# Properties

 Label 624.4.a.r.1.2 Level $624$ Weight $4$ Character 624.1 Self dual yes Analytic conductor $36.817$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [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: $$2$$ Coefficient field: $$\Q(\sqrt{14})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 14$$ x^2 - 14 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 39) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$3.74166$$ of defining polynomial 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)

 $$f(q)$$ $$=$$ $$q+3.00000 q^{3} +19.4833 q^{5} -7.48331 q^{7} +9.00000 q^{9} +O(q^{10})$$ $$q+3.00000 q^{3} +19.4833 q^{5} -7.48331 q^{7} +9.00000 q^{9} -22.8999 q^{11} -13.0000 q^{13} +58.4499 q^{15} +67.0334 q^{17} -16.5167 q^{19} -22.4499 q^{21} +175.600 q^{23} +254.600 q^{25} +27.0000 q^{27} +291.800 q^{29} -117.283 q^{31} -68.6997 q^{33} -145.800 q^{35} -154.766 q^{37} -39.0000 q^{39} -251.716 q^{41} +502.566 q^{43} +175.350 q^{45} +281.733 q^{47} -287.000 q^{49} +201.100 q^{51} +366.999 q^{53} -446.166 q^{55} -49.5501 q^{57} +79.6663 q^{59} -194.865 q^{61} -67.3498 q^{63} -253.283 q^{65} -400.082 q^{67} +526.799 q^{69} -528.299 q^{71} -734.366 q^{73} +763.799 q^{75} +171.367 q^{77} -113.266 q^{79} +81.0000 q^{81} +933.466 q^{83} +1306.03 q^{85} +875.399 q^{87} +1190.91 q^{89} +97.2831 q^{91} -351.849 q^{93} -321.800 q^{95} +557.165 q^{97} -206.099 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{3} + 24 q^{5} + 18 q^{9}+O(q^{10})$$ 2 * q + 6 * q^3 + 24 * q^5 + 18 * q^9 $$2 q + 6 q^{3} + 24 q^{5} + 18 q^{9} + 44 q^{11} - 26 q^{13} + 72 q^{15} + 164 q^{17} - 48 q^{19} - 8 q^{23} + 150 q^{25} + 54 q^{27} + 404 q^{29} - 40 q^{31} + 132 q^{33} - 112 q^{35} - 100 q^{37} - 78 q^{39} + 200 q^{41} + 616 q^{43} + 216 q^{45} + 324 q^{47} - 574 q^{49} + 492 q^{51} - 164 q^{53} - 144 q^{55} - 144 q^{57} - 140 q^{59} + 628 q^{61} - 312 q^{65} + 472 q^{67} - 24 q^{69} - 428 q^{71} - 900 q^{73} + 450 q^{75} + 672 q^{77} + 432 q^{79} + 162 q^{81} + 1388 q^{83} + 1744 q^{85} + 1212 q^{87} + 960 q^{89} - 120 q^{93} - 464 q^{95} - 532 q^{97} + 396 q^{99}+O(q^{100})$$ 2 * q + 6 * q^3 + 24 * q^5 + 18 * q^9 + 44 * q^11 - 26 * q^13 + 72 * q^15 + 164 * q^17 - 48 * q^19 - 8 * q^23 + 150 * q^25 + 54 * q^27 + 404 * q^29 - 40 * q^31 + 132 * q^33 - 112 * q^35 - 100 * q^37 - 78 * q^39 + 200 * q^41 + 616 * q^43 + 216 * q^45 + 324 * q^47 - 574 * q^49 + 492 * q^51 - 164 * q^53 - 144 * q^55 - 144 * q^57 - 140 * q^59 + 628 * q^61 - 312 * q^65 + 472 * q^67 - 24 * q^69 - 428 * q^71 - 900 * q^73 + 450 * q^75 + 672 * q^77 + 432 * q^79 + 162 * q^81 + 1388 * q^83 + 1744 * q^85 + 1212 * q^87 + 960 * q^89 - 120 * q^93 - 464 * q^95 - 532 * q^97 + 396 * 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
$$5$$ 19.4833 1.74264 0.871320 0.490715i $$-0.163264\pi$$
0.871320 + 0.490715i $$0.163264\pi$$
$$6$$ 0 0
$$7$$ −7.48331 −0.404061 −0.202031 0.979379i $$-0.564754\pi$$
−0.202031 + 0.979379i $$0.564754\pi$$
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ −22.8999 −0.627689 −0.313844 0.949474i $$-0.601617\pi$$
−0.313844 + 0.949474i $$0.601617\pi$$
$$12$$ 0 0
$$13$$ −13.0000 −0.277350
$$14$$ 0 0
$$15$$ 58.4499 1.00611
$$16$$ 0 0
$$17$$ 67.0334 0.956352 0.478176 0.878264i $$-0.341298\pi$$
0.478176 + 0.878264i $$0.341298\pi$$
$$18$$ 0 0
$$19$$ −16.5167 −0.199431 −0.0997155 0.995016i $$-0.531793\pi$$
−0.0997155 + 0.995016i $$0.531793\pi$$
$$20$$ 0 0
$$21$$ −22.4499 −0.233285
$$22$$ 0 0
$$23$$ 175.600 1.59196 0.795979 0.605324i $$-0.206956\pi$$
0.795979 + 0.605324i $$0.206956\pi$$
$$24$$ 0 0
$$25$$ 254.600 2.03680
$$26$$ 0 0
$$27$$ 27.0000 0.192450
$$28$$ 0 0
$$29$$ 291.800 1.86848 0.934239 0.356648i $$-0.116080\pi$$
0.934239 + 0.356648i $$0.116080\pi$$
$$30$$ 0 0
$$31$$ −117.283 −0.679505 −0.339753 0.940515i $$-0.610343\pi$$
−0.339753 + 0.940515i $$0.610343\pi$$
$$32$$ 0 0
$$33$$ −68.6997 −0.362396
$$34$$ 0 0
$$35$$ −145.800 −0.704133
$$36$$ 0 0
$$37$$ −154.766 −0.687661 −0.343830 0.939032i $$-0.611724\pi$$
−0.343830 + 0.939032i $$0.611724\pi$$
$$38$$ 0 0
$$39$$ −39.0000 −0.160128
$$40$$ 0 0
$$41$$ −251.716 −0.958815 −0.479407 0.877592i $$-0.659148\pi$$
−0.479407 + 0.877592i $$0.659148\pi$$
$$42$$ 0 0
$$43$$ 502.566 1.78234 0.891170 0.453669i $$-0.149885\pi$$
0.891170 + 0.453669i $$0.149885\pi$$
$$44$$ 0 0
$$45$$ 175.350 0.580880
$$46$$ 0 0
