# Properties

 Label 624.4.a.f.1.1 Level $624$ Weight $4$ Character 624.1 Self dual yes Analytic conductor $36.817$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [624,4,Mod(1,624)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(624, 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("624.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$624 = 2^{4} \cdot 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 624.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: $$36.8171918436$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 78) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 624.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.00000 q^{3} -16.0000 q^{5} -28.0000 q^{7} +9.00000 q^{9} +O(q^{10})$$ $$q+3.00000 q^{3} -16.0000 q^{5} -28.0000 q^{7} +9.00000 q^{9} -34.0000 q^{11} -13.0000 q^{13} -48.0000 q^{15} +138.000 q^{17} -108.000 q^{19} -84.0000 q^{21} +52.0000 q^{23} +131.000 q^{25} +27.0000 q^{27} -190.000 q^{29} +176.000 q^{31} -102.000 q^{33} +448.000 q^{35} +342.000 q^{37} -39.0000 q^{39} +240.000 q^{41} +140.000 q^{43} -144.000 q^{45} -454.000 q^{47} +441.000 q^{49} +414.000 q^{51} +198.000 q^{53} +544.000 q^{55} -324.000 q^{57} +154.000 q^{59} +34.0000 q^{61} -252.000 q^{63} +208.000 q^{65} +656.000 q^{67} +156.000 q^{69} -550.000 q^{71} +614.000 q^{73} +393.000 q^{75} +952.000 q^{77} -8.00000 q^{79} +81.0000 q^{81} -762.000 q^{83} -2208.00 q^{85} -570.000 q^{87} -444.000 q^{89} +364.000 q^{91} +528.000 q^{93} +1728.00 q^{95} +1022.00 q^{97} -306.000 q^{99} +O(q^{100})$$

## 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
$$5$$ −16.0000 −1.43108 −0.715542 0.698570i $$-0.753820\pi$$
−0.715542 + 0.698570i $$0.753820\pi$$
$$6$$ 0 0
$$7$$ −28.0000 −1.51186 −0.755929 0.654654i $$-0.772814\pi$$
−0.755929 + 0.654654i $$0.772814\pi$$
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ −34.0000 −0.931944 −0.465972 0.884799i $$-0.654295\pi$$
−0.465972 + 0.884799i $$0.654295\pi$$
$$12$$ 0 0
$$13$$ −13.0000 −0.277350
$$14$$ 0 0
$$15$$ −48.0000 −0.826236
$$16$$ 0 0
$$17$$ 138.000 1.96882 0.984409 0.175893i $$-0.0562813\pi$$
0.984409 + 0.175893i $$0.0562813\pi$$
$$18$$ 0 0
$$19$$ −108.000 −1.30405 −0.652024 0.758199i $$-0.726080\pi$$
−0.652024 + 0.758199i $$0.726080\pi$$
$$20$$ 0 0
$$21$$ −84.0000 −0.872872
$$22$$ 0 0
$$23$$ 52.0000 0.471424 0.235712 0.971823i $$-0.424258\pi$$
0.235712 + 0.971823i $$0.424258\pi$$
$$24$$ 0 0
$$25$$ 131.000 1.04800
$$26$$ 0 0
$$27$$ 27.0000 0.192450
$$28$$ 0 0
$$29$$ −190.000 −1.21662 −0.608312 0.793698i $$-0.708153\pi$$
−0.608312 + 0.793698i $$0.708153\pi$$
$$30$$ 0 0
$$31$$ 176.000 1.01969 0.509847 0.860265i $$-0.329702\pi$$
0.509847 + 0.860265i $$0.329702\pi$$
$$32$$ 0 0
$$33$$ −102.000 −0.538058
$$34$$ 0 0
$$35$$ 448.000 2.16359
$$36$$ 0 0
$$37$$ 342.000 1.51958 0.759790 0.650169i $$-0.225302\pi$$
0.759790 + 0.650169i $$0.225302\pi$$
$$38$$ 0 0
$$39$$ −39.0000 −0.160128
$$40$$ 0 0
$$41$$ 240.000 0.914188 0.457094 0.889418i $$-0.348890\pi$$
0.457094 + 0.889418i $$0.348890\pi$$
$$42$$ 0 0
$$43$$ 140.000 0.496507 0.248253 0.968695i $$-0.420143\pi$$
0.248253 + 0.968695i $$0.420143\pi$$
$$44$$ 0 0
$$45$$ −144.000 −0.477028
$$46$$ 0 0
$$47$$ −454.000 −1.40899 −0.704497 0.709707i $$-0.748827\pi$$
−0.704497 + 0.709707i $$0.748827\pi$$
$$48$$ 0 0
$$49$$ 441.000 1.28571
$$50$$ 0 0
$$51$$ 414.000 1.13670
$$52$$ 0 0
$$53$$ 198.000 0.513158 0.256579 0.966523i $$-0.417405\pi$$
0.256579 + 0.966523i $$0.417405\pi$$
$$54$$ 0 0
$$55$$ 544.000 1.33369
$$56$$ 0 0
$$57$$ −324.000 −0.752892
$$58$$ 0 0
$$59$$ 154.000 0.339815 0.169908 0.985460i $$-0.445653\pi$$
0.169908 + 0.985460i $$0.445653\pi$$
$$60$$ 0 0
$$61$$ 34.0000 0.0713648 0.0356824 0.999363i $$-0.488640\pi$$
0.0356824 + 0.999363i $$0.488640\pi$$
$$62$$ 0 0
$$63$$ −252.000 −0.503953
$$64$$ 0 0
$$65$$ 208.000 0.396911
$$66$$ 0 0
$$67$$ 656.000 1.19617 0.598083 0.801434i $$-0.295929\pi$$
0.598083 + 0.801434i $$0.295929\pi$$
$$68$$ 0 0
$$69$$ 156.000 0.272177
$$70$$ 0 0
$$71$$ −550.000 −0.919338 −0.459669 0.888090i $$-0.652032\pi$$
−0.459669 + 0.888090i $$0.652032\pi$$
$$72$$ 0 0
$$73$$ 614.000 0.984428 0.492214 0.870474i $$-0.336188\pi$$
0.492214 + 0.870474i $$0.336188\pi$$
