# Properties

 Label 363.4.a.q.1.1 Level $363$ Weight $4$ Character 363.1 Self dual yes Analytic conductor $21.418$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [363,4,Mod(1,363)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(363, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("363.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$363 = 3 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 363.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: $$21.4176933321$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 28x^{2} - 2x + 72$$ x^4 - x^3 - 28*x^2 - 2*x + 72 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$5.63177$$ of defining polynomial Character $$\chi$$ $$=$$ 363.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-5.63177 q^{2} +3.00000 q^{3} +23.7168 q^{4} +5.58204 q^{5} -16.8953 q^{6} +16.3245 q^{7} -88.5134 q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q-5.63177 q^{2} +3.00000 q^{3} +23.7168 q^{4} +5.58204 q^{5} -16.8953 q^{6} +16.3245 q^{7} -88.5134 q^{8} +9.00000 q^{9} -31.4368 q^{10} +71.1504 q^{12} +36.6875 q^{13} -91.9358 q^{14} +16.7461 q^{15} +308.752 q^{16} +2.22540 q^{17} -50.6859 q^{18} +89.8470 q^{19} +132.388 q^{20} +48.9735 q^{21} +133.825 q^{23} -265.540 q^{24} -93.8408 q^{25} -206.615 q^{26} +27.0000 q^{27} +387.165 q^{28} +79.6014 q^{29} -94.3103 q^{30} -167.723 q^{31} -1030.71 q^{32} -12.5329 q^{34} +91.1241 q^{35} +213.451 q^{36} +38.2217 q^{37} -505.998 q^{38} +110.062 q^{39} -494.086 q^{40} -189.462 q^{41} -275.807 q^{42} +114.242 q^{43} +50.2384 q^{45} -753.672 q^{46} +162.032 q^{47} +926.257 q^{48} -76.5108 q^{49} +528.489 q^{50} +6.67620 q^{51} +870.110 q^{52} -211.325 q^{53} -152.058 q^{54} -1444.94 q^{56} +269.541 q^{57} -448.296 q^{58} -478.011 q^{59} +397.165 q^{60} +134.260 q^{61} +944.576 q^{62} +146.920 q^{63} +3334.73 q^{64} +204.791 q^{65} +809.611 q^{67} +52.7794 q^{68} +401.475 q^{69} -513.190 q^{70} +190.954 q^{71} -796.621 q^{72} +745.929 q^{73} -215.255 q^{74} -281.522 q^{75} +2130.89 q^{76} -619.846 q^{78} +56.5345 q^{79} +1723.47 q^{80} +81.0000 q^{81} +1067.01 q^{82} -595.935 q^{83} +1161.49 q^{84} +12.4223 q^{85} -643.382 q^{86} +238.804 q^{87} +1434.99 q^{89} -282.931 q^{90} +598.905 q^{91} +3173.90 q^{92} -503.168 q^{93} -912.527 q^{94} +501.530 q^{95} -3092.14 q^{96} -622.609 q^{97} +430.891 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - q^{2} + 12 q^{3} + 25 q^{4} + 14 q^{5} - 3 q^{6} + 20 q^{7} - 75 q^{8} + 36 q^{9}+O(q^{10})$$ 4 * q - q^2 + 12 * q^3 + 25 * q^4 + 14 * q^5 - 3 * q^6 + 20 * q^7 - 75 * q^8 + 36 * q^9 $$4 q - q^{2} + 12 q^{3} + 25 q^{4} + 14 q^{5} - 3 q^{6} + 20 q^{7} - 75 q^{8} + 36 q^{9} + 3 q^{10} + 75 q^{12} + 32 q^{13} + 62 q^{14} + 42 q^{15} + 289 q^{16} + 92 q^{17} - 9 q^{18} - 34 q^{19} + 391 q^{20} + 60 q^{21} - 26 q^{23} - 225 q^{24} + 334 q^{25} - 181 q^{26} + 108 q^{27} + 692 q^{28} + 174 q^{29} + 9 q^{30} + 422 q^{31} - 1271 q^{32} - 477 q^{34} - 82 q^{35} + 225 q^{36} + 518 q^{37} - 798 q^{38} + 96 q^{39} + 423 q^{40} + 428 q^{41} + 186 q^{42} - 550 q^{43} + 126 q^{45} - 2004 q^{46} + 556 q^{47} + 867 q^{48} + 282 q^{49} + 2074 q^{50} + 276 q^{51} + 467 q^{52} + 882 q^{53} - 27 q^{54} - 2112 q^{56} - 102 q^{57} - 225 q^{58} - 158 q^{59} + 1173 q^{60} + 290 q^{61} + 1142 q^{62} + 180 q^{63} + 3097 q^{64} - 1636 q^{65} + 992 q^{67} - 2033 q^{68} - 78 q^{69} + 948 q^{70} - 42 q^{71} - 675 q^{72} + 1274 q^{73} + 1509 q^{74} + 1002 q^{75} + 632 q^{76} - 543 q^{78} + 362 q^{79} + 1423 q^{80} + 324 q^{81} + 2403 q^{82} + 1500 q^{83} + 2076 q^{84} - 2388 q^{85} - 1272 q^{86} + 522 q^{87} + 1428 q^{89} + 27 q^{90} + 1750 q^{91} + 896 q^{92} + 1266 q^{93} - 1728 q^{94} - 4452 q^{95} - 3813 q^{96} + 1052 q^{97} - 615 q^{98}+O(q^{100})$$ 4 * q - q^2 + 12 * q^3 + 25 * q^4 + 14 * q^5 - 3 * q^6 + 20 * q^7 - 75 * q^8 + 36 * q^9 + 3 * q^10 + 75 * q^12 + 32 * q^13 + 62 * q^14 + 42 * q^15 + 289 * q^16 + 92 * q^17 - 9 * q^18 - 34 * q^19 + 391 * q^20 + 60 * q^21 - 26 * q^23 - 225 * q^24 + 334 * q^25 - 181 * q^26 + 108 * q^27 + 692 * q^28 + 174 * q^29 + 9 * q^30 + 422 * q^31 - 1271 * q^32 - 477 * q^34 - 82 * q^35 + 225 * q^36 + 518 * q^37 - 798 * q^38 + 96 * q^39 + 423 * q^40 + 428 * q^41 + 186 * q^42 - 550 * q^43 + 126 * q^45 - 2004 * q^46 + 556 * q^47 + 867 * q^48 + 282 * q^49 + 2074 * q^50 + 276 * q^51 + 467 * q^52 + 882 * q^53 - 27 * q^54 - 2112 * q^56 - 102 * q^57 - 225 * q^58 - 158 * q^59 + 1173 * q^60 + 290 * q^61 + 1142 * q^62 + 180 * q^63 + 3097 * q^64 - 1636 * q^65 + 992 * q^67 - 2033 * q^68 - 78 * q^69 + 948 * q^70 - 42 * q^71 - 675 * q^72 + 1274 * q^73 + 1509 * q^74 + 1002 * q^75 + 632 * q^76 - 543 * q^78 + 362 * q^79 + 1423 * q^80 + 324 * q^81 + 2403 * q^82 + 1500 * q^83 + 2076 * q^84 - 2388 * q^85 - 1272 * q^86 + 522 * q^87 + 1428 * q^89 + 27 * q^90 + 1750 * q^91 + 896 * q^92 + 1266 * q^93 - 1728 * q^94 - 4452 * q^95 - 3813 * q^96 + 1052 * q^97 - 615 * q^98

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −5.63177 −1.99113 −0.995565 0.0940734i $$-0.970011\pi$$
−0.995565 + 0.0940734i $$0.970011\pi$$
$$3$$ 3.00000 0.577350
$$4$$ 23.7168 2.96460
$$5$$ 5.58204 0.499273 0.249637 0.968340i $$-0.419689\pi$$
0.249637 + 0.968340i $$0.419689\pi$$
$$6$$ −16.8953 −1.14958
$$7$$ 16.3245 0.881440 0.440720 0.897645i $$-0.354723\pi$$