$$47$$ 281.733 0.874361 0.437181 0.899374i $$-0.355977\pi$$
0.437181 + 0.899374i $$0.355977\pi$$
$$48$$ 0 0
$$49$$ −287.000 −0.836735
$$50$$ 0 0
$$51$$ 201.100 0.552150
$$52$$ 0 0
$$53$$ 366.999 0.951154 0.475577 0.879674i $$-0.342239\pi$$
0.475577 + 0.879674i $$0.342239\pi$$
$$54$$ 0 0
$$55$$ −446.166 −1.09384
$$56$$ 0 0
$$57$$ −49.5501 −0.115141
$$58$$ 0 0
$$59$$ 79.6663 0.175791 0.0878955 0.996130i $$-0.471986\pi$$
0.0878955 + 0.996130i $$0.471986\pi$$
$$60$$ 0 0
$$61$$ −194.865 −0.409016 −0.204508 0.978865i $$-0.565559\pi$$
−0.204508 + 0.978865i $$0.565559\pi$$
$$62$$ 0 0
$$63$$ −67.3498 −0.134687
$$64$$ 0 0
$$65$$ −253.283 −0.483322
$$66$$ 0 0
$$67$$ −400.082 −0.729519 −0.364759 0.931102i $$-0.618849\pi$$
−0.364759 + 0.931102i $$0.618849\pi$$
$$68$$ 0 0
$$69$$ 526.799 0.919117
$$70$$ 0 0
$$71$$ −528.299 −0.883065 −0.441532 0.897245i $$-0.645565\pi$$
−0.441532 + 0.897245i $$0.645565\pi$$
$$72$$ 0 0
$$73$$ −734.366 −1.17741 −0.588706 0.808347i $$-0.700362\pi$$
−0.588706 + 0.808347i $$0.700362\pi$$
$$74$$ 0 0
$$75$$ 763.799 1.17594
$$76$$ 0 0
$$77$$ 171.367 0.253625
$$78$$ 0 0
$$79$$ −113.266 −0.161309 −0.0806545 0.996742i $$-0.525701\pi$$
−0.0806545 + 0.996742i $$0.525701\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ 933.466 1.23447 0.617236 0.786778i $$-0.288252\pi$$
0.617236 + 0.786778i $$0.288252\pi$$
$$84$$ 0 0
$$85$$ 1306.03 1.66658
$$86$$ 0 0
$$87$$ 875.399 1.07877
$$88$$ 0 0
$$89$$ 1190.91 1.41839 0.709195 0.705012i $$-0.249059\pi$$
0.709195 + 0.705012i $$0.249059\pi$$
$$90$$ 0 0
$$91$$ 97.2831 0.112066
$$92$$ 0 0
$$93$$ −351.849 −0.392313
$$94$$ 0 0
$$95$$ −321.800 −0.347536
$$96$$ 0 0
$$97$$ 557.165 0.583211 0.291606 0.956539i $$-0.405811\pi$$
0.291606 + 0.956539i $$0.405811\pi$$
$$98$$ 0 0
$$99$$ −206.099 −0.209230
$$100$$ 0 0
$$101$$ −286.766 −0.282518 −0.141259 0.989973i $$-0.545115\pi$$
−0.141259 + 0.989973i $$0.545115\pi$$
$$102$$ 0 0
$$103$$ 1911.36 1.82847 0.914234 0.405187i $$-0.132794\pi$$
0.914234 + 0.405187i $$0.132794\pi$$
$$104$$ 0 0
$$105$$ −437.399 −0.406531
$$106$$ 0 0
$$107$$ −834.334 −0.753814 −0.376907 0.926251i $$-0.623012\pi$$
−0.376907 + 0.926251i $$0.623012\pi$$
$$108$$ 0 0
$$109$$ −1077.66 −0.946986 −0.473493 0.880798i $$-0.657007\pi$$
−0.473493 + 0.880798i $$0.657007\pi$$
$$110$$ 0 0
$$111$$ −464.299 −0.397021
$$112$$ 0 0
$$113$$ −166.065 −0.138248 −0.0691241 0.997608i $$-0.522020\pi$$
−0.0691241 + 0.997608i $$0.522020\pi$$
$$114$$ 0 0
$$115$$ 3421.26 2.77421
$$116$$ 0 0
$$117$$ −117.000 −0.0924500
$$118$$ 0 0
$$119$$ −501.632 −0.386424
$$120$$ 0 0
$$121$$ −806.595 −0.606007
$$122$$ 0 0
$$123$$ −755.147 −0.553572
$$124$$ 0 0
$$125$$ 2525.03 1.80676
$$126$$ 0 0
$$127$$ −1296.16 −0.905637 −0.452819 0.891603i $$-0.649581\pi$$
−0.452819 + 0.891603i $$0.649581\pi$$
$$128$$ 0 0
$$129$$ 1507.70 1.02903
$$130$$ 0 0
$$131$$ 197.201 0.131523 0.0657617 0.997835i $$-0.479052\pi$$
0.0657617 + 0.997835i $$0.479052\pi$$
$$132$$ 0 0
$$133$$ 123.600 0.0805823
$$134$$ 0 0
$$135$$ 526.049 0.335371
$$136$$ 0 0
$$137$$ −546.915 −0.341066 −0.170533 0.985352i $$-0.554549\pi$$
−0.170533 + 0.985352i $$0.554549\pi$$
$$138$$ 0 0
$$139$$ −609.666 −0.372023 −0.186012 0.982548i $$-0.559556\pi$$
−0.186012 + 0.982548i $$0.559556\pi$$
$$140$$ 0 0
$$141$$ 845.199 0.504813
$$142$$ 0 0
$$143$$ 297.699 0.174090
$$144$$ 0 0
$$145$$ 5685.23 3.25609
$$146$$ 0 0
$$147$$ −861.000 −0.483089
$$148$$ 0 0
$$149$$ −2165.08 −1.19040 −0.595202 0.803576i $$-0.702928\pi$$
−0.595202 + 0.803576i $$0.702928\pi$$
$$150$$ 0 0
$$151$$ 846.549 0.456233 0.228116 0.973634i $$-0.426743\pi$$
0.228116 + 0.973634i $$0.426743\pi$$
$$152$$ 0 0
$$153$$ 603.300 0.318784
$$154$$ 0 0
$$155$$ −2285.06 −1.18413
$$156$$ 0 0
$$157$$ 1653.60 0.840581 0.420291 0.907390i $$-0.361928\pi$$
0.420291 + 0.907390i $$0.361928\pi$$
$$158$$ 0 0
$$159$$ 1101.00 0.549149
$$160$$ 0 0
$$161$$ −1314.07 −0.643248
$$162$$ 0 0
$$163$$ 2866.51 1.37744 0.688720 0.725027i $$-0.258173\pi$$
0.688720 + 0.725027i $$0.258173\pi$$
$$164$$ 0 0
$$165$$ −1338.50 −0.631526
$$166$$ 0 0
$$167$$ −729.066 −0.337825 −0.168913 0.985631i $$-0.554026\pi$$
−0.168913 + 0.985631i $$0.554026\pi$$
$$168$$ 0 0
$$169$$ 169.000 0.0769231
$$170$$ 0 0
$$171$$ −148.650 −0.0664770
$$172$$ 0 0
$$173$$ −3834.83 −1.68530 −0.842650 0.538462i $$-0.819005\pi$$
−0.842650 + 0.538462i $$0.819005\pi$$
$$174$$ 0 0
$$175$$ −1905.25 −0.822990
