# Properties

 Label 2112.4.a.bh.1.2 Level $2112$ Weight $4$ Character 2112.1 Self dual yes Analytic conductor $124.612$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2112,4,Mod(1,2112)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2112, 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("2112.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2112 = 2^{6} \cdot 3 \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2112.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: $$124.612033932$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{33})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 8$$ x^2 - x - 8 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 33) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-2.37228$$ of defining polynomial Character $$\chi$$ $$=$$ 2112.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.00000 q^{3} +3.48913 q^{5} +4.74456 q^{7} +9.00000 q^{9} +O(q^{10})$$ $$q+3.00000 q^{3} +3.48913 q^{5} +4.74456 q^{7} +9.00000 q^{9} +11.0000 q^{11} +15.0217 q^{13} +10.4674 q^{15} +73.1684 q^{17} -78.7011 q^{19} +14.2337 q^{21} -112.000 q^{23} -112.826 q^{25} +27.0000 q^{27} -243.125 q^{29} -278.717 q^{31} +33.0000 q^{33} +16.5544 q^{35} -102.380 q^{37} +45.0652 q^{39} -241.255 q^{41} -280.016 q^{43} +31.4021 q^{45} +169.870 q^{47} -320.489 q^{49} +219.505 q^{51} +409.652 q^{53} +38.3804 q^{55} -236.103 q^{57} +196.000 q^{59} +701.359 q^{61} +42.7011 q^{63} +52.4128 q^{65} +900.587 q^{67} -336.000 q^{69} -756.500 q^{71} -1019.81 q^{73} -338.478 q^{75} +52.1902 q^{77} +327.549 q^{79} +81.0000 q^{81} -756.619 q^{83} +255.294 q^{85} -729.375 q^{87} +508.978 q^{89} +71.2716 q^{91} -836.152 q^{93} -274.598 q^{95} +614.358 q^{97} +99.0000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{3} - 16 q^{5} - 2 q^{7} + 18 q^{9}+O(q^{10})$$ 2 * q + 6 * q^3 - 16 * q^5 - 2 * q^7 + 18 * q^9 $$2 q + 6 q^{3} - 16 q^{5} - 2 q^{7} + 18 q^{9} + 22 q^{11} + 76 q^{13} - 48 q^{15} - 26 q^{17} - 54 q^{19} - 6 q^{21} - 224 q^{23} + 142 q^{25} + 54 q^{27} - 222 q^{29} + 40 q^{31} + 66 q^{33} + 148 q^{35} + 48 q^{37} + 228 q^{39} - 494 q^{41} - 66 q^{43} - 144 q^{45} + 64 q^{47} - 618 q^{49} - 78 q^{51} + 84 q^{53} - 176 q^{55} - 162 q^{57} + 392 q^{59} + 1104 q^{61} - 18 q^{63} - 1136 q^{65} + 928 q^{67} - 672 q^{69} - 456 q^{71} - 592 q^{73} + 426 q^{75} - 22 q^{77} + 230 q^{79} + 162 q^{81} + 348 q^{83} + 2188 q^{85} - 666 q^{87} + 972 q^{89} - 340 q^{91} + 120 q^{93} - 756 q^{95} - 1184 q^{97} + 198 q^{99}+O(q^{100})$$ 2 * q + 6 * q^3 - 16 * q^5 - 2 * q^7 + 18 * q^9 + 22 * q^11 + 76 * q^13 - 48 * q^15 - 26 * q^17 - 54 * q^19 - 6 * q^21 - 224 * q^23 + 142 * q^25 + 54 * q^27 - 222 * q^29 + 40 * q^31 + 66 * q^33 + 148 * q^35 + 48 * q^37 + 228 * q^39 - 494 * q^41 - 66 * q^43 - 144 * q^45 + 64 * q^47 - 618 * q^49 - 78 * q^51 + 84 * q^53 - 176 * q^55 - 162 * q^57 + 392 * q^59 + 1104 * q^61 - 18 * q^63 - 1136 * q^65 + 928 * q^67 - 672 * q^69 - 456 * q^71 - 592 * q^73 + 426 * q^75 - 22 * q^77 + 230 * q^79 + 162 * q^81 + 348 * q^83 + 2188 * q^85 - 666 * q^87 + 972 * q^89 - 340 * q^91 + 120 * q^93 - 756 * q^95 - 1184 * q^97 + 198 * q^99