0.440720 + 0.897645i $$0.354723\pi$$
$$8$$ −88.5134 −3.91178
$$9$$ 9.00000 0.333333
$$10$$ −31.4368 −0.994118
$$11$$ 0 0
$$12$$ 71.1504 1.71161
$$13$$ 36.6875 0.782713 0.391357 0.920239i $$-0.372006\pi$$
0.391357 + 0.920239i $$0.372006\pi$$
$$14$$ −91.9358 −1.75506
$$15$$ 16.7461 0.288256
$$16$$ 308.752 4.82426
$$17$$ 2.22540 0.0317493 0.0158747 0.999874i $$-0.494947\pi$$
0.0158747 + 0.999874i $$0.494947\pi$$
$$18$$ −50.6859 −0.663710
$$19$$ 89.8470 1.08486 0.542430 0.840101i $$-0.317505\pi$$
0.542430 + 0.840101i $$0.317505\pi$$
$$20$$ 132.388 1.48015
$$21$$ 48.9735 0.508900
$$22$$ 0 0
$$23$$ 133.825 1.21324 0.606619 0.794993i $$-0.292525\pi$$
0.606619 + 0.794993i $$0.292525\pi$$
$$24$$ −265.540 −2.25847
$$25$$ −93.8408 −0.750726
$$26$$ −206.615 −1.55848
$$27$$ 27.0000 0.192450
$$28$$ 387.165 2.61312
$$29$$ 79.6014 0.509711 0.254855 0.966979i $$-0.417972\pi$$
0.254855 + 0.966979i $$0.417972\pi$$
$$30$$ −94.3103 −0.573954
$$31$$ −167.723 −0.971739 −0.485869 0.874031i $$-0.661497\pi$$
−0.485869 + 0.874031i $$0.661497\pi$$
$$32$$ −1030.71 −5.69395
$$33$$ 0 0
$$34$$ −12.5329 −0.0632171
$$35$$ 91.1241 0.440079
$$36$$ 213.451 0.988200
$$37$$ 38.2217 0.169827 0.0849135 0.996388i $$-0.472939\pi$$
0.0849135 + 0.996388i $$0.472939\pi$$
$$38$$ −505.998 −2.16010
$$39$$ 110.062 0.451900
$$40$$ −494.086 −1.95305
$$41$$ −189.462 −0.721684 −0.360842 0.932627i $$-0.617511\pi$$
−0.360842 + 0.932627i $$0.617511\pi$$
$$42$$ −275.807 −1.01329
$$43$$ 114.242 0.405155 0.202578 0.979266i $$-0.435068\pi$$
0.202578 + 0.979266i $$0.435068\pi$$
$$44$$ 0 0
$$45$$ 50.2384 0.166424
$$46$$ −753.672 −2.41571
$$47$$ 162.032 0.502868 0.251434 0.967874i $$-0.419098\pi$$
0.251434 + 0.967874i $$0.419098\pi$$
$$48$$ 926.257 2.78529
$$49$$ −76.5108 −0.223064
$$50$$ 528.489 1.49479
$$51$$ 6.67620 0.0183305
$$52$$ 870.110 2.32043
$$53$$ −211.325 −0.547693 −0.273847 0.961773i $$-0.588296\pi$$
−0.273847 + 0.961773i $$0.588296\pi$$
$$54$$ −152.058 −0.383193
$$55$$ 0 0
$$56$$ −1444.94 −3.44800
$$57$$ 269.541 0.626344
$$58$$ −448.296 −1.01490
$$59$$ −478.011 −1.05477 −0.527387 0.849625i $$-0.676828\pi$$
−0.527387 + 0.849625i $$0.676828\pi$$
$$60$$ 397.165 0.854563
$$61$$ 134.260 0.281808 0.140904 0.990023i $$-0.454999\pi$$
0.140904 + 0.990023i $$0.454999\pi$$
$$62$$ 944.576 1.93486
$$63$$ 146.920 0.293813
$$64$$ 3334.73 6.51314
$$65$$ 204.791 0.390788
$$66$$ 0 0
$$67$$ 809.611 1.47626 0.738132 0.674657i $$-0.235708\pi$$
0.738132 + 0.674657i $$0.235708\pi$$
$$68$$ 52.7794 0.0941241
$$69$$ 401.475 0.700463
$$70$$ −513.190 −0.876256
$$71$$ 190.954 0.319184 0.159592 0.987183i $$-0.448982\pi$$
0.159592 + 0.987183i $$0.448982\pi$$
$$72$$ −796.621 −1.30393
$$73$$ 745.929 1.19595 0.597975 0.801514i $$-0.295972\pi$$
0.597975 + 0.801514i $$0.295972\pi$$
$$74$$ −215.255 −0.338148
$$75$$ −281.522 −0.433432
$$76$$ 2130.89 3.21617
$$77$$ 0 0
$$78$$ −619.846 −0.899792
$$79$$ 56.5345 0.0805143 0.0402572 0.999189i $$-0.487182\pi$$
0.0402572 + 0.999189i $$0.487182\pi$$
$$80$$ 1723.47 2.40862
$$81$$ 81.0000 0.111111
$$82$$ 1067.01 1.43697
$$83$$ −595.935 −0.788101 −0.394050 0.919089i $$-0.628926\pi$$
−0.394050 + 0.919089i $$0.628926\pi$$
$$84$$ 1161.49 1.50868
$$85$$ 12.4223 0.0158516
$$86$$ −643.382 −0.806717
$$87$$ 238.804 0.294282
$$88$$ 0 0
$$89$$ 1434.99 1.70908 0.854540 0.519386i $$-0.173839\pi$$
0.854540 + 0.519386i $$0.173839\pi$$
$$90$$ −282.931 −0.331373
$$91$$ 598.905 0.689915
$$92$$ 3173.90 3.59677
$$93$$ −503.168 −0.561034
$$94$$ −912.527 −1.00128
$$95$$ 501.530 0.541641
$$96$$ −3092.14 −3.28740
$$97$$ −622.609 −0.651715 −0.325857 0.945419i $$-0.605653\pi$$
−0.325857 + 0.945419i $$0.605653\pi$$
$$98$$ 430.891 0.444149
$$99$$ 0 0
$$100$$ −2225.60 −2.22560
$$101$$ −726.647 −0.715881 −0.357941 0.933744i $$-0.616521\pi$$
−0.357941 + 0.933744i $$0.616521\pi$$
$$102$$ −37.5988 −0.0364984
$$103$$ −1689.51 −1.61624 −0.808121 0.589017i $$-0.799515\pi$$
−0.808121 + 0.589017i $$0.799515\pi$$
$$104$$ −3247.33 −3.06180
$$105$$ 273.372 0.254080
$$106$$ 1190.13 1.09053
$$107$$ 1031.70 0.932133 0.466067 0.884750i $$-0.345671\pi$$
0.466067 + 0.884750i $$0.345671\pi$$
$$108$$ 640.354 0.570538
$$109$$ −1479.61 −1.30020 −0.650098 0.759851i $$-0.725272\pi$$
−0.650098 + 0.759851i $$0.725272\pi$$
$$110$$ 0 0
$$111$$ 114.665 0.0980497
$$112$$ 5040.23 4.25229
$$113$$ 1903.03 1.58427 0.792134 0.610347i $$-0.208970\pi$$
0.792134 + 0.610347i $$0.208970\pi$$
$$114$$ −1517.99 −1.24713
$$115$$ 747.018 0.605737
$$116$$ 1887.89 1.51109
$$117$$ 330.187 0.260904
$$118$$ 2692.05 2.10019
$$119$$ 36.3285 0.0279851
$$120$$ −1482.26 −1.12759
$$121$$ 0 0
$$122$$ −756.123 −0.561116
$$123$$ −568.387 −0.416665
$$124$$ −3977.85 −2.88082
$$125$$ −1221.58 −0.874091
$$126$$ −827.422 −0.585021
$$127$$ 2650.07 1.85162 0.925811 0.377987i $$-0.123384\pi$$
0.925811 + 0.377987i $$0.123384\pi$$
$$128$$ −10534.7 −7.27456
$$129$$ 342.725 0.233916
$$130$$ −1153.34 −0.778110
$$131$$ 2339.06 1.56003 0.780016 0.625759i $$-0.215211\pi$$
0.780016 + 0.625759i $$0.215211\pi$$
$$132$$ 0 0
$$133$$ 1466.71 0.956238
$$134$$ −4559.54 −2.93943
$$135$$ 150.715 0.0960852
$$136$$ −196.978 −0.124196
$$137$$ 1201.25 0.749125 0.374562 0.927202i $$-0.377793\pi$$
0.374562 + 0.927202i $$0.377793\pi$$
$$138$$ −2261.02 −1.39471
$$139$$ −1840.90 −1.12333 −0.561665 0.827365i $$-0.689839\pi$$