$$176$$ 0 0
$$177$$ 238.999 0.101493
$$178$$ 0 0
$$179$$ 283.862 0.118530 0.0592649 0.998242i $$-0.481124\pi$$
0.0592649 + 0.998242i $$0.481124\pi$$
$$180$$ 0 0
$$181$$ 2363.60 0.970634 0.485317 0.874338i $$-0.338704\pi$$
0.485317 + 0.874338i $$0.338704\pi$$
$$182$$ 0 0
$$183$$ −584.596 −0.236145
$$184$$ 0 0
$$185$$ −3015.36 −1.19835
$$186$$ 0 0
$$187$$ −1535.06 −0.600291
$$188$$ 0 0
$$189$$ −202.049 −0.0777616
$$190$$ 0 0
$$191$$ −2514.26 −0.952491 −0.476246 0.879312i $$-0.658003\pi$$
−0.476246 + 0.879312i $$0.658003\pi$$
$$192$$ 0 0
$$193$$ 2420.73 0.902839 0.451420 0.892312i $$-0.350918\pi$$
0.451420 + 0.892312i $$0.350918\pi$$
$$194$$ 0 0
$$195$$ −759.849 −0.279046
$$196$$ 0 0
$$197$$ −4633.65 −1.67581 −0.837903 0.545819i $$-0.816219\pi$$
−0.837903 + 0.545819i $$0.816219\pi$$
$$198$$ 0 0
$$199$$ −3054.17 −1.08796 −0.543980 0.839098i $$-0.683083\pi$$
−0.543980 + 0.839098i $$0.683083\pi$$
$$200$$ 0 0
$$201$$ −1200.25 −0.421188
$$202$$ 0 0
$$203$$ −2183.63 −0.754979
$$204$$ 0 0
$$205$$ −4904.26 −1.67087
$$206$$ 0 0
$$207$$ 1580.40 0.530653
$$208$$ 0 0
$$209$$ 378.230 0.125181
$$210$$ 0 0
$$211$$ 4031.60 1.31539 0.657694 0.753285i $$-0.271532\pi$$
0.657694 + 0.753285i $$0.271532\pi$$
$$212$$ 0 0
$$213$$ −1584.90 −0.509838
$$214$$ 0 0
$$215$$ 9791.66 3.10598
$$216$$ 0 0
$$217$$ 877.666 0.274562
$$218$$ 0 0
$$219$$ −2203.10 −0.679779
$$220$$ 0 0
$$221$$ −871.434 −0.265244
$$222$$ 0 0
$$223$$ −3784.95 −1.13659 −0.568294 0.822826i $$-0.692396\pi$$
−0.568294 + 0.822826i $$0.692396\pi$$
$$224$$ 0 0
$$225$$ 2291.40 0.678932
$$226$$ 0 0
$$227$$ −2013.83 −0.588821 −0.294411 0.955679i $$-0.595123\pi$$
−0.294411 + 0.955679i $$0.595123\pi$$
$$228$$ 0 0
$$229$$ −3050.73 −0.880340 −0.440170 0.897915i $$-0.645082\pi$$
−0.440170 + 0.897915i $$0.645082\pi$$
$$230$$ 0 0
$$231$$ 514.101 0.146430
$$232$$ 0 0
$$233$$ 5587.49 1.57103 0.785513 0.618846i $$-0.212399\pi$$
0.785513 + 0.618846i $$0.212399\pi$$
$$234$$ 0 0
$$235$$ 5489.09 1.52370
$$236$$ 0 0
$$237$$ −339.798 −0.0931317
$$238$$ 0 0
$$239$$ 1335.69 0.361501 0.180750 0.983529i $$-0.442147\pi$$
0.180750 + 0.983529i $$0.442147\pi$$
$$240$$ 0 0
$$241$$ −571.558 −0.152769 −0.0763845 0.997078i $$-0.524338\pi$$
−0.0763845 + 0.997078i $$0.524338\pi$$
$$242$$ 0 0
$$243$$ 243.000 0.0641500
$$244$$ 0 0
$$245$$ −5591.71 −1.45813
$$246$$ 0 0
$$247$$ 214.717 0.0553122
$$248$$ 0 0
$$249$$ 2800.40 0.712723
$$250$$ 0 0
$$251$$ −4088.60 −1.02817 −0.514084 0.857740i $$-0.671868\pi$$
−0.514084 + 0.857740i $$0.671868\pi$$
$$252$$ 0 0
$$253$$ −4021.21 −0.999254
$$254$$ 0 0
$$255$$ 3918.10 0.962199
$$256$$ 0 0
$$257$$ 3050.23 0.740342 0.370171 0.928964i $$-0.379299\pi$$
0.370171 + 0.928964i $$0.379299\pi$$
$$258$$ 0 0
$$259$$ 1158.17 0.277857
$$260$$ 0 0
$$261$$ 2626.20 0.622826
$$262$$ 0 0
$$263$$ −5770.99 −1.35306 −0.676530 0.736415i $$-0.736517\pi$$
−0.676530 + 0.736415i $$0.736517\pi$$
$$264$$ 0 0
$$265$$ 7150.35 1.65752
$$266$$ 0 0
$$267$$ 3572.74 0.818908
$$268$$ 0 0
$$269$$ −2079.40 −0.471314 −0.235657 0.971836i $$-0.575724\pi$$
−0.235657 + 0.971836i $$0.575724\pi$$
$$270$$ 0 0
$$271$$ −6012.00 −1.34761 −0.673807 0.738908i $$-0.735342\pi$$
−0.673807 + 0.738908i $$0.735342\pi$$
$$272$$ 0 0
$$273$$ 291.849 0.0647015
$$274$$ 0 0
$$275$$ −5830.30 −1.27847
$$276$$ 0 0
$$277$$ −735.201 −0.159473 −0.0797364 0.996816i $$-0.525408\pi$$
−0.0797364 + 0.996816i $$0.525408\pi$$
$$278$$ 0 0
$$279$$ −1055.55 −0.226502
$$280$$ 0 0
$$281$$ −1902.92 −0.403981 −0.201990 0.979387i $$-0.564741\pi$$
−0.201990 + 0.979387i $$0.564741\pi$$
$$282$$ 0 0
$$283$$ −2125.71 −0.446502 −0.223251 0.974761i $$-0.571667\pi$$
−0.223251 + 0.974761i $$0.571667\pi$$
$$284$$ 0 0
$$285$$ −965.399 −0.200650
$$286$$ 0 0
$$287$$ 1883.67 0.387420
$$288$$ 0 0
$$289$$ −419.527 −0.0853913
$$290$$ 0 0
$$291$$ 1671.49 0.336717
$$292$$ 0 0
$$293$$ −1641.03 −0.327200 −0.163600 0.986527i $$-0.552311\pi$$
−0.163600 + 0.986527i $$0.552311\pi$$
$$294$$ 0 0
$$295$$ 1552.16 0.306341
$$296$$ 0 0
$$297$$ −618.297 −0.120799
$$298$$ 0 0
$$299$$ −2282.79 −0.441530
$$300$$ 0 0
$$301$$ −3760.86 −0.720174
$$302$$ 0 0
$$303$$ −860.299 −0.163112
$$304$$ 0 0
$$305$$ −3796.62 −0.712767
$$306$$ 0 0
$$307$$ 3373.27 0.627111 0.313555 0.949570i $$-0.398480\pi$$
0.313555 + 0.949570i $$0.398480\pi$$
$$308$$ 0 0
$$309$$ 5734.09 1.05567
$$310$$ 0 0
$$311$$ 868.525 0.158359 0.0791793 0.996860i $$-0.474770\pi$$
