# Properties

 Label 528.4.a.p.1.1 Level $528$ Weight $4$ Character 528.1 Self dual yes Analytic conductor $31.153$ Analytic rank $1$ 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] = [528,4,Mod(1,528)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(528, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("528.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$528 = 2^{4} \cdot 3 \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 528.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: $$31.1530084830$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{97})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 24$$ x^2 - x - 24 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 33) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$5.42443$$ of defining polynomial Character $$\chi$$ $$=$$ 528.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.00000 q^{3} -16.8489 q^{5} +7.69772 q^{7} +9.00000 q^{9} +O(q^{10})$$ $$q+3.00000 q^{3} -16.8489 q^{5} +7.69772 q^{7} +9.00000 q^{9} +11.0000 q^{11} +24.8489 q^{13} -50.5466 q^{15} -15.9420 q^{17} -15.1511 q^{19} +23.0931 q^{21} -17.7557 q^{23} +158.884 q^{25} +27.0000 q^{27} -128.547 q^{29} -219.395 q^{31} +33.0000 q^{33} -129.698 q^{35} +92.0703 q^{37} +74.5466 q^{39} -459.942 q^{41} -64.9648 q^{43} -151.640 q^{45} -497.408 q^{47} -283.745 q^{49} -47.8260 q^{51} -526.919 q^{53} -185.337 q^{55} -45.4534 q^{57} +578.443 q^{59} -221.569 q^{61} +69.2794 q^{63} -418.675 q^{65} +860.745 q^{67} -53.2671 q^{69} -580.919 q^{71} +510.116 q^{73} +476.652 q^{75} +84.6749 q^{77} -1035.12 q^{79} +81.0000 q^{81} -606.211 q^{83} +268.605 q^{85} -385.640 q^{87} -23.4411 q^{89} +191.279 q^{91} -658.186 q^{93} +255.279 q^{95} +719.490 q^{97} +99.0000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{3} - 14 q^{5} - 24 q^{7} + 18 q^{9}+O(q^{10})$$ 2 * q + 6 * q^3 - 14 * q^5 - 24 * q^7 + 18 * q^9 $$2 q + 6 q^{3} - 14 q^{5} - 24 q^{7} + 18 q^{9} + 22 q^{11} + 30 q^{13} - 42 q^{15} + 106 q^{17} - 50 q^{19} - 72 q^{21} - 134 q^{23} + 42 q^{25} + 54 q^{27} - 198 q^{29} - 360 q^{31} + 66 q^{33} - 220 q^{35} - 328 q^{37} + 90 q^{39} - 782 q^{41} - 386 q^{43} - 126 q^{45} - 266 q^{47} + 378 q^{49} + 318 q^{51} - 522 q^{53} - 154 q^{55} - 150 q^{57} + 172 q^{59} - 778 q^{61} - 216 q^{63} - 404 q^{65} + 776 q^{67} - 402 q^{69} - 630 q^{71} + 1296 q^{73} + 126 q^{75} - 264 q^{77} - 652 q^{79} + 162 q^{81} + 324 q^{83} + 616 q^{85} - 594 q^{87} - 756 q^{89} + 28 q^{91} - 1080 q^{93} + 156 q^{95} - 452 q^{97} + 198 q^{99}+O(q^{100})$$ 2 * q + 6 * q^3 - 14 * q^5 - 24 * q^7 + 18 * q^9 + 22 * q^11 + 30 * q^13 - 42 * q^15 + 106 * q^17 - 50 * q^19 - 72 * q^21 - 134 * q^23 + 42 * q^25 + 54 * q^27 - 198 * q^29 - 360 * q^31 + 66 * q^33 - 220 * q^35 - 328 * q^37 + 90 * q^39 - 782 * q^41 - 386 * q^43 - 126 * q^45 - 266 * q^47 + 378 * q^49 + 318 * q^51 - 522 * q^53 - 154 * q^55 - 150 * q^57 + 172 * q^59 - 778 * q^61 - 216 * q^63 - 404 * q^65 + 776 * q^67 - 402 * q^69 - 630 * q^71 + 1296 * q^73 + 126 * q^75 - 264 * q^77 - 652 * q^79 + 162 * q^81 + 324 * q^83 + 616 * q^85 - 594 * q^87 - 756 * q^89 + 28 * q^91 - 1080 * q^93 + 156 * q^95 - 452 * q^97 + 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 3.00000 0.577350
$$4$$ 0 0
$$5$$ −16.8489 −1.50701 −0.753504 0.657444i $$-0.771638\pi$$
−0.753504 + 0.657444i $$0.771638\pi$$
$$6$$ 0 0
$$7$$ 7.69772 0.415638 0.207819 0.978167i $$-0.433364\pi$$
0.207819 + 0.978167i $$0.433364\pi$$
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ 11.0000 0.301511
$$12$$ 0 0
$$13$$ 24.8489 0.530141 0.265071 0.964229i $$-0.414605\pi$$
0.265071 + 0.964229i $$0.414605\pi$$
$$14$$ 0 0
$$15$$ −50.5466 −0.870071
$$16$$ 0 0
$$17$$ −15.9420 −0.227441 −0.113721 0.993513i $$-0.536277\pi$$
−0.113721 + 0.993513i $$0.536277\pi$$
$$18$$ 0 0
$$19$$ −15.1511 −0.182943 −0.0914713 0.995808i $$-0.529157\pi$$
−0.0914713 + 0.995808i $$0.529157\pi$$
$$20$$ 0 0
$$21$$ 23.0931 0.239968
$$22$$ 0 0
$$23$$ −17.7557 −0.160971 −0.0804853 0.996756i $$-0.525647\pi$$
−0.0804853 + 0.996756i $$0.525647\pi$$
$$24$$ 0 0
$$25$$ 158.884 1.27107
$$26$$ 0 0
$$27$$ 27.0000 0.192450
$$28$$ 0 0
$$29$$ −128.547 −0.823121 −0.411560 0.911383i $$-0.635016\pi$$
−0.411560 + 0.911383i $$0.635016\pi$$
$$30$$ 0 0
$$31$$ −219.395 −1.27112 −0.635558 0.772053i $$-0.719230\pi$$
−0.635558 + 0.772053i $$0.719230\pi$$
$$32$$ 0 0
$$33$$ 33.0000 0.174078
$$34$$ 0 0
$$35$$ −129.698 −0.626369
$$36$$ 0 0