−0.561665 + 0.827365i $$0.689839\pi$$
$$140$$ 2161.17 1.30466
$$141$$ 486.096 0.290331
$$142$$ −1075.41 −0.635537
$$143$$ 0 0
$$144$$ 2778.77 1.60809
$$145$$ 444.339 0.254485
$$146$$ −4200.90 −2.38129
$$147$$ −229.532 −0.128786
$$148$$ 906.496 0.503469
$$149$$ 828.112 0.455313 0.227656 0.973742i $$-0.426894\pi$$
0.227656 + 0.973742i $$0.426894\pi$$
$$150$$ 1585.47 0.863020
$$151$$ −883.706 −0.476258 −0.238129 0.971233i $$-0.576534\pi$$
−0.238129 + 0.971233i $$0.576534\pi$$
$$152$$ −7952.67 −4.24373
$$153$$ 20.0286 0.0105831
$$154$$ 0 0
$$155$$ −936.236 −0.485163
$$156$$ 2610.33 1.33970
$$157$$ 1235.42 0.628010 0.314005 0.949421i $$-0.398329\pi$$
0.314005 + 0.949421i $$0.398329\pi$$
$$158$$ −318.389 −0.160315
$$159$$ −633.975 −0.316211
$$160$$ −5753.50 −2.84284
$$161$$ 2184.63 1.06940
$$162$$ −456.173 −0.221237
$$163$$ −1148.84 −0.552050 −0.276025 0.961150i $$-0.589017\pi$$
−0.276025 + 0.961150i $$0.589017\pi$$
$$164$$ −4493.44 −2.13951
$$165$$ 0 0
$$166$$ 3356.17 1.56921
$$167$$ 2316.46 1.07337 0.536686 0.843782i $$-0.319676\pi$$
0.536686 + 0.843782i $$0.319676\pi$$
$$168$$ −4334.81 −1.99070
$$169$$ −851.029 −0.387360
$$170$$ −69.9594 −0.0315626
$$171$$ 808.623 0.361620
$$172$$ 2709.44 1.20112
$$173$$ −1949.29 −0.856659 −0.428330 0.903623i $$-0.640898\pi$$
−0.428330 + 0.903623i $$0.640898\pi$$
$$174$$ −1344.89 −0.585953
$$175$$ −1531.90 −0.661720
$$176$$ 0 0
$$177$$ −1434.03 −0.608974
$$178$$ −8081.50 −3.40300
$$179$$ 1889.42 0.788947 0.394474 0.918907i $$-0.370927\pi$$
0.394474 + 0.918907i $$0.370927\pi$$
$$180$$ 1191.49 0.493382
$$181$$ 1180.63 0.484839 0.242419 0.970172i $$-0.422059\pi$$
0.242419 + 0.970172i $$0.422059\pi$$
$$182$$ −3372.89 −1.37371
$$183$$ 402.781 0.162702
$$184$$ −11845.3 −4.74591
$$185$$ 213.355 0.0847901
$$186$$ 2833.73 1.11709
$$187$$ 0 0
$$188$$ 3842.88 1.49080
$$189$$ 440.761 0.169633
$$190$$ −2824.50 −1.07848
$$191$$ 2773.75 1.05079 0.525396 0.850858i $$-0.323917\pi$$
0.525396 + 0.850858i $$0.323917\pi$$
$$192$$ 10004.2 3.76036
$$193$$ 2035.09 0.759010 0.379505 0.925190i $$-0.376094\pi$$
0.379505 + 0.925190i $$0.376094\pi$$
$$194$$ 3506.39 1.29765
$$195$$ 614.373 0.225622
$$196$$ −1814.59 −0.661294
$$197$$ −48.0363 −0.0173728 −0.00868640 0.999962i $$-0.502765\pi$$
−0.00868640 + 0.999962i $$0.502765\pi$$
$$198$$ 0 0
$$199$$ 3307.72 1.17828 0.589141 0.808030i $$-0.299466\pi$$
0.589141 + 0.808030i $$0.299466\pi$$
$$200$$ 8306.17 2.93667
$$201$$ 2428.83 0.852321
$$202$$ 4092.30 1.42541
$$203$$ 1299.45 0.449279
$$204$$ 158.338 0.0543426
$$205$$ −1057.59 −0.360318
$$206$$ 9514.95 3.21815
$$207$$ 1204.43 0.404413
$$208$$ 11327.3 3.77601
$$209$$ 0 0
$$210$$ −1539.57 −0.505906
$$211$$ −1334.95 −0.435554 −0.217777 0.975999i $$-0.569881\pi$$
−0.217777 + 0.975999i $$0.569881\pi$$
$$212$$ −5011.96 −1.62369
$$213$$ 572.862 0.184281
$$214$$ −5810.30 −1.85600
$$215$$ 637.701 0.202283
$$216$$ −2389.86 −0.752822
$$217$$ −2737.99 −0.856529
$$218$$ 8332.84 2.58886
$$219$$ 2237.79 0.690482
$$220$$ 0 0
$$221$$ 81.6443 0.0248506
$$222$$ −645.766 −0.195230
$$223$$ −3251.94 −0.976528 −0.488264 0.872696i $$-0.662370\pi$$
−0.488264 + 0.872696i $$0.662370\pi$$
$$224$$ −16825.9 −5.01887
$$225$$ −844.567 −0.250242
$$226$$ −10717.4 −3.15449
$$227$$ −1315.55 −0.384652 −0.192326 0.981331i $$-0.561603\pi$$
−0.192326 + 0.981331i $$0.561603\pi$$
$$228$$ 6392.66 1.85686
$$229$$ 2562.22 0.739372 0.369686 0.929157i $$-0.379465\pi$$
0.369686 + 0.929157i $$0.379465\pi$$
$$230$$ −4207.03 −1.20610
$$231$$ 0 0
$$232$$ −7045.79 −1.99387
$$233$$ 5033.07 1.41514 0.707570 0.706643i $$-0.249791\pi$$
0.707570 + 0.706643i $$0.249791\pi$$
$$234$$ −1859.54 −0.519495
$$235$$ 904.470 0.251069
$$236$$ −11336.9 −3.12699
$$237$$ 169.604 0.0464850
$$238$$ −204.594 −0.0557221
$$239$$ −4246.79 −1.14938 −0.574690 0.818371i $$-0.694877\pi$$
−0.574690 + 0.818371i $$0.694877\pi$$
$$240$$ 5170.41 1.39062
$$241$$ −5203.06 −1.39070 −0.695350 0.718671i $$-0.744751\pi$$
−0.695350 + 0.718671i $$0.744751\pi$$
$$242$$ 0 0
$$243$$ 243.000 0.0641500
$$244$$ 3184.22 0.835447
$$245$$ −427.087 −0.111370
$$246$$ 3201.03 0.829634
$$247$$ 3296.26 0.849134
$$248$$ 14845.7 3.80122
$$249$$ −1787.80 −0.455010
$$250$$ 6879.65 1.74043
$$251$$ −232.010 −0.0583440 −0.0291720 0.999574i $$-0.509287\pi$$
−0.0291720 + 0.999574i $$0.509287\pi$$
$$252$$ 3484.48 0.871039
$$253$$ 0 0
$$254$$ −14924.6 −3.68682
$$255$$ 37.2669 0.00915192
$$256$$ 32651.1 7.97146
$$257$$ −1974.93 −0.479349 −0.239675 0.970853i $$-0.577041\pi$$
−0.239675 + 0.970853i $$0.577041\pi$$
$$258$$ −1930.15 −0.465758
$$259$$ 623.949 0.149692
$$260$$ 4856.99 1.15853
$$261$$ 716.412 0.169904
$$262$$ −13173.0 −3.10623
$$263$$ 6288.51 1.47440 0.737198 0.675677i $$-0.236149\pi$$
0.737198 + 0.675677i $$0.236149\pi$$
$$264$$ 0 0
$$265$$ −1179.63 −0.273449
$$266$$ −8260.16 −1.90400
$$267$$ 4304.96 0.986738
$$268$$ 19201.4 4.37653
$$269$$ −6900.18 −1.56398 −0.781991 0.623290i $$-0.785796\pi$$
−0.781991 + 0.623290i $$0.785796\pi$$
$$270$$ −848.793 −0.191318
$$271$$ −5503.35 −1.23360 −0.616798 0.787121i $$-0.711571\pi$$
−0.616798 + 0.787121i $$0.711571\pi$$
$$272$$ 687.098 0.153167
$$273$$ 1796.71 0.398323
$$274$$ −6765.18 −1.49160
$$275$$ 0 0
$$276$$ 9521.71 2.07659
$$277$$ −181.153 −0.0392939 −0.0196470 0.999807i $$-0.506254\pi$$