0.0791793 + 0.996860i $$0.474770\pi$$
$$312$$ 0 0
$$313$$ −4343.19 −0.784319 −0.392159 0.919897i $$-0.628272\pi$$
−0.392159 + 0.919897i $$0.628272\pi$$
$$314$$ 0 0
$$315$$ −1312.20 −0.234711
$$316$$ 0 0
$$317$$ −3277.65 −0.580730 −0.290365 0.956916i $$-0.593777\pi$$
−0.290365 + 0.956916i $$0.593777\pi$$
$$318$$ 0 0
$$319$$ −6682.18 −1.17282
$$320$$ 0 0
$$321$$ −2503.00 −0.435215
$$322$$ 0 0
$$323$$ −1107.17 −0.190726
$$324$$ 0 0
$$325$$ −3309.79 −0.564906
$$326$$ 0 0
$$327$$ −3232.99 −0.546743
$$328$$ 0 0
$$329$$ −2108.30 −0.353295
$$330$$ 0 0
$$331$$ −5589.62 −0.928197 −0.464099 0.885784i $$-0.653622\pi$$
−0.464099 + 0.885784i $$0.653622\pi$$
$$332$$ 0 0
$$333$$ −1392.90 −0.229220
$$334$$ 0 0
$$335$$ −7794.92 −1.27129
$$336$$ 0 0
$$337$$ 901.544 0.145728 0.0728638 0.997342i $$-0.476786\pi$$
0.0728638 + 0.997342i $$0.476786\pi$$
$$338$$ 0 0
$$339$$ −498.194 −0.0798176
$$340$$ 0 0
$$341$$ 2685.77 0.426518
$$342$$ 0 0
$$343$$ 4714.49 0.742153
$$344$$ 0 0
$$345$$ 10263.8 1.60169
$$346$$ 0 0
$$347$$ 812.318 0.125670 0.0628350 0.998024i $$-0.479986\pi$$
0.0628350 + 0.998024i $$0.479986\pi$$
$$348$$ 0 0
$$349$$ 4437.96 0.680683 0.340342 0.940302i $$-0.389457\pi$$
0.340342 + 0.940302i $$0.389457\pi$$
$$350$$ 0 0
$$351$$ −351.000 −0.0533761
$$352$$ 0 0
$$353$$ 7115.35 1.07284 0.536419 0.843952i $$-0.319777\pi$$
0.536419 + 0.843952i $$0.319777\pi$$
$$354$$ 0 0
$$355$$ −10293.0 −1.53886
$$356$$ 0 0
$$357$$ −1504.90 −0.223102
$$358$$ 0 0
$$359$$ −4693.98 −0.690081 −0.345040 0.938588i $$-0.612135\pi$$
−0.345040 + 0.938588i $$0.612135\pi$$
$$360$$ 0 0
$$361$$ −6586.20 −0.960227
$$362$$ 0 0
$$363$$ −2419.79 −0.349878
$$364$$ 0 0
$$365$$ −14307.9 −2.05181
$$366$$ 0 0
$$367$$ −9243.98 −1.31480 −0.657400 0.753542i $$-0.728344\pi$$
−0.657400 + 0.753542i $$0.728344\pi$$
$$368$$ 0 0
$$369$$ −2265.44 −0.319605
$$370$$ 0 0
$$371$$ −2746.37 −0.384324
$$372$$ 0 0
$$373$$ −4311.99 −0.598569 −0.299285 0.954164i $$-0.596748\pi$$
−0.299285 + 0.954164i $$0.596748\pi$$
$$374$$ 0 0
$$375$$ 7575.09 1.04314
$$376$$ 0 0
$$377$$ −3793.40 −0.518223
$$378$$ 0 0
$$379$$ 2382.73 0.322936 0.161468 0.986878i $$-0.448377\pi$$
0.161468 + 0.986878i $$0.448377\pi$$
$$380$$ 0 0
$$381$$ −3888.49 −0.522870
$$382$$ 0 0
$$383$$ −4845.81 −0.646499 −0.323250 0.946314i $$-0.604775\pi$$
−0.323250 + 0.946314i $$0.604775\pi$$
$$384$$ 0 0
$$385$$ 3338.80 0.441976
$$386$$ 0 0
$$387$$ 4523.10 0.594113
$$388$$ 0 0
$$389$$ 9561.50 1.24624 0.623120 0.782127i $$-0.285865\pi$$
0.623120 + 0.782127i $$0.285865\pi$$
$$390$$ 0 0
$$391$$ 11771.0 1.52247
$$392$$ 0 0
$$393$$ 591.604 0.0759350
$$394$$ 0 0
$$395$$ −2206.79 −0.281103
$$396$$ 0 0
$$397$$ −7440.11 −0.940575 −0.470287 0.882513i $$-0.655850\pi$$
−0.470287 + 0.882513i $$0.655850\pi$$
$$398$$ 0 0
$$399$$ 370.799 0.0465242
$$400$$ 0 0
$$401$$ −8687.80 −1.08192 −0.540958 0.841050i $$-0.681938\pi$$
−0.540958 + 0.841050i $$0.681938\pi$$
$$402$$ 0 0
$$403$$ 1524.68 0.188461
$$404$$ 0 0
$$405$$ 1578.15 0.193627
$$406$$ 0 0
$$407$$ 3544.13 0.431637
$$408$$ 0 0
$$409$$ 2556.10 0.309024 0.154512 0.987991i $$-0.450619\pi$$
0.154512 + 0.987991i $$0.450619\pi$$
$$410$$ 0 0
$$411$$ −1640.74 −0.196915
$$412$$ 0 0
$$413$$ −596.168 −0.0710303
$$414$$ 0 0
$$415$$ 18187.0 2.15124
$$416$$ 0 0
$$417$$ −1829.00 −0.214788
$$418$$ 0 0
$$419$$ 3347.46 0.390296 0.195148 0.980774i $$-0.437481\pi$$
0.195148 + 0.980774i $$0.437481\pi$$
$$420$$ 0 0
$$421$$ −1854.48 −0.214684 −0.107342 0.994222i $$-0.534234\pi$$
−0.107342 + 0.994222i $$0.534234\pi$$
$$422$$ 0 0
$$423$$ 2535.60 0.291454
$$424$$ 0 0
$$425$$ 17066.7 1.94789
$$426$$ 0 0
$$427$$ 1458.24 0.165267
$$428$$ 0 0
$$429$$ 893.096 0.100511
$$430$$ 0 0
$$431$$ 14043.1 1.56945 0.784725 0.619844i $$-0.212804\pi$$
0.784725 + 0.619844i $$0.212804\pi$$
$$432$$ 0 0
$$433$$ 3086.47 0.342555 0.171278 0.985223i $$-0.445210\pi$$
0.171278 + 0.985223i $$0.445210\pi$$
$$434$$ 0 0
$$435$$ 17055.7 1.87990
$$436$$ 0 0
$$437$$ −2900.32 −0.317486
$$438$$ 0 0
$$439$$ −2837.68 −0.308508 −0.154254 0.988031i $$-0.549297\pi$$
−0.154254 + 0.988031i $$0.549297\pi$$
$$440$$ 0 0
$$441$$ −2583.00 −0.278912
$$442$$ 0 0
$$443$$ −18309.4 −1.96367 −0.981834 0.189744i $$-0.939234\pi$$
−0.981834 + 0.189744i $$0.939234\pi$$
$$444$$ 0 0
$$445$$ 23203.0 2.47174
$$446$$ 0 0
$$447$$ −6495.24 −0.687281
$$448$$ 0 0
$$449$$ 13861.2 1.45690 0.728451 0.685098i $$-0.240241\pi$$