$$37$$ 92.0703 0.409088 0.204544 0.978857i $$-0.434429\pi$$
0.204544 + 0.978857i $$0.434429\pi$$
$$38$$ 0 0
$$39$$ 74.5466 0.306077
$$40$$ 0 0
$$41$$ −459.942 −1.75197 −0.875986 0.482336i $$-0.839788\pi$$
−0.875986 + 0.482336i $$0.839788\pi$$
$$42$$ 0 0
$$43$$ −64.9648 −0.230396 −0.115198 0.993343i $$-0.536750\pi$$
−0.115198 + 0.993343i $$0.536750\pi$$
$$44$$ 0 0
$$45$$ −151.640 −0.502336
$$46$$ 0 0
$$47$$ −497.408 −1.54371 −0.771855 0.635799i $$-0.780671\pi$$
−0.771855 + 0.635799i $$0.780671\pi$$
$$48$$ 0 0
$$49$$ −283.745 −0.827245
$$50$$ 0 0
$$51$$ −47.8260 −0.131313
$$52$$ 0 0
$$53$$ −526.919 −1.36562 −0.682811 0.730596i $$-0.739243\pi$$
−0.682811 + 0.730596i $$0.739243\pi$$
$$54$$ 0 0
$$55$$ −185.337 −0.454380
$$56$$ 0 0
$$57$$ −45.4534 −0.105622
$$58$$ 0 0
$$59$$ 578.443 1.27639 0.638194 0.769876i $$-0.279682\pi$$
0.638194 + 0.769876i $$0.279682\pi$$
$$60$$ 0 0
$$61$$ −221.569 −0.465067 −0.232533 0.972588i $$-0.574701\pi$$
−0.232533 + 0.972588i $$0.574701\pi$$
$$62$$ 0 0
$$63$$ 69.2794 0.138546
$$64$$ 0 0
$$65$$ −418.675 −0.798927
$$66$$ 0 0
$$67$$ 860.745 1.56950 0.784752 0.619810i $$-0.212790\pi$$
0.784752 + 0.619810i $$0.212790\pi$$
$$68$$ 0 0
$$69$$ −53.2671 −0.0929364
$$70$$ 0 0
$$71$$ −580.919 −0.971020 −0.485510 0.874231i $$-0.661366\pi$$
−0.485510 + 0.874231i $$0.661366\pi$$
$$72$$ 0 0
$$73$$ 510.116 0.817871 0.408935 0.912563i $$-0.365900\pi$$
0.408935 + 0.912563i $$0.365900\pi$$
$$74$$ 0 0
$$75$$ 476.652 0.733854
$$76$$ 0 0
$$77$$ 84.6749 0.125319
$$78$$ 0 0
$$79$$ −1035.12 −1.47418 −0.737088 0.675797i $$-0.763800\pi$$
−0.737088 + 0.675797i $$0.763800\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ −606.211 −0.801690 −0.400845 0.916146i $$-0.631283\pi$$
−0.400845 + 0.916146i $$0.631283\pi$$
$$84$$ 0 0
$$85$$ 268.605 0.342756
$$86$$ 0 0
$$87$$ −385.640 −0.475229
$$88$$ 0 0
$$89$$ −23.4411 −0.0279186 −0.0139593 0.999903i $$-0.504444\pi$$
−0.0139593 + 0.999903i $$0.504444\pi$$
$$90$$ 0 0
$$91$$ 191.279 0.220347
$$92$$ 0 0
$$93$$ −658.186 −0.733879
$$94$$ 0 0
$$95$$ 255.279 0.275696
$$96$$ 0 0
$$97$$ 719.490 0.753126 0.376563 0.926391i $$-0.377106\pi$$
0.376563 + 0.926391i $$0.377106\pi$$
$$98$$ 0 0
$$99$$ 99.0000 0.100504
$$100$$ 0 0
$$101$$ 1871.27 1.84355 0.921774 0.387727i $$-0.126740\pi$$
0.921774 + 0.387727i $$0.126740\pi$$
$$102$$ 0 0
$$103$$ 428.745 0.410151 0.205075 0.978746i $$-0.434256\pi$$
0.205075 + 0.978746i $$0.434256\pi$$
$$104$$ 0 0
$$105$$ −389.093 −0.361634
$$106$$ 0 0
$$107$$ −1148.02 −1.03723 −0.518616 0.855008i $$-0.673552\pi$$
−0.518616 + 0.855008i $$0.673552\pi$$
$$108$$ 0 0
$$109$$ −1828.32 −1.60662 −0.803308 0.595564i $$-0.796929\pi$$
−0.803308 + 0.595564i $$0.796929\pi$$
$$110$$ 0 0
$$111$$ 276.211 0.236187
$$112$$ 0 0
$$113$$ 1126.40 0.937722 0.468861 0.883272i $$-0.344665\pi$$
0.468861 + 0.883272i $$0.344665\pi$$
$$114$$ 0 0
$$115$$ 299.163 0.242584
$$116$$ 0 0
$$117$$ 223.640 0.176714
$$118$$ 0 0
$$119$$ −122.717 −0.0945332
$$120$$ 0 0
$$121$$ 121.000 0.0909091
$$122$$ 0 0
$$123$$ −1379.83 −1.01150
$$124$$ 0 0
$$125$$ −570.907 −0.408508
$$126$$ 0 0
$$127$$ −661.304 −0.462057 −0.231029 0.972947i $$-0.574209\pi$$
−0.231029 + 0.972947i $$0.574209\pi$$
$$128$$ 0 0
$$129$$ −194.895 −0.133019
$$130$$ 0 0
$$131$$ −622.186 −0.414967 −0.207483 0.978239i $$-0.566527\pi$$
−0.207483 + 0.978239i $$0.566527\pi$$
$$132$$ 0 0
$$133$$ −116.629 −0.0760378
$$134$$ 0 0
$$135$$ −454.919 −0.290024
$$136$$ 0 0
$$137$$ −1872.84 −1.16794 −0.583969 0.811776i $$-0.698501\pi$$
−0.583969 + 0.811776i $$0.698501\pi$$
$$138$$ 0 0
$$139$$ 954.058 0.582174 0.291087 0.956697i $$-0.405983\pi$$
0.291087 + 0.956697i $$0.405983\pi$$
$$140$$ 0 0
$$141$$ −1492.22 −0.891261
$$142$$ 0 0
$$143$$ 273.337 0.159844
$$144$$ 0 0
$$145$$ 2165.86 1.24045
$$146$$ 0 0
$$147$$ −851.236 −0.477610
$$148$$ 0 0
$$149$$ −2047.01 −1.12549 −0.562745 0.826631i $$-0.690255\pi$$
−0.562745 + 0.826631i $$0.690255\pi$$
$$150$$ 0 0
$$151$$ −475.863 −0.256458 −0.128229 0.991745i $$-0.540929\pi$$
−0.128229 + 0.991745i $$0.540929\pi$$
$$152$$ 0 0
$$153$$ −143.478 −0.0758138
$$154$$ 0 0
$$155$$ 3696.56 1.91558
$$156$$ 0 0
$$157$$ −647.466 −0.329130 −0.164565 0.986366i $$-0.552622\pi$$
−0.164565 + 0.986366i $$0.552622\pi$$
$$158$$ 0 0
$$159$$ −1580.76 −0.788442
$$160$$ 0 0
$$161$$ −136.678 −0.0669054
$$162$$ 0 0
$$163$$ −1093.23 −0.525329 −0.262665 0.964887i $$-0.584601\pi$$
−0.262665 + 0.964887i $$0.584601\pi$$
$$164$$ 0 0
$$165$$ −556.012 −0.262336