−0.0196470 + 0.999807i $$0.506254\pi$$
$$278$$ 10367.5 2.23669
$$279$$ −1509.50 −0.323913
$$280$$ −8065.70 −1.72149
$$281$$ −4829.86 −1.02536 −0.512678 0.858581i $$-0.671347\pi$$
−0.512678 + 0.858581i $$0.671347\pi$$
$$282$$ −2737.58 −0.578087
$$283$$ −3157.70 −0.663272 −0.331636 0.943408i $$-0.607600\pi$$
−0.331636 + 0.943408i $$0.607600\pi$$
$$284$$ 4528.82 0.946253
$$285$$ 1504.59 0.312717
$$286$$ 0 0
$$287$$ −3092.88 −0.636121
$$288$$ −9276.43 −1.89798
$$289$$ −4908.05 −0.998992
$$290$$ −2502.41 −0.506713
$$291$$ −1867.83 −0.376268
$$292$$ 17691.1 3.54552
$$293$$ −5930.76 −1.18252 −0.591260 0.806481i $$-0.701369\pi$$
−0.591260 + 0.806481i $$0.701369\pi$$
$$294$$ 1292.67 0.256429
$$295$$ −2668.28 −0.526621
$$296$$ −3383.13 −0.664326
$$297$$ 0 0
$$298$$ −4663.74 −0.906587
$$299$$ 4909.71 0.949617
$$300$$ −6676.81 −1.28495
$$301$$ 1864.94 0.357120
$$302$$ 4976.83 0.948293
$$303$$ −2179.94 −0.413314
$$304$$ 27740.5 5.23364
$$305$$ 749.447 0.140699
$$306$$ −112.796 −0.0210724
$$307$$ −2344.18 −0.435797 −0.217898 0.975971i $$-0.569920\pi$$
−0.217898 + 0.975971i $$0.569920\pi$$
$$308$$ 0 0
$$309$$ −5068.54 −0.933137
$$310$$ 5272.66 0.966023
$$311$$ −2943.21 −0.536637 −0.268318 0.963330i $$-0.586468\pi$$
−0.268318 + 0.963330i $$0.586468\pi$$
$$312$$ −9742.00 −1.76773
$$313$$ 7534.35 1.36060 0.680298 0.732935i $$-0.261850\pi$$
0.680298 + 0.732935i $$0.261850\pi$$
$$314$$ −6957.63 −1.25045
$$315$$ 820.117 0.146693
$$316$$ 1340.82 0.238693
$$317$$ 7918.10 1.40292 0.701458 0.712710i $$-0.252533\pi$$
0.701458 + 0.712710i $$0.252533\pi$$
$$318$$ 3570.40 0.629617
$$319$$ 0 0
$$320$$ 18614.6 3.25184
$$321$$ 3095.10 0.538167
$$322$$ −12303.3 −2.12931
$$323$$ 199.946 0.0344436
$$324$$ 1921.06 0.329400
$$325$$ −3442.78 −0.587604
$$326$$ 6470.00 1.09920
$$327$$ −4438.84 −0.750668
$$328$$ 16770.0 2.82307
$$329$$ 2645.09 0.443248
$$330$$ 0 0
$$331$$ −4941.26 −0.820533 −0.410267 0.911966i $$-0.634564\pi$$
−0.410267 + 0.911966i $$0.634564\pi$$
$$332$$ −14133.7 −2.33640
$$333$$ 343.995 0.0566090
$$334$$ −13045.8 −2.13722
$$335$$ 4519.28 0.737059
$$336$$ 15120.7 2.45506
$$337$$ −10324.7 −1.66891 −0.834457 0.551073i $$-0.814218\pi$$
−0.834457 + 0.551073i $$0.814218\pi$$
$$338$$ 4792.80 0.771284
$$339$$ 5709.10 0.914678
$$340$$ 294.617 0.0469937
$$341$$ 0 0
$$342$$ −4553.98 −0.720032
$$343$$ −6848.30 −1.07806
$$344$$ −10111.9 −1.58488
$$345$$ 2241.05 0.349723
$$346$$ 10978.0 1.70572
$$347$$ −3806.42 −0.588875 −0.294437 0.955671i $$-0.595132\pi$$
−0.294437 + 0.955671i $$0.595132\pi$$
$$348$$ 5663.67 0.872427
$$349$$ 337.780 0.0518079 0.0259039 0.999664i $$-0.491754\pi$$
0.0259039 + 0.999664i $$0.491754\pi$$
$$350$$ 8627.32 1.31757
$$351$$ 990.562 0.150633
$$352$$ 0 0
$$353$$ 5488.35 0.827523 0.413761 0.910385i $$-0.364215\pi$$
0.413761 + 0.910385i $$0.364215\pi$$
$$354$$ 8076.14 1.21255
$$355$$ 1065.91 0.159360
$$356$$ 34033.3 5.06674
$$357$$ 108.986 0.0161572
$$358$$ −10640.8 −1.57090
$$359$$ −9157.77 −1.34632 −0.673159 0.739497i $$-0.735063\pi$$
−0.673159 + 0.739497i $$0.735063\pi$$
$$360$$ −4446.77 −0.651015
$$361$$ 1213.49 0.176920
$$362$$ −6649.06 −0.965377
$$363$$ 0 0
$$364$$ 14204.1 2.04532
$$365$$ 4163.81 0.597106
$$366$$ −2268.37 −0.323960
$$367$$ 2339.69 0.332782 0.166391 0.986060i $$-0.446789\pi$$
0.166391 + 0.986060i $$0.446789\pi$$
$$368$$ 41318.8 5.85297
$$369$$ −1705.16 −0.240561
$$370$$ −1201.57 −0.168828
$$371$$ −3449.78 −0.482759
$$372$$ −11933.5 −1.66324
$$373$$ −10249.9 −1.42284 −0.711421 0.702766i $$-0.751948\pi$$
−0.711421 + 0.702766i $$0.751948\pi$$
$$374$$ 0 0
$$375$$ −3664.74 −0.504657
$$376$$ −14342.0 −1.96711
$$377$$ 2920.37 0.398957
$$378$$ −2482.27 −0.337762
$$379$$ −1256.76 −0.170331 −0.0851654 0.996367i $$-0.527142\pi$$
−0.0851654 + 0.996367i $$0.527142\pi$$
$$380$$ 11894.7 1.60575
$$381$$ 7950.22 1.06903
$$382$$ −15621.1 −2.09226
$$383$$ −7948.33 −1.06042 −0.530210 0.847866i $$-0.677887\pi$$
−0.530210 + 0.847866i $$0.677887\pi$$
$$384$$ −31604.1 −4.19997
$$385$$ 0 0
$$386$$ −11461.1 −1.51129
$$387$$ 1028.17 0.135052
$$388$$ −14766.3 −1.93207
$$389$$ 952.983 0.124211 0.0621056 0.998070i $$-0.480218\pi$$
0.0621056 + 0.998070i $$0.480218\pi$$
$$390$$ −3460.01 −0.449242
$$391$$ 297.814 0.0385195
$$392$$ 6772.23 0.872575
$$393$$ 7017.17 0.900685
$$394$$ 270.529 0.0345915
$$395$$ 315.578 0.0401987
$$396$$ 0 0
$$397$$ 13876.1 1.75421 0.877105 0.480299i $$-0.159472\pi$$
0.877105 + 0.480299i $$0.159472\pi$$
$$398$$ −18628.3 −2.34612
$$399$$ 4400.12 0.552084
$$400$$ −28973.6 −3.62170
$$401$$ −13450.7 −1.67505 −0.837526 0.546397i $$-0.815999\pi$$
−0.837526 + 0.546397i $$0.815999\pi$$
$$402$$ −13678.6 −1.69708
$$403$$ −6153.32 −0.760593
$$404$$ −17233.7 −2.12230
$$405$$ 452.146 0.0554748
$$406$$ −7318.21 −0.894574
$$407$$ 0 0
$$408$$ −590.933 −0.0717048
$$409$$ 11241.4 1.35905 0.679525 0.733653i $$-0.262186\pi$$
0.679525 + 0.733653i $$0.262186\pi$$
$$410$$ 5956.09 0.717440
$$411$$ 3603.76 0.432507
$$412$$ −40069.9 −4.79151
$$413$$ −7803.29 −0.929721
$$414$$ −6783.05 −0.805238
$$415$$ −3326.54 −0.393478
$$416$$ −37814.3 −4.45673
$$417$$ −5522.69 −0.648554
$$418$$ 0 0
$$419$$ −13757.6 −1.60407 −0.802033 0.597280i $$-0.796248\pi$$
−0.802033 + 0.597280i $$0.796248\pi$$
$$420$$ 6483.52 0.753246
$$421$$ −7423.26 −0.859353 −0.429676 0.902983i $$-0.641372\pi$$
