Properties

 Label 525.4.a.i.1.1 Level $525$ Weight $4$ Character 525.1 Self dual yes Analytic conductor $30.976$ 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] = [525,4,Mod(1,525)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(525, 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("525.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$525 = 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 525.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: $$30.9760027530$$ 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: no (minimal twist has level 105) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.1 Root $$3.70156$$ of defining polynomial Character $$\chi$$ $$=$$ 525.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-4.70156 q^{2} -3.00000 q^{3} +14.1047 q^{4} +14.1047 q^{6} -7.00000 q^{7} -28.7016 q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q-4.70156 q^{2} -3.00000 q^{3} +14.1047 q^{4} +14.1047 q^{6} -7.00000 q^{7} -28.7016 q^{8} +9.00000 q^{9} +24.5969 q^{11} -42.3141 q^{12} +35.0156 q^{13} +32.9109 q^{14} +22.1047 q^{16} +18.4187 q^{17} -42.3141 q^{18} -67.4031 q^{19} +21.0000 q^{21} -115.644 q^{22} +145.675 q^{23} +86.1047 q^{24} -164.628 q^{26} -27.0000 q^{27} -98.7328 q^{28} +214.419 q^{29} -88.6594 q^{31} +125.686 q^{32} -73.7906 q^{33} -86.5969 q^{34} +126.942 q^{36} -162.125 q^{37} +316.900 q^{38} -105.047 q^{39} -337.769 q^{41} -98.7328 q^{42} -122.156 q^{43} +346.931 q^{44} -684.900 q^{46} -354.219 q^{47} -66.3141 q^{48} +49.0000 q^{49} -55.2562 q^{51} +493.884 q^{52} -676.691 q^{53} +126.942 q^{54} +200.911 q^{56} +202.209 q^{57} -1008.10 q^{58} +501.319 q^{59} -708.931 q^{61} +416.837 q^{62} -63.0000 q^{63} -767.758 q^{64} +346.931 q^{66} +907.956 q^{67} +259.791 q^{68} -437.025 q^{69} +430.334 q^{71} -258.314 q^{72} -41.3406 q^{73} +762.241 q^{74} -950.700 q^{76} -172.178 q^{77} +493.884 q^{78} +890.388 q^{79} +81.0000 q^{81} +1588.04 q^{82} +1057.15 q^{83} +296.198 q^{84} +574.325 q^{86} -643.256 q^{87} -705.969 q^{88} +1473.72 q^{89} -245.109 q^{91} +2054.70 q^{92} +265.978 q^{93} +1665.38 q^{94} -377.058 q^{96} -555.034 q^{97} -230.377 q^{98} +221.372 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{2} - 6 q^{3} + 9 q^{4} + 9 q^{6} - 14 q^{7} - 51 q^{8} + 18 q^{9}+O(q^{10})$$ 2 * q - 3 * q^2 - 6 * q^3 + 9 * q^4 + 9 * q^6 - 14 * q^7 - 51 * q^8 + 18 * q^9 $$2 q - 3 q^{2} - 6 q^{3} + 9 q^{4} + 9 q^{6} - 14 q^{7} - 51 q^{8} + 18 q^{9} + 62 q^{11} - 27 q^{12} + 6 q^{13} + 21 q^{14} + 25 q^{16} - 40 q^{17} - 27 q^{18} - 122 q^{19} + 42 q^{21} - 52 q^{22} - 16 q^{23} + 153 q^{24} - 214 q^{26} - 54 q^{27} - 63 q^{28} + 352 q^{29} + 66 q^{31} + 309 q^{32} - 186 q^{33} - 186 q^{34} + 81 q^{36} + 188 q^{37} + 224 q^{38} - 18 q^{39} + 16 q^{41} - 63 q^{42} + 396 q^{43} + 156 q^{44} - 960 q^{46} + 188 q^{47} - 75 q^{48} + 98 q^{49} + 120 q^{51} + 642 q^{52} - 982 q^{53} + 81 q^{54} + 357 q^{56} + 366 q^{57} - 774 q^{58} + 516 q^{59} - 880 q^{61} + 680 q^{62} - 126 q^{63} - 479 q^{64} + 156 q^{66} + 356 q^{67} + 558 q^{68} + 48 q^{69} + 310 q^{71} - 459 q^{72} - 326 q^{73} + 1358 q^{74} - 672 q^{76} - 434 q^{77} + 642 q^{78} + 1832 q^{79} + 162 q^{81} + 2190 q^{82} + 680 q^{83} + 189 q^{84} + 1456 q^{86} - 1056 q^{87} - 1540 q^{88} + 796 q^{89} - 42 q^{91} + 2880 q^{92} - 198 q^{93} + 2588 q^{94} - 927 q^{96} + 670 q^{97} - 147 q^{98} + 558 q^{99}+O(q^{100})$$ 2 * q - 3 * q^2 - 6 * q^3 + 9 * q^4 + 9 * q^6 - 14 * q^7 - 51 * q^8 + 18 * q^9 + 62 * q^11 - 27 * q^12 + 6 * q^13 + 21 * q^14 + 25 * q^16 - 40 * q^17 - 27 * q^18 - 122 * q^19 + 42 * q^21 - 52 * q^22 - 16 * q^23 + 153 * q^24 - 214 * q^26 - 54 * q^27 - 63 * q^28 + 352 * q^29 + 66 * q^31 + 309 * q^32 - 186 * q^33 - 186 * q^34 + 81 * q^36 + 188 * q^37 + 224 * q^38 - 18 * q^39 + 16 * q^41 - 63 * q^42 + 396 * q^43 + 156 * q^44 - 960 * q^46 + 188 * q^47 - 75 * q^48 + 98 * q^49 + 120 * q^51 + 642 * q^52 - 982 * q^53 + 81 * q^54 + 357 * q^56 + 366 * q^57 - 774 * q^58 + 516 * q^59 - 880 * q^61 + 680 * q^62 - 126 * q^63 - 479 * q^64 + 156 * q^66 + 356 * q^67 + 558 * q^68 + 48 * q^69 + 310 * q^71 - 459 * q^72 - 326 * q^73 + 1358 * q^74 - 672 * q^76 - 434 * q^77 + 642 * q^78 + 1832 * q^79 + 162 * q^81 + 2190 * q^82 + 680 * q^83 + 189 * q^84 + 1456 * q^86 - 1056 * q^87 - 1540 * q^88 + 796 * q^89 - 42 * q^91 + 2880 * q^92 - 198 * q^93 + 2588 * q^94 - 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 $$-0.812304\pi$$