$$74$$ 0 0
$$75$$ 393.000 0.605063
$$76$$ 0 0
$$77$$ 952.000 1.40897
$$78$$ 0 0
$$79$$ −8.00000 −0.0113933 −0.00569665 0.999984i $$-0.501813\pi$$
−0.00569665 + 0.999984i $$0.501813\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ −762.000 −1.00772 −0.503858 0.863787i $$-0.668086\pi$$
−0.503858 + 0.863787i $$0.668086\pi$$
$$84$$ 0 0
$$85$$ −2208.00 −2.81754
$$86$$ 0 0
$$87$$ −570.000 −0.702419
$$88$$ 0 0
$$89$$ −444.000 −0.528808 −0.264404 0.964412i $$-0.585175\pi$$
−0.264404 + 0.964412i $$0.585175\pi$$
$$90$$ 0 0
$$91$$ 364.000 0.419314
$$92$$ 0 0
$$93$$ 528.000 0.588721
$$94$$ 0 0
$$95$$ 1728.00 1.86620
$$96$$ 0 0
$$97$$ 1022.00 1.06978 0.534889 0.844923i $$-0.320354\pi$$
0.534889 + 0.844923i $$0.320354\pi$$
$$98$$ 0 0
$$99$$ −306.000 −0.310648
$$100$$ 0 0
$$101$$ −1190.00 −1.17237 −0.586185 0.810177i $$-0.699371\pi$$
−0.586185 + 0.810177i $$0.699371\pi$$
$$102$$ 0 0
$$103$$ 224.000 0.214285 0.107143 0.994244i $$-0.465830\pi$$
0.107143 + 0.994244i $$0.465830\pi$$
$$104$$ 0 0
$$105$$ 1344.00 1.24915
$$106$$ 0 0
$$107$$ 640.000 0.578235 0.289117 0.957294i $$-0.406638\pi$$
0.289117 + 0.957294i $$0.406638\pi$$
$$108$$ 0 0
$$109$$ −1934.00 −1.69948 −0.849741 0.527200i $$-0.823242\pi$$
−0.849741 + 0.527200i $$0.823242\pi$$
$$110$$ 0 0
$$111$$ 1026.00 0.877330
$$112$$ 0 0
$$113$$ −418.000 −0.347983 −0.173992 0.984747i $$-0.555667\pi$$
−0.173992 + 0.984747i $$0.555667\pi$$
$$114$$ 0 0
$$115$$ −832.000 −0.674647
$$116$$ 0 0
$$117$$ −117.000 −0.0924500
$$118$$ 0 0
$$119$$ −3864.00 −2.97657
$$120$$ 0 0
$$121$$ −175.000 −0.131480
$$122$$ 0 0
$$123$$ 720.000 0.527807
$$124$$ 0 0
$$125$$ −96.0000 −0.0686920
$$126$$ 0 0
$$127$$ 1040.00 0.726654 0.363327 0.931662i $$-0.381641\pi$$
0.363327 + 0.931662i $$0.381641\pi$$
$$128$$ 0 0
$$129$$ 420.000 0.286658
$$130$$ 0 0
$$131$$ 568.000 0.378827 0.189414 0.981897i $$-0.439341\pi$$
0.189414 + 0.981897i $$0.439341\pi$$
$$132$$ 0 0
$$133$$ 3024.00 1.97153
$$134$$ 0 0
$$135$$ −432.000 −0.275412
$$136$$ 0 0
$$137$$ 528.000 0.329271 0.164635 0.986355i $$-0.447355\pi$$
0.164635 + 0.986355i $$0.447355\pi$$
$$138$$ 0 0
$$139$$ 1556.00 0.949483 0.474742 0.880125i $$-0.342541\pi$$
0.474742 + 0.880125i $$0.342541\pi$$
$$140$$ 0 0
$$141$$ −1362.00 −0.813483
$$142$$ 0 0
$$143$$ 442.000 0.258475
$$144$$ 0 0
$$145$$ 3040.00 1.74109
$$146$$ 0 0
$$147$$ 1323.00 0.742307
$$148$$ 0 0
$$149$$ 1524.00 0.837926 0.418963 0.908003i $$-0.362394\pi$$
0.418963 + 0.908003i $$0.362394\pi$$
$$150$$ 0 0
$$151$$ 3024.00 1.62973 0.814866 0.579649i $$-0.196810\pi$$
0.814866 + 0.579649i $$0.196810\pi$$
$$152$$ 0 0
$$153$$ 1242.00 0.656273
$$154$$ 0 0
$$155$$ −2816.00 −1.45927
$$156$$ 0 0
$$157$$ 2198.00 1.11732 0.558661 0.829396i $$-0.311315\pi$$
0.558661 + 0.829396i $$0.311315\pi$$
$$158$$ 0 0
$$159$$ 594.000 0.296272
$$160$$ 0 0
$$161$$ −1456.00 −0.712726
$$162$$ 0 0
$$163$$ 268.000 0.128781 0.0643907 0.997925i $$-0.479490\pi$$
0.0643907 + 0.997925i $$0.479490\pi$$
$$164$$ 0 0
$$165$$ 1632.00 0.770006
$$166$$ 0 0
$$167$$ −702.000 −0.325284 −0.162642 0.986685i $$-0.552002\pi$$
−0.162642 + 0.986685i $$0.552002\pi$$
$$168$$ 0 0
$$169$$ 169.000 0.0769231
$$170$$ 0 0
$$171$$ −972.000 −0.434682
$$172$$ 0 0
$$173$$ 2066.00 0.907948 0.453974 0.891015i $$-0.350006\pi$$
0.453974 + 0.891015i $$0.350006\pi$$
$$174$$ 0 0
$$175$$ −3668.00 −1.58443
$$176$$ 0 0
$$177$$ 462.000 0.196192
$$178$$ 0 0
$$179$$ 276.000 0.115247 0.0576235 0.998338i $$-0.481648\pi$$
0.0576235 + 0.998338i $$0.481648\pi$$
$$180$$ 0 0
$$181$$ −3474.00 −1.42663 −0.713316 0.700843i $$-0.752808\pi$$
−0.713316 + 0.700843i $$0.752808\pi$$
$$182$$ 0 0
$$183$$ 102.000 0.0412025
$$184$$ 0 0
$$185$$ −5472.00 −2.17465
$$186$$ 0 0
$$187$$ −4692.00 −1.83483
$$188$$ 0 0
$$189$$ −756.000 −0.290957
$$190$$ 0 0
$$191$$ 3920.00 1.48503 0.742516 0.669828i $$-0.233632\pi$$
0.742516 + 0.669828i $$0.233632\pi$$
$$192$$ 0 0
$$193$$ 2186.00 0.815294 0.407647 0.913140i $$-0.366349\pi$$
0.407647 + 0.913140i $$0.366349\pi$$
$$194$$ 0 0
$$195$$ 624.000 0.229157
$$196$$ 0 0
$$197$$ −1368.00 −0.494751 −0.247376 0.968920i $$-0.579568\pi$$
−0.247376 + 0.968920i $$0.579568\pi$$
$$198$$ 0 0
$$199$$ 1072.00 0.381870 0.190935 0.981603i $$-0.438848\pi$$