−0.429676 + 0.902983i $$0.641372\pi$$
$$422$$ 7518.15 0.867246
$$423$$ 1458.29 0.167623
$$424$$ 18705.1 2.14245
$$425$$ −208.833 −0.0238351
$$426$$ −3226.23 −0.366928
$$427$$ 2191.73 0.248397
$$428$$ 24468.6 2.76340
$$429$$ 0 0
$$430$$ −3591.39 −0.402772
$$431$$ −3919.09 −0.437995 −0.218997 0.975725i $$-0.570279\pi$$
−0.218997 + 0.975725i $$0.570279\pi$$
$$432$$ 8336.32 0.928429
$$433$$ 2196.59 0.243791 0.121896 0.992543i $$-0.461103\pi$$
0.121896 + 0.992543i $$0.461103\pi$$
$$434$$ 15419.7 1.70546
$$435$$ 1333.02 0.146927
$$436$$ −35091.7 −3.85456
$$437$$ 12023.8 1.31619
$$438$$ −12602.7 −1.37484
$$439$$ −7985.51 −0.868173 −0.434086 0.900871i $$-0.642929\pi$$
−0.434086 + 0.900871i $$0.642929\pi$$
$$440$$ 0 0
$$441$$ −688.597 −0.0743545
$$442$$ −459.802 −0.0494809
$$443$$ 2056.68 0.220577 0.110289 0.993900i $$-0.464822\pi$$
0.110289 + 0.993900i $$0.464822\pi$$
$$444$$ 2719.49 0.290678
$$445$$ 8010.15 0.853298
$$446$$ 18314.2 1.94440
$$447$$ 2484.34 0.262875
$$448$$ 54437.7 5.74094
$$449$$ −2956.70 −0.310769 −0.155384 0.987854i $$-0.549662\pi$$
−0.155384 + 0.987854i $$0.549662\pi$$
$$450$$ 4756.40 0.498265
$$451$$ 0 0
$$452$$ 45133.9 4.69672
$$453$$ −2651.12 −0.274968
$$454$$ 7408.87 0.765893
$$455$$ 3343.11 0.344456
$$456$$ −23858.0 −2.45012
$$457$$ 5264.75 0.538894 0.269447 0.963015i $$-0.413159\pi$$
0.269447 + 0.963015i $$0.413159\pi$$
$$458$$ −14429.8 −1.47219
$$459$$ 60.0858 0.00611016
$$460$$ 17716.9 1.79577
$$461$$ 6127.96 0.619105 0.309552 0.950882i $$-0.399821\pi$$
0.309552 + 0.950882i $$0.399821\pi$$
$$462$$ 0 0
$$463$$ −1362.20 −0.136732 −0.0683658 0.997660i $$-0.521778\pi$$
−0.0683658 + 0.997660i $$0.521778\pi$$
$$464$$ 24577.1 2.45897
$$465$$ −2808.71 −0.280109
$$466$$ −28345.1 −2.81773
$$467$$ −10269.1 −1.01756 −0.508778 0.860898i $$-0.669903\pi$$
−0.508778 + 0.860898i $$0.669903\pi$$
$$468$$ 7830.99 0.773478
$$469$$ 13216.5 1.30124
$$470$$ −5093.76 −0.499910
$$471$$ 3706.27 0.362582
$$472$$ 42310.4 4.12604
$$473$$ 0 0
$$474$$ −955.168 −0.0925577
$$475$$ −8431.32 −0.814432
$$476$$ 861.597 0.0829648
$$477$$ −1901.93 −0.182564
$$478$$ 23916.9 2.28856
$$479$$ 1487.98 0.141936 0.0709680 0.997479i $$-0.477391\pi$$
0.0709680 + 0.997479i $$0.477391\pi$$
$$480$$ −17260.5 −1.64131
$$481$$ 1402.26 0.132926
$$482$$ 29302.4 2.76906
$$483$$ 6553.88 0.617416
$$484$$ 0 0
$$485$$ −3475.43 −0.325384
$$486$$ −1368.52 −0.127731
$$487$$ −1712.51 −0.159346 −0.0796728 0.996821i $$-0.525388\pi$$
−0.0796728 + 0.996821i $$0.525388\pi$$
$$488$$ −11883.8 −1.10237
$$489$$ −3446.52 −0.318726
$$490$$ 2405.25 0.221752
$$491$$ 11924.4 1.09601 0.548005 0.836475i $$-0.315387\pi$$
0.548005 + 0.836475i $$0.315387\pi$$
$$492$$ −13480.3 −1.23524
$$493$$ 177.145 0.0161830
$$494$$ −18563.8 −1.69074
$$495$$ 0 0
$$496$$ −51784.8 −4.68792
$$497$$ 3117.23 0.281342
$$498$$ 10068.5 0.905985
$$499$$ 3853.81 0.345732 0.172866 0.984945i $$-0.444697\pi$$
0.172866 + 0.984945i $$0.444697\pi$$
$$500$$ −28972.0 −2.59133
$$501$$ 6949.38 0.619712
$$502$$ 1306.63 0.116171
$$503$$ 5342.09 0.473543 0.236771 0.971565i $$-0.423911\pi$$
0.236771 + 0.971565i $$0.423911\pi$$
$$504$$ −13004.4 −1.14933
$$505$$ −4056.17 −0.357420
$$506$$ 0 0
$$507$$ −2553.09 −0.223642
$$508$$ 62851.3 5.48932
$$509$$ −2310.50 −0.201201 −0.100600 0.994927i $$-0.532076\pi$$
−0.100600 + 0.994927i $$0.532076\pi$$
$$510$$ −209.878 −0.0182227
$$511$$ 12176.9 1.05416
$$512$$ −99605.9 −8.59766
$$513$$ 2425.87 0.208781
$$514$$ 11122.3 0.954447
$$515$$ −9430.95 −0.806946
$$516$$ 8128.33 0.693469
$$517$$ 0 0
$$518$$ −3513.94 −0.298057
$$519$$ −5847.88 −0.494592
$$520$$ −18126.8 −1.52868
$$521$$ −6312.10 −0.530783 −0.265391 0.964141i $$-0.585501\pi$$
−0.265391 + 0.964141i $$0.585501\pi$$
$$522$$ −4034.67 −0.338300
$$523$$ 21924.3 1.83305 0.916523 0.399982i $$-0.130984\pi$$
0.916523 + 0.399982i $$0.130984\pi$$
$$524$$ 55474.9 4.62487
$$525$$ −4595.71 −0.382044
$$526$$ −35415.4 −2.93571
$$527$$ −373.250 −0.0308521
$$528$$ 0 0
$$529$$ 5742.16 0.471946
$$530$$ 6643.38 0.544472
$$531$$ −4302.10 −0.351592
$$532$$ 34785.6 2.83486
$$533$$ −6950.90 −0.564872
$$534$$ −24244.5 −1.96472
$$535$$ 5759.00 0.465389
$$536$$ −71661.4 −5.77481
$$537$$ 5668.25 0.455499
$$538$$ 38860.2 3.11409
$$539$$ 0 0
$$540$$ 3574.48 0.284854
$$541$$ −12039.3 −0.956765 −0.478382 0.878152i $$-0.658777\pi$$
−0.478382 + 0.878152i $$0.658777\pi$$
$$542$$ 30993.6 2.45625
$$543$$ 3541.90 0.279922
$$544$$ −2293.75 −0.180779
$$545$$ −8259.27 −0.649153
$$546$$ −10118.7 −0.793112
$$547$$ −2517.28 −0.196766 −0.0983829 0.995149i $$-0.531367\pi$$
−0.0983829 + 0.995149i $$0.531367\pi$$
$$548$$ 28489.9 2.22086
$$549$$ 1208.34 0.0939359
$$550$$ 0 0
$$551$$ 7151.95 0.552964
$$552$$ −35535.9 −2.74006
$$553$$ 922.898 0.0709686
$$554$$ 1020.21 0.0782394
$$555$$ 640.065 0.0489536
$$556$$ −43660.2 −3.33022
$$557$$ −6507.93 −0.495062 −0.247531 0.968880i $$-0.579619\pi$$
−0.247531 + 0.968880i $$0.579619\pi$$
$$558$$ 8501.18 0.644953
$$559$$ 4191.23 0.317120
$$560$$ 28134.8 2.12306
$$561$$ 0 0
$$562$$ 27200.6 2.04162
$$563$$ −627.184 −0.0469497 −0.0234748 0.999724i $$-0.507473\pi$$
−0.0234748 + 0.999724i $$0.507473\pi$$
$$564$$ 11528.6 0.860715
$$565$$ 10622.8 0.790983
$$566$$ 17783.4 1.32066
$$567$$ 1322.28 0.0979378
$$568$$ −16902.0 −1.24858