−0.831127 + 0.556083i $$0.812304\pi$$
$$3$$ −3.00000 −0.577350
$$4$$ 14.1047 1.76309
$$5$$ 0 0
$$6$$ 14.1047 0.959702
$$7$$ −7.00000 −0.377964
$$8$$ −28.7016 −1.26844
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ 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 $$-0.378150\pi$$
0.373523 + 0.927621i $$0.378150\pi$$
$$14$$ 32.9109 0.628273
$$15$$ 0 0
$$16$$ 22.1047 0.345386
$$17$$ 18.4187 0.262777 0.131388 0.991331i $$-0.458057\pi$$
0.131388 + 0.991331i $$0.458057\pi$$
$$18$$ −42.3141 −0.554084
$$19$$ −67.4031 −0.813860 −0.406930 0.913459i $$-0.633401\pi$$
−0.406930 + 0.913459i $$0.633401\pi$$
$$20$$ 0 0
$$21$$ 21.0000 0.218218
$$22$$ −115.644 −1.12070
$$23$$ 145.675 1.32067 0.660333 0.750973i $$-0.270415\pi$$
0.660333 + 0.750973i $$0.270415\pi$$
$$24$$ 86.1047 0.732335
$$25$$ 0 0
$$26$$ −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 $$0.259149\pi$$
$$30$$ 0 0
$$31$$ −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
$$35$$ 0 0
$$36$$ 126.942 0.587695
$$37$$ −162.125 −0.720356 −0.360178 0.932884i $$-0.617284\pi$$
−0.360178 + 0.932884i $$0.617284\pi$$
$$38$$ 316.900 1.35284
$$39$$ −105.047 −0.431307
$$40$$ 0 0
$$41$$ −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 $$-0.569501\pi$$
−0.216612 + 0.976258i $$0.569501\pi$$
$$44$$ 346.931 1.18868
$$45$$ 0 0
$$46$$ −684.900 −2.19528
$$47$$ −354.219 −1.09932 −0.549661 0.835388i $$-0.685243\pi$$
−0.549661 + 0.835388i $$0.685243\pi$$
$$48$$ −66.3141 −0.199409
$$49$$ 49.0000 0.142857
$$50$$ 0 0
$$51$$ −55.2562 −0.151714
$$52$$ 493.884 1.31710
$$53$$ −676.691 −1.75378 −0.876892 0.480687i $$-0.840387\pi$$
−0.876892 + 0.480687i $$0.840387\pi$$
$$54$$ 126.942 0.319901
$$55$$ 0 0
$$56$$ 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 $$0.313444\pi$$
$$60$$ 0 0
$$61$$ −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
$$65$$ 0 0
$$66$$ 346.931 0.647035
$$67$$ 907.956 1.65559 0.827795 0.561031i $$-0.189595\pi$$
0.827795 + 0.561031i $$0.189595\pi$$
$$68$$ 259.791 0.463298
$$69$$ −437.025 −0.762487
$$70$$ 0 0
$$71$$ 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 $$-0.510551\pi$$
−0.0331408 + 0.999451i $$0.510551\pi$$
$$74$$ 762.241 1.19741
$$75$$ 0 0
$$76$$ −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 $$0.281400\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 1588.04 2.13866
$$83$$ 1057.15 1.39804 0.699020 0.715102i $$-0.253620\pi$$
0.699020 + 0.715102i $$0.253620\pi$$
$$84$$ 296.198 0.384737
$$85$$ 0 0
$$86$$ 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 $$0.159137\pi$$
$$90$$ 0 0
$$91$$ −245.109 −0.282356
$$92$$ 2054.70 2.32845
$$93$$ 265.978 0.296566
$$94$$ 1665.38 1.82735
$$95$$ 0 0
$$96$$ −377.058 −0.400868
$$97$$ −555.034 −0.580981 −0.290491 0.956878i $$-0.593819\pi$$
−0.290491 + 0.956878i $$0.593819\pi$$
$$98$$ −230.377 −0.237465
$$99$$ 221.372 0.224734
$$100$$ 0 0
$$101$$ 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 $$-0.602669\pi$$
−0.316979 + 0.948432i $$0.602669\pi$$
$$104$$ −1005.00 −0.947583
$$105$$ 0 0
$$106$$ 3181.50 2.91523
$$107$$ −1614.53 −1.45872 −0.729358 0.684132i $$-0.760181\pi$$
−0.729358 + 0.684132i $$0.760181\pi$$
$$108$$ −380.827 −0.339306
$$109$$ 217.206 0.190868 0.0954339 0.995436i $$-0.469576\pi$$
0.0954339 + 0.995436i $$0.469576\pi$$
$$110$$ 0 0
$$111$$ 486.375 0.415898
$$112$$ −154.733 −0.130544
$$113$$ 1658.20 1.38044 0.690221 0.723598i $$-0.257513\pi$$
0.690221 + 0.723598i $$0.257513\pi$$
$$114$$ −950.700 −0.781063
$$115$$ 0 0
$$116$$ 3024.31 2.42069
$$117$$ 315.141 0.249015
$$118$$ −2356.98 −1.83879
$$119$$ −128.931 −0.0993202
$$120$$ 0 0
$$121$$ −725.994 −0.545450
$$122$$ 3333.08 2.47347
$$123$$ 1013.31 0.742819
$$124$$ −1250.51 −0.905640
$$125$$ 0 0
$$126$$ 296.198 0.209424
$$127$$ 1108.81 0.774734 0.387367 0.921926i $$-0.373385\pi$$
0.387367 + 0.921926i $$0.373385\pi$$