0.190935 + 0.981603i $$0.438848\pi$$
$$200$$ 0 0
$$201$$ 1968.00 0.690607
$$202$$ 0 0
$$203$$ 5320.00 1.83936
$$204$$ 0 0
$$205$$ −3840.00 −1.30828
$$206$$ 0 0
$$207$$ 468.000 0.157141
$$208$$ 0 0
$$209$$ 3672.00 1.21530
$$210$$ 0 0
$$211$$ −5444.00 −1.77621 −0.888105 0.459640i $$-0.847978\pi$$
−0.888105 + 0.459640i $$0.847978\pi$$
$$212$$ 0 0
$$213$$ −1650.00 −0.530780
$$214$$ 0 0
$$215$$ −2240.00 −0.710543
$$216$$ 0 0
$$217$$ −4928.00 −1.54163
$$218$$ 0 0
$$219$$ 1842.00 0.568360
$$220$$ 0 0
$$221$$ −1794.00 −0.546052
$$222$$ 0 0
$$223$$ −96.0000 −0.0288280 −0.0144140 0.999896i $$-0.504588\pi$$
−0.0144140 + 0.999896i $$0.504588\pi$$
$$224$$ 0 0
$$225$$ 1179.00 0.349333
$$226$$ 0 0
$$227$$ −198.000 −0.0578930 −0.0289465 0.999581i $$-0.509215\pi$$
−0.0289465 + 0.999581i $$0.509215\pi$$
$$228$$ 0 0
$$229$$ 5922.00 1.70889 0.854447 0.519538i $$-0.173896\pi$$
0.854447 + 0.519538i $$0.173896\pi$$
$$230$$ 0 0
$$231$$ 2856.00 0.813468
$$232$$ 0 0
$$233$$ −5114.00 −1.43789 −0.718947 0.695065i $$-0.755376\pi$$
−0.718947 + 0.695065i $$0.755376\pi$$
$$234$$ 0 0
$$235$$ 7264.00 2.01639
$$236$$ 0 0
$$237$$ −24.0000 −0.00657792
$$238$$ 0 0
$$239$$ 5226.00 1.41440 0.707200 0.707013i $$-0.249958\pi$$
0.707200 + 0.707013i $$0.249958\pi$$
$$240$$ 0 0
$$241$$ −762.000 −0.203671 −0.101836 0.994801i $$-0.532472\pi$$
−0.101836 + 0.994801i $$0.532472\pi$$
$$242$$ 0 0
$$243$$ 243.000 0.0641500
$$244$$ 0 0
$$245$$ −7056.00 −1.83996
$$246$$ 0 0
$$247$$ 1404.00 0.361678
$$248$$ 0 0
$$249$$ −2286.00 −0.581805
$$250$$ 0 0
$$251$$ −3240.00 −0.814769 −0.407384 0.913257i $$-0.633559\pi$$
−0.407384 + 0.913257i $$0.633559\pi$$
$$252$$ 0 0
$$253$$ −1768.00 −0.439341
$$254$$ 0 0
$$255$$ −6624.00 −1.62671
$$256$$ 0 0
$$257$$ −1386.00 −0.336406 −0.168203 0.985752i $$-0.553796\pi$$
−0.168203 + 0.985752i $$0.553796\pi$$
$$258$$ 0 0
$$259$$ −9576.00 −2.29739
$$260$$ 0 0
$$261$$ −1710.00 −0.405542
$$262$$ 0 0
$$263$$ −3300.00 −0.773714 −0.386857 0.922140i $$-0.626439\pi$$
−0.386857 + 0.922140i $$0.626439\pi$$
$$264$$ 0 0
$$265$$ −3168.00 −0.734372
$$266$$ 0 0
$$267$$ −1332.00 −0.305307
$$268$$ 0 0
$$269$$ 4290.00 0.972364 0.486182 0.873858i $$-0.338389\pi$$
0.486182 + 0.873858i $$0.338389\pi$$
$$270$$ 0 0
$$271$$ −2452.00 −0.549625 −0.274813 0.961498i $$-0.588616\pi$$
−0.274813 + 0.961498i $$0.588616\pi$$
$$272$$ 0 0
$$273$$ 1092.00 0.242091
$$274$$ 0 0
$$275$$ −4454.00 −0.976677
$$276$$ 0 0
$$277$$ −42.0000 −0.00911024 −0.00455512 0.999990i $$-0.501450\pi$$
−0.00455512 + 0.999990i $$0.501450\pi$$
$$278$$ 0 0
$$279$$ 1584.00 0.339898
$$280$$ 0 0
$$281$$ −2288.00 −0.485732 −0.242866 0.970060i $$-0.578088\pi$$
−0.242866 + 0.970060i $$0.578088\pi$$
$$282$$ 0 0
$$283$$ −1156.00 −0.242816 −0.121408 0.992603i $$-0.538741\pi$$
−0.121408 + 0.992603i $$0.538741\pi$$
$$284$$ 0 0
$$285$$ 5184.00 1.07745
$$286$$ 0 0
$$287$$ −6720.00 −1.38212
$$288$$ 0 0
$$289$$ 14131.0 2.87625
$$290$$ 0 0
$$291$$ 3066.00 0.617636
$$292$$ 0 0
$$293$$ −8684.00 −1.73148 −0.865742 0.500491i $$-0.833153\pi$$
−0.865742 + 0.500491i $$0.833153\pi$$
$$294$$ 0 0
$$295$$ −2464.00 −0.486304
$$296$$ 0 0
$$297$$ −918.000 −0.179353
$$298$$ 0 0
$$299$$ −676.000 −0.130749
$$300$$ 0 0
$$301$$ −3920.00 −0.750648
$$302$$ 0 0
$$303$$ −3570.00 −0.676868
$$304$$ 0 0
$$305$$ −544.000 −0.102129
$$306$$ 0 0
$$307$$ 7552.00 1.40396 0.701979 0.712197i $$-0.252300\pi$$
0.701979 + 0.712197i $$0.252300\pi$$
$$308$$ 0 0
$$309$$ 672.000 0.123718
$$310$$ 0 0
$$311$$ −2652.00 −0.483541 −0.241770 0.970334i $$-0.577728\pi$$
−0.241770 + 0.970334i $$0.577728\pi$$
$$312$$ 0 0
$$313$$ −4426.00 −0.799273 −0.399636 0.916674i $$-0.630864\pi$$
−0.399636 + 0.916674i $$0.630864\pi$$
$$314$$ 0 0
$$315$$ 4032.00 0.721198
$$316$$ 0 0
$$317$$ −4944.00 −0.875971 −0.437985 0.898982i $$-0.644308\pi$$
−0.437985 + 0.898982i $$0.644308\pi$$
$$318$$ 0 0
$$319$$ 6460.00 1.13383
$$320$$ 0 0
$$321$$ 1920.00 0.333844
$$322$$ 0 0
$$323$$ −14904.0 −2.56743
$$324$$ 0 0
$$325$$ −1703.00 −0.290663
$$326$$ 0 0
$$327$$ −5802.00 −0.981197
$$328$$ 0 0
$$329$$ 12712.0 2.13020
$$330$$ 0 0
$$331$$ 6088.00 1.01096 0.505478 0.862839i $$-0.331316\pi$$
0.505478 + 0.862839i $$0.331316\pi$$
$$332$$ 0 0