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 3.00000 0.577350
$$4$$ 0 0
$$5$$ 3.48913 0.312077 0.156038 0.987751i $$-0.450128\pi$$
0.156038 + 0.987751i $$0.450128\pi$$
$$6$$ 0 0
$$7$$ 4.74456 0.256182 0.128091 0.991762i $$-0.459115\pi$$
0.128091 + 0.991762i $$0.459115\pi$$
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ 11.0000 0.301511
$$12$$ 0 0
$$13$$ 15.0217 0.320483 0.160242 0.987078i $$-0.448773\pi$$
0.160242 + 0.987078i $$0.448773\pi$$
$$14$$ 0 0
$$15$$ 10.4674 0.180178
$$16$$ 0 0
$$17$$ 73.1684 1.04388 0.521940 0.852982i $$-0.325209\pi$$
0.521940 + 0.852982i $$0.325209\pi$$
$$18$$ 0 0
$$19$$ −78.7011 −0.950277 −0.475138 0.879911i $$-0.657602\pi$$
−0.475138 + 0.879911i $$0.657602\pi$$
$$20$$ 0 0
$$21$$ 14.2337 0.147907
$$22$$ 0 0
$$23$$ −112.000 −1.01537 −0.507687 0.861541i $$-0.669499\pi$$
−0.507687 + 0.861541i $$0.669499\pi$$
$$24$$ 0 0
$$25$$ −112.826 −0.902608
$$26$$ 0 0
$$27$$ 27.0000 0.192450
$$28$$ 0 0
$$29$$ −243.125 −1.55680 −0.778399 0.627769i $$-0.783968\pi$$
−0.778399 + 0.627769i $$0.783968\pi$$
$$30$$ 0 0
$$31$$ −278.717 −1.61481 −0.807405 0.589998i $$-0.799129\pi$$
−0.807405 + 0.589998i $$0.799129\pi$$
$$32$$ 0 0
$$33$$ 33.0000 0.174078
$$34$$ 0 0
$$35$$ 16.5544 0.0799486
$$36$$ 0 0
$$37$$ −102.380 −0.454898 −0.227449 0.973790i $$-0.573039\pi$$
−0.227449 + 0.973790i $$0.573039\pi$$
$$38$$ 0 0
$$39$$ 45.0652 0.185031
$$40$$ 0 0
$$41$$ −241.255 −0.918970 −0.459485 0.888186i $$-0.651966\pi$$
−0.459485 + 0.888186i $$0.651966\pi$$
$$42$$ 0 0
$$43$$ −280.016 −0.993071 −0.496536 0.868016i $$-0.665395\pi$$
−0.496536 + 0.868016i $$0.665395\pi$$
$$44$$ 0 0
$$45$$ 31.4021 0.104026
$$46$$ 0 0
$$47$$ 169.870 0.527192 0.263596 0.964633i $$-0.415091\pi$$
0.263596 + 0.964633i $$0.415091\pi$$
$$48$$ 0 0
$$49$$ −320.489 −0.934371
$$50$$ 0 0
$$51$$ 219.505 0.602684
$$52$$ 0 0
$$53$$ 409.652 1.06170 0.530849 0.847466i $$-0.321873\pi$$
0.530849 + 0.847466i $$0.321873\pi$$
$$54$$ 0 0
$$55$$ 38.3804 0.0940947
$$56$$ 0 0
$$57$$ −236.103 −0.548643
$$58$$ 0 0
$$59$$ 196.000 0.432492 0.216246 0.976339i $$-0.430619\pi$$
0.216246 + 0.976339i $$0.430619\pi$$
$$60$$ 0 0
$$61$$ 701.359 1.47213 0.736064 0.676912i $$-0.236682\pi$$
0.736064 + 0.676912i $$0.236682\pi$$
$$62$$ 0 0
$$63$$ 42.7011 0.0853941
$$64$$ 0 0
$$65$$ 52.4128 0.100015
$$66$$ 0 0
$$67$$ 900.587 1.64215 0.821076 0.570819i $$-0.193374\pi$$
0.821076 + 0.570819i $$0.193374\pi$$
$$68$$ 0 0
$$69$$ −336.000 −0.586227
$$70$$ 0 0
$$71$$ −756.500 −1.26451 −0.632254 0.774762i $$-0.717870\pi$$
−0.632254 + 0.774762i $$0.717870\pi$$
$$72$$ 0 0
$$73$$ −1019.81 −1.63507 −0.817536 0.575877i $$-0.804661\pi$$
−0.817536 + 0.575877i $$0.804661\pi$$
$$74$$ 0 0
$$75$$ −338.478 −0.521121
$$76$$ 0 0
$$77$$ 52.1902 0.0772419
$$78$$ 0 0
$$79$$ 327.549 0.466483 0.233241 0.972419i $$-0.425067\pi$$
0.233241 + 0.972419i $$0.425067\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ −756.619 −1.00060 −0.500300 0.865852i $$-0.666777\pi$$