$$128$$ 2604.17 1.79827
$$129$$ 366.469 0.250122
$$130$$ 0 0
$$131$$ 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
$$135$$ 0 0
$$136$$ −528.647 −0.333317
$$137$$ 37.9907 0.0236917 0.0118458 0.999930i $$-0.496229\pi$$
0.0118458 + 0.999930i $$0.496229\pi$$
$$138$$ 2054.70 1.26745
$$139$$ 183.609 0.112040 0.0560199 0.998430i $$-0.482159\pi$$
0.0560199 + 0.998430i $$0.482159\pi$$
$$140$$ 0 0
$$141$$ 1062.66 0.634694
$$142$$ −2023.24 −1.19568
$$143$$ 861.275 0.503660
$$144$$ 198.942 0.115129
$$145$$ 0 0
$$146$$ 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 $$0.624177\pi$$
$$150$$ 0 0
$$151$$ 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
$$155$$ 0 0
$$156$$ −1481.65 −0.760431
$$157$$ 2366.76 1.20311 0.601554 0.798832i $$-0.294548\pi$$
0.601554 + 0.798832i $$0.294548\pi$$
$$158$$ −4186.21 −2.10783
$$159$$ 2030.07 1.01255
$$160$$ 0 0
$$161$$ −1019.72 −0.499165
$$162$$ −380.827 −0.184695
$$163$$ 3137.69 1.50775 0.753875 0.657018i $$-0.228183\pi$$
0.753875 + 0.657018i $$0.228183\pi$$
$$164$$ −4764.12 −2.26839
$$165$$ 0 0
$$166$$ −4970.26 −2.32390
$$167$$ −146.469 −0.0678688 −0.0339344 0.999424i $$-0.510804\pi$$
−0.0339344 + 0.999424i $$0.510804\pi$$
$$168$$ −602.733 −0.276797
$$169$$ −970.906 −0.441924
$$170$$ 0 0
$$171$$ −606.628 −0.271287
$$172$$ −1722.98 −0.763812
$$173$$ 1424.12 0.625860 0.312930 0.949776i $$-0.398689\pi$$
0.312930 + 0.949776i $$0.398689\pi$$
$$174$$ 3024.31 1.31766
$$175$$ 0 0
$$176$$ 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 $$0.416321\pi$$
$$180$$ 0 0
$$181$$ −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
$$185$$ 0 0
$$186$$ −1250.51 −0.492968
$$187$$ 453.044 0.177165
$$188$$ −4996.14 −1.93820
$$189$$ 189.000 0.0727393
$$190$$ 0 0
$$191$$ 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 $$-0.150305\pi$$
0.890572 + 0.454843i $$0.150305\pi$$
$$194$$ 2609.53 0.965738
$$195$$ 0 0
$$196$$ 691.130 0.251869
$$197$$ 2803.58 1.01394 0.506971 0.861963i $$-0.330765\pi$$
0.506971 + 0.861963i $$0.330765\pi$$
$$198$$ −1040.79 −0.373566
$$199$$ 4102.92 1.46155 0.730774 0.682620i $$-0.239159\pi$$
0.730774 + 0.682620i $$0.239159\pi$$
$$200$$ 0 0
$$201$$ −2723.87 −0.955855
$$202$$ −8886.63 −3.09535
$$203$$ −1500.93 −0.518940
$$204$$ −779.372 −0.267485
$$205$$ 0 0
$$206$$ 3115.72 1.05380
$$207$$ 1311.07 0.440222
$$208$$ 774.009 0.258019
$$209$$ −1657.91 −0.548707
$$210$$ 0 0
$$211$$ −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
$$215$$ 0 0
$$216$$ 774.942 0.244112
$$217$$ 620.616 0.194148
$$218$$ −1021.21 −0.317271
$$219$$ 124.022 0.0382677
$$220$$ 0 0
$$221$$ 644.944 0.196306
$$222$$ −2286.72 −0.691328
$$223$$ −817.194 −0.245396 −0.122698 0.992444i $$-0.539155\pi$$
−0.122698 + 0.992444i $$0.539155\pi$$
$$224$$ −879.802 −0.262430
$$225$$ 0 0
$$226$$ −7796.12 −2.29465
$$227$$ −3655.85 −1.06893 −0.534465 0.845190i $$-0.679487\pi$$
−0.534465 + 0.845190i $$0.679487\pi$$
$$228$$ 2852.10 0.828443
$$229$$ 939.393 0.271078 0.135539 0.990772i $$-0.456723\pi$$
0.135539 + 0.990772i $$0.456723\pi$$
$$230$$ 0 0
$$231$$ 516.534 0.147123
$$232$$ −6154.15 −1.74155
$$233$$ 7.64701 0.00215010 0.00107505 0.999999i $$-0.499658\pi$$
0.00107505 + 0.999999i $$0.499658\pi$$
$$234$$ −1481.65 −0.413926
$$235$$ 0 0
$$236$$ 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 $$0.538391\pi$$
$$240$$ 0 0
$$241$$ 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
$$245$$ 0 0
$$246$$ −4764.12 −1.23475
$$247$$ −2360.16 −0.607990
$$248$$ 2544.66 0.651557
$$249$$ −3171.45 −0.807158
$$250$$ 0 0
$$251$$ −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
$$255$$ 0 0
$$256$$ −6101.62 −1.48965
$$257$$ −3068.64 −0.744811 −0.372405 0.928070i $$-0.621467\pi$$
−0.372405 + 0.928070i $$0.621467\pi$$
$$258$$ −1722.98 −0.415766
$$259$$ 1134.87 0.272269
$$260$$ 0 0
$$261$$ 1929.77 0.457662
$$262$$ −872.081 −0.205639
$$263$$ 4674.12 1.09589 0.547944 0.836515i $$-0.315411\pi$$
0.547944 + 0.836515i $$0.315411\pi$$
$$264$$ 2117.91 0.493743
$$265$$ 0 0
$$266$$ −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 $$0.411667\pi$$
$$270$$ 0 0
$$271$$ 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