$$333$$ 3078.00 0.506527
$$334$$ 0 0
$$335$$ −10496.0 −1.71181
$$336$$ 0 0
$$337$$ 6638.00 1.07298 0.536491 0.843906i $$-0.319750\pi$$
0.536491 + 0.843906i $$0.319750\pi$$
$$338$$ 0 0
$$339$$ −1254.00 −0.200908
$$340$$ 0 0
$$341$$ −5984.00 −0.950298
$$342$$ 0 0
$$343$$ −2744.00 −0.431959
$$344$$ 0 0
$$345$$ −2496.00 −0.389508
$$346$$ 0 0
$$347$$ 2292.00 0.354585 0.177293 0.984158i $$-0.443266\pi$$
0.177293 + 0.984158i $$0.443266\pi$$
$$348$$ 0 0
$$349$$ −9866.00 −1.51322 −0.756612 0.653865i $$-0.773147\pi$$
−0.756612 + 0.653865i $$0.773147\pi$$
$$350$$ 0 0
$$351$$ −351.000 −0.0533761
$$352$$ 0 0
$$353$$ 2368.00 0.357042 0.178521 0.983936i $$-0.442869\pi$$
0.178521 + 0.983936i $$0.442869\pi$$
$$354$$ 0 0
$$355$$ 8800.00 1.31565
$$356$$ 0 0
$$357$$ −11592.0 −1.71853
$$358$$ 0 0
$$359$$ −5070.00 −0.745360 −0.372680 0.927960i $$-0.621561\pi$$
−0.372680 + 0.927960i $$0.621561\pi$$
$$360$$ 0 0
$$361$$ 4805.00 0.700539
$$362$$ 0 0
$$363$$ −525.000 −0.0759101
$$364$$ 0 0
$$365$$ −9824.00 −1.40880
$$366$$ 0 0
$$367$$ 8584.00 1.22093 0.610465 0.792043i $$-0.290983\pi$$
0.610465 + 0.792043i $$0.290983\pi$$
$$368$$ 0 0
$$369$$ 2160.00 0.304729
$$370$$ 0 0
$$371$$ −5544.00 −0.775822
$$372$$ 0 0
$$373$$ 4994.00 0.693243 0.346621 0.938005i $$-0.387329\pi$$
0.346621 + 0.938005i $$0.387329\pi$$
$$374$$ 0 0
$$375$$ −288.000 −0.0396593
$$376$$ 0 0
$$377$$ 2470.00 0.337431
$$378$$ 0 0
$$379$$ −1300.00 −0.176191 −0.0880957 0.996112i $$-0.528078\pi$$
−0.0880957 + 0.996112i $$0.528078\pi$$
$$380$$ 0 0
$$381$$ 3120.00 0.419534
$$382$$ 0 0
$$383$$ 4590.00 0.612371 0.306185 0.951972i $$-0.400947\pi$$
0.306185 + 0.951972i $$0.400947\pi$$
$$384$$ 0 0
$$385$$ −15232.0 −2.01635
$$386$$ 0 0
$$387$$ 1260.00 0.165502
$$388$$ 0 0
$$389$$ 3510.00 0.457491 0.228746 0.973486i $$-0.426538\pi$$
0.228746 + 0.973486i $$0.426538\pi$$
$$390$$ 0 0
$$391$$ 7176.00 0.928148
$$392$$ 0 0
$$393$$ 1704.00 0.218716
$$394$$ 0 0
$$395$$ 128.000 0.0163048
$$396$$ 0 0
$$397$$ 6230.00 0.787594 0.393797 0.919197i $$-0.371161\pi$$
0.393797 + 0.919197i $$0.371161\pi$$
$$398$$ 0 0
$$399$$ 9072.00 1.13827
$$400$$ 0 0
$$401$$ −7500.00 −0.933995 −0.466998 0.884259i $$-0.654664\pi$$
−0.466998 + 0.884259i $$0.654664\pi$$
$$402$$ 0 0
$$403$$ −2288.00 −0.282812
$$404$$ 0 0
$$405$$ −1296.00 −0.159009
$$406$$ 0 0
$$407$$ −11628.0 −1.41616
$$408$$ 0 0
$$409$$ 8254.00 0.997883 0.498941 0.866636i $$-0.333722\pi$$
0.498941 + 0.866636i $$0.333722\pi$$
$$410$$ 0 0
$$411$$ 1584.00 0.190105
$$412$$ 0 0
$$413$$ −4312.00 −0.513752
$$414$$ 0 0
$$415$$ 12192.0 1.44212
$$416$$ 0 0
$$417$$ 4668.00 0.548185
$$418$$ 0 0
$$419$$ 14808.0 1.72653 0.863267 0.504747i $$-0.168414\pi$$
0.863267 + 0.504747i $$0.168414\pi$$
$$420$$ 0 0
$$421$$ 10354.0 1.19863 0.599315 0.800513i $$-0.295440\pi$$
0.599315 + 0.800513i $$0.295440\pi$$
$$422$$ 0 0
$$423$$ −4086.00 −0.469665
$$424$$ 0 0
$$425$$ 18078.0 2.06332
$$426$$ 0 0
$$427$$ −952.000 −0.107893
$$428$$ 0 0
$$429$$ 1326.00 0.149230
$$430$$ 0 0
$$431$$ 15486.0 1.73071 0.865353 0.501163i $$-0.167094\pi$$
0.865353 + 0.501163i $$0.167094\pi$$
$$432$$ 0 0
$$433$$ −2018.00 −0.223970 −0.111985 0.993710i $$-0.535721\pi$$
−0.111985 + 0.993710i $$0.535721\pi$$
$$434$$ 0 0
$$435$$ 9120.00 1.00522
$$436$$ 0 0
$$437$$ −5616.00 −0.614759
$$438$$ 0 0
$$439$$ −8792.00 −0.955853 −0.477926 0.878400i $$-0.658611\pi$$
−0.477926 + 0.878400i $$0.658611\pi$$
$$440$$ 0 0
$$441$$ 3969.00 0.428571
$$442$$ 0 0
$$443$$ 2760.00 0.296008 0.148004 0.988987i $$-0.452715\pi$$
0.148004 + 0.988987i $$0.452715\pi$$
$$444$$ 0 0
$$445$$ 7104.00 0.756768
$$446$$ 0 0
$$447$$ 4572.00 0.483777
$$448$$ 0 0
$$449$$ 9532.00 1.00188 0.500939 0.865483i $$-0.332988\pi$$
0.500939 + 0.865483i $$0.332988\pi$$
$$450$$ 0 0
$$451$$ −8160.00 −0.851972
$$452$$ 0 0
$$453$$ 9072.00 0.940927
$$454$$ 0 0
$$455$$ −5824.00 −0.600073
$$456$$ 0 0
$$457$$ 12862.0 1.31654 0.658270 0.752782i $$-0.271288\pi$$
0.658270 + 0.752782i $$0.271288\pi$$
$$458$$ 0 0
$$459$$ 3726.00 0.378899
$$460$$ 0 0
$$461$$ −6744.00 −0.681344 −0.340672 0.940182i $$-0.610654\pi$$
−0.340672 + 0.940182i $$0.610654\pi$$
$$462$$ 0 0
$$463$$ 9572.00 0.960796 0.480398 0.877051i $$-0.340492\pi$$
0.480398 + 0.877051i $$0.340492\pi$$