$$166$$ 0 0
$$167$$ −1123.25 −0.520479 −0.260240 0.965544i $$-0.583802\pi$$
−0.260240 + 0.965544i $$0.583802\pi$$
$$168$$ 0 0
$$169$$ −1579.53 −0.718951
$$170$$ 0 0
$$171$$ −136.360 −0.0609809
$$172$$ 0 0
$$173$$ 46.0123 0.0202211 0.0101106 0.999949i $$-0.496782\pi$$
0.0101106 + 0.999949i $$0.496782\pi$$
$$174$$ 0 0
$$175$$ 1223.04 0.528305
$$176$$ 0 0
$$177$$ 1735.33 0.736923
$$178$$ 0 0
$$179$$ 831.975 0.347401 0.173700 0.984799i $$-0.444428\pi$$
0.173700 + 0.984799i $$0.444428\pi$$
$$180$$ 0 0
$$181$$ −1810.63 −0.743553 −0.371776 0.928322i $$-0.621251\pi$$
−0.371776 + 0.928322i $$0.621251\pi$$
$$182$$ 0 0
$$183$$ −664.708 −0.268506
$$184$$ 0 0
$$185$$ −1551.28 −0.616499
$$186$$ 0 0
$$187$$ −175.362 −0.0685762
$$188$$ 0 0
$$189$$ 207.838 0.0799895
$$190$$ 0 0
$$191$$ −458.898 −0.173847 −0.0869233 0.996215i $$-0.527704\pi$$
−0.0869233 + 0.996215i $$0.527704\pi$$
$$192$$ 0 0
$$193$$ 1778.91 0.663465 0.331733 0.943373i $$-0.392367\pi$$
0.331733 + 0.943373i $$0.392367\pi$$
$$194$$ 0 0
$$195$$ −1256.02 −0.461260
$$196$$ 0 0
$$197$$ 5304.53 1.91844 0.959218 0.282666i $$-0.0912188\pi$$
0.959218 + 0.282666i $$0.0912188\pi$$
$$198$$ 0 0
$$199$$ 5138.40 1.83041 0.915205 0.402989i $$-0.132029\pi$$
0.915205 + 0.402989i $$0.132029\pi$$
$$200$$ 0 0
$$201$$ 2582.24 0.906153
$$202$$ 0 0
$$203$$ −989.515 −0.342120
$$204$$ 0 0
$$205$$ 7749.50 2.64024
$$206$$ 0 0
$$207$$ −159.801 −0.0536568
$$208$$ 0 0
$$209$$ −166.663 −0.0551593
$$210$$ 0 0
$$211$$ 4262.36 1.39068 0.695339 0.718682i $$-0.255254\pi$$
0.695339 + 0.718682i $$0.255254\pi$$
$$212$$ 0 0
$$213$$ −1742.76 −0.560619
$$214$$ 0 0
$$215$$ 1094.58 0.347209
$$216$$ 0 0
$$217$$ −1688.84 −0.528323
$$218$$ 0 0
$$219$$ 1530.35 0.472198
$$220$$ 0 0
$$221$$ −396.141 −0.120576
$$222$$ 0 0
$$223$$ 1377.80 0.413740 0.206870 0.978368i $$-0.433672\pi$$
0.206870 + 0.978368i $$0.433672\pi$$
$$224$$ 0 0
$$225$$ 1429.96 0.423691
$$226$$ 0 0
$$227$$ 1227.28 0.358843 0.179422 0.983772i $$-0.442577\pi$$
0.179422 + 0.983772i $$0.442577\pi$$
$$228$$ 0 0
$$229$$ 3890.28 1.12261 0.561304 0.827610i $$-0.310300\pi$$
0.561304 + 0.827610i $$0.310300\pi$$
$$230$$ 0 0
$$231$$ 254.025 0.0723532
$$232$$ 0 0
$$233$$ −3218.14 −0.904837 −0.452419 0.891806i $$-0.649439\pi$$
−0.452419 + 0.891806i $$0.649439\pi$$
$$234$$ 0 0
$$235$$ 8380.75 2.32638
$$236$$ 0 0
$$237$$ −3105.35 −0.851115
$$238$$ 0 0
$$239$$ 428.098 0.115864 0.0579318 0.998321i $$-0.481549\pi$$
0.0579318 + 0.998321i $$0.481549\pi$$
$$240$$ 0 0
$$241$$ 1231.16 0.329070 0.164535 0.986371i $$-0.447388\pi$$
0.164535 + 0.986371i $$0.447388\pi$$
$$242$$ 0 0
$$243$$ 243.000 0.0641500
$$244$$ 0 0
$$245$$ 4780.78 1.24667
$$246$$ 0 0
$$247$$ −376.489 −0.0969854
$$248$$ 0 0
$$249$$ −1818.63 −0.462856
$$250$$ 0 0
$$251$$ −2838.22 −0.713732 −0.356866 0.934156i $$-0.616155\pi$$
−0.356866 + 0.934156i $$0.616155\pi$$
$$252$$ 0 0
$$253$$ −195.313 −0.0485344
$$254$$ 0 0
$$255$$ 805.814 0.197890
$$256$$ 0 0
$$257$$ 342.007 0.0830110 0.0415055 0.999138i $$-0.486785\pi$$
0.0415055 + 0.999138i $$0.486785\pi$$
$$258$$ 0 0
$$259$$ 708.731 0.170032
$$260$$ 0 0
$$261$$ −1156.92 −0.274374
$$262$$ 0 0
$$263$$ −5895.00 −1.38213 −0.691067 0.722791i $$-0.742859\pi$$
−0.691067 + 0.722791i $$0.742859\pi$$
$$264$$ 0 0
$$265$$ 8877.99 2.05800
$$266$$ 0 0
$$267$$ −70.3234 −0.0161188
$$268$$ 0 0
$$269$$ −2496.18 −0.565779 −0.282890 0.959152i $$-0.591293\pi$$
−0.282890 + 0.959152i $$0.591293\pi$$
$$270$$ 0 0
$$271$$ 2249.68 0.504274 0.252137 0.967692i $$-0.418867\pi$$
0.252137 + 0.967692i $$0.418867\pi$$
$$272$$ 0 0
$$273$$ 573.838 0.127217
$$274$$ 0 0
$$275$$ 1747.72 0.383243
$$276$$ 0 0
$$277$$ 4082.59 0.885556 0.442778 0.896631i $$-0.353993\pi$$
0.442778 + 0.896631i $$0.353993\pi$$
$$278$$ 0 0
$$279$$ −1974.56 −0.423705
$$280$$ 0 0
$$281$$ −1033.79 −0.219468 −0.109734 0.993961i $$-0.535000\pi$$
−0.109734 + 0.993961i $$0.535000\pi$$
$$282$$ 0 0
$$283$$ 7809.14 1.64030 0.820150 0.572148i $$-0.193890\pi$$
0.820150 + 0.572148i $$0.193890\pi$$
$$284$$ 0 0
$$285$$ 765.838 0.159173
$$286$$ 0 0
$$287$$ −3540.50 −0.728186
$$288$$ 0 0
$$289$$ −4658.85 −0.948270
$$290$$ 0 0
$$291$$ 2158.47 0.434817
$$292$$ 0 0
$$293$$ −1949.19 −0.388645 −0.194323 0.980938i $$-0.562251\pi$$
−0.194323 + 0.980938i $$0.562251\pi$$
$$294$$ 0 0
$$295$$ −9746.10 −1.92353
$$296$$ 0 0
$$297$$ 297.000 0.0580259
$$298$$ 0 0
$$299$$ −441.209 −0.0853371
$$300$$ 0 0
$$301$$ −500.081 −0.0957614
$$302$$ 0 0
