# Properties

 Label 2496.4.a.g.1.1 Level $2496$ Weight $4$ Character 2496.1 Self dual yes Analytic conductor $147.269$ 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] = [2496,4,Mod(1,2496)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2496, 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("2496.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2496 = 2^{6} \cdot 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2496.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: $$147.268767374$$ 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 $$\chi$$ $$=$$ 2496.1

## $q$-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 $$-0.246180\pi$$
0.715542 + 0.698570i $$0.246180\pi$$
$$6$$ 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 $$-0.345705\pi$$
0.465972 + 0.884799i $$0.345705\pi$$
$$12$$ 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 $$-0.273920\pi$$
0.652024 + 0.758199i $$0.273920\pi$$
$$20$$ 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 $$-0.291847\pi$$
0.608312 + 0.793698i $$0.291847\pi$$
$$30$$ 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 $$-0.774698\pi$$
−0.759790 + 0.650169i $$0.774698\pi$$
$$38$$ 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 $$-0.579857\pi$$
−0.248253 + 0.968695i $$0.579857\pi$$
$$44$$ 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 $$-0.582595\pi$$
−0.256579 + 0.966523i $$0.582595\pi$$
$$54$$ 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 $$-0.554347\pi$$
−0.169908 + 0.985460i $$0.554347\pi$$
$$60$$ 0 0
$$61$$ −34.0000 −0.0713648 −0.0356824 0.999363i $$-0.511360\pi$$
−0.0356824 + 0.999363i $$0.511360\pi$$
$$62$$ 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 $$-0.704071\pi$$
−0.598083 + 0.801434i $$0.704071\pi$$
$$68$$ 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 $$-0.331914\pi$$
0.503858 + 0.863787i $$0.331914\pi$$
$$84$$ 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 $$-0.300629\pi$$
0.586185 + 0.810177i $$0.300629\pi$$
$$102$$ 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 $$-0.593362\pi$$
−0.289117 + 0.957294i $$0.593362\pi$$
$$108$$ 0 0
$$109$$ 1934.00 1.69948 0.849741 0.527200i $$-0.176758\pi$$
0.849741 + 0.527200i $$0.176758\pi$$
$$110$$ 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 $$-0.560659\pi$$
−0.189414 + 0.981897i $$0.560659\pi$$
$$132$$ 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 $$-0.657459\pi$$
−0.474742 + 0.880125i $$0.657459\pi$$
$$140$$ 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 $$-0.637606\pi$$
−0.418963 + 0.908003i $$0.637606\pi$$
$$150$$ 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 $$-0.688685\pi$$
−0.558661 + 0.829396i $$0.688685\pi$$
$$158$$ 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 $$-0.520510\pi$$
−0.0643907 + 0.997925i $$0.520510\pi$$
$$164$$ 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 $$-0.649994\pi$$
−0.453974 + 0.891015i $$0.649994\pi$$
$$174$$ 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 $$-0.518352\pi$$
−0.0576235 + 0.998338i $$0.518352\pi$$
$$180$$ 0 0
$$181$$ 3474.00 1.42663 0.713316 0.700843i $$-0.247192\pi$$
0.713316 + 0.700843i $$0.247192\pi$$
$$182$$ 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 $$-0.420432\pi$$
0.247376 + 0.968920i $$0.420432\pi$$
$$198$$ 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 $$-0.152022\pi$$
0.888105 + 0.459640i $$0.152022\pi$$
$$212$$ 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 $$-0.490785\pi$$
0.0289465 + 0.999581i $$0.490785\pi$$
$$228$$ 0 0
$$229$$ −5922.00 −1.70889 −0.854447 0.519538i $$-0.826104\pi$$
−0.854447 + 0.519538i $$0.826104\pi$$
$$230$$ 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 $$-0.366441\pi$$
0.407384 + 0.913257i $$0.366441\pi$$