$$569$$ −2336.62 −0.172155 −0.0860775 0.996288i $$-0.527433\pi$$
−0.0860775 + 0.996288i $$0.527433\pi$$
$$570$$ −8473.51 −0.622660
$$571$$ 12354.5 0.905462 0.452731 0.891647i $$-0.350450\pi$$
0.452731 + 0.891647i $$0.350450\pi$$
$$572$$ 0 0
$$573$$ 8321.24 0.606675
$$574$$ 17418.4 1.26660
$$575$$ −12558.3 −0.910809
$$576$$ 30012.5 2.17105
$$577$$ −2391.41 −0.172540 −0.0862699 0.996272i $$-0.527495\pi$$
−0.0862699 + 0.996272i $$0.527495\pi$$
$$578$$ 27641.0 1.98912
$$579$$ 6105.26 0.438214
$$580$$ 10538.3 0.754446
$$581$$ −9728.34 −0.694663
$$582$$ 10519.2 0.749198
$$583$$ 0 0
$$584$$ −66024.7 −4.67829
$$585$$ 1843.12 0.130263
$$586$$ 33400.6 2.35455
$$587$$ −24107.9 −1.69513 −0.847563 0.530694i $$-0.821931\pi$$
−0.847563 + 0.530694i $$0.821931\pi$$
$$588$$ −5443.78 −0.381799
$$589$$ −15069.4 −1.05420
$$590$$ 15027.1 1.04857
$$591$$ −144.109 −0.0100302
$$592$$ 11801.0 0.819289
$$593$$ −607.390 −0.0420616 −0.0210308 0.999779i $$-0.506695\pi$$
−0.0210308 + 0.999779i $$0.506695\pi$$
$$594$$ 0 0
$$595$$ 202.788 0.0139722
$$596$$ 19640.2 1.34982
$$597$$ 9923.17 0.680282
$$598$$ −27650.3 −1.89081
$$599$$ 10687.8 0.729035 0.364517 0.931197i $$-0.381234\pi$$
0.364517 + 0.931197i $$0.381234\pi$$
$$600$$ 24918.5 1.69549
$$601$$ 21602.2 1.46618 0.733088 0.680134i $$-0.238078\pi$$
0.733088 + 0.680134i $$0.238078\pi$$
$$602$$ −10502.9 −0.711072
$$603$$ 7286.49 0.492088
$$604$$ −20958.7 −1.41192
$$605$$ 0 0
$$606$$ 12276.9 0.822963
$$607$$ −19316.5 −1.29165 −0.645824 0.763486i $$-0.723486\pi$$
−0.645824 + 0.763486i $$0.723486\pi$$
$$608$$ −92606.7 −6.17713
$$609$$ 3898.36 0.259392
$$610$$ −4220.71 −0.280150
$$611$$ 5944.55 0.393602
$$612$$ 475.014 0.0313747
$$613$$ 25466.8 1.67797 0.838983 0.544158i $$-0.183151\pi$$
0.838983 + 0.544158i $$0.183151\pi$$
$$614$$ 13201.9 0.867728
$$615$$ −3172.76 −0.208030
$$616$$ 0 0
$$617$$ 19548.6 1.27552 0.637760 0.770235i $$-0.279861\pi$$
0.637760 + 0.770235i $$0.279861\pi$$
$$618$$ 28544.9 1.85800
$$619$$ −7285.93 −0.473096 −0.236548 0.971620i $$-0.576016\pi$$
−0.236548 + 0.971620i $$0.576016\pi$$
$$620$$ −22204.5 −1.43831
$$621$$ 3613.28 0.233488
$$622$$ 16575.5 1.06851
$$623$$ 23425.4 1.50645
$$624$$ 33982.0 2.18008
$$625$$ 4911.19 0.314316
$$626$$ −42431.7 −2.70912
$$627$$ 0 0
$$628$$ 29300.3 1.86180
$$629$$ 85.0585 0.00539190
$$630$$ −4618.71 −0.292085
$$631$$ −21199.2 −1.33744 −0.668722 0.743513i $$-0.733158\pi$$
−0.668722 + 0.743513i $$0.733158\pi$$
$$632$$ −5004.06 −0.314954
$$633$$ −4004.86 −0.251467
$$634$$ −44592.9 −2.79339
$$635$$ 14792.8 0.924465
$$636$$ −15035.9 −0.937439
$$637$$ −2806.99 −0.174595
$$638$$ 0 0
$$639$$ 1718.59 0.106395
$$640$$ −58805.1 −3.63199
$$641$$ 10801.4 0.665568 0.332784 0.943003i $$-0.392012\pi$$
0.332784 + 0.943003i $$0.392012\pi$$
$$642$$ −17430.9 −1.07156
$$643$$ −23096.4 −1.41654 −0.708269 0.705943i $$-0.750523\pi$$
−0.708269 + 0.705943i $$0.750523\pi$$
$$644$$ 51812.4 3.17033
$$645$$ 1913.10 0.116788
$$646$$ −1126.05 −0.0685816
$$647$$ −23521.2 −1.42923 −0.714616 0.699517i $$-0.753399\pi$$
−0.714616 + 0.699517i $$0.753399\pi$$
$$648$$ −7169.59 −0.434642
$$649$$ 0 0
$$650$$ 19388.9 1.17000
$$651$$ −8213.97 −0.494517
$$652$$ −27246.8 −1.63661
$$653$$ 2419.38 0.144989 0.0724945 0.997369i $$-0.476904\pi$$
0.0724945 + 0.997369i $$0.476904\pi$$
$$654$$ 24998.5 1.49468
$$655$$ 13056.7 0.778882
$$656$$ −58497.0 −3.48159
$$657$$ 6713.36 0.398650
$$658$$ −14896.5 −0.882565
$$659$$ 17627.3 1.04197 0.520987 0.853564i $$-0.325564\pi$$
0.520987 + 0.853564i $$0.325564\pi$$
$$660$$ 0 0
$$661$$ 750.997 0.0441912 0.0220956 0.999756i $$-0.492966\pi$$
0.0220956 + 0.999756i $$0.492966\pi$$
$$662$$ 27828.1 1.63379
$$663$$ 244.933 0.0143475
$$664$$ 52748.2 3.08287
$$665$$ 8187.23 0.477424
$$666$$ −1937.30 −0.112716
$$667$$ 10652.7 0.618400
$$668$$ 54939.1 3.18212
$$669$$ −9755.81 −0.563799
$$670$$ −25451.5 −1.46758
$$671$$ 0 0
$$672$$ −50477.7 −2.89765
$$673$$ −18959.8 −1.08595 −0.542977 0.839747i $$-0.682703\pi$$
−0.542977 + 0.839747i $$0.682703\pi$$
$$674$$ 58146.5 3.32302
$$675$$ −2533.70 −0.144477
$$676$$ −20183.7 −1.14837
$$677$$ −13858.9 −0.786767 −0.393383 0.919374i $$-0.628696\pi$$
−0.393383 + 0.919374i $$0.628696\pi$$
$$678$$ −32152.3 −1.82124
$$679$$ −10163.8 −0.574448
$$680$$ −1099.54 −0.0620079
$$681$$ −3946.65 −0.222079
$$682$$ 0 0
$$683$$ −15562.5 −0.871862 −0.435931 0.899980i $$-0.643581\pi$$
−0.435931 + 0.899980i $$0.643581\pi$$
$$684$$ 19178.0 1.07206
$$685$$ 6705.45 0.374018
$$686$$ 38568.0 2.14655
$$687$$ 7686.65 0.426876
$$688$$ 35272.4 1.95457
$$689$$ −7752.99 −0.428687
$$690$$ −12621.1 −0.696343
$$691$$ −18362.3 −1.01091 −0.505453 0.862854i $$-0.668675\pi$$
−0.505453 + 0.862854i $$0.668675\pi$$
$$692$$ −46231.0 −2.53965
$$693$$ 0 0
$$694$$ 21436.9 1.17253
$$695$$ −10276.0 −0.560848
$$696$$ −21137.4 −1.15116
$$697$$ −421.630 −0.0229130
$$698$$ −1902.30 −0.103156
$$699$$ 15099.2 0.817031
$$700$$ −36331.9 −1.96174
$$701$$ 7705.40 0.415163 0.207581 0.978218i $$-0.433441\pi$$
0.207581 + 0.978218i $$0.433441\pi$$
$$702$$ −5578.61 −0.299931
$$703$$ 3434.10 0.184238
$$704$$ 0 0
$$705$$ 2713.41 0.144955
$$706$$ −30909.1 −1.64771
$$707$$ −11862.1 −0.631007
$$708$$ −34010.7 −1.80537
$$709$$ −15368.2 −0.814053 −0.407027 0.913416i $$-0.633434\pi$$
−0.407027 + 0.913416i $$0.633434\pi$$
$$710$$ −6002.98 −0.317307