0.728451 + 0.685098i $$0.240241\pi$$
$$450$$ 0 0
$$451$$ 5764.26 0.601837
$$452$$ 0 0
$$453$$ 2539.65 0.263406
$$454$$ 0 0
$$455$$ 1895.40 0.195291
$$456$$ 0 0
$$457$$ −8990.36 −0.920243 −0.460122 0.887856i $$-0.652194\pi$$
−0.460122 + 0.887856i $$0.652194\pi$$
$$458$$ 0 0
$$459$$ 1809.90 0.184050
$$460$$ 0 0
$$461$$ −3406.90 −0.344198 −0.172099 0.985080i $$-0.555055\pi$$
−0.172099 + 0.985080i $$0.555055\pi$$
$$462$$ 0 0
$$463$$ 7498.45 0.752662 0.376331 0.926485i $$-0.377186\pi$$
0.376331 + 0.926485i $$0.377186\pi$$
$$464$$ 0 0
$$465$$ −6855.19 −0.683660
$$466$$ 0 0
$$467$$ −7711.38 −0.764112 −0.382056 0.924139i $$-0.624784\pi$$
−0.382056 + 0.924139i $$0.624784\pi$$
$$468$$ 0 0
$$469$$ 2993.94 0.294770
$$470$$ 0 0
$$471$$ 4960.79 0.485310
$$472$$ 0 0
$$473$$ −11508.7 −1.11875
$$474$$ 0 0
$$475$$ −4205.14 −0.406200
$$476$$ 0 0
$$477$$ 3302.99 0.317051
$$478$$ 0 0
$$479$$ 9439.82 0.900451 0.450226 0.892915i $$-0.351344\pi$$
0.450226 + 0.892915i $$0.351344\pi$$
$$480$$ 0 0
$$481$$ 2011.96 0.190723
$$482$$ 0 0
$$483$$ −3942.20 −0.371380
$$484$$ 0 0
$$485$$ 10855.4 1.01633
$$486$$ 0 0
$$487$$ 6156.20 0.572821 0.286411 0.958107i $$-0.407538\pi$$
0.286411 + 0.958107i $$0.407538\pi$$
$$488$$ 0 0
$$489$$ 8599.54 0.795265
$$490$$ 0 0
$$491$$ −3842.74 −0.353198 −0.176599 0.984283i $$-0.556510\pi$$
−0.176599 + 0.984283i $$0.556510\pi$$
$$492$$ 0 0
$$493$$ 19560.3 1.78692
$$494$$ 0 0
$$495$$ −4015.49 −0.364612
$$496$$ 0 0
$$497$$ 3953.43 0.356812
$$498$$ 0 0
$$499$$ 12842.4 1.15211 0.576056 0.817410i $$-0.304591\pi$$
0.576056 + 0.817410i $$0.304591\pi$$
$$500$$ 0 0
$$501$$ −2187.20 −0.195043
$$502$$ 0 0
$$503$$ −8580.11 −0.760573 −0.380287 0.924869i $$-0.624175\pi$$
−0.380287 + 0.924869i $$0.624175\pi$$
$$504$$ 0 0
$$505$$ −5587.16 −0.492327
$$506$$ 0 0
$$507$$ 507.000 0.0444116
$$508$$ 0 0
$$509$$ −43.5957 −0.00379635 −0.00189818 0.999998i $$-0.500604\pi$$
−0.00189818 + 0.999998i $$0.500604\pi$$
$$510$$ 0 0
$$511$$ 5495.49 0.475746
$$512$$ 0 0
$$513$$ −445.951 −0.0383805
$$514$$ 0 0
$$515$$ 37239.7 3.18636
$$516$$ 0 0
$$517$$ −6451.66 −0.548827
$$518$$ 0 0
$$519$$ −11504.5 −0.973008
$$520$$ 0 0
$$521$$ 11368.1 0.955939 0.477969 0.878377i $$-0.341373\pi$$
0.477969 + 0.878377i $$0.341373\pi$$
$$522$$ 0 0
$$523$$ 5229.53 0.437230 0.218615 0.975811i $$-0.429846\pi$$
0.218615 + 0.975811i $$0.429846\pi$$
$$524$$ 0 0
$$525$$ −5715.75 −0.475154
$$526$$ 0 0
$$527$$ −7861.88 −0.649846
$$528$$ 0 0
$$529$$ 18668.2 1.53433
$$530$$ 0 0
$$531$$ 716.997 0.0585970
$$532$$ 0 0
$$533$$ 3272.31 0.265927
$$534$$ 0 0
$$535$$ −16255.6 −1.31363
$$536$$ 0 0
$$537$$ 851.586 0.0684333
$$538$$ 0 0
$$539$$ 6572.27 0.525209
$$540$$ 0 0
$$541$$ −6567.99 −0.521959 −0.260980 0.965344i $$-0.584046\pi$$
−0.260980 + 0.965344i $$0.584046\pi$$
$$542$$ 0 0
$$543$$ 7090.79 0.560396
$$544$$ 0 0
$$545$$ −20996.5 −1.65026
$$546$$ 0 0
$$547$$ 13675.7 1.06897 0.534487 0.845177i $$-0.320505\pi$$
0.534487 + 0.845177i $$0.320505\pi$$
$$548$$ 0 0
$$549$$ −1753.79 −0.136339
$$550$$ 0 0
$$551$$ −4819.57 −0.372632
$$552$$ 0 0
$$553$$ 847.604 0.0651786
$$554$$ 0 0
$$555$$ −9046.09 −0.691865
$$556$$ 0 0
$$557$$ 4527.96 0.344445 0.172222 0.985058i $$-0.444905\pi$$
0.172222 + 0.985058i $$0.444905\pi$$
$$558$$ 0 0
$$559$$ −6533.36 −0.494332
$$560$$ 0 0
$$561$$ −4605.17 −0.346578
$$562$$ 0 0
$$563$$ −18441.8 −1.38051 −0.690256 0.723566i $$-0.742502\pi$$
−0.690256 + 0.723566i $$0.742502\pi$$
$$564$$ 0 0
$$565$$ −3235.49 −0.240917
$$566$$ 0 0
$$567$$ −606.148 −0.0448957
$$568$$ 0 0
$$569$$ −13553.5 −0.998578 −0.499289 0.866436i $$-0.666405\pi$$
−0.499289 + 0.866436i $$0.666405\pi$$
$$570$$ 0 0
$$571$$ −14815.5 −1.08583 −0.542915 0.839788i $$-0.682679\pi$$
−0.542915 + 0.839788i $$0.682679\pi$$
$$572$$ 0 0
$$573$$ −7542.79 −0.549921
$$574$$ 0 0
$$575$$ 44707.6 3.24249
$$576$$ 0 0
$$577$$ 21596.2 1.55816 0.779081 0.626923i $$-0.215686\pi$$
0.779081 + 0.626923i $$0.215686\pi$$
$$578$$ 0 0
$$579$$ 7262.19 0.521254
$$580$$ 0 0
$$581$$ −6985.42 −0.498802
$$582$$ 0 0
$$583$$ −8404.23 −0.597029
$$584$$ 0 0
$$585$$ −2279.55 −0.161107
$$586$$ 0 0
$$587$$ 918.801 0.0646047 0.0323024 0.999478i $$-0.489716\pi$$
0.0323024 + 0.999478i $$0.489716\pi$$
$$588$$ 0 0
$$589$$ 1937.13 0.135514
$$590$$ 0 0
$$591$$ −13900.9 −0.967527
$$592$$ 0 0
$$593$$ 19816.0 1.37226 0.686128 0.727481i $$-0.259309\pi$$