$$275$$ 0 0
$$276$$ −6164.10 −1.34433
$$277$$ 4576.17 0.992620 0.496310 0.868145i $$-0.334688\pi$$
0.496310 + 0.868145i $$0.334688\pi$$
$$278$$ −863.250 −0.186239
$$279$$ −797.934 −0.171222
$$280$$ 0 0
$$281$$ −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 $$-0.633782\pi$$
−0.408024 + 0.912971i $$0.633782\pi$$
$$284$$ 6069.73 1.26821
$$285$$ 0 0
$$286$$ −4049.34 −0.837211
$$287$$ 2364.38 0.486289
$$288$$ 1131.17 0.231441
$$289$$ −4573.75 −0.930948
$$290$$ 0 0
$$291$$ 1665.10 0.335430
$$292$$ −583.097 −0.116860
$$293$$ 4033.91 0.804312 0.402156 0.915571i $$-0.368261\pi$$
0.402156 + 0.915571i $$0.368261\pi$$
$$294$$ 691.130 0.137100
$$295$$ 0 0
$$296$$ 4653.24 0.913730
$$297$$ −664.116 −0.129750
$$298$$ 6503.85 1.26429
$$299$$ 5100.90 0.986598
$$300$$ 0 0
$$301$$ 855.093 0.163743
$$302$$ −3597.90 −0.685549
$$303$$ −5670.43 −1.07511
$$304$$ −1489.92 −0.281096
$$305$$ 0 0
$$306$$ −779.372 −0.145600
$$307$$ 4620.36 0.858950 0.429475 0.903079i $$-0.358699\pi$$
0.429475 + 0.903079i $$0.358699\pi$$
$$308$$ −2428.52 −0.449278
$$309$$ 1988.10 0.366016
$$310$$ 0 0
$$311$$ 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 $$-0.582451\pi$$
−0.256141 + 0.966639i $$0.582451\pi$$
$$314$$ −11127.5 −1.99987
$$315$$ 0 0
$$316$$ 12558.6 2.23569
$$317$$ −4010.63 −0.710597 −0.355299 0.934753i $$-0.615621\pi$$
−0.355299 + 0.934753i $$0.615621\pi$$
$$318$$ −9544.51 −1.68311
$$319$$ 5274.03 0.925671
$$320$$ 0 0
$$321$$ 4843.59 0.842190
$$322$$ 4794.30 0.829739
$$323$$ −1241.48 −0.213863
$$324$$ 1142.48 0.195898
$$325$$ 0 0
$$326$$ −14752.1 −2.50626
$$327$$ −651.619 −0.110198
$$328$$ 9694.49 1.63198
$$329$$ 2479.53 0.415504
$$330$$ 0 0
$$331$$ 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
$$335$$ 0 0
$$336$$ 464.198 0.0753693
$$337$$ −12118.7 −1.95890 −0.979450 0.201689i $$-0.935357\pi$$
−0.979450 + 0.201689i $$0.935357\pi$$
$$338$$ 4564.78 0.734589
$$339$$ −4974.59 −0.796999
$$340$$ 0 0
$$341$$ −2180.74 −0.346316
$$342$$ 2852.10 0.450947
$$343$$ −343.000 −0.0539949
$$344$$ 3506.07 0.549520
$$345$$ 0 0
$$346$$ −6695.58 −1.04034
$$347$$ 6361.22 0.984116 0.492058 0.870562i $$-0.336245\pi$$
0.492058 + 0.870562i $$0.336245\pi$$
$$348$$ −9072.93 −1.39759
$$349$$ −3115.18 −0.477799 −0.238899 0.971044i $$-0.576787\pi$$
−0.238899 + 0.971044i $$0.576787\pi$$
$$350$$ 0 0
$$351$$ −945.422 −0.143769
$$352$$ 3091.48 0.468115
$$353$$ 11927.4 1.79839 0.899194 0.437550i $$-0.144154\pi$$
0.899194 + 0.437550i $$0.144154\pi$$
$$354$$ 7070.94 1.06163
$$355$$ 0 0
$$356$$ 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 $$0.649155\pi$$
$$360$$ 0 0
$$361$$ −2315.82 −0.337632
$$362$$ 18237.8 2.64794
$$363$$ 2177.98 0.314916
$$364$$ −3457.19 −0.497819
$$365$$ 0 0
$$366$$ −9999.25 −1.42806
$$367$$ 1927.67 0.274178 0.137089 0.990559i $$-0.456225\pi$$
0.137089 + 0.990559i $$0.456225\pi$$
$$368$$ 3220.10 0.456139
$$369$$ −3039.92 −0.428867
$$370$$ 0 0
$$371$$ 4736.83 0.662868
$$372$$ 3751.54 0.522871
$$373$$ −10452.0 −1.45090 −0.725449 0.688276i $$-0.758368\pi$$
−0.725449 + 0.688276i $$0.758368\pi$$
$$374$$ −2130.01 −0.294493
$$375$$ 0 0
$$376$$ 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 $$0.341049\pi$$
$$380$$ 0 0
$$381$$ −3326.44 −0.447293
$$382$$ −7404.51 −0.991747
$$383$$ −7168.04 −0.956318 −0.478159 0.878273i $$-0.658696\pi$$
−0.478159 + 0.878273i $$0.658696\pi$$
$$384$$ −7812.52 −1.03823
$$385$$ 0 0
$$386$$ −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 $$0.660515\pi$$
$$390$$ 0 0
$$391$$ 2683.15 0.347040
$$392$$ −1406.38 −0.181206
$$393$$ −556.463 −0.0714245
$$394$$ −13181.2 −1.68543
$$395$$ 0 0
$$396$$ 3122.38 0.396226
$$397$$ 8936.01 1.12969 0.564843 0.825198i $$-0.308937\pi$$
0.564843 + 0.825198i $$0.308937\pi$$
$$398$$ −19290.1 −2.42946
$$399$$ −1415.47 −0.177599
$$400$$ 0 0
$$401$$ 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
$$405$$ 0 0
$$406$$ 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 $$0.677629\pi$$
$$410$$ 0 0
$$411$$ −113.972 −0.0136784
$$412$$ −9347.17 −1.11772
$$413$$ −3509.23 −0.418106
$$414$$ −6164.10 −0.731761
$$415$$ 0 0
$$416$$ 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 $$0.559972\pi$$