$$303$$ 5613.81 1.06437
$$304$$ 0 0
$$305$$ 3733.19 0.700859
$$306$$ 0 0
$$307$$ −2364.09 −0.439497 −0.219748 0.975557i $$-0.570524\pi$$
−0.219748 + 0.975557i $$0.570524\pi$$
$$308$$ 0 0
$$309$$ 1286.24 0.236801
$$310$$ 0 0
$$311$$ 1989.17 0.362686 0.181343 0.983420i $$-0.441956\pi$$
0.181343 + 0.983420i $$0.441956\pi$$
$$312$$ 0 0
$$313$$ −3878.67 −0.700433 −0.350216 0.936669i $$-0.613892\pi$$
−0.350216 + 0.936669i $$0.613892\pi$$
$$314$$ 0 0
$$315$$ −1167.28 −0.208790
$$316$$ 0 0
$$317$$ 2913.73 0.516251 0.258126 0.966111i $$-0.416895\pi$$
0.258126 + 0.966111i $$0.416895\pi$$
$$318$$ 0 0
$$319$$ −1414.01 −0.248180
$$320$$ 0 0
$$321$$ −3444.07 −0.598846
$$322$$ 0 0
$$323$$ 241.540 0.0416087
$$324$$ 0 0
$$325$$ 3948.09 0.673847
$$326$$ 0 0
$$327$$ −5484.95 −0.927580
$$328$$ 0 0
$$329$$ −3828.90 −0.641624
$$330$$ 0 0
$$331$$ −8104.46 −1.34580 −0.672902 0.739731i $$-0.734953\pi$$
−0.672902 + 0.739731i $$0.734953\pi$$
$$332$$ 0 0
$$333$$ 828.633 0.136363
$$334$$ 0 0
$$335$$ −14502.6 −2.36525
$$336$$ 0 0
$$337$$ 5919.19 0.956792 0.478396 0.878144i $$-0.341218\pi$$
0.478396 + 0.878144i $$0.341218\pi$$
$$338$$ 0 0
$$339$$ 3379.19 0.541394
$$340$$ 0 0
$$341$$ −2413.35 −0.383256
$$342$$ 0 0
$$343$$ −4824.51 −0.759472
$$344$$ 0 0
$$345$$ 897.490 0.140056
$$346$$ 0 0
$$347$$ 8540.59 1.32128 0.660638 0.750705i $$-0.270286\pi$$
0.660638 + 0.750705i $$0.270286\pi$$
$$348$$ 0 0
$$349$$ 937.337 0.143767 0.0718833 0.997413i $$-0.477099\pi$$
0.0718833 + 0.997413i $$0.477099\pi$$
$$350$$ 0 0
$$351$$ 670.919 0.102026
$$352$$ 0 0
$$353$$ −211.118 −0.0318319 −0.0159160 0.999873i $$-0.505066\pi$$
−0.0159160 + 0.999873i $$0.505066\pi$$
$$354$$ 0 0
$$355$$ 9787.82 1.46333
$$356$$ 0 0
$$357$$ −368.151 −0.0545788
$$358$$ 0 0
$$359$$ 1376.31 0.202337 0.101169 0.994869i $$-0.467742\pi$$
0.101169 + 0.994869i $$0.467742\pi$$
$$360$$ 0 0
$$361$$ −6629.44 −0.966532
$$362$$ 0 0
$$363$$ 363.000 0.0524864
$$364$$ 0 0
$$365$$ −8594.87 −1.23254
$$366$$ 0 0
$$367$$ −1030.45 −0.146564 −0.0732821 0.997311i $$-0.523347\pi$$
−0.0732821 + 0.997311i $$0.523347\pi$$
$$368$$ 0 0
$$369$$ −4139.48 −0.583991
$$370$$ 0 0
$$371$$ −4056.07 −0.567603
$$372$$ 0 0
$$373$$ 9365.39 1.30006 0.650029 0.759909i $$-0.274757\pi$$
0.650029 + 0.759909i $$0.274757\pi$$
$$374$$ 0 0
$$375$$ −1712.72 −0.235852
$$376$$ 0 0
$$377$$ −3194.24 −0.436370
$$378$$ 0 0
$$379$$ 7120.23 0.965017 0.482509 0.875891i $$-0.339726\pi$$
0.482509 + 0.875891i $$0.339726\pi$$
$$380$$ 0 0
$$381$$ −1983.91 −0.266769
$$382$$ 0 0
$$383$$ 1163.56 0.155235 0.0776176 0.996983i $$-0.475269\pi$$
0.0776176 + 0.996983i $$0.475269\pi$$
$$384$$ 0 0
$$385$$ −1426.67 −0.188857
$$386$$ 0 0
$$387$$ −584.684 −0.0767988
$$388$$ 0 0
$$389$$ −10958.9 −1.42838 −0.714188 0.699954i $$-0.753204\pi$$
−0.714188 + 0.699954i $$0.753204\pi$$
$$390$$ 0 0
$$391$$ 283.062 0.0366114
$$392$$ 0 0
$$393$$ −1866.56 −0.239581
$$394$$ 0 0
$$395$$ 17440.6 2.22159
$$396$$ 0 0
$$397$$ −2172.09 −0.274595 −0.137298 0.990530i $$-0.543842\pi$$
−0.137298 + 0.990530i $$0.543842\pi$$
$$398$$ 0 0
$$399$$ −349.888 −0.0439005
$$400$$ 0 0
$$401$$ 7830.71 0.975180 0.487590 0.873073i $$-0.337876\pi$$
0.487590 + 0.873073i $$0.337876\pi$$
$$402$$ 0 0
$$403$$ −5451.73 −0.673870
$$404$$ 0 0
$$405$$ −1364.76 −0.167445
$$406$$ 0 0
$$407$$ 1012.77 0.123345
$$408$$ 0 0
$$409$$ −10731.2 −1.29736 −0.648682 0.761060i $$-0.724679\pi$$
−0.648682 + 0.761060i $$0.724679\pi$$
$$410$$ 0 0
$$411$$ −5618.52 −0.674309
$$412$$ 0 0
$$413$$ 4452.69 0.530515
$$414$$ 0 0
$$415$$ 10214.0 1.20815
$$416$$ 0 0
$$417$$ 2862.17 0.336118
$$418$$ 0 0
$$419$$ 7315.88 0.852994 0.426497 0.904489i $$-0.359748\pi$$
0.426497 + 0.904489i $$0.359748\pi$$
$$420$$ 0 0
$$421$$ −12495.7 −1.44657 −0.723284 0.690551i $$-0.757368\pi$$
−0.723284 + 0.690551i $$0.757368\pi$$
$$422$$ 0 0
$$423$$ −4476.67 −0.514570
$$424$$ 0 0
$$425$$ −2532.93 −0.289094
$$426$$ 0 0
$$427$$ −1705.58 −0.193299
$$428$$ 0 0
$$429$$ 820.012 0.0922857
$$430$$ 0 0
$$431$$ −6075.01 −0.678939 −0.339470 0.940617i $$-0.610248\pi$$
−0.339470 + 0.940617i $$0.610248\pi$$
$$432$$ 0 0
$$433$$ 5641.79 0.626160 0.313080 0.949727i $$-0.398639\pi$$
0.313080 + 0.949727i $$0.398639\pi$$
$$434$$ 0 0
$$435$$ 6497.59 0.716174
$$436$$ 0 0
$$437$$ 269.019 0.0294484
$$438$$ 0 0
$$439$$ −10897.0 −1.18470 −0.592351 0.805680i $$-0.701800\pi$$
−0.592351 + 0.805680i $$0.701800\pi$$
$$440$$ 0 0
$$441$$ −2553.71 −0.275748
$$442$$ 0 0