$$711$$ 508.811 0.0268381
$$712$$ −127015. −6.68554
$$713$$ −22445.5 −1.17895
$$714$$ −613.782 −0.0321712
$$715$$ 0 0
$$716$$ 44810.9 2.33891
$$717$$ −12740.4 −0.663595
$$718$$ 51574.4 2.68070
$$719$$ −9397.60 −0.487443 −0.243721 0.969845i $$-0.578368\pi$$
−0.243721 + 0.969845i $$0.578368\pi$$
$$720$$ 15511.2 0.802874
$$721$$ −27580.5 −1.42462
$$722$$ −6834.11 −0.352270
$$723$$ −15609.2 −0.802921
$$724$$ 28000.9 1.43735
$$725$$ −7469.86 −0.382653
$$726$$ 0 0
$$727$$ −28278.0 −1.44260 −0.721302 0.692620i $$-0.756456\pi$$
−0.721302 + 0.692620i $$0.756456\pi$$
$$728$$ −53011.1 −2.69879
$$729$$ 729.000 0.0370370
$$730$$ −23449.6 −1.18892
$$731$$ 254.233 0.0128634
$$732$$ 9552.67 0.482346
$$733$$ 33183.9 1.67214 0.836069 0.548625i $$-0.184848\pi$$
0.836069 + 0.548625i $$0.184848\pi$$
$$734$$ −13176.6 −0.662613
$$735$$ −1281.26 −0.0642993
$$736$$ −137936. −6.90811
$$737$$ 0 0
$$738$$ 9603.08 0.478989
$$739$$ 13324.8 0.663277 0.331638 0.943407i $$-0.392399\pi$$
0.331638 + 0.943407i $$0.392399\pi$$
$$740$$ 5060.10 0.251369
$$741$$ 9888.78 0.490248
$$742$$ 19428.3 0.961236
$$743$$ 1953.28 0.0964455 0.0482227 0.998837i $$-0.484644\pi$$
0.0482227 + 0.998837i $$0.484644\pi$$
$$744$$ 44537.1 2.19464
$$745$$ 4622.56 0.227326
$$746$$ 57725.1 2.83306
$$747$$ −5363.41 −0.262700
$$748$$ 0 0
$$749$$ 16842.0 0.821619
$$750$$ 20638.9 1.00484
$$751$$ 2848.62 0.138412 0.0692062 0.997602i $$-0.477953\pi$$
0.0692062 + 0.997602i $$0.477953\pi$$
$$752$$ 50027.8 2.42596
$$753$$ −696.031 −0.0336849
$$754$$ −16446.9 −0.794376
$$755$$ −4932.89 −0.237783
$$756$$ 10453.5 0.502895
$$757$$ −17909.1 −0.859864 −0.429932 0.902861i $$-0.641463\pi$$
−0.429932 + 0.902861i $$0.641463\pi$$
$$758$$ 7077.78 0.339151
$$759$$ 0 0
$$760$$ −44392.1 −2.11878
$$761$$ 15984.0 0.761392 0.380696 0.924700i $$-0.375684\pi$$
0.380696 + 0.924700i $$0.375684\pi$$
$$762$$ −44773.8 −2.12859
$$763$$ −24154.0 −1.14604
$$764$$ 65784.4 3.11518
$$765$$ 111.801 0.00528387
$$766$$ 44763.2 2.11143
$$767$$ −17537.0 −0.825586
$$768$$ 97953.3 4.60232
$$769$$ 12888.9 0.604402 0.302201 0.953244i $$-0.402279\pi$$
0.302201 + 0.953244i $$0.402279\pi$$
$$770$$ 0 0
$$771$$ −5924.79 −0.276752
$$772$$ 48265.8 2.25016
$$773$$ −20981.4 −0.976258 −0.488129 0.872772i $$-0.662320\pi$$
−0.488129 + 0.872772i $$0.662320\pi$$
$$774$$ −5790.44 −0.268906
$$775$$ 15739.2 0.729510
$$776$$ 55109.2 2.54936
$$777$$ 1871.85 0.0864249
$$778$$ −5366.98 −0.247321
$$779$$ −17022.6 −0.782926
$$780$$ 14571.0 0.668878
$$781$$ 0 0
$$782$$ −1677.22 −0.0766973
$$783$$ 2149.24 0.0980938
$$784$$ −23622.9 −1.07612
$$785$$ 6896.20 0.313549
$$786$$ −39519.1 −1.79338
$$787$$ 5493.86 0.248837 0.124419 0.992230i $$-0.460293\pi$$
0.124419 + 0.992230i $$0.460293\pi$$
$$788$$ −1139.27 −0.0515034
$$789$$ 18865.5 0.851242
$$790$$ −1777.26 −0.0800408
$$791$$ 31066.1 1.39644
$$792$$ 0 0
$$793$$ 4925.67 0.220575
$$794$$ −78146.9 −3.49286
$$795$$ −3538.88 −0.157876
$$796$$ 78448.6 3.49314
$$797$$ 15558.3 0.691474 0.345737 0.938331i $$-0.387629\pi$$
0.345737 + 0.938331i $$0.387629\pi$$
$$798$$ −24780.5 −1.09927
$$799$$ 360.586 0.0159657
$$800$$ 96723.1 4.27460
$$801$$ 12914.9 0.569693
$$802$$ 75751.2 3.33525
$$803$$ 0 0
$$804$$ 57604.1 2.52679
$$805$$ 12194.7 0.533921
$$806$$ 34654.1 1.51444
$$807$$ −20700.5 −0.902965
$$808$$ 64318.0 2.80037
$$809$$ −8185.89 −0.355748 −0.177874 0.984053i $$-0.556922\pi$$
−0.177874 + 0.984053i $$0.556922\pi$$
$$810$$ −2546.38 −0.110458
$$811$$ 10532.1 0.456021 0.228011 0.973659i $$-0.426778\pi$$
0.228011 + 0.973659i $$0.426778\pi$$
$$812$$ 30818.9 1.33193
$$813$$ −16510.1 −0.712218
$$814$$ 0 0
$$815$$ −6412.88 −0.275624
$$816$$ 2061.29 0.0884310
$$817$$ 10264.3 0.439536
$$818$$ −63308.9 −2.70604
$$819$$ 5390.14 0.229972
$$820$$ −25082.6 −1.06820
$$821$$ −20996.0 −0.892527 −0.446263 0.894902i $$-0.647246\pi$$
−0.446263 + 0.894902i $$0.647246\pi$$
$$822$$ −20295.6 −0.861178
$$823$$ 27292.2 1.15595 0.577974 0.816055i $$-0.303843\pi$$
0.577974 + 0.816055i $$0.303843\pi$$
$$824$$ 149545. 6.32237
$$825$$ 0 0
$$826$$ 43946.3 1.85120
$$827$$ −27817.4 −1.16966 −0.584829 0.811157i $$-0.698838\pi$$
−0.584829 + 0.811157i $$0.698838\pi$$
$$828$$ 28565.1 1.19892
$$829$$ 29670.5 1.24306 0.621531 0.783389i $$-0.286511\pi$$
0.621531 + 0.783389i $$0.286511\pi$$
$$830$$ 18734.3 0.783465
$$831$$ −543.459 −0.0226864
$$832$$ 122343. 5.09792
$$833$$ −170.267 −0.00708212
$$834$$ 31102.5 1.29136
$$835$$ 12930.6 0.535906
$$836$$ 0 0
$$837$$ −4528.51 −0.187011
$$838$$ 77479.7 3.19390
$$839$$ −47358.6 −1.94875 −0.974374 0.224932i $$-0.927784\pi$$
−0.974374 + 0.224932i $$0.927784\pi$$
$$840$$ −24197.1 −0.993904
$$841$$ −18052.6 −0.740195
$$842$$ 41806.1 1.71108
$$843$$ −14489.6 −0.591990
$$844$$ −31660.8 −1.29125
$$845$$ −4750.48 −0.193398
$$846$$ −8212.74 −0.333759
$$847$$ 0 0
$$848$$ −65247.1 −2.64221
$$849$$ −9473.10 −0.382940
$$850$$ 1176.10 0.0474587
$$851$$ 5115.02 0.206041
$$852$$ 13586.5 0.546320
$$853$$ 10456.2 0.419709 0.209855 0.977733i $$-0.432701\pi$$
0.209855 + 0.977733i $$0.432701\pi$$
$$854$$ −12343.3 −0.494590
$$855$$ 4513.77 0.180547
$$856$$ −91319.3 −3.64630
$$857$$ −33631.2 −1.34051 −0.670256 0.742130i $$-0.733816\pi$$
−0.670256 + 0.742130i $$0.733816\pi$$
$$858$$ 0 0
$$859$$ −41703.9 −1.65648 −0.828241 0.560372i $$-0.810658\pi$$