$$252$$ 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 $$-0.661611\pi$$
−0.486182 + 0.873858i $$0.661611\pi$$
$$270$$ 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 $$-0.498550\pi$$
0.00455512 + 0.999990i $$0.498550\pi$$
$$278$$ 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 $$-0.461259\pi$$
0.121408 + 0.992603i $$0.461259\pi$$
$$284$$ 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 $$-0.166847\pi$$
0.865742 + 0.500491i $$0.166847\pi$$
$$294$$ 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 $$-0.747700\pi$$
−0.701979 + 0.712197i $$0.747700\pi$$
$$308$$ 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 $$-0.355692\pi$$
0.437985 + 0.898982i $$0.355692\pi$$
$$318$$ 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 $$-0.668684\pi$$
−0.505478 + 0.862839i $$0.668684\pi$$
$$332$$ 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 $$-0.556734\pi$$
−0.177293 + 0.984158i $$0.556734\pi$$
$$348$$ 0 0
$$349$$ 9866.00 1.51322 0.756612 0.653865i $$-0.226853\pi$$
0.756612 + 0.653865i $$0.226853\pi$$
$$350$$ 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 $$-0.612671\pi$$
−0.346621 + 0.938005i $$0.612671\pi$$
$$374$$ 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 $$-0.471922\pi$$
0.0880957 + 0.996112i $$0.471922\pi$$
$$380$$ 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 $$-0.573462\pi$$
−0.228746 + 0.973486i $$0.573462\pi$$
$$390$$ 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 $$-0.628839\pi$$
−0.393797 + 0.919197i $$0.628839\pi$$
$$398$$ 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 $$-0.831586\pi$$
−0.863267 + 0.504747i $$0.831586\pi$$
$$420$$ 0 0
$$421$$ −10354.0 −1.19863 −0.599315 0.800513i $$-0.704560\pi$$
−0.599315 + 0.800513i $$0.704560\pi$$
$$422$$ 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 $$-0.547285\pi$$
−0.148004 + 0.988987i $$0.547285\pi$$
$$444$$ 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 $$-0.389346\pi$$
0.340672 + 0.940182i $$0.389346\pi$$
$$462$$ 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 $$-0.351049\pi$$
0.451052 + 0.892498i $$0.351049\pi$$
$$468$$ 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 $$-0.289568\pi$$
0.613980 + 0.789322i $$0.289568\pi$$
$$492$$ 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 $$-0.215698\pi$$
0.779057 + 0.626954i $$0.215698\pi$$
$$500$$ 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 $$-0.654953\pi$$
−0.467799 + 0.883835i $$0.654953\pi$$
$$510$$ 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 $$-0.475865\pi$$
0.0757488 + 0.997127i $$0.475865\pi$$
$$524$$ 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 $$-0.577903\pi$$
−0.242305 + 0.970200i $$0.577903\pi$$
$$542$$ 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 $$-0.754244\pi$$
−0.716471 + 0.697616i $$0.754244\pi$$
$$548$$ 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 $$-0.224778\pi$$
0.760859 + 0.648917i $$0.224778\pi$$
$$558$$ 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 $$-0.365086\pi$$
0.411269 + 0.911514i $$0.365086\pi$$
$$564$$ 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 $$-0.508259\pi$$
−0.0259447 + 0.999663i $$0.508259\pi$$
$$572$$ 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 $$-0.716001\pi$$
−0.627695 + 0.778460i $$0.716001\pi$$
$$588$$ 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 $$-0.278945\pi$$
0.639975 + 0.768396i $$0.278945\pi$$
$$614$$ 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 $$-0.733863\pi$$
−0.670366 + 0.742031i $$0.733863\pi$$
$$620$$ 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 $$-0.729413\pi$$
−0.659927 + 0.751330i $$0.729413\pi$$
$$644$$ 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 $$-0.529514\pi$$
−0.0925889 + 0.995704i $$0.529514\pi$$