$$443$$ 7720.83 0.828054 0.414027 0.910265i $$-0.364122\pi$$
0.414027 + 0.910265i $$0.364122\pi$$
$$444$$ 0 0
$$445$$ 394.956 0.0420735
$$446$$ 0 0
$$447$$ −6141.04 −0.649802
$$448$$ 0 0
$$449$$ −7473.86 −0.785553 −0.392776 0.919634i $$-0.628485\pi$$
−0.392776 + 0.919634i $$0.628485\pi$$
$$450$$ 0 0
$$451$$ −5059.36 −0.528240
$$452$$ 0 0
$$453$$ −1427.59 −0.148066
$$454$$ 0 0
$$455$$ −3222.84 −0.332064
$$456$$ 0 0
$$457$$ 11140.5 1.14033 0.570167 0.821529i $$-0.306879\pi$$
0.570167 + 0.821529i $$0.306879\pi$$
$$458$$ 0 0
$$459$$ −430.434 −0.0437711
$$460$$ 0 0
$$461$$ 14328.8 1.44763 0.723817 0.689992i $$-0.242386\pi$$
0.723817 + 0.689992i $$0.242386\pi$$
$$462$$ 0 0
$$463$$ −11760.7 −1.18049 −0.590246 0.807223i $$-0.700969\pi$$
−0.590246 + 0.807223i $$0.700969\pi$$
$$464$$ 0 0
$$465$$ 11089.7 1.10596
$$466$$ 0 0
$$467$$ −11854.9 −1.17469 −0.587343 0.809338i $$-0.699826\pi$$
−0.587343 + 0.809338i $$0.699826\pi$$
$$468$$ 0 0
$$469$$ 6625.77 0.652345
$$470$$ 0 0
$$471$$ −1942.40 −0.190023
$$472$$ 0 0
$$473$$ −714.613 −0.0694671
$$474$$ 0 0
$$475$$ −2407.27 −0.232533
$$476$$ 0 0
$$477$$ −4742.27 −0.455207
$$478$$ 0 0
$$479$$ 1324.68 0.126359 0.0631796 0.998002i $$-0.479876\pi$$
0.0631796 + 0.998002i $$0.479876\pi$$
$$480$$ 0 0
$$481$$ 2287.84 0.216874
$$482$$ 0 0
$$483$$ −410.035 −0.0386278
$$484$$ 0 0
$$485$$ −12122.6 −1.13497
$$486$$ 0 0
$$487$$ −18636.4 −1.73408 −0.867040 0.498239i $$-0.833980\pi$$
−0.867040 + 0.498239i $$0.833980\pi$$
$$488$$ 0 0
$$489$$ −3279.70 −0.303299
$$490$$ 0 0
$$491$$ −124.552 −0.0114480 −0.00572398 0.999984i $$-0.501822\pi$$
−0.00572398 + 0.999984i $$0.501822\pi$$
$$492$$ 0 0
$$493$$ 2049.29 0.187212
$$494$$ 0 0
$$495$$ −1668.04 −0.151460
$$496$$ 0 0
$$497$$ −4471.75 −0.403592
$$498$$ 0 0
$$499$$ −10230.2 −0.917768 −0.458884 0.888496i $$-0.651751\pi$$
−0.458884 + 0.888496i $$0.651751\pi$$
$$500$$ 0 0
$$501$$ −3369.76 −0.300499
$$502$$ 0 0
$$503$$ −5150.81 −0.456587 −0.228294 0.973592i $$-0.573315\pi$$
−0.228294 + 0.973592i $$0.573315\pi$$
$$504$$ 0 0
$$505$$ −31528.8 −2.77824
$$506$$ 0 0
$$507$$ −4738.60 −0.415086
$$508$$ 0 0
$$509$$ −22.7715 −0.00198296 −0.000991481 1.00000i $$-0.500316\pi$$
−0.000991481 1.00000i $$0.500316\pi$$
$$510$$ 0 0
$$511$$ 3926.73 0.339938
$$512$$ 0 0
$$513$$ −409.081 −0.0352073
$$514$$ 0 0
$$515$$ −7223.87 −0.618100
$$516$$ 0 0
$$517$$ −5471.49 −0.465446
$$518$$ 0 0
$$519$$ 138.037 0.0116747
$$520$$ 0 0
$$521$$ 21521.7 1.80976 0.904879 0.425669i $$-0.139961\pi$$
0.904879 + 0.425669i $$0.139961\pi$$
$$522$$ 0 0
$$523$$ −2923.36 −0.244416 −0.122208 0.992504i $$-0.538998\pi$$
−0.122208 + 0.992504i $$0.538998\pi$$
$$524$$ 0 0
$$525$$ 3669.13 0.305017
$$526$$ 0 0
$$527$$ 3497.60 0.289104
$$528$$ 0 0
$$529$$ −11851.7 −0.974088
$$530$$ 0 0
$$531$$ 5205.99 0.425462
$$532$$ 0 0
$$533$$ −11429.0 −0.928792
$$534$$ 0 0
$$535$$ 19342.9 1.56312
$$536$$ 0 0
$$537$$ 2495.93 0.200572
$$538$$ 0 0
$$539$$ −3121.20 −0.249424
$$540$$ 0 0
$$541$$ 21272.8 1.69056 0.845278 0.534327i $$-0.179435\pi$$
0.845278 + 0.534327i $$0.179435\pi$$
$$542$$ 0 0
$$543$$ −5431.89 −0.429290
$$544$$ 0 0
$$545$$ 30805.1 2.42118
$$546$$ 0 0
$$547$$ 18730.5 1.46409 0.732046 0.681256i $$-0.238566\pi$$
0.732046 + 0.681256i $$0.238566\pi$$
$$548$$ 0 0
$$549$$ −1994.12 −0.155022
$$550$$ 0 0
$$551$$ 1947.63 0.150584
$$552$$ 0 0
$$553$$ −7968.04 −0.612723
$$554$$ 0 0
$$555$$ −4653.84 −0.355936
$$556$$ 0 0
$$557$$ −18885.0 −1.43659 −0.718297 0.695736i $$-0.755078\pi$$
−0.718297 + 0.695736i $$0.755078\pi$$
$$558$$ 0 0
$$559$$ −1614.30 −0.122143
$$560$$ 0 0
$$561$$ −526.086 −0.0395925
$$562$$ 0 0
$$563$$ −10285.1 −0.769922 −0.384961 0.922933i $$-0.625785\pi$$
−0.384961 + 0.922933i $$0.625785\pi$$
$$564$$ 0 0
$$565$$ −18978.5 −1.41315
$$566$$ 0 0
$$567$$ 623.515 0.0461820
$$568$$ 0 0
$$569$$ 18008.8 1.32683 0.663415 0.748251i $$-0.269106\pi$$
0.663415 + 0.748251i $$0.269106\pi$$
$$570$$ 0 0
$$571$$ 7010.79 0.513822 0.256911 0.966435i $$-0.417295\pi$$
0.256911 + 0.966435i $$0.417295\pi$$
$$572$$ 0 0
$$573$$ −1376.69 −0.100370
$$574$$ 0 0
$$575$$ −2821.10 −0.204605
$$576$$ 0 0
$$577$$ −16398.9 −1.18318 −0.591589 0.806240i $$-0.701499\pi$$
−0.591589 + 0.806240i $$0.701499\pi$$
$$578$$ 0 0
$$579$$ 5336.73 0.383052
$$580$$ 0 0
$$581$$ −4666.44 −0.333213
$$582$$ 0 0
$$583$$ −5796.11 −0.411750
$$584$$ 0 0
$$585$$ −3768.07 −0.266309
$$586$$ 0 0
$$587$$ −12823.5 −0.901671 −0.450836 0.892607i $$-0.648874\pi$$
−0.450836 + 0.892607i $$0.648874\pi$$