$$464$$ 0 0
$$465$$ −8448.00 −0.842509
$$466$$ 0 0
$$467$$ −9104.00 −0.902105 −0.451052 0.892498i $$-0.648951\pi$$
−0.451052 + 0.892498i $$0.648951\pi$$
$$468$$ 0 0
$$469$$ −18368.0 −1.80843
$$470$$ 0 0
$$471$$ 6594.00 0.645086
$$472$$ 0 0
$$473$$ −4760.00 −0.462717
$$474$$ 0 0
$$475$$ −14148.0 −1.36664
$$476$$ 0 0
$$477$$ 1782.00 0.171053
$$478$$ 0 0
$$479$$ 18870.0 1.79998 0.899992 0.435906i $$-0.143572\pi$$
0.899992 + 0.435906i $$0.143572\pi$$
$$480$$ 0 0
$$481$$ −4446.00 −0.421456
$$482$$ 0 0
$$483$$ −4368.00 −0.411493
$$484$$ 0 0
$$485$$ −16352.0 −1.53094
$$486$$ 0 0
$$487$$ 1744.00 0.162276 0.0811378 0.996703i $$-0.474145\pi$$
0.0811378 + 0.996703i $$0.474145\pi$$
$$488$$ 0 0
$$489$$ 804.000 0.0743520
$$490$$ 0 0
$$491$$ −13360.0 −1.22796 −0.613980 0.789322i $$-0.710432\pi$$
−0.613980 + 0.789322i $$0.710432\pi$$
$$492$$ 0 0
$$493$$ −26220.0 −2.39531
$$494$$ 0 0
$$495$$ 4896.00 0.444563
$$496$$ 0 0
$$497$$ 15400.0 1.38991
$$498$$ 0 0
$$499$$ −17368.0 −1.55811 −0.779057 0.626954i $$-0.784302\pi$$
−0.779057 + 0.626954i $$0.784302\pi$$
$$500$$ 0 0
$$501$$ −2106.00 −0.187803
$$502$$ 0 0
$$503$$ 5828.00 0.516616 0.258308 0.966063i $$-0.416835\pi$$
0.258308 + 0.966063i $$0.416835\pi$$
$$504$$ 0 0
$$505$$ 19040.0 1.67776
$$506$$ 0 0
$$507$$ 507.000 0.0444116
$$508$$ 0 0
$$509$$ 10744.0 0.935598 0.467799 0.883835i $$-0.345047\pi$$
0.467799 + 0.883835i $$0.345047\pi$$
$$510$$ 0 0
$$511$$ −17192.0 −1.48832
$$512$$ 0 0
$$513$$ −2916.00 −0.250964
$$514$$ 0 0
$$515$$ −3584.00 −0.306660
$$516$$ 0 0
$$517$$ 15436.0 1.31310
$$518$$ 0 0
$$519$$ 6198.00 0.524204
$$520$$ 0 0
$$521$$ −12234.0 −1.02875 −0.514377 0.857564i $$-0.671977\pi$$
−0.514377 + 0.857564i $$0.671977\pi$$
$$522$$ 0 0
$$523$$ −1812.00 −0.151498 −0.0757488 0.997127i $$-0.524135\pi$$
−0.0757488 + 0.997127i $$0.524135\pi$$
$$524$$ 0 0
$$525$$ −11004.0 −0.914769
$$526$$ 0 0
$$527$$ 24288.0 2.00759
$$528$$ 0 0
$$529$$ −9463.00 −0.777760
$$530$$ 0 0
$$531$$ 1386.00 0.113272
$$532$$ 0 0
$$533$$ −3120.00 −0.253550
$$534$$ 0 0
$$535$$ −10240.0 −0.827502
$$536$$ 0 0
$$537$$ 828.000 0.0665379
$$538$$ 0 0
$$539$$ −14994.0 −1.19821
$$540$$ 0 0
$$541$$ 6098.00 0.484609 0.242305 0.970200i $$-0.422097\pi$$
0.242305 + 0.970200i $$0.422097\pi$$
$$542$$ 0 0
$$543$$ −10422.0 −0.823666
$$544$$ 0 0
$$545$$ 30944.0 2.43210
$$546$$ 0 0
$$547$$ 18332.0 1.43294 0.716471 0.697616i $$-0.245756\pi$$
0.716471 + 0.697616i $$0.245756\pi$$
$$548$$ 0 0
$$549$$ 306.000 0.0237883
$$550$$ 0 0
$$551$$ 20520.0 1.58654
$$552$$ 0 0
$$553$$ 224.000 0.0172250
$$554$$ 0 0
$$555$$ −16416.0 −1.25553
$$556$$ 0 0
$$557$$ −20004.0 −1.52172 −0.760859 0.648917i $$-0.775222\pi$$
−0.760859 + 0.648917i $$0.775222\pi$$
$$558$$ 0 0
$$559$$ −1820.00 −0.137706
$$560$$ 0 0
$$561$$ −14076.0 −1.05934
$$562$$ 0 0
$$563$$ −10988.0 −0.822538 −0.411269 0.911514i $$-0.634914\pi$$
−0.411269 + 0.911514i $$0.634914\pi$$
$$564$$ 0 0
$$565$$ 6688.00 0.497993
$$566$$ 0 0
$$567$$ −2268.00 −0.167984
$$568$$ 0 0
$$569$$ 11062.0 0.815014 0.407507 0.913202i $$-0.366398\pi$$
0.407507 + 0.913202i $$0.366398\pi$$
$$570$$ 0 0
$$571$$ 708.000 0.0518895 0.0259447 0.999663i $$-0.491741\pi$$
0.0259447 + 0.999663i $$0.491741\pi$$
$$572$$ 0 0
$$573$$ 11760.0 0.857384
$$574$$ 0 0
$$575$$ 6812.00 0.494052
$$576$$ 0 0
$$577$$ −2094.00 −0.151082 −0.0755410 0.997143i $$-0.524068\pi$$
−0.0755410 + 0.997143i $$0.524068\pi$$
$$578$$ 0 0
$$579$$ 6558.00 0.470710
$$580$$ 0 0
$$581$$ 21336.0 1.52352
$$582$$ 0 0
$$583$$ −6732.00 −0.478235
$$584$$ 0 0
$$585$$ 1872.00 0.132304
$$586$$ 0 0
$$587$$ 17854.0 1.25539 0.627695 0.778460i $$-0.283999\pi$$
0.627695 + 0.778460i $$0.283999\pi$$
$$588$$ 0 0
$$589$$ −19008.0 −1.32973
$$590$$ 0 0
$$591$$ −4104.00 −0.285645
$$592$$ 0 0
$$593$$ 23948.0 1.65839 0.829196 0.558958i $$-0.188799\pi$$
0.829196 + 0.558958i $$0.188799\pi$$
$$594$$ 0 0
$$595$$ 61824.0 4.25973
$$596$$ 0 0
$$597$$ 3216.00 0.220473
$$598$$ 0 0
$$599$$ 18068.0 1.23245 0.616226 0.787570i $$-0.288661\pi$$
0.616226 + 0.787570i $$0.288661\pi$$
$$600$$ 0 0
$$601$$ 19942.0 1.35350 0.676748 0.736215i $$-0.263389\pi$$
0.676748 + 0.736215i $$0.263389\pi$$
$$602$$ 0 0
$$603$$ 5904.00 0.398722
$$604$$ 0 0
$$605$$ 2800.00 0.188159