$$420$$ 0 0
$$421$$ 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
$$425$$ 0 0
$$426$$ 6069.73 0.690327
$$427$$ 4962.52 0.562419
$$428$$ −22772.5 −2.57184
$$429$$ −2583.82 −0.290788
$$430$$ 0 0
$$431$$ −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 $$-0.356622\pi$$
0.435357 + 0.900258i $$0.356622\pi$$
$$434$$ −2917.86 −0.322723
$$435$$ 0 0
$$436$$ 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 $$0.492680\pi$$
$$440$$ 0 0
$$441$$ 441.000 0.0476190
$$442$$ −3032.24 −0.326310
$$443$$ 16058.7 1.72229 0.861143 0.508362i $$-0.169749\pi$$
0.861143 + 0.508362i $$0.169749\pi$$
$$444$$ 6860.17 0.733264
$$445$$ 0 0
$$446$$ 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 $$0.463338\pi$$
$$450$$ 0 0
$$451$$ −8308.05 −0.867430
$$452$$ 23388.3 2.43384
$$453$$ −2295.77 −0.238112
$$454$$ 17188.2 1.77683
$$455$$ 0 0
$$456$$ −5803.72 −0.596018
$$457$$ 5799.22 0.593602 0.296801 0.954939i $$-0.404080\pi$$
0.296801 + 0.954939i $$0.404080\pi$$
$$458$$ −4416.62 −0.450600
$$459$$ −497.306 −0.0505714
$$460$$ 0 0
$$461$$ 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 $$-0.399586\pi$$
0.310253 + 0.950654i $$0.399586\pi$$
$$464$$ 4739.66 0.474209
$$465$$ 0 0
$$466$$ −35.9529 −0.00357400
$$467$$ −6145.50 −0.608950 −0.304475 0.952520i $$-0.598481\pi$$
−0.304475 + 0.952520i $$0.598481\pi$$
$$468$$ 4444.96 0.439035
$$469$$ −6355.69 −0.625754
$$470$$ 0 0
$$471$$ −7100.28 −0.694615
$$472$$ −14388.6 −1.40316
$$473$$ −3004.66 −0.292081
$$474$$ 12558.6 1.21696
$$475$$ 0 0
$$476$$ −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 $$0.326347\pi$$
$$480$$ 0 0
$$481$$ −5676.91 −0.538139
$$482$$ −10062.4 −0.950894
$$483$$ 3059.17 0.288193
$$484$$ −10239.9 −0.961675
$$485$$ 0 0
$$486$$ 1142.48 0.106634
$$487$$ −8087.51 −0.752526 −0.376263 0.926513i $$-0.622791\pi$$
−0.376263 + 0.926513i $$0.622791\pi$$
$$488$$ 20347.4 1.88747
$$489$$ −9413.08 −0.870499
$$490$$ 0 0
$$491$$ −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
$$495$$ 0 0
$$496$$ −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 $$0.185030\pi$$
$$500$$ 0 0
$$501$$ 439.406 0.0391840
$$502$$ 31734.6 2.82149
$$503$$ −4627.62 −0.410209 −0.205105 0.978740i $$-0.565753\pi$$
−0.205105 + 0.978740i $$0.565753\pi$$
$$504$$ 1808.20 0.159809
$$505$$ 0 0
$$506$$ −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 $$0.664562\pi$$
$$510$$ 0 0
$$511$$ 289.384 0.0250521
$$512$$ 7853.76 0.677911
$$513$$ 1819.88 0.156627
$$514$$ 14427.4 1.23806
$$515$$ 0 0
$$516$$ 5168.93 0.440987
$$517$$ −8712.67 −0.741166
$$518$$ −5335.68 −0.452580
$$519$$ −4272.36 −0.361340
$$520$$ 0 0
$$521$$ 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 $$-0.458028\pi$$
0.131478 + 0.991319i $$0.458028\pi$$
$$524$$ 2616.24 0.218113
$$525$$ 0 0
$$526$$ −21975.7 −1.82164
$$527$$ −1632.99 −0.134980
$$528$$ −1631.12 −0.134442
$$529$$ 9054.20 0.744160
$$530$$ 0 0
$$531$$ 4511.87 0.368735
$$532$$ 6654.90 0.542343
$$533$$ −11827.2 −0.961148
$$534$$ 20786.4 1.68449
$$535$$ 0 0
$$536$$ −26059.8 −2.10002
$$537$$ −3734.09 −0.300071
$$538$$ −11365.5 −0.910781
$$539$$ 1205.25 0.0963148
$$540$$ 0 0
$$541$$ 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
$$545$$ 0 0
$$546$$ −3457.19 −0.270978
$$547$$ 13695.1 1.07049 0.535247 0.844696i $$-0.320219\pi$$
0.535247 + 0.844696i $$0.320219\pi$$
$$548$$ 535.847 0.0417705
$$549$$ −6380.38 −0.496007
$$550$$ 0 0
$$551$$ −14452.5 −1.11742
$$552$$ 12543.3 0.967171
$$553$$ −6232.71 −0.479280
$$554$$ −21515.2 −1.64999
$$555$$ 0 0
$$556$$ 2589.75 0.197536
$$557$$ −7850.44 −0.597188 −0.298594 0.954380i $$-0.596518\pi$$
−0.298594 + 0.954380i $$0.596518\pi$$
$$558$$ 3751.54 0.284615
$$559$$ −4277.38 −0.323638
$$560$$ 0 0
$$561$$ −1359.13 −0.102286
$$562$$ 6387.33 0.479419
$$563$$ 4948.81 0.370457 0.185229 0.982695i $$-0.440697\pi$$
0.185229 + 0.982695i $$0.440697\pi$$
$$564$$ 14988.4 1.11902
$$565$$ 0 0
$$566$$ 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 $$0.596644\pi$$
$$570$$ 0 0
$$571$$ 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
$$575$$ 0 0
$$576$$ −6909.82 −0.499842
$$577$$ 9536.77 0.688078 0.344039 0.938955i $$-0.388205\pi$$
