 0.520021
$$977$$ 46620.4 1.52663 0.763316 0.646025i $$-0.223570\pi$$
0.763316 + 0.646025i $$0.223570\pi$$
$$978$$ −23079.9 −0.754615
$$979$$ −768.957 −0.0251031
$$980$$ 1094.87 0.0356882
$$981$$ −7027.70 −0.228723
$$982$$ 19906.6 0.646889
$$983$$ −2824.37 −0.0916414 −0.0458207 0.998950i $$-0.514590\pi$$
−0.0458207 + 0.998950i $$0.514590\pi$$
$$984$$ −18370.9 −0.595165
$$985$$ −25701.1 −0.831377
$$986$$ −41306.6 −1.33415
$$987$$ −9943.38 −0.320670
$$988$$ −9152.11 −0.294704
$$989$$ −9057.59 −0.291218
$$990$$ 466.810 0.0149860
$$991$$ 16951.4 0.543370 0.271685 0.962386i $$-0.412419\pi$$
0.271685 + 0.962386i $$0.412419\pi$$
$$992$$ −17160.5 −0.549239
$$993$$ 3665.00 0.117125
$$994$$ 16454.5 0.525056
$$995$$ 2928.15 0.0932952
$$996$$ −5452.79 −0.173472
$$997$$ 23847.8 0.757540 0.378770 0.925491i $$-0.376347\pi$$
0.378770 + 0.925491i $$0.376347\pi$$
$$998$$ 61703.4 1.95710
$$999$$ 2622.26 0.0830476
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.f.1.1 2
3.2 odd 2 315.4.a.i.1.2 2
4.3 odd 2 1680.4.a.bg.1.1 2
5.2 odd 4 525.4.d.h.274.2 4
5.3 odd 4 525.4.d.h.274.3 4
5.4 even 2 525.4.a.k.1.2 2
7.6 odd 2 735.4.a.p.1.1 2
15.14 odd 2 1575.4.a.w.1.1 2
21.20 even 2 2205.4.a.z.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
105.4.a.f.1.1 2 1.1 even 1 trivial
315.4.a.i.1.2 2 3.2 odd 2
525.4.a.k.1.2 2 5.4 even 2
525.4.d.h.274.2 4 5.2 odd 4
525.4.d.h.274.3 4 5.3 odd 4
735.4.a.p.1.1 2 7.6 odd 2
1575.4.a.w.1.1 2 15.14 odd 2
1680.4.a.bg.1.1 2 4.3 odd 2
2205.4.a.z.1.2 2 21.20 even 2