0.686128 + 0.727481i $$0.259309\pi$$
$$594$$ 0 0
$$595$$ −9773.45 −0.673399
$$596$$ 0 0
$$597$$ −9162.50 −0.628134
$$598$$ 0 0
$$599$$ 5141.86 0.350736 0.175368 0.984503i $$-0.443889\pi$$
0.175368 + 0.984503i $$0.443889\pi$$
$$600$$ 0 0
$$601$$ 12380.9 0.840312 0.420156 0.907452i $$-0.361975\pi$$
0.420156 + 0.907452i $$0.361975\pi$$
$$602$$ 0 0
$$603$$ −3600.74 −0.243173
$$604$$ 0 0
$$605$$ −15715.1 −1.05605
$$606$$ 0 0
$$607$$ 23717.0 1.58590 0.792951 0.609286i $$-0.208544\pi$$
0.792951 + 0.609286i $$0.208544\pi$$
$$608$$ 0 0
$$609$$ −6550.89 −0.435887
$$610$$ 0 0
$$611$$ −3662.53 −0.242504
$$612$$ 0 0
$$613$$ −26157.1 −1.72345 −0.861726 0.507373i $$-0.830617\pi$$
−0.861726 + 0.507373i $$0.830617\pi$$
$$614$$ 0 0
$$615$$ −14712.8 −0.964677
$$616$$ 0 0
$$617$$ 23613.9 1.54077 0.770387 0.637576i $$-0.220063\pi$$
0.770387 + 0.637576i $$0.220063\pi$$
$$618$$ 0 0
$$619$$ −23345.4 −1.51588 −0.757940 0.652324i $$-0.773794\pi$$
−0.757940 + 0.652324i $$0.773794\pi$$
$$620$$ 0 0
$$621$$ 4741.19 0.306372
$$622$$ 0 0
$$623$$ −8911.99 −0.573116
$$624$$ 0 0
$$625$$ 17371.0 1.11174
$$626$$ 0 0
$$627$$ 1134.69 0.0722730
$$628$$ 0 0
$$629$$ −10374.5 −0.657645
$$630$$ 0 0
$$631$$ −15245.7 −0.961841 −0.480921 0.876764i $$-0.659698\pi$$
−0.480921 + 0.876764i $$0.659698\pi$$
$$632$$ 0 0
$$633$$ 12094.8 0.759439
$$634$$ 0 0
$$635$$ −25253.6 −1.57820
$$636$$ 0 0
$$637$$ 3731.00 0.232068
$$638$$ 0 0
$$639$$ −4754.69 −0.294355
$$640$$ 0 0
$$641$$ 10192.7 0.628063 0.314032 0.949413i $$-0.398320\pi$$
0.314032 + 0.949413i $$0.398320\pi$$
$$642$$ 0 0
$$643$$ 5506.31 0.337710 0.168855 0.985641i $$-0.445993\pi$$
0.168855 + 0.985641i $$0.445993\pi$$
$$644$$ 0 0
$$645$$ 29375.0 1.79324
$$646$$ 0 0
$$647$$ 13297.5 0.808005 0.404003 0.914758i $$-0.367619\pi$$
0.404003 + 0.914758i $$0.367619\pi$$
$$648$$ 0 0
$$649$$ −1824.35 −0.110342
$$650$$ 0 0
$$651$$ 2633.00 0.158518
$$652$$ 0 0
$$653$$ −12440.2 −0.745519 −0.372760 0.927928i $$-0.621588\pi$$
−0.372760 + 0.927928i $$0.621588\pi$$
$$654$$ 0 0
$$655$$ 3842.14 0.229198
$$656$$ 0 0
$$657$$ −6609.29 −0.392470
$$658$$ 0 0
$$659$$ 9562.87 0.565276 0.282638 0.959227i $$-0.408791\pi$$
0.282638 + 0.959227i $$0.408791\pi$$
$$660$$ 0 0
$$661$$ 2409.69 0.141795 0.0708973 0.997484i $$-0.477414\pi$$
0.0708973 + 0.997484i $$0.477414\pi$$
$$662$$ 0 0
$$663$$ −2614.30 −0.153139
$$664$$ 0 0
$$665$$ 2408.13 0.140426
$$666$$ 0 0
$$667$$ 51239.9 2.97454
$$668$$ 0 0
$$669$$ −11354.9 −0.656209
$$670$$ 0 0
$$671$$ 4462.40 0.256735
$$672$$ 0 0
$$673$$ 7929.02 0.454147 0.227074 0.973878i $$-0.427084\pi$$
0.227074 + 0.973878i $$0.427084\pi$$
$$674$$ 0 0
$$675$$ 6874.19 0.391982
$$676$$ 0 0
$$677$$ −2628.26 −0.149206 −0.0746030 0.997213i $$-0.523769\pi$$
−0.0746030 + 0.997213i $$0.523769\pi$$
$$678$$ 0 0
$$679$$ −4169.44 −0.235653
$$680$$ 0 0
$$681$$ −6041.48 −0.339956
$$682$$ 0 0
$$683$$ −10021.5 −0.561437 −0.280719 0.959790i $$-0.590573\pi$$
−0.280719 + 0.959790i $$0.590573\pi$$
$$684$$ 0 0
$$685$$ −10655.7 −0.594356
$$686$$ 0 0
$$687$$ −9152.18 −0.508264
$$688$$ 0 0
$$689$$ −4770.99 −0.263803
$$690$$ 0 0
$$691$$ −23987.2 −1.32057 −0.660286 0.751014i $$-0.729565\pi$$
−0.660286 + 0.751014i $$0.729565\pi$$
$$692$$ 0 0
$$693$$ 1542.30 0.0845415
$$694$$ 0 0
$$695$$ −11878.3 −0.648303
$$696$$ 0 0
$$697$$ −16873.4 −0.916964
$$698$$ 0 0
$$699$$ 16762.5 0.907032
$$700$$ 0 0
$$701$$ −3763.71 −0.202787 −0.101393 0.994846i $$-0.532330\pi$$
−0.101393 + 0.994846i $$0.532330\pi$$
$$702$$ 0 0
$$703$$ 2556.23 0.137141
$$704$$ 0 0
$$705$$ 16467.3 0.879707
$$706$$ 0 0
$$707$$ 2145.96 0.114155
$$708$$ 0 0
$$709$$ −36047.8 −1.90946 −0.954728 0.297479i $$-0.903854\pi$$
−0.954728 + 0.297479i $$0.903854\pi$$
$$710$$ 0 0
$$711$$ −1019.39 −0.0537696
$$712$$ 0 0
$$713$$ −20594.9 −1.08174
$$714$$ 0 0
$$715$$ 5800.15 0.303376
$$716$$ 0 0
$$717$$ 4007.08 0.208713
$$718$$ 0 0
$$719$$ 3944.18 0.204580 0.102290 0.994755i $$-0.467383\pi$$
0.102290 + 0.994755i $$0.467383\pi$$
$$720$$ 0 0
$$721$$ −14303.3 −0.738812
$$722$$ 0 0
$$723$$ −1714.68 −0.0882012
$$724$$ 0 0
$$725$$ 74292.1 3.80571
$$726$$ 0 0
$$727$$ 20447.8 1.04315 0.521573 0.853206i $$-0.325345\pi$$
0.521573 + 0.853206i $$0.325345\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ 33688.7 1.70454
$$732$$ 0 0
$$733$$ −13536.2 −0.682089 −0.341045 0.940047i $$-0.610781\pi$$
−0.341045 + 0.940047i $$0.610781\pi$$
$$734$$ 0 0