0.344039 + 0.938955i $$0.388205\pi$$
$$578$$ 21503.8 1.54747
$$579$$ −14327.0 −1.02834
$$580$$ 0 0
$$581$$ −7400.05 −0.528409
$$582$$ −7828.58 −0.557569
$$583$$ −16644.5 −1.18241
$$584$$ 1186.54 0.0840743
$$585$$ 0 0
$$586$$ −18965.7 −1.33697
$$587$$ 13089.6 0.920383 0.460191 0.887820i $$-0.347781\pi$$
0.460191 + 0.887820i $$0.347781\pi$$
$$588$$ −2073.39 −0.145417
$$589$$ 5975.92 0.418053
$$590$$ 0 0
$$591$$ −8410.73 −0.585400
$$592$$ −3583.72 −0.248801
$$593$$ −4281.96 −0.296524 −0.148262 0.988948i $$-0.547368\pi$$
−0.148262 + 0.988948i $$0.547368\pi$$
$$594$$ 3122.38 0.215678
$$595$$ 0 0
$$596$$ −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 $$0.459725\pi$$
$$600$$ 0 0
$$601$$ −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
$$605$$ 0 0
$$606$$ 26659.9 1.78710
$$607$$ −14456.7 −0.966689 −0.483344 0.875430i $$-0.660578\pi$$
−0.483344 + 0.875430i $$0.660578\pi$$
$$608$$ −8471.63 −0.565082
$$609$$ 4502.79 0.299610
$$610$$ 0 0
$$611$$ −12403.2 −0.821243
$$612$$ 2338.12 0.154433
$$613$$ −17981.9 −1.18480 −0.592400 0.805644i $$-0.701819\pi$$
−0.592400 + 0.805644i $$0.701819\pi$$
$$614$$ −21722.9 −1.42779
$$615$$ 0 0
$$616$$ 4941.78 0.323231
$$617$$ −19614.7 −1.27983 −0.639916 0.768445i $$-0.721031\pi$$
−0.639916 + 0.768445i $$0.721031\pi$$
$$618$$ −9347.17 −0.608412
$$619$$ −10462.9 −0.679385 −0.339692 0.940537i $$-0.610323\pi$$
−0.339692 + 0.940537i $$0.610323\pi$$
$$620$$ 0 0
$$621$$ −3933.22 −0.254162
$$622$$ −31387.1 −2.02332
$$623$$ −10316.1 −0.663411
$$624$$ −2322.03 −0.148967
$$625$$ 0 0
$$626$$ 13337.3 0.851544
$$627$$ 4973.72 0.316796
$$628$$ 33382.4 2.12118
$$629$$ −2986.14 −0.189293
$$630$$ 0 0
$$631$$ 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
$$635$$ 0 0
$$636$$ 28633.5 1.78521
$$637$$ 1715.77 0.106721
$$638$$ −24796.2 −1.53870
$$639$$ 3873.01 0.239771
$$640$$ 0 0
$$641$$ −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 $$-0.874620\pi$$
−0.923422 + 0.383787i $$0.874620\pi$$
$$644$$ −14382.9 −0.880071
$$645$$ 0 0
$$646$$ 5836.90 0.355495
$$647$$ 4260.27 0.258869 0.129435 0.991588i $$-0.458684\pi$$
0.129435 + 0.991588i $$0.458684\pi$$
$$648$$ −2324.83 −0.140938
$$649$$ 12330.9 0.745808
$$650$$ 0 0
$$651$$ −1861.85 −0.112091
$$652$$ 44256.2 2.65829
$$653$$ 10576.8 0.633844 0.316922 0.948452i $$-0.397351\pi$$
0.316922 + 0.948452i $$0.397351\pi$$
$$654$$ 3063.63 0.183176
$$655$$ 0 0
$$656$$ −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 $$0.468009\pi$$
$$660$$ 0 0
$$661$$ −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
$$665$$ 0 0
$$666$$ 6860.17 0.399138
$$667$$ 31235.4 1.81326
$$668$$ −2065.89 −0.119658
$$669$$ 2451.58 0.141680
$$670$$ 0 0
$$671$$ −17437.5 −1.00323
$$672$$ 2639.40 0.151514
$$673$$ −753.881 −0.0431797 −0.0215899 0.999767i $$-0.506873\pi$$
−0.0215899 + 0.999767i $$0.506873\pi$$
$$674$$ 56976.9 3.25619
$$675$$ 0 0
$$676$$ −13694.3 −0.779149
$$677$$ −15668.8 −0.889511 −0.444756 0.895652i $$-0.646709\pi$$
−0.444756 + 0.895652i $$0.646709\pi$$
$$678$$ 23388.3 1.32481
$$679$$ 3885.24 0.219590
$$680$$ 0 0
$$681$$ 10967.5 0.617147
$$682$$ 10252.9 0.575665
$$683$$ 11557.4 0.647485 0.323742 0.946145i $$-0.395059\pi$$
0.323742 + 0.946145i $$0.395059\pi$$
$$684$$ −8556.30 −0.478302
$$685$$ 0 0
$$686$$ 1612.64 0.0897532
$$687$$ −2818.18 −0.156507
$$688$$ −2700.22 −0.149630
$$689$$ −23694.7 −1.31016
$$690$$ 0 0
$$691$$ −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
$$695$$ 0 0
$$696$$ 18462.5 1.00549
$$697$$ −6221.28 −0.338088
$$698$$ 14646.2 0.794223
$$699$$ −22.9410 −0.00124136
$$700$$ 0 0
$$701$$ 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
$$705$$ 0 0
$$706$$ −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 $$0.758805\pi$$
$$710$$ 0 0
$$711$$ 8013.49 0.422685
$$712$$ −42298.2 −2.22639
$$713$$ −12915.5 −0.678383
$$714$$ −1818.53 −0.0953178
$$715$$ 0 0
$$716$$ 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 $$0.332344\pi$$
$$720$$ 0 0
$$721$$ 4638.90 0.239614
$$722$$ 10888.0 0.561230
$$723$$ −6420.69 −0.330274
$$724$$ −54713.3 −2.80857
$$725$$ 0 0
$$726$$ −10239.9 −0.523469