$$588$$ 0 0
$$589$$ 3324.09 0.232541
$$590$$ 0 0
$$591$$ 15913.6 1.10761
$$592$$ 0 0
$$593$$ 16899.5 1.17029 0.585144 0.810929i $$-0.301038\pi$$
0.585144 + 0.810929i $$0.301038\pi$$
$$594$$ 0 0
$$595$$ 2067.64 0.142462
$$596$$ 0 0
$$597$$ 15415.2 1.05679
$$598$$ 0 0
$$599$$ 15074.9 1.02829 0.514143 0.857704i $$-0.328110\pi$$
0.514143 + 0.857704i $$0.328110\pi$$
$$600$$ 0 0
$$601$$ −11418.8 −0.775014 −0.387507 0.921867i $$-0.626664\pi$$
−0.387507 + 0.921867i $$0.626664\pi$$
$$602$$ 0 0
$$603$$ 7746.71 0.523168
$$604$$ 0 0
$$605$$ −2038.71 −0.137001
$$606$$ 0 0
$$607$$ 17952.8 1.20046 0.600232 0.799826i $$-0.295075\pi$$
0.600232 + 0.799826i $$0.295075\pi$$
$$608$$ 0 0
$$609$$ −2968.54 −0.197523
$$610$$ 0 0
$$611$$ −12360.0 −0.818384
$$612$$ 0 0
$$613$$ 12528.9 0.825507 0.412753 0.910843i $$-0.364567\pi$$
0.412753 + 0.910843i $$0.364567\pi$$
$$614$$ 0 0
$$615$$ 23248.5 1.52434
$$616$$ 0 0
$$617$$ −8586.10 −0.560232 −0.280116 0.959966i $$-0.590373\pi$$
−0.280116 + 0.959966i $$0.590373\pi$$
$$618$$ 0 0
$$619$$ −18415.4 −1.19576 −0.597882 0.801584i $$-0.703991\pi$$
−0.597882 + 0.801584i $$0.703991\pi$$
$$620$$ 0 0
$$621$$ −479.404 −0.0309788
$$622$$ 0 0
$$623$$ −180.443 −0.0116040
$$624$$ 0 0
$$625$$ −10241.4 −0.655448
$$626$$ 0 0
$$627$$ −499.988 −0.0318462
$$628$$ 0 0
$$629$$ −1467.79 −0.0930436
$$630$$ 0 0
$$631$$ −2374.38 −0.149798 −0.0748989 0.997191i $$-0.523863\pi$$
−0.0748989 + 0.997191i $$0.523863\pi$$
$$632$$ 0 0
$$633$$ 12787.1 0.802909
$$634$$ 0 0
$$635$$ 11142.2 0.696324
$$636$$ 0 0
$$637$$ −7050.74 −0.438557
$$638$$ 0 0
$$639$$ −5228.27 −0.323673
$$640$$ 0 0
$$641$$ 11086.0 0.683104 0.341552 0.939863i $$-0.389047\pi$$
0.341552 + 0.939863i $$0.389047\pi$$
$$642$$ 0 0
$$643$$ −19934.1 −1.22259 −0.611294 0.791403i $$-0.709351\pi$$
−0.611294 + 0.791403i $$0.709351\pi$$
$$644$$ 0 0
$$645$$ 3283.75 0.200461
$$646$$ 0 0
$$647$$ 30634.8 1.86148 0.930739 0.365684i $$-0.119165\pi$$
0.930739 + 0.365684i $$0.119165\pi$$
$$648$$ 0 0
$$649$$ 6362.87 0.384845
$$650$$ 0 0
$$651$$ −5066.53 −0.305028
$$652$$ 0 0
$$653$$ −9818.07 −0.588378 −0.294189 0.955747i $$-0.595049\pi$$
−0.294189 + 0.955747i $$0.595049\pi$$
$$654$$ 0 0
$$655$$ 10483.1 0.625358
$$656$$ 0 0
$$657$$ 4591.04 0.272624
$$658$$ 0 0
$$659$$ −16478.5 −0.974070 −0.487035 0.873383i $$-0.661922\pi$$
−0.487035 + 0.873383i $$0.661922\pi$$
$$660$$ 0 0
$$661$$ 2958.12 0.174066 0.0870328 0.996205i $$-0.472262\pi$$
0.0870328 + 0.996205i $$0.472262\pi$$
$$662$$ 0 0
$$663$$ −1188.42 −0.0696146
$$664$$ 0 0
$$665$$ 1965.07 0.114590
$$666$$ 0 0
$$667$$ 2282.44 0.132498
$$668$$ 0 0
$$669$$ 4133.39 0.238873
$$670$$ 0 0
$$671$$ −2437.26 −0.140223
$$672$$ 0 0
$$673$$ −29960.3 −1.71602 −0.858012 0.513630i $$-0.828300\pi$$
−0.858012 + 0.513630i $$0.828300\pi$$
$$674$$ 0 0
$$675$$ 4289.87 0.244618
$$676$$ 0 0
$$677$$ −4514.73 −0.256300 −0.128150 0.991755i $$-0.540904\pi$$
−0.128150 + 0.991755i $$0.540904\pi$$
$$678$$ 0 0
$$679$$ 5538.43 0.313027
$$680$$ 0 0
$$681$$ 3681.84 0.207178
$$682$$ 0 0
$$683$$ 13555.7 0.759438 0.379719 0.925102i $$-0.376021\pi$$
0.379719 + 0.925102i $$0.376021\pi$$
$$684$$ 0 0
$$685$$ 31555.2 1.76009
$$686$$ 0 0
$$687$$ 11670.8 0.648137
$$688$$ 0 0
$$689$$ −13093.3 −0.723972
$$690$$ 0 0
$$691$$ 11471.3 0.631535 0.315768 0.948837i $$-0.397738\pi$$
0.315768 + 0.948837i $$0.397738\pi$$
$$692$$ 0 0
$$693$$ 762.074 0.0417731
$$694$$ 0 0
$$695$$ −16074.8 −0.877340
$$696$$ 0 0
$$697$$ 7332.40 0.398471
$$698$$ 0 0
$$699$$ −9654.41 −0.522408
$$700$$ 0 0
$$701$$ 22229.0 1.19769 0.598843 0.800866i $$-0.295627\pi$$
0.598843 + 0.800866i $$0.295627\pi$$
$$702$$ 0 0
$$703$$ −1394.97 −0.0748397
$$704$$ 0 0
$$705$$ 25142.3 1.34314
$$706$$ 0 0
$$707$$ 14404.5 0.766248
$$708$$ 0 0
$$709$$ −15081.2 −0.798851 −0.399426 0.916766i $$-0.630790\pi$$
−0.399426 + 0.916766i $$0.630790\pi$$
$$710$$ 0 0
$$711$$ −9316.06 −0.491392
$$712$$ 0 0
$$713$$ 3895.52 0.204612
$$714$$ 0 0
$$715$$ −4605.42 −0.240885
$$716$$ 0 0
$$717$$ 1284.30 0.0668938
$$718$$ 0 0
$$719$$ 7399.80 0.383819 0.191910 0.981413i $$-0.438532\pi$$
0.191910 + 0.981413i $$0.438532\pi$$
$$720$$ 0 0
$$721$$ 3300.36 0.170474
$$722$$ 0 0
$$723$$ 3693.48 0.189989
$$724$$ 0 0
$$725$$ −20424.0 −1.04625
$$726$$ 0 0
$$727$$ −1705.77 −0.0870202 −0.0435101 0.999053i $$-0.513854\pi$$
−0.0435101 + 0.999053i $$0.513854\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ 1035.67 0.0524017
$$732$$ 0 0
$$733$$ 37122.6 1.87061 0.935303 0.353847i $$-0.115127\pi$$