$$735$$ −16775.1 −0.841851
$$736$$ 0 0
$$737$$ 9161.83 0.457911
$$738$$ 0 0
$$739$$ −15839.1 −0.788433 −0.394217 0.919018i $$-0.628984\pi$$
−0.394217 + 0.919018i $$0.628984\pi$$
$$740$$ 0 0
$$741$$ 644.151 0.0319345
$$742$$ 0 0
$$743$$ 1664.92 0.0822075 0.0411037 0.999155i $$-0.486913\pi$$
0.0411037 + 0.999155i $$0.486913\pi$$
$$744$$ 0 0
$$745$$ −42182.9 −2.07445
$$746$$ 0 0
$$747$$ 8401.19 0.411491
$$748$$ 0 0
$$749$$ 6243.58 0.304587
$$750$$ 0 0
$$751$$ −22399.1 −1.08835 −0.544177 0.838970i $$-0.683158\pi$$
−0.544177 + 0.838970i $$0.683158\pi$$
$$752$$ 0 0
$$753$$ −12265.8 −0.593613
$$754$$ 0 0
$$755$$ 16493.6 0.795050
$$756$$ 0 0
$$757$$ 23798.9 1.14265 0.571326 0.820723i $$-0.306429\pi$$
0.571326 + 0.820723i $$0.306429\pi$$
$$758$$ 0 0
$$759$$ −12063.6 −0.576920
$$760$$ 0 0
$$761$$ 13693.5 0.652285 0.326142 0.945321i $$-0.394251\pi$$
0.326142 + 0.945321i $$0.394251\pi$$
$$762$$ 0 0
$$763$$ 8064.50 0.382640
$$764$$ 0 0
$$765$$ 11754.3 0.555526
$$766$$ 0 0
$$767$$ −1035.66 −0.0487556
$$768$$ 0 0
$$769$$ 16299.9 0.764358 0.382179 0.924088i $$-0.375174\pi$$
0.382179 + 0.924088i $$0.375174\pi$$
$$770$$ 0 0
$$771$$ 9150.68 0.427437
$$772$$ 0 0
$$773$$ 33532.2 1.56024 0.780122 0.625628i $$-0.215157\pi$$
0.780122 + 0.625628i $$0.215157\pi$$
$$774$$ 0 0
$$775$$ −29860.2 −1.38401
$$776$$ 0 0
$$777$$ 3474.50 0.160421
$$778$$ 0 0
$$779$$ 4157.51 0.191217
$$780$$ 0 0
$$781$$ 12098.0 0.554290
$$782$$ 0 0
$$783$$ 7878.59 0.359589
$$784$$ 0 0
$$785$$ 32217.5 1.46483
$$786$$ 0 0
$$787$$ −16163.3 −0.732097 −0.366049 0.930596i $$-0.619290\pi$$
−0.366049 + 0.930596i $$0.619290\pi$$
$$788$$ 0 0
$$789$$ −17313.0 −0.781189
$$790$$ 0 0
$$791$$ 1242.71 0.0558607
$$792$$ 0 0
$$793$$ 2533.25 0.113441
$$794$$ 0 0
$$795$$ 21451.1 0.956970
$$796$$ 0 0
$$797$$ −39636.4 −1.76160 −0.880798 0.473492i $$-0.842993\pi$$
−0.880798 + 0.473492i $$0.842993\pi$$
$$798$$ 0 0
$$799$$ 18885.5 0.836197
$$800$$ 0 0
$$801$$ 10718.2 0.472797
$$802$$ 0 0
$$803$$ 16816.9 0.739048
$$804$$ 0 0
$$805$$ −25602.4 −1.12095
$$806$$ 0 0
$$807$$ −6238.21 −0.272113
$$808$$ 0 0
$$809$$ −23811.2 −1.03481 −0.517403 0.855742i $$-0.673101\pi$$
−0.517403 + 0.855742i $$0.673101\pi$$
$$810$$ 0 0
$$811$$ −27218.6 −1.17851 −0.589256 0.807946i $$-0.700579\pi$$
−0.589256 + 0.807946i $$0.700579\pi$$
$$812$$ 0 0
$$813$$ −18036.0 −0.778045
$$814$$ 0 0
$$815$$ 55849.2 2.40038
$$816$$ 0 0
$$817$$ −8300.73 −0.355454
$$818$$ 0 0
$$819$$ 875.548 0.0373555
$$820$$ 0 0
$$821$$ −43094.8 −1.83193 −0.915967 0.401253i $$-0.868575\pi$$
−0.915967 + 0.401253i $$0.868575\pi$$
$$822$$ 0 0
$$823$$ −26541.1 −1.12414 −0.562068 0.827091i $$-0.689994\pi$$
−0.562068 + 0.827091i $$0.689994\pi$$
$$824$$ 0 0
$$825$$ −17490.9 −0.738127
$$826$$ 0 0
$$827$$ 44898.7 1.88788 0.943942 0.330112i $$-0.107087\pi$$
0.943942 + 0.330112i $$0.107087\pi$$
$$828$$ 0 0
$$829$$ −7137.48 −0.299029 −0.149514 0.988760i $$-0.547771\pi$$
−0.149514 + 0.988760i $$0.547771\pi$$
$$830$$ 0 0
$$831$$ −2205.60 −0.0920717
$$832$$ 0 0
$$833$$ −19238.6 −0.800213
$$834$$ 0 0
$$835$$ −14204.6 −0.588708
$$836$$ 0 0
$$837$$ −3166.64 −0.130771
$$838$$ 0 0
$$839$$ 4387.17 0.180527 0.0902634 0.995918i $$-0.471229\pi$$
0.0902634 + 0.995918i $$0.471229\pi$$
$$840$$ 0 0
$$841$$ 60758.1 2.49121
$$842$$ 0 0
$$843$$ −5708.76 −0.233239
$$844$$ 0 0
$$845$$ 3292.68 0.134049
$$846$$ 0 0
$$847$$ 6036.01 0.244864
$$848$$ 0 0
$$849$$ −6377.12 −0.257788
$$850$$ 0 0
$$851$$ −27176.9 −1.09473
$$852$$ 0 0
$$853$$ −9328.85 −0.374459 −0.187230 0.982316i $$-0.559951\pi$$
−0.187230 + 0.982316i $$0.559951\pi$$
$$854$$ 0 0
$$855$$ −2896.20 −0.115845
$$856$$ 0 0
$$857$$ −5010.39 −0.199710 −0.0998552 0.995002i $$-0.531838\pi$$
−0.0998552 + 0.995002i $$0.531838\pi$$
$$858$$ 0 0
$$859$$ −30233.4 −1.20088 −0.600438 0.799672i $$-0.705007\pi$$
−0.600438 + 0.799672i $$0.705007\pi$$
$$860$$ 0 0
$$861$$ 5651.01 0.223677
$$862$$ 0 0
$$863$$ −4334.93 −0.170988 −0.0854940 0.996339i $$-0.527247\pi$$
−0.0854940 + 0.996339i $$0.527247\pi$$
$$864$$ 0 0
$$865$$ −74715.2 −2.93687
$$866$$ 0 0
$$867$$ −1258.58 −0.0493007
$$868$$ 0 0
$$869$$ 2593.78 0.101252
$$870$$ 0 0
$$871$$ 5201.06 0.202332
$$872$$ 0 0
$$873$$ 5014.48 0.194404
$$874$$ 0 0
$$875$$ −18895.6 −0.730043
$$876$$ 0 0
$$877$$ 34683.3 1.33543 0.667716 0.744416i $$-0.267272\pi$$
0.667716 + 0.744416i $$0.267272\pi$$
$$878$$ 0 0
$$879$$ −4923.08 −0.188909
$$880$$ 0 0