$$727$$ 12317.3 0.628368 0.314184 0.949362i $$-0.398269\pi$$
0.314184 + 0.949362i $$0.398269\pi$$
$$728$$ 7035.02 0.358153
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ −2249.96 −0.113841
$$732$$ 29997.8 1.51468
$$733$$ −1234.02 −0.0621822 −0.0310911 0.999517i $$-0.509898\pi$$
−0.0310911 + 0.999517i $$0.509898\pi$$
$$734$$ −9063.05 −0.455754
$$735$$ 0 0
$$736$$ 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 $$0.623985\pi$$
$$740$$ 0 0
$$741$$ 7080.49 0.351023
$$742$$ −22270.5 −1.10186
$$743$$ 35565.1 1.75606 0.878032 0.478602i $$-0.158856\pi$$
0.878032 + 0.478602i $$0.158856\pi$$
$$744$$ −7633.99 −0.376177
$$745$$ 0 0
$$746$$ 49140.8 2.41176
$$747$$ 9514.35 0.466013
$$748$$ 6390.04 0.312357
$$749$$ 11301.7 0.551343
$$750$$ 0 0
$$751$$ 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
$$755$$ 0 0
$$756$$ 2665.79 0.128246
$$757$$ 15927.9 0.764744 0.382372 0.924009i $$-0.375107\pi$$
0.382372 + 0.924009i $$0.375107\pi$$
$$758$$ −33223.3 −1.59198
$$759$$ −10749.4 −0.514071
$$760$$ 0 0
$$761$$ −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
$$765$$ 0 0
$$766$$ 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 $$0.390112\pi$$
$$770$$ 0 0
$$771$$ 9205.91 0.430017
$$772$$ 67359.4 3.14031
$$773$$ 29443.2 1.36999 0.684993 0.728550i $$-0.259805\pi$$
0.684993 + 0.728550i $$0.259805\pi$$
$$774$$ 5168.93 0.240043
$$775$$ 0 0
$$776$$ 15930.4 0.736941
$$777$$ −3404.62 −0.157195
$$778$$ 34857.7 1.60631
$$779$$ 22766.7 1.04711
$$780$$ 0 0
$$781$$ 10584.9 0.484964
$$782$$ −12615.0 −0.576869
$$783$$ −5789.31 −0.264231
$$784$$ 1083.13 0.0493408
$$785$$ 0 0
$$786$$ 2616.24 0.118726
$$787$$ −26390.6 −1.19533 −0.597664 0.801747i $$-0.703904\pi$$
−0.597664 + 0.801747i $$0.703904\pi$$
$$788$$ 39543.6 1.78767
$$789$$ −14022.4 −0.632711
$$790$$ 0 0
$$791$$ −11607.4 −0.521758
$$792$$ −6353.72 −0.285063
$$793$$ −24823.7 −1.11162
$$794$$ −42013.2 −1.87783
$$795$$ 0 0
$$796$$ 57870.3 2.57683
$$797$$ 3738.33 0.166146 0.0830730 0.996543i $$-0.473527\pi$$
0.0830730 + 0.996543i $$0.473527\pi$$
$$798$$ 6654.90 0.295214
$$799$$ −6524.26 −0.288876
$$800$$ 0 0
$$801$$ 13263.5 0.585073
$$802$$ −8382.48 −0.369072
$$803$$ −1016.85 −0.0446873
$$804$$ −38419.3 −1.68525
$$805$$ 0 0
$$806$$ 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 $$0.111941\pi$$
$$810$$ 0 0
$$811$$ −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
$$815$$ 0 0
$$816$$ −1221.42 −0.0523999
$$817$$ 8233.71 0.352584
$$818$$ 41185.3 1.76041
$$819$$ −2205.98 −0.0941188
$$820$$ 0 0
$$821$$ −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 $$-0.679061\pi$$
−0.533334 + 0.845905i $$0.679061\pi$$
$$824$$ 19020.5 0.804140
$$825$$ 0 0
$$826$$ 16498.9 0.694999
$$827$$ 38941.7 1.63741 0.818703 0.574218i $$-0.194694\pi$$
0.818703 + 0.574218i $$0.194694\pi$$
$$828$$ 18492.3 0.776150
$$829$$ −8327.05 −0.348867 −0.174433 0.984669i $$-0.555809\pi$$
−0.174433 + 0.984669i $$0.555809\pi$$
$$830$$ 0 0
$$831$$ −13728.5 −0.573089
$$832$$ −26883.5 −1.12021
$$833$$ 902.519 0.0375395
$$834$$ 2589.75 0.107525
$$835$$ 0 0
$$836$$ −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 $$0.442153\pi$$
$$840$$ 0 0
$$841$$ 21586.4 0.885087
$$842$$ −74086.4 −3.03229
$$843$$ 4075.67 0.166517
$$844$$ −11615.4 −0.473718
$$845$$ 0 0
$$846$$ 14988.4 0.609117
$$847$$ 5081.96 0.206161
$$848$$ −14958.0 −0.605732
$$849$$ 11655.1 0.471145
$$850$$ 0 0
$$851$$ −23617.6 −0.951350
$$852$$ −18209.2 −0.732203
$$853$$ 9076.15 0.364316 0.182158 0.983269i $$-0.441692\pi$$
0.182158 + 0.983269i $$0.441692\pi$$
$$854$$ −23331.6 −0.934884
$$855$$ 0 0
$$856$$ 46339.6 1.85030
$$857$$ −36396.7 −1.45074 −0.725372 0.688357i $$-0.758332\pi$$
−0.725372 + 0.688357i $$0.758332\pi$$
$$858$$ 12148.0 0.483364
$$859$$ 8915.27 0.354115 0.177058 0.984200i $$-0.443342\pi$$
0.177058 + 0.984200i $$0.443342\pi$$
$$860$$ 0 0
$$861$$ −7093.14 −0.280759
$$862$$ 1908.44 0.0754081
$$863$$ 6148.26 0.242514 0.121257 0.992621i $$-0.461308\pi$$
0.121257 + 0.992621i $$0.461308\pi$$
$$864$$ −3393.52 −0.133623
$$865$$ 0 0
$$866$$ −36884.9 −1.44735
$$867$$ 13721.2 0.537483
$$868$$ 8753.59 0.342300
$$869$$ 21900.8 0.854928
$$870$$ 0 0
$$871$$ 31792.6 1.23680