$$654$$ 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 $$-0.629905\pi$$
−0.396875 + 0.917873i $$0.629905\pi$$
$$660$$ 0 0
$$661$$ −22598.0 −1.32974 −0.664872 0.746958i $$-0.731514\pi$$
−0.664872 + 0.746958i $$0.731514\pi$$
$$662$$ 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 $$-0.719315\pi$$
−0.635764 + 0.771883i $$0.719315\pi$$
$$678$$ 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 $$-0.396447\pi$$
0.319613 + 0.947548i $$0.396447\pi$$
$$684$$ 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 $$-0.147685\pi$$
0.894285 + 0.447498i $$0.147685\pi$$
$$692$$ 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 $$-0.543820\pi$$
−0.137231 + 0.990539i $$0.543820\pi$$
$$702$$ 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 $$-0.735079\pi$$
−0.673197 + 0.739463i $$0.735079\pi$$
$$710$$ 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 $$-0.498669\pi$$
0.00418237 + 0.999991i $$0.498669\pi$$
$$734$$ 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 $$-0.716286\pi$$
−0.628392 + 0.777897i $$0.716286\pi$$
$$740$$ 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 $$-0.253108\pi$$
0.700169 + 0.713977i $$0.253108\pi$$
$$758$$ 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 $$-0.572943\pi$$
−0.227158 + 0.973858i $$0.572943\pi$$
$$774$$ 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 $$-0.803620\pi$$
−0.815649 + 0.578547i $$0.803620\pi$$
$$788$$ 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 $$-0.335048\pi$$
0.495328 + 0.868706i $$0.335048\pi$$
$$798$$ 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 $$-0.275895\pi$$
0.647306 + 0.762230i $$0.275895\pi$$
$$812$$ 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 $$-0.613422\pi$$
−0.348832 + 0.937185i $$0.613422\pi$$
$$822$$ 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 $$-0.705863\pi$$
−0.602585 + 0.798055i $$0.705863\pi$$
$$828$$ 0 0
$$829$$ 3686.00 0.154427 0.0772136 0.997015i $$-0.475398\pi$$
0.0772136 + 0.997015i $$0.475398\pi$$
$$830$$ 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 $$-0.573466\pi$$
−0.228757 + 0.973483i $$0.573466\pi$$
$$854$$ 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 $$-0.451439\pi$$
0.151969 + 0.988385i $$0.451439\pi$$
$$860$$ 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 $$-0.403251\pi$$
0.299288 + 0.954163i $$0.403251\pi$$
$$878$$ 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 $$-0.393321\pi$$
0.328904 + 0.944363i $$0.393321\pi$$
$$884$$ 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 $$-0.302667\pi$$
0.580986 + 0.813913i $$0.302667\pi$$
$$908$$ 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 $$-0.846785\pi$$
−0.886376 + 0.462966i $$0.846785\pi$$
$$942$$ 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 $$-0.479637\pi$$
0.0639275 + 0.997955i $$0.479637\pi$$
$$948$$ 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 $$-0.545794\pi$$
−0.143371 + 0.989669i $$0.545794\pi$$
$$972$$ 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 $$-0.530939\pi$$
−0.0970440 + 0.995280i $$0.530939\pi$$
$$998$$ 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 2496.4.a.g.1.1 1
4.3 odd 2 2496.4.a.q.1.1 1
8.3 odd 2 78.4.a.a.1.1 1
8.5 even 2 624.4.a.f.1.1 1
24.5 odd 2 1872.4.a.o.1.1 1
24.11 even 2 234.4.a.k.1.1 1
40.19 odd 2 1950.4.a.o.1.1 1
104.51 odd 2 1014.4.a.i.1.1 1
104.83 even 4 1014.4.b.a.337.2 2
104.99 even 4 1014.4.b.a.337.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
78.4.a.a.1.1 1 8.3 odd 2
234.4.a.k.1.1 1 24.11 even 2
624.4.a.f.1.1 1 8.5 even 2
1014.4.a.i.1.1 1 104.51 odd 2
1014.4.b.a.337.1 2 104.99 even 4
1014.4.b.a.337.2 2 104.83 even 4
1872.4.a.o.1.1 1 24.5 odd 2
1950.4.a.o.1.1 1 40.19 odd 2
2496.4.a.g.1.1 1 1.1 even 1 trivial
2496.4.a.q.1.1 1 4.3 odd 2