$$606$$ 0 0
$$607$$ −26376.0 −1.76370 −0.881852 0.471526i $$-0.843704\pi$$
−0.881852 + 0.471526i $$0.843704\pi$$
$$608$$ 0 0
$$609$$ 15960.0 1.06196
$$610$$ 0 0
$$611$$ 5902.00 0.390785
$$612$$ 0 0
$$613$$ −19426.0 −1.27995 −0.639975 0.768396i $$-0.721055\pi$$
−0.639975 + 0.768396i $$0.721055\pi$$
$$614$$ 0 0
$$615$$ −11520.0 −0.755335
$$616$$ 0 0
$$617$$ −8024.00 −0.523556 −0.261778 0.965128i $$-0.584309\pi$$
−0.261778 + 0.965128i $$0.584309\pi$$
$$618$$ 0 0
$$619$$ 20648.0 1.34073 0.670366 0.742031i $$-0.266137\pi$$
0.670366 + 0.742031i $$0.266137\pi$$
$$620$$ 0 0
$$621$$ 1404.00 0.0907256
$$622$$ 0 0
$$623$$ 12432.0 0.799482
$$624$$ 0 0
$$625$$ −14839.0 −0.949696
$$626$$ 0 0
$$627$$ 11016.0 0.701653
$$628$$ 0 0
$$629$$ 47196.0 2.99178
$$630$$ 0 0
$$631$$ −12280.0 −0.774737 −0.387369 0.921925i $$-0.626616\pi$$
−0.387369 + 0.921925i $$0.626616\pi$$
$$632$$ 0 0
$$633$$ −16332.0 −1.02550
$$634$$ 0 0
$$635$$ −16640.0 −1.03990
$$636$$ 0 0
$$637$$ −5733.00 −0.356593
$$638$$ 0 0
$$639$$ −4950.00 −0.306446
$$640$$ 0 0
$$641$$ −15878.0 −0.978383 −0.489191 0.872176i $$-0.662708\pi$$
−0.489191 + 0.872176i $$0.662708\pi$$
$$642$$ 0 0
$$643$$ 21520.0 1.31985 0.659927 0.751330i $$-0.270587\pi$$
0.659927 + 0.751330i $$0.270587\pi$$
$$644$$ 0 0
$$645$$ −6720.00 −0.410232
$$646$$ 0 0
$$647$$ −7312.00 −0.444304 −0.222152 0.975012i $$-0.571308\pi$$
−0.222152 + 0.975012i $$0.571308\pi$$
$$648$$ 0 0
$$649$$ −5236.00 −0.316689
$$650$$ 0 0
$$651$$ −14784.0 −0.890062
$$652$$ 0 0
$$653$$ 3090.00 0.185178 0.0925889 0.995704i $$-0.470486\pi$$
0.0925889 + 0.995704i $$0.470486\pi$$
$$654$$ 0 0
$$655$$ −9088.00 −0.542134
$$656$$ 0 0
$$657$$ 5526.00 0.328143
$$658$$ 0 0
$$659$$ 13428.0 0.793749 0.396875 0.917873i $$-0.370095\pi$$
0.396875 + 0.917873i $$0.370095\pi$$
$$660$$ 0 0
$$661$$ 22598.0 1.32974 0.664872 0.746958i $$-0.268486\pi$$
0.664872 + 0.746958i $$0.268486\pi$$
$$662$$ 0 0
$$663$$ −5382.00 −0.315263
$$664$$ 0 0
$$665$$ −48384.0 −2.82143
$$666$$ 0 0
$$667$$ −9880.00 −0.573546
$$668$$ 0 0
$$669$$ −288.000 −0.0166438
$$670$$ 0 0
$$671$$ −1156.00 −0.0665080
$$672$$ 0 0
$$673$$ 6178.00 0.353855 0.176927 0.984224i $$-0.443384\pi$$
0.176927 + 0.984224i $$0.443384\pi$$
$$674$$ 0 0
$$675$$ 3537.00 0.201688
$$676$$ 0 0
$$677$$ 22398.0 1.27153 0.635764 0.771883i $$-0.280685\pi$$
0.635764 + 0.771883i $$0.280685\pi$$
$$678$$ 0 0
$$679$$ −28616.0 −1.61735
$$680$$ 0 0
$$681$$ −594.000 −0.0334246
$$682$$ 0 0
$$683$$ −11410.0 −0.639226 −0.319613 0.947548i $$-0.603553\pi$$
−0.319613 + 0.947548i $$0.603553\pi$$
$$684$$ 0 0
$$685$$ −8448.00 −0.471214
$$686$$ 0 0
$$687$$ 17766.0 0.986631
$$688$$ 0 0
$$689$$ −2574.00 −0.142325
$$690$$ 0 0
$$691$$ −32488.0 −1.78857 −0.894285 0.447498i $$-0.852315\pi$$
−0.894285 + 0.447498i $$0.852315\pi$$
$$692$$ 0 0
$$693$$ 8568.00 0.469656
$$694$$ 0 0
$$695$$ −24896.0 −1.35879
$$696$$ 0 0
$$697$$ 33120.0 1.79987
$$698$$ 0 0
$$699$$ −15342.0 −0.830168
$$700$$ 0 0
$$701$$ 5094.00 0.274462 0.137231 0.990539i $$-0.456180\pi$$
0.137231 + 0.990539i $$0.456180\pi$$
$$702$$ 0 0
$$703$$ −36936.0 −1.98160
$$704$$ 0 0
$$705$$ 21792.0 1.16416
$$706$$ 0 0
$$707$$ 33320.0 1.77246
$$708$$ 0 0
$$709$$ 25418.0 1.34639 0.673197 0.739463i $$-0.264921\pi$$
0.673197 + 0.739463i $$0.264921\pi$$
$$710$$ 0 0
$$711$$ −72.0000 −0.00379777
$$712$$ 0 0
$$713$$ 9152.00 0.480708
$$714$$ 0 0
$$715$$ −7072.00 −0.369899
$$716$$ 0 0
$$717$$ 15678.0 0.816605
$$718$$ 0 0
$$719$$ 20428.0 1.05958 0.529788 0.848130i $$-0.322271\pi$$
0.529788 + 0.848130i $$0.322271\pi$$
$$720$$ 0 0
$$721$$ −6272.00 −0.323969
$$722$$ 0 0
$$723$$ −2286.00 −0.117590
$$724$$ 0 0
$$725$$ −24890.0 −1.27502
$$726$$ 0 0
$$727$$ 38336.0 1.95571 0.977857 0.209276i $$-0.0671107\pi$$
0.977857 + 0.209276i $$0.0671107\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ 19320.0 0.977532
$$732$$ 0 0
$$733$$ −166.000 −0.00836473 −0.00418237 0.999991i $$-0.501331\pi$$
−0.00418237 + 0.999991i $$0.501331\pi$$
$$734$$ 0 0
$$735$$ −21168.0 −1.06230
$$736$$ 0 0
$$737$$ −22304.0 −1.11476
$$738$$ 0 0
$$739$$ 25248.0 1.25678 0.628392 0.777897i $$-0.283714\pi$$
0.628392 + 0.777897i $$0.283714\pi$$
$$740$$ 0 0
$$741$$ 4212.00 0.208815
$$742$$ 0 0