$$872$$ −6234.16 −0.242105
$$873$$ −4995.31 −0.193660
$$874$$ 46164.4 1.78665
$$875$$ 0 0
$$876$$ 1749.29 0.0674692
$$877$$ 14287.0 0.550101 0.275050 0.961430i $$-0.411306\pi$$
0.275050 + 0.961430i $$0.411306\pi$$
$$878$$ −1988.90 −0.0764488
$$879$$ −12101.7 −0.464370
$$880$$ 0 0
$$881$$ −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 $$-0.467969\pi$$
0.100458 + 0.994941i $$0.467969\pi$$
$$884$$ 9096.73 0.346104
$$885$$ 0 0
$$886$$ −75501.1 −2.86288
$$887$$ −2606.07 −0.0986507 −0.0493253 0.998783i $$-0.515707\pi$$
−0.0493253 + 0.998783i $$0.515707\pi$$
$$888$$ −13959.7 −0.527542
$$889$$ −7761.69 −0.292822
$$890$$ 0 0
$$891$$ 1992.35 0.0749115
$$892$$ −11526.3 −0.432654
$$893$$ 23875.4 0.894694
$$894$$ −19511.5 −0.729937
$$895$$ 0 0
$$896$$ −18229.2 −0.679682
$$897$$ −15302.7 −0.569612
$$898$$ −10281.1 −0.382056
$$899$$ −19010.2 −0.705258
$$900$$ 0 0
$$901$$ −12463.8 −0.460854
$$902$$ 39060.8 1.44189
$$903$$ −2565.28 −0.0945373
$$904$$ −47592.8 −1.75101
$$905$$ 0 0
$$906$$ 10793.7 0.395802
$$907$$ −18610.6 −0.681317 −0.340659 0.940187i $$-0.610650\pi$$
−0.340659 + 0.940187i $$0.610650\pi$$
$$908$$ −51564.6 −1.88462
$$909$$ 17011.3 0.620714
$$910$$ 0 0
$$911$$ 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
$$915$$ 0 0
$$916$$ 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 $$0.254078\pi$$
$$920$$ 0 0
$$921$$ −13861.1 −0.495915
$$922$$ −46420.2 −1.65810
$$923$$ 15068.4 0.537360
$$924$$ 7285.56 0.259391
$$925$$ 0 0
$$926$$ −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 $$0.392885\pi$$
$$930$$ 0 0
$$931$$ −3302.75 −0.116266
$$932$$ 107.859 0.00379080
$$933$$ −20027.7 −0.702761
$$934$$ 28893.4 1.01223
$$935$$ 0 0
$$936$$ −9045.03 −0.315861
$$937$$ −21509.6 −0.749933 −0.374967 0.927038i $$-0.622346\pi$$
−0.374967 + 0.927038i $$0.622346\pi$$
$$938$$ 29881.7 1.04016
$$939$$ 8510.35 0.295767
$$940$$ 0 0
$$941$$ −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
$$945$$ 0 0
$$946$$ 14126.6 0.485513
$$947$$ 36556.3 1.25441 0.627203 0.778856i $$-0.284200\pi$$
0.627203 + 0.778856i $$0.284200\pi$$
$$948$$ −37675.9 −1.29078
$$949$$ −1447.57 −0.0495153
$$950$$ 0 0
$$951$$ 12031.9 0.410263
$$952$$ 3700.53 0.125982
$$953$$ 36633.4 1.24520 0.622598 0.782542i $$-0.286077\pi$$
0.622598 + 0.782542i $$0.286077\pi$$
$$954$$ 28633.5 0.971745
$$955$$ 0 0
$$956$$ −12540.7 −0.424263
$$957$$ −15822.1 −0.534436
$$958$$ −51150.3 −1.72504
$$959$$ −265.935 −0.00895462
$$960$$ 0 0
$$961$$ −21930.5 −0.736146
$$962$$ 26690.3 0.894523
$$963$$ −14530.8 −0.486239
$$964$$ 30187.3 1.00858
$$965$$ 0 0
$$966$$ −14382.9 −0.479050
$$967$$ 35515.8 1.18109 0.590544 0.807006i $$-0.298913\pi$$
0.590544 + 0.807006i $$0.298913\pi$$
$$968$$ 20837.2 0.691871
$$969$$ 3724.44 0.123474
$$970$$ 0 0
$$971$$ −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
$$975$$ 0 0
$$976$$ −15670.7 −0.513942
$$977$$ −50325.3 −1.64795 −0.823977 0.566624i $$-0.808249\pi$$
−0.823977 + 0.566624i $$0.808249\pi$$
$$978$$ 44256.2 1.44699
$$979$$ 36249.0 1.18337
$$980$$ 0 0
$$981$$ 1954.86 0.0636226
$$982$$ 32722.4 1.06335
$$983$$ −51189.0 −1.66091 −0.830456 0.557084i $$-0.811920\pi$$
−0.830456 + 0.557084i $$0.811920\pi$$
$$984$$ −29083.5 −0.942223
$$985$$ 0 0
$$986$$ −18568.0 −0.599721
$$987$$ −7438.59 −0.239892
$$988$$ −33289.3 −1.07194
$$989$$ −17795.1 −0.572145
$$990$$ 0 0
$$991$$ −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
$$995$$ 0 0
$$996$$ −44732.3 −1.42309
$$997$$ −41606.5 −1.32166 −0.660828 0.750537i $$-0.729795\pi$$
−0.660828 + 0.750537i $$0.729795\pi$$
$$998$$ −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 525.4.a.i.1.1 2
3.2 odd 2 1575.4.a.y.1.2 2
5.2 odd 4 525.4.d.j.274.1 4
5.3 odd 4 525.4.d.j.274.4 4
5.4 even 2 105.4.a.g.1.2 2
15.14 odd 2 315.4.a.g.1.1 2
20.19 odd 2 1680.4.a.y.1.2 2
35.34 odd 2 735.4.a.q.1.2 2
105.104 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 5.4 even 2
315.4.a.g.1.1 2 15.14 odd 2
525.4.a.i.1.1 2 1.1 even 1 trivial
525.4.d.j.274.1 4 5.2 odd 4
525.4.d.j.274.4 4 5.3 odd 4
735.4.a.q.1.2 2 35.34 odd 2
1575.4.a.y.1.2 2 3.2 odd 2
1680.4.a.y.1.2 2 20.19 odd 2
2205.4.a.v.1.1 2 105.104 even 2