$$881$$ −18269.2 −0.698642 −0.349321 0.937003i $$-0.613588\pi$$
−0.349321 + 0.937003i $$0.613588\pi$$
$$882$$ 0 0
$$883$$ 14592.0 0.556128 0.278064 0.960563i $$-0.410307\pi$$
0.278064 + 0.960563i $$0.410307\pi$$
$$884$$ 0 0
$$885$$ 4656.49 0.176866
$$886$$ 0 0
$$887$$ −30459.3 −1.15301 −0.576507 0.817092i $$-0.695585\pi$$
−0.576507 + 0.817092i $$0.695585\pi$$
$$888$$ 0 0
$$889$$ 9699.60 0.365933
$$890$$ 0 0
$$891$$ −1854.89 −0.0697432
$$892$$ 0 0
$$893$$ −4653.30 −0.174375
$$894$$ 0 0
$$895$$ 5530.57 0.206555
$$896$$ 0 0
$$897$$ −6848.38 −0.254917
$$898$$ 0 0
$$899$$ −34223.2 −1.26964
$$900$$ 0 0
$$901$$ 24601.2 0.909638
$$902$$ 0 0
$$903$$ −11282.6 −0.415793
$$904$$ 0 0
$$905$$ 46050.7 1.69147
$$906$$ 0 0
$$907$$ 9364.89 0.342840 0.171420 0.985198i $$-0.445164\pi$$
0.171420 + 0.985198i $$0.445164\pi$$
$$908$$ 0 0
$$909$$ −2580.90 −0.0941727
$$910$$ 0 0
$$911$$ −32479.8 −1.18123 −0.590616 0.806952i $$-0.701115\pi$$
−0.590616 + 0.806952i $$0.701115\pi$$
$$912$$ 0 0
$$913$$ −21376.3 −0.774864
$$914$$ 0 0
$$915$$ −11389.9 −0.411516
$$916$$ 0 0
$$917$$ −1475.72 −0.0531435
$$918$$ 0 0
$$919$$ −295.958 −0.0106232 −0.00531161 0.999986i $$-0.501691\pi$$
−0.00531161 + 0.999986i $$0.501691\pi$$
$$920$$ 0 0
$$921$$ 10119.8 0.362062
$$922$$ 0 0
$$923$$ 6867.89 0.244918
$$924$$ 0 0
$$925$$ −39403.5 −1.40062
$$926$$ 0 0
$$927$$ 17202.3 0.609489
$$928$$ 0 0
$$929$$ −5620.38 −0.198492 −0.0992458 0.995063i $$-0.531643\pi$$
−0.0992458 + 0.995063i $$0.531643\pi$$
$$930$$ 0 0
$$931$$ 4740.29 0.166871
$$932$$ 0 0
$$933$$ 2605.58 0.0914284
$$934$$ 0 0
$$935$$ −29908.0 −1.04609
$$936$$ 0 0
$$937$$ −32583.1 −1.13601 −0.568006 0.823024i $$-0.692285\pi$$
−0.568006 + 0.823024i $$0.692285\pi$$
$$938$$ 0 0
$$939$$ −13029.6 −0.452827
$$940$$ 0 0
$$941$$ 8812.99 0.305308 0.152654 0.988280i $$-0.451218\pi$$
0.152654 + 0.988280i $$0.451218\pi$$
$$942$$ 0 0
$$943$$ −44201.2 −1.52639
$$944$$ 0 0
$$945$$ −3936.59 −0.135510
$$946$$ 0 0
$$947$$ −13426.8 −0.460732 −0.230366 0.973104i $$-0.573992\pi$$
−0.230366 + 0.973104i $$0.573992\pi$$
$$948$$ 0 0
$$949$$ 9546.76 0.326555
$$950$$ 0 0
$$951$$ −9832.96 −0.335285
$$952$$ 0 0
$$953$$ −13394.6 −0.455293 −0.227647 0.973744i $$-0.573103\pi$$
−0.227647 + 0.973744i $$0.573103\pi$$
$$954$$ 0 0
$$955$$ −48986.2 −1.65985
$$956$$ 0 0
$$957$$ −20046.5 −0.677129
$$958$$ 0 0
$$959$$ 4092.74 0.137812
$$960$$ 0 0
$$961$$ −16035.7 −0.538273
$$962$$ 0 0
$$963$$ −7509.00 −0.251271
$$964$$ 0 0
$$965$$ 47163.8 1.57332
$$966$$ 0 0
$$967$$ −45590.8 −1.51613 −0.758066 0.652178i $$-0.773856\pi$$
−0.758066 + 0.652178i $$0.773856\pi$$
$$968$$ 0 0
$$969$$ −3321.51 −0.110116
$$970$$ 0 0
$$971$$ −264.763 −0.00875041 −0.00437521 0.999990i $$-0.501393\pi$$
−0.00437521 + 0.999990i $$0.501393\pi$$
$$972$$ 0 0
$$973$$ 4562.32 0.150320
$$974$$ 0 0
$$975$$ −9929.38 −0.326148
$$976$$ 0 0
$$977$$ 610.521 0.0199921 0.00999606 0.999950i $$-0.496818\pi$$
0.00999606 + 0.999950i $$0.496818\pi$$
$$978$$ 0 0
$$979$$ −27271.8 −0.890308
$$980$$ 0 0
$$981$$ −9698.98 −0.315662
$$982$$ 0 0
$$983$$ 57829.7 1.87638 0.938190 0.346121i $$-0.112501\pi$$
0.938190 + 0.346121i $$0.112501\pi$$
$$984$$ 0 0
$$985$$ −90278.8 −2.92033
$$986$$ 0 0
$$987$$ −6324.89 −0.203975
$$988$$ 0 0
$$989$$ 88250.4 2.83741
$$990$$ 0 0
$$991$$ 56780.7 1.82008 0.910039 0.414522i $$-0.136051\pi$$
0.910039 + 0.414522i $$0.136051\pi$$
$$992$$ 0 0
$$993$$ −16768.9 −0.535895
$$994$$ 0 0
$$995$$ −59505.3 −1.89592
$$996$$ 0 0
$$997$$ 18616.6 0.591369 0.295684 0.955286i $$-0.404452\pi$$
0.295684 + 0.955286i $$0.404452\pi$$
$$998$$ 0 0
$$999$$ −4178.69 −0.132340
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.r.1.2 2
3.2 odd 2 1872.4.a.t.1.1 2
4.3 odd 2 39.4.a.b.1.1 2
8.3 odd 2 2496.4.a.bc.1.1 2
8.5 even 2 2496.4.a.s.1.1 2
12.11 even 2 117.4.a.c.1.2 2
20.19 odd 2 975.4.a.j.1.2 2
28.27 even 2 1911.4.a.h.1.1 2
52.31 even 4 507.4.b.f.337.3 4
52.47 even 4 507.4.b.f.337.2 4
52.51 odd 2 507.4.a.f.1.2 2
156.155 even 2 1521.4.a.s.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.a.b.1.1 2 4.3 odd 2
117.4.a.c.1.2 2 12.11 even 2
507.4.a.f.1.2 2 52.51 odd 2
507.4.b.f.337.2 4 52.47 even 4
507.4.b.f.337.3 4 52.31 even 4
624.4.a.r.1.2 2 1.1 even 1 trivial
975.4.a.j.1.2 2 20.19 odd 2
1521.4.a.s.1.1 2 156.155 even 2
1872.4.a.t.1.1 2 3.2 odd 2
1911.4.a.h.1.1 2 28.27 even 2
2496.4.a.s.1.1 2 8.5 even 2
2496.4.a.bc.1.1 2 8.3 odd 2