$$743$$ 4442.00 0.219329 0.109664 0.993969i $$-0.465022\pi$$
0.109664 + 0.993969i $$0.465022\pi$$
$$744$$ 0 0
$$745$$ −24384.0 −1.19914
$$746$$ 0 0
$$747$$ −6858.00 −0.335905
$$748$$ 0 0
$$749$$ −17920.0 −0.874209
$$750$$ 0 0
$$751$$ 19848.0 0.964399 0.482200 0.876061i $$-0.339838\pi$$
0.482200 + 0.876061i $$0.339838\pi$$
$$752$$ 0 0
$$753$$ −9720.00 −0.470407
$$754$$ 0 0
$$755$$ −48384.0 −2.33228
$$756$$ 0 0
$$757$$ −29166.0 −1.40034 −0.700169 0.713977i $$-0.746892\pi$$
−0.700169 + 0.713977i $$0.746892\pi$$
$$758$$ 0 0
$$759$$ −5304.00 −0.253653
$$760$$ 0 0
$$761$$ −6240.00 −0.297240 −0.148620 0.988894i $$-0.547483\pi$$
−0.148620 + 0.988894i $$0.547483\pi$$
$$762$$ 0 0
$$763$$ 54152.0 2.56938
$$764$$ 0 0
$$765$$ −19872.0 −0.939181
$$766$$ 0 0
$$767$$ −2002.00 −0.0942478
$$768$$ 0 0
$$769$$ −39750.0 −1.86401 −0.932004 0.362449i $$-0.881941\pi$$
−0.932004 + 0.362449i $$0.881941\pi$$
$$770$$ 0 0
$$771$$ −4158.00 −0.194224
$$772$$ 0 0
$$773$$ 9764.00 0.454317 0.227158 0.973858i $$-0.427057\pi$$
0.227158 + 0.973858i $$0.427057\pi$$
$$774$$ 0 0
$$775$$ 23056.0 1.06864
$$776$$ 0 0
$$777$$ −28728.0 −1.32640
$$778$$ 0 0
$$779$$ −25920.0 −1.19214
$$780$$ 0 0
$$781$$ 18700.0 0.856772
$$782$$ 0 0
$$783$$ −5130.00 −0.234140
$$784$$ 0 0
$$785$$ −35168.0 −1.59898
$$786$$ 0 0
$$787$$ 36016.0 1.63130 0.815649 0.578547i $$-0.196380\pi$$
0.815649 + 0.578547i $$0.196380\pi$$
$$788$$ 0 0
$$789$$ −9900.00 −0.446704
$$790$$ 0 0
$$791$$ 11704.0 0.526102
$$792$$ 0 0
$$793$$ −442.000 −0.0197930
$$794$$ 0 0
$$795$$ −9504.00 −0.423990
$$796$$ 0 0
$$797$$ −22290.0 −0.990655 −0.495328 0.868706i $$-0.664952\pi$$
−0.495328 + 0.868706i $$0.664952\pi$$
$$798$$ 0 0
$$799$$ −62652.0 −2.77405
$$800$$ 0 0
$$801$$ −3996.00 −0.176269
$$802$$ 0 0
$$803$$ −20876.0 −0.917432
$$804$$ 0 0
$$805$$ 23296.0 1.01997
$$806$$ 0 0
$$807$$ 12870.0 0.561395
$$808$$ 0 0
$$809$$ −25578.0 −1.11159 −0.555794 0.831320i $$-0.687586\pi$$
−0.555794 + 0.831320i $$0.687586\pi$$
$$810$$ 0 0
$$811$$ −29900.0 −1.29461 −0.647306 0.762230i $$-0.724105\pi$$
−0.647306 + 0.762230i $$0.724105\pi$$
$$812$$ 0 0
$$813$$ −7356.00 −0.317326
$$814$$ 0 0
$$815$$ −4288.00 −0.184297
$$816$$ 0 0
$$817$$ −15120.0 −0.647469
$$818$$ 0 0
$$819$$ 3276.00 0.139771
$$820$$ 0 0
$$821$$ 16412.0 0.697665 0.348832 0.937185i $$-0.386578\pi$$
0.348832 + 0.937185i $$0.386578\pi$$
$$822$$ 0 0
$$823$$ −18552.0 −0.785762 −0.392881 0.919589i $$-0.628522\pi$$
−0.392881 + 0.919589i $$0.628522\pi$$
$$824$$ 0 0
$$825$$ −13362.0 −0.563885
$$826$$ 0 0
$$827$$ 28662.0 1.20517 0.602585 0.798055i $$-0.294137\pi$$
0.602585 + 0.798055i $$0.294137\pi$$
$$828$$ 0 0
$$829$$ −3686.00 −0.154427 −0.0772136 0.997015i $$-0.524602\pi$$
−0.0772136 + 0.997015i $$0.524602\pi$$
$$830$$ 0 0
$$831$$ −126.000 −0.00525980
$$832$$ 0 0
$$833$$ 60858.0 2.53134
$$834$$ 0 0
$$835$$ 11232.0 0.465508
$$836$$ 0 0
$$837$$ 4752.00 0.196240
$$838$$ 0 0
$$839$$ −13370.0 −0.550159 −0.275080 0.961421i $$-0.588704\pi$$
−0.275080 + 0.961421i $$0.588704\pi$$
$$840$$ 0 0
$$841$$ 11711.0 0.480175
$$842$$ 0 0
$$843$$ −6864.00 −0.280437
$$844$$ 0 0
$$845$$ −2704.00 −0.110083
$$846$$ 0 0
$$847$$ 4900.00 0.198779
$$848$$ 0 0
$$849$$ −3468.00 −0.140190
$$850$$ 0 0
$$851$$ 17784.0 0.716366
$$852$$ 0 0
$$853$$ 11398.0 0.457515 0.228757 0.973483i $$-0.426534\pi$$
0.228757 + 0.973483i $$0.426534\pi$$
$$854$$ 0 0
$$855$$ 15552.0 0.622067
$$856$$ 0 0
$$857$$ 7990.00 0.318475 0.159238 0.987240i $$-0.449096\pi$$
0.159238 + 0.987240i $$0.449096\pi$$
$$858$$ 0 0
$$859$$ −7652.00 −0.303938 −0.151969 0.988385i $$-0.548561\pi$$
−0.151969 + 0.988385i $$0.548561\pi$$
$$860$$ 0 0
$$861$$ −20160.0 −0.797969
$$862$$ 0 0
$$863$$ 1022.00 0.0403120 0.0201560 0.999797i $$-0.493584\pi$$
0.0201560 + 0.999797i $$0.493584\pi$$
$$864$$ 0 0
$$865$$ −33056.0 −1.29935
$$866$$ 0 0
$$867$$ 42393.0 1.66060
$$868$$ 0 0
$$869$$ 272.000 0.0106179
$$870$$ 0 0
$$871$$ −8528.00 −0.331757
$$872$$ 0 0
$$873$$ 9198.00 0.356592
$$874$$ 0 0
$$875$$ 2688.00 0.103853
$$876$$ 0 0
$$877$$ −15546.0 −0.598576 −0.299288 0.954163i $$-0.596749\pi$$
−0.299288 + 0.954163i $$0.596749\pi$$
$$878$$ 0 0
$$879$$ −26052.0 −0.999673
$$880$$ 0 0
$$881$$ 11310.0 0.432513 0.216256 0.976337i $$-0.430615\pi$$