0.935303 + 0.353847i $$0.115127\pi$$
$$734$$ 0 0
$$735$$ 14342.3 0.719762
$$736$$ 0 0
$$737$$ 9468.20 0.473223
$$738$$ 0 0
$$739$$ −34256.3 −1.70520 −0.852598 0.522568i $$-0.824974\pi$$
−0.852598 + 0.522568i $$0.824974\pi$$
$$740$$ 0 0
$$741$$ −1129.47 −0.0559945
$$742$$ 0 0
$$743$$ −1567.88 −0.0774160 −0.0387080 0.999251i $$-0.512324\pi$$
−0.0387080 + 0.999251i $$0.512324\pi$$
$$744$$ 0 0
$$745$$ 34489.8 1.69612
$$746$$ 0 0
$$747$$ −5455.90 −0.267230
$$748$$ 0 0
$$749$$ −8837.17 −0.431112
$$750$$ 0 0
$$751$$ 955.613 0.0464325 0.0232163 0.999730i $$-0.492609\pi$$
0.0232163 + 0.999730i $$0.492609\pi$$
$$752$$ 0 0
$$753$$ −8514.65 −0.412073
$$754$$ 0 0
$$755$$ 8017.75 0.386484
$$756$$ 0 0
$$757$$ −14015.4 −0.672918 −0.336459 0.941698i $$-0.609229\pi$$
−0.336459 + 0.941698i $$0.609229\pi$$
$$758$$ 0 0
$$759$$ −585.938 −0.0280214
$$760$$ 0 0
$$761$$ −36271.0 −1.72776 −0.863879 0.503699i $$-0.831972\pi$$
−0.863879 + 0.503699i $$0.831972\pi$$
$$762$$ 0 0
$$763$$ −14073.9 −0.667770
$$764$$ 0 0
$$765$$ 2417.44 0.114252
$$766$$ 0 0
$$767$$ 14373.6 0.676665
$$768$$ 0 0
$$769$$ −18163.6 −0.851749 −0.425874 0.904782i $$-0.640033\pi$$
−0.425874 + 0.904782i $$0.640033\pi$$
$$770$$ 0 0
$$771$$ 1026.02 0.0479264
$$772$$ 0 0
$$773$$ −8345.65 −0.388321 −0.194160 0.980970i $$-0.562198\pi$$
−0.194160 + 0.980970i $$0.562198\pi$$
$$774$$ 0 0
$$775$$ −34858.4 −1.61568
$$776$$ 0 0
$$777$$ 2126.19 0.0981683
$$778$$ 0 0
$$779$$ 6968.65 0.320510
$$780$$ 0 0
$$781$$ −6390.11 −0.292774
$$782$$ 0 0
$$783$$ −3470.76 −0.158410
$$784$$ 0 0
$$785$$ 10909.1 0.496001
$$786$$ 0 0
$$787$$ 22996.2 1.04158 0.520791 0.853684i $$-0.325637\pi$$
0.520791 + 0.853684i $$0.325637\pi$$
$$788$$ 0 0
$$789$$ −17685.0 −0.797975
$$790$$ 0 0
$$791$$ 8670.69 0.389752
$$792$$ 0 0
$$793$$ −5505.75 −0.246551
$$794$$ 0 0
$$795$$ 26634.0 1.18819
$$796$$ 0 0
$$797$$ −2743.82 −0.121946 −0.0609730 0.998139i $$-0.519420\pi$$
−0.0609730 + 0.998139i $$0.519420\pi$$
$$798$$ 0 0
$$799$$ 7929.68 0.351104
$$800$$ 0 0
$$801$$ −210.970 −0.00930619
$$802$$ 0 0
$$803$$ 5611.28 0.246597
$$804$$ 0 0
$$805$$ 2302.88 0.100827
$$806$$ 0 0
$$807$$ −7488.53 −0.326653
$$808$$ 0 0
$$809$$ 41241.7 1.79231 0.896156 0.443738i $$-0.146348\pi$$
0.896156 + 0.443738i $$0.146348\pi$$
$$810$$ 0 0
$$811$$ −12832.9 −0.555641 −0.277820 0.960633i $$-0.589612\pi$$
−0.277820 + 0.960633i $$0.589612\pi$$
$$812$$ 0 0
$$813$$ 6749.03 0.291142
$$814$$ 0 0
$$815$$ 18419.7 0.791675
$$816$$ 0 0
$$817$$ 984.292 0.0421493
$$818$$ 0 0
$$819$$ 1721.51 0.0734488
$$820$$ 0 0
$$821$$ 16368.5 0.695817 0.347908 0.937529i $$-0.386892\pi$$
0.347908 + 0.937529i $$0.386892\pi$$
$$822$$ 0 0
$$823$$ −3869.53 −0.163892 −0.0819461 0.996637i $$-0.526114\pi$$
−0.0819461 + 0.996637i $$0.526114\pi$$
$$824$$ 0 0
$$825$$ 5243.17 0.221265
$$826$$ 0 0
$$827$$ −7388.69 −0.310677 −0.155339 0.987861i $$-0.549647\pi$$
−0.155339 + 0.987861i $$0.549647\pi$$
$$828$$ 0 0
$$829$$ 23990.1 1.00508 0.502539 0.864554i $$-0.332399\pi$$
0.502539 + 0.864554i $$0.332399\pi$$
$$830$$ 0 0
$$831$$ 12247.8 0.511276
$$832$$ 0 0
$$833$$ 4523.47 0.188150
$$834$$ 0 0
$$835$$ 18925.6 0.784367
$$836$$ 0 0
$$837$$ −5923.68 −0.244626
$$838$$ 0 0
$$839$$ 18228.3 0.750074 0.375037 0.927010i $$-0.377630\pi$$
0.375037 + 0.927010i $$0.377630\pi$$
$$840$$ 0 0
$$841$$ −7864.78 −0.322472
$$842$$ 0 0
$$843$$ −3101.36 −0.126710
$$844$$ 0 0
$$845$$ 26613.3 1.08346
$$846$$ 0 0
$$847$$ 931.424 0.0377852
$$848$$ 0 0
$$849$$ 23427.4 0.947028
$$850$$ 0 0
$$851$$ −1634.77 −0.0658511
$$852$$ 0 0
$$853$$ −21737.3 −0.872534 −0.436267 0.899817i $$-0.643700\pi$$
−0.436267 + 0.899817i $$0.643700\pi$$
$$854$$ 0 0
$$855$$ 2297.51 0.0918987
$$856$$ 0 0
$$857$$ −18712.2 −0.745852 −0.372926 0.927861i $$-0.621645\pi$$
−0.372926 + 0.927861i $$0.621645\pi$$
$$858$$ 0 0
$$859$$ −30527.6 −1.21256 −0.606279 0.795252i $$-0.707339\pi$$
−0.606279 + 0.795252i $$0.707339\pi$$
$$860$$ 0 0
$$861$$ −10621.5 −0.420418
$$862$$ 0 0
$$863$$ −10906.4 −0.430196 −0.215098 0.976592i $$-0.569007\pi$$
−0.215098 + 0.976592i $$0.569007\pi$$
$$864$$ 0 0
$$865$$ −775.255 −0.0304734
$$866$$ 0 0
$$867$$ −13976.6 −0.547484
$$868$$ 0 0
$$869$$ −11386.3 −0.444481
$$870$$ 0 0
$$871$$ 21388.5 0.832058
$$872$$ 0 0
$$873$$ 6475.41 0.251042
$$874$$ 0 0
$$875$$ −4394.68 −0.169791
$$876$$ 0 0
$$877$$ 21770.9 0.838256 0.419128 0.907927i $$-0.362336\pi$$
0.419128 + 0.907927i $$0.362336\pi$$
$$878$$ 0 0