−0.828241 + 0.560372i $$0.810658\pi$$
$$860$$ 15124.2 0.599689
$$861$$ −9278.64 −0.367265
$$862$$ 22071.4 0.872105
$$863$$ −18699.3 −0.737579 −0.368790 0.929513i $$-0.620228\pi$$
−0.368790 + 0.929513i $$0.620228\pi$$
$$864$$ −27829.3 −1.09580
$$865$$ −10881.0 −0.427707
$$866$$ −12370.7 −0.485420
$$867$$ −14724.1 −0.576768
$$868$$ −64936.4 −2.53927
$$869$$ 0 0
$$870$$ −7507.23 −0.292551
$$871$$ 29702.6 1.15549
$$872$$ 130966. 5.08607
$$873$$ −5603.48 −0.217238
$$874$$ −67715.2 −2.62071
$$875$$ −19941.7 −0.770459
$$876$$ 53073.2 2.04700
$$877$$ 24095.9 0.927777 0.463889 0.885894i $$-0.346454\pi$$
0.463889 + 0.885894i $$0.346454\pi$$
$$878$$ 44972.6 1.72864
$$879$$ −17792.3 −0.682728
$$880$$ 0 0
$$881$$ 37466.9 1.43280 0.716398 0.697692i $$-0.245790\pi$$
0.716398 + 0.697692i $$0.245790\pi$$
$$882$$ 3878.02 0.148050
$$883$$ 28314.9 1.07913 0.539565 0.841944i $$-0.318589\pi$$
0.539565 + 0.841944i $$0.318589\pi$$
$$884$$ 1936.34 0.0736722
$$885$$ −8004.83 −0.304045
$$886$$ −11582.7 −0.439198
$$887$$ 3094.24 0.117130 0.0585651 0.998284i $$-0.481347\pi$$
0.0585651 + 0.998284i $$0.481347\pi$$
$$888$$ −10149.4 −0.383549
$$889$$ 43261.1 1.63209
$$890$$ −45111.3 −1.69903
$$891$$ 0 0
$$892$$ −77125.6 −2.89502
$$893$$ 14558.1 0.545541
$$894$$ −13991.2 −0.523418
$$895$$ 10546.8 0.393900
$$896$$ −171973. −6.41209
$$897$$ 14729.1 0.548262
$$898$$ 16651.4 0.618781
$$899$$ −13351.0 −0.495305
$$900$$ −20030.4 −0.741868
$$901$$ −470.283 −0.0173889
$$902$$ 0 0
$$903$$ 5594.81 0.206183
$$904$$ −168444. −6.19730
$$905$$ 6590.35 0.242067
$$906$$ 14930.5 0.547497
$$907$$ −3604.15 −0.131945 −0.0659723 0.997821i $$-0.521015\pi$$
−0.0659723 + 0.997821i $$0.521015\pi$$
$$908$$ −31200.6 −1.14034
$$909$$ −6539.82 −0.238627
$$910$$ −18827.6 −0.685857
$$911$$ 20320.4 0.739016 0.369508 0.929228i $$-0.379526\pi$$
0.369508 + 0.929228i $$0.379526\pi$$
$$912$$ 83221.5 3.02164
$$913$$ 0 0
$$914$$ −29649.8 −1.07301
$$915$$ 2248.34 0.0812326
$$916$$ 60767.6 2.19194
$$917$$ 38183.9 1.37507
$$918$$ −338.389 −0.0121661
$$919$$ −5615.52 −0.201566 −0.100783 0.994908i $$-0.532135\pi$$
−0.100783 + 0.994908i $$0.532135\pi$$
$$920$$ −66121.1 −2.36951
$$921$$ −7032.55 −0.251607
$$922$$ −34511.2 −1.23272
$$923$$ 7005.62 0.249830
$$924$$ 0 0
$$925$$ −3586.75 −0.127494
$$926$$ 7671.58 0.272250
$$927$$ −15205.6 −0.538747
$$928$$ −82046.3 −2.90227
$$929$$ −32968.4 −1.16433 −0.582163 0.813072i $$-0.697793\pi$$
−0.582163 + 0.813072i $$0.697793\pi$$
$$930$$ 15818.0 0.557734
$$931$$ −6874.27 −0.241993
$$932$$ 119368. 4.19532
$$933$$ −8829.63 −0.309827
$$934$$ 57833.4 2.02609
$$935$$ 0 0
$$936$$ −29226.0 −1.02060
$$937$$ 9150.95 0.319049 0.159524 0.987194i $$-0.449004\pi$$
0.159524 + 0.987194i $$0.449004\pi$$
$$938$$ −74432.2 −2.59093
$$939$$ 22603.0 0.785541
$$940$$ 21451.1 0.744318
$$941$$ 5093.14 0.176442 0.0882209 0.996101i $$-0.471882\pi$$
0.0882209 + 0.996101i $$0.471882\pi$$
$$942$$ −20872.9 −0.721948
$$943$$ −25354.8 −0.875575
$$944$$ −147587. −5.08850
$$945$$ 2460.35 0.0846933
$$946$$ 0 0
$$947$$ −3803.68 −0.130520 −0.0652602 0.997868i $$-0.520788\pi$$
−0.0652602 + 0.997868i $$0.520788\pi$$
$$948$$ 4022.46 0.137809
$$949$$ 27366.3 0.936087
$$950$$ 47483.2 1.62164
$$951$$ 23754.3 0.809974
$$952$$ −3215.56 −0.109472
$$953$$ 21263.6 0.722765 0.361383 0.932418i $$-0.382305\pi$$
0.361383 + 0.932418i $$0.382305\pi$$
$$954$$ 10711.2 0.363510
$$955$$ 15483.2 0.524632
$$956$$ −100720. −3.40745
$$957$$ 0 0
$$958$$ −8379.94 −0.282613
$$959$$ 19609.9 0.660308
$$960$$ 55843.8 1.87745
$$961$$ −1660.08 −0.0557241
$$962$$ −7897.18 −0.264673
$$963$$ 9285.30 0.310711
$$964$$ −123400. −4.12287
$$965$$ 11360.0 0.378953
$$966$$ −36909.9 −1.22936
$$967$$ 3365.02 0.111905 0.0559523 0.998433i $$-0.482181\pi$$
0.0559523 + 0.998433i $$0.482181\pi$$
$$968$$ 0 0
$$969$$ 599.837 0.0198860
$$970$$ 19572.8 0.647882
$$971$$ −18906.7 −0.624867 −0.312433 0.949940i $$-0.601144\pi$$
−0.312433 + 0.949940i $$0.601144\pi$$
$$972$$ 5763.18 0.190179
$$973$$ −30051.7 −0.990147
$$974$$ 9644.46 0.317278
$$975$$ −10328.3 −0.339253
$$976$$ 41453.2 1.35951
$$977$$ 7122.77 0.233242 0.116621 0.993176i $$-0.462794\pi$$
0.116621 + 0.993176i $$0.462794\pi$$
$$978$$ 19410.0 0.634625
$$979$$ 0 0
$$980$$ −10129.1 −0.330167
$$981$$ −13316.5 −0.433399
$$982$$ −67155.5 −2.18230
$$983$$ −10427.0 −0.338320 −0.169160 0.985589i $$-0.554105\pi$$
−0.169160 + 0.985589i $$0.554105\pi$$
$$984$$ 50309.9 1.62990
$$985$$ −268.141 −0.00867378
$$986$$ −997.639 −0.0322224
$$987$$ 7935.27 0.255909
$$988$$ 78176.8 2.51734
$$989$$ 15288.4 0.491549
$$990$$ 0 0
$$991$$ −12245.8 −0.392532 −0.196266 0.980551i $$-0.562882\pi$$
−0.196266 + 0.980551i $$0.562882\pi$$
$$992$$ 172874. 5.53303
$$993$$ −14823.8 −0.473735
$$994$$ −17555.5 −0.560188
$$995$$ 18463.9 0.588285
$$996$$ −42401.0 −1.34892
$$997$$ −46890.4 −1.48950 −0.744751 0.667342i $$-0.767432\pi$$
−0.744751 + 0.667342i $$0.767432\pi$$
$$998$$ −21703.8 −0.688397
$$999$$ 1031.98 0.0326832
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 363.4.a.q.1.1 4
3.2 odd 2 1089.4.a.bf.1.4 4
11.10 odd 2 363.4.a.s.1.4 yes 4
33.32 even 2 1089.4.a.ba.1.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
363.4.a.q.1.1 4 1.1 even 1 trivial
363.4.a.s.1.4 yes 4 11.10 odd 2
1089.4.a.ba.1.1 4 33.32 even 2
1089.4.a.bf.1.4 4 3.2 odd 2