0.216256 + 0.976337i $$0.430615\pi$$
$$882$$ 0 0
$$883$$ −17260.0 −0.657809 −0.328904 0.944363i $$-0.606679\pi$$
−0.328904 + 0.944363i $$0.606679\pi$$
$$884$$ 0 0
$$885$$ −7392.00 −0.280768
$$886$$ 0 0
$$887$$ −832.000 −0.0314947 −0.0157474 0.999876i $$-0.505013\pi$$
−0.0157474 + 0.999876i $$0.505013\pi$$
$$888$$ 0 0
$$889$$ −29120.0 −1.09860
$$890$$ 0 0
$$891$$ −2754.00 −0.103549
$$892$$ 0 0
$$893$$ 49032.0 1.83739
$$894$$ 0 0
$$895$$ −4416.00 −0.164928
$$896$$ 0 0
$$897$$ −2028.00 −0.0754882
$$898$$ 0 0
$$899$$ −33440.0 −1.24059
$$900$$ 0 0
$$901$$ 27324.0 1.01032
$$902$$ 0 0
$$903$$ −11760.0 −0.433387
$$904$$ 0 0
$$905$$ 55584.0 2.04163
$$906$$ 0 0
$$907$$ −31740.0 −1.16197 −0.580986 0.813913i $$-0.697333\pi$$
−0.580986 + 0.813913i $$0.697333\pi$$
$$908$$ 0 0
$$909$$ −10710.0 −0.390790
$$910$$ 0 0
$$911$$ 23568.0 0.857127 0.428563 0.903512i $$-0.359020\pi$$
0.428563 + 0.903512i $$0.359020\pi$$
$$912$$ 0 0
$$913$$ 25908.0 0.939134
$$914$$ 0 0
$$915$$ −1632.00 −0.0589642
$$916$$ 0 0
$$917$$ −15904.0 −0.572733
$$918$$ 0 0
$$919$$ 18864.0 0.677112 0.338556 0.940946i $$-0.390062\pi$$
0.338556 + 0.940946i $$0.390062\pi$$
$$920$$ 0 0
$$921$$ 22656.0 0.810576
$$922$$ 0 0
$$923$$ 7150.00 0.254978
$$924$$ 0 0
$$925$$ 44802.0 1.59252
$$926$$ 0 0
$$927$$ 2016.00 0.0714284
$$928$$ 0 0
$$929$$ −19536.0 −0.689941 −0.344971 0.938613i $$-0.612111\pi$$
−0.344971 + 0.938613i $$0.612111\pi$$
$$930$$ 0 0
$$931$$ −47628.0 −1.67663
$$932$$ 0 0
$$933$$ −7956.00 −0.279172
$$934$$ 0 0
$$935$$ 75072.0 2.62579
$$936$$ 0 0
$$937$$ 18174.0 0.633638 0.316819 0.948486i $$-0.397385\pi$$
0.316819 + 0.948486i $$0.397385\pi$$
$$938$$ 0 0
$$939$$ −13278.0 −0.461460
$$940$$ 0 0
$$941$$ 51172.0 1.77275 0.886376 0.462966i $$-0.153215\pi$$
0.886376 + 0.462966i $$0.153215\pi$$
$$942$$ 0 0
$$943$$ 12480.0 0.430970
$$944$$ 0 0
$$945$$ 12096.0 0.416384
$$946$$ 0 0
$$947$$ −3726.00 −0.127855 −0.0639275 0.997955i $$-0.520363\pi$$
−0.0639275 + 0.997955i $$0.520363\pi$$
$$948$$ 0 0
$$949$$ −7982.00 −0.273031
$$950$$ 0 0
$$951$$ −14832.0 −0.505742
$$952$$ 0 0
$$953$$ −40498.0 −1.37656 −0.688279 0.725447i $$-0.741633\pi$$
−0.688279 + 0.725447i $$0.741633\pi$$
$$954$$ 0 0
$$955$$ −62720.0 −2.12521
$$956$$ 0 0
$$957$$ 19380.0 0.654615
$$958$$ 0 0
$$959$$ −14784.0 −0.497810
$$960$$ 0 0
$$961$$ 1185.00 0.0397771
$$962$$ 0 0
$$963$$ 5760.00 0.192745
$$964$$ 0 0
$$965$$ −34976.0 −1.16675
$$966$$ 0 0
$$967$$ −28568.0 −0.950036 −0.475018 0.879976i $$-0.657558\pi$$
−0.475018 + 0.879976i $$0.657558\pi$$
$$968$$ 0 0
$$969$$ −44712.0 −1.48231
$$970$$ 0 0
$$971$$ 8676.00 0.286742 0.143371 0.989669i $$-0.454206\pi$$
0.143371 + 0.989669i $$0.454206\pi$$
$$972$$ 0 0
$$973$$ −43568.0 −1.43548
$$974$$ 0 0
$$975$$ −5109.00 −0.167814
$$976$$ 0 0
$$977$$ −2796.00 −0.0915578 −0.0457789 0.998952i $$-0.514577\pi$$
−0.0457789 + 0.998952i $$0.514577\pi$$
$$978$$ 0 0
$$979$$ 15096.0 0.492819
$$980$$ 0 0
$$981$$ −17406.0 −0.566494
$$982$$ 0 0
$$983$$ 406.000 0.0131733 0.00658667 0.999978i $$-0.497903\pi$$
0.00658667 + 0.999978i $$0.497903\pi$$
$$984$$ 0 0
$$985$$ 21888.0 0.708030
$$986$$ 0 0
$$987$$ 38136.0 1.22987
$$988$$ 0 0
$$989$$ 7280.00 0.234065
$$990$$ 0 0
$$991$$ 23232.0 0.744691 0.372346 0.928094i $$-0.378554\pi$$
0.372346 + 0.928094i $$0.378554\pi$$
$$992$$ 0 0
$$993$$ 18264.0 0.583676
$$994$$ 0 0
$$995$$ −17152.0 −0.546487
$$996$$ 0 0
$$997$$ 6110.00 0.194088 0.0970440 0.995280i $$-0.469061\pi$$
0.0970440 + 0.995280i $$0.469061\pi$$
$$998$$ 0 0
$$999$$ 9234.00 0.292443
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 624.4.a.f.1.1 1
3.2 odd 2 1872.4.a.o.1.1 1
4.3 odd 2 78.4.a.a.1.1 1
8.3 odd 2 2496.4.a.q.1.1 1
8.5 even 2 2496.4.a.g.1.1 1
12.11 even 2 234.4.a.k.1.1 1
20.19 odd 2 1950.4.a.o.1.1 1
52.31 even 4 1014.4.b.a.337.2 2
52.47 even 4 1014.4.b.a.337.1 2
52.51 odd 2 1014.4.a.i.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
78.4.a.a.1.1 1 4.3 odd 2
234.4.a.k.1.1 1 12.11 even 2
624.4.a.f.1.1 1 1.1 even 1 trivial
1014.4.a.i.1.1 1 52.51 odd 2
1014.4.b.a.337.1 2 52.47 even 4
1014.4.b.a.337.2 2 52.31 even 4
1872.4.a.o.1.1 1 3.2 odd 2
1950.4.a.o.1.1 1 20.19 odd 2
2496.4.a.g.1.1 1 8.5 even 2
2496.4.a.q.1.1 1 8.3 odd 2