$$879$$ −5847.58 −0.224384
$$880$$ 0 0
$$881$$ 47206.9 1.80527 0.902634 0.430409i $$-0.141631\pi$$
0.902634 + 0.430409i $$0.141631\pi$$
$$882$$ 0 0
$$883$$ 6059.68 0.230945 0.115473 0.993311i $$-0.463162\pi$$
0.115473 + 0.993311i $$0.463162\pi$$
$$884$$ 0 0
$$885$$ −29238.3 −1.11055
$$886$$ 0 0
$$887$$ 37130.2 1.40553 0.702767 0.711420i $$-0.251948\pi$$
0.702767 + 0.711420i $$0.251948\pi$$
$$888$$ 0 0
$$889$$ −5090.53 −0.192048
$$890$$ 0 0
$$891$$ 891.000 0.0335013
$$892$$ 0 0
$$893$$ 7536.30 0.282410
$$894$$ 0 0
$$895$$ −14017.8 −0.523536
$$896$$ 0 0
$$897$$ −1323.63 −0.0492694
$$898$$ 0 0
$$899$$ 28202.5 1.04628
$$900$$ 0 0
$$901$$ 8400.15 0.310599
$$902$$ 0 0
$$903$$ −1500.24 −0.0552879
$$904$$ 0 0
$$905$$ 30507.0 1.12054
$$906$$ 0 0
$$907$$ −1182.94 −0.0433064 −0.0216532 0.999766i $$-0.506893\pi$$
−0.0216532 + 0.999766i $$0.506893\pi$$
$$908$$ 0 0
$$909$$ 16841.4 0.614516
$$910$$ 0 0
$$911$$ −37676.4 −1.37022 −0.685112 0.728438i $$-0.740247\pi$$
−0.685112 + 0.728438i $$0.740247\pi$$
$$912$$ 0 0
$$913$$ −6668.32 −0.241719
$$914$$ 0 0
$$915$$ 11199.6 0.404641
$$916$$ 0 0
$$917$$ −4789.41 −0.172476
$$918$$ 0 0
$$919$$ −8697.82 −0.312203 −0.156101 0.987741i $$-0.549893\pi$$
−0.156101 + 0.987741i $$0.549893\pi$$
$$920$$ 0 0
$$921$$ −7092.26 −0.253744
$$922$$ 0 0
$$923$$ −14435.2 −0.514778
$$924$$ 0 0
$$925$$ 14628.5 0.519981
$$926$$ 0 0
$$927$$ 3858.71 0.136717
$$928$$ 0 0
$$929$$ −17247.5 −0.609119 −0.304559 0.952493i $$-0.598509\pi$$
−0.304559 + 0.952493i $$0.598509\pi$$
$$930$$ 0 0
$$931$$ 4299.06 0.151338
$$932$$ 0 0
$$933$$ 5967.51 0.209397
$$934$$ 0 0
$$935$$ 2954.65 0.103345
$$936$$ 0 0
$$937$$ −41812.4 −1.45779 −0.728896 0.684624i $$-0.759966\pi$$
−0.728896 + 0.684624i $$0.759966\pi$$
$$938$$ 0 0
$$939$$ −11636.0 −0.404395
$$940$$ 0 0
$$941$$ 37655.9 1.30451 0.652257 0.757998i $$-0.273822\pi$$
0.652257 + 0.757998i $$0.273822\pi$$
$$942$$ 0 0
$$943$$ 8166.60 0.282016
$$944$$ 0 0
$$945$$ −3501.84 −0.120545
$$946$$ 0 0
$$947$$ 21244.4 0.728986 0.364493 0.931206i $$-0.381242\pi$$
0.364493 + 0.931206i $$0.381242\pi$$
$$948$$ 0 0
$$949$$ 12675.8 0.433587
$$950$$ 0 0
$$951$$ 8741.20 0.298058
$$952$$ 0 0
$$953$$ 1324.27 0.0450130 0.0225065 0.999747i $$-0.492835\pi$$
0.0225065 + 0.999747i $$0.492835\pi$$
$$954$$ 0 0
$$955$$ 7731.91 0.261988
$$956$$ 0 0
$$957$$ −4242.04 −0.143287
$$958$$ 0 0
$$959$$ −14416.6 −0.485439
$$960$$ 0 0
$$961$$ 18343.4 0.615735
$$962$$ 0 0
$$963$$ −10332.2 −0.345744
$$964$$ 0 0
$$965$$ −29972.6 −0.999847
$$966$$ 0 0
$$967$$ −52267.1 −1.73815 −0.869077 0.494676i $$-0.835287\pi$$
−0.869077 + 0.494676i $$0.835287\pi$$
$$968$$ 0 0
$$969$$ 724.619 0.0240228
$$970$$ 0 0
$$971$$ 52489.8 1.73479 0.867394 0.497622i $$-0.165793\pi$$
0.867394 + 0.497622i $$0.165793\pi$$
$$972$$ 0 0
$$973$$ 7344.07 0.241973
$$974$$ 0 0
$$975$$ 11844.3 0.389046
$$976$$ 0 0
$$977$$ 8324.11 0.272581 0.136291 0.990669i $$-0.456482\pi$$
0.136291 + 0.990669i $$0.456482\pi$$
$$978$$ 0 0
$$979$$ −257.852 −0.00841777
$$980$$ 0 0
$$981$$ −16454.9 −0.535539
$$982$$ 0 0
$$983$$ 44407.1 1.44086 0.720431 0.693527i $$-0.243944\pi$$
0.720431 + 0.693527i $$0.243944\pi$$
$$984$$ 0 0
$$985$$ −89375.3 −2.89110
$$986$$ 0 0
$$987$$ −11486.7 −0.370442
$$988$$ 0 0
$$989$$ 1153.50 0.0370870
$$990$$ 0 0
$$991$$ 45124.7 1.44645 0.723226 0.690612i $$-0.242659\pi$$
0.723226 + 0.690612i $$0.242659\pi$$
$$992$$ 0 0
$$993$$ −24313.4 −0.777001
$$994$$ 0 0
$$995$$ −86576.2 −2.75844
$$996$$ 0 0
$$997$$ 5480.61 0.174095 0.0870474 0.996204i $$-0.472257\pi$$
0.0870474 + 0.996204i $$0.472257\pi$$
$$998$$ 0 0
$$999$$ 2485.90 0.0787291
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 528.4.a.p.1.1 2
3.2 odd 2 1584.4.a.bj.1.2 2
4.3 odd 2 33.4.a.c.1.2 2
8.3 odd 2 2112.4.a.bn.1.2 2
8.5 even 2 2112.4.a.bg.1.2 2
12.11 even 2 99.4.a.f.1.1 2
20.3 even 4 825.4.c.h.199.1 4
20.7 even 4 825.4.c.h.199.4 4
20.19 odd 2 825.4.a.l.1.1 2
28.27 even 2 1617.4.a.k.1.2 2
44.43 even 2 363.4.a.i.1.1 2
60.59 even 2 2475.4.a.p.1.2 2
132.131 odd 2 1089.4.a.u.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
33.4.a.c.1.2 2 4.3 odd 2
99.4.a.f.1.1 2 12.11 even 2
363.4.a.i.1.1 2 44.43 even 2
528.4.a.p.1.1 2 1.1 even 1 trivial
825.4.a.l.1.1 2 20.19 odd 2
825.4.c.h.199.1 4 20.3 even 4
825.4.c.h.199.4 4 20.7 even 4
1089.4.a.u.1.2 2 132.131 odd 2
1584.4.a.bj.1.2 2 3.2 odd 2
1617.4.a.k.1.2 2 28.27 even 2
2112.4.a.bg.1.2 2 8.5 even 2
2112.4.a.bn.1.2 2 8.3 odd 2
2475.4.a.p.1.2 2 60.59 even 2