−0.500300 + 0.865852i $$0.666777\pi$$
$$84$$ 0 0
$$85$$ 255.294 0.325771
$$86$$ 0 0
$$87$$ −729.375 −0.898818
$$88$$ 0 0
$$89$$ 508.978 0.606198 0.303099 0.952959i $$-0.401979\pi$$
0.303099 + 0.952959i $$0.401979\pi$$
$$90$$ 0 0
$$91$$ 71.2716 0.0821022
$$92$$ 0 0
$$93$$ −836.152 −0.932311
$$94$$ 0 0
$$95$$ −274.598 −0.296559
$$96$$ 0 0
$$97$$ 614.358 0.643079 0.321539 0.946896i $$-0.395800\pi$$
0.321539 + 0.946896i $$0.395800\pi$$
$$98$$ 0 0
$$99$$ 99.0000 0.100504
$$100$$ 0 0
$$101$$ 1015.92 1.00087 0.500434 0.865775i $$-0.333174\pi$$
0.500434 + 0.865775i $$0.333174\pi$$
$$102$$ 0 0
$$103$$ −1102.16 −1.05436 −0.527181 0.849753i $$-0.676751\pi$$
−0.527181 + 0.849753i $$0.676751\pi$$
$$104$$ 0 0
$$105$$ 49.6631 0.0461583
$$106$$ 0 0
$$107$$ 1377.58 1.24463 0.622315 0.782767i $$-0.286192\pi$$
0.622315 + 0.782767i $$0.286192\pi$$
$$108$$ 0 0
$$109$$ −320.217 −0.281388 −0.140694 0.990053i $$-0.544933\pi$$
−0.140694 + 0.990053i $$0.544933\pi$$
$$110$$ 0 0
$$111$$ −307.141 −0.262636
$$112$$ 0 0
$$113$$ −1629.45 −1.35651 −0.678254 0.734828i $$-0.737263\pi$$
−0.678254 + 0.734828i $$0.737263\pi$$
$$114$$ 0 0
$$115$$ −390.782 −0.316875
$$116$$ 0 0
$$117$$ 135.196 0.106828
$$118$$ 0 0
$$119$$ 347.152 0.267423
$$120$$ 0 0
$$121$$ 121.000 0.0909091
$$122$$ 0 0
$$123$$ −723.766 −0.530568
$$124$$ 0 0
$$125$$ −829.805 −0.593760
$$126$$ 0 0
$$127$$ −2291.26 −1.60091 −0.800457 0.599390i $$-0.795410\pi$$
−0.800457 + 0.599390i $$0.795410\pi$$
$$128$$ 0 0
$$129$$ −840.049 −0.573350
$$130$$ 0 0
$$131$$ −1147.41 −0.765267 −0.382633 0.923900i $$-0.624983\pi$$
−0.382633 + 0.923900i $$0.624983\pi$$
$$132$$ 0 0
$$133$$ −373.402 −0.243444
$$134$$ 0 0
$$135$$ 94.2064 0.0600592
$$136$$ 0 0
$$137$$ 1268.60 0.791121 0.395561 0.918440i $$-0.370550\pi$$
0.395561 + 0.918440i $$0.370550\pi$$
$$138$$ 0 0
$$139$$ −486.288 −0.296737 −0.148368 0.988932i $$-0.547402\pi$$
−0.148368 + 0.988932i $$0.547402\pi$$
$$140$$ 0 0
$$141$$ 509.609 0.304374
$$142$$ 0 0
$$143$$ 165.239 0.0966294
$$144$$ 0 0
$$145$$ −848.293 −0.485841
$$146$$ 0 0
$$147$$ −961.467 −0.539459
$$148$$ 0 0
$$149$$ −2354.11 −1.29434 −0.647169 0.762346i $$-0.724047\pi$$
−0.647169 + 0.762346i $$0.724047\pi$$
$$150$$ 0 0
$$151$$ 570.070 0.307229 0.153615 0.988131i $$-0.450909\pi$$
0.153615 + 0.988131i $$0.450909\pi$$
$$152$$ 0 0
$$153$$ 658.516 0.347960
$$154$$ 0 0
$$155$$ −972.479 −0.503945
$$156$$ 0 0
$$157$$ 2072.67 1.05361 0.526807 0.849985i $$-0.323389\pi$$
0.526807 + 0.849985i $$0.323389\pi$$
$$158$$ 0 0
$$159$$ 1228.96 0.612972
$$160$$ 0 0
$$161$$ −531.391 −0.260121
$$162$$ 0 0
$$163$$ 2676.51 1.28614 0.643069 0.765808i $$-0.277661\pi$$
0.643069 + 0.765808i $$0.277661\pi$$
$$164$$ 0 0
$$165$$ 115.141 0.0543256
$$166$$ 0 0
$$167$$ 1188.12 0.550536 0.275268 0.961368i $$-0.411233\pi$$
0.275268 + 0.961368i $$0.411233\pi$$
$$168$$ 0 0
$$169$$ −1971.35 −0.897290
$$170$$ 0 0
$$171$$ −708.310 −0.316759
$$172$$ 0 0
$$173$$ −807.147 −0.354718 −0.177359 0.984146i $$-0.556755\pi$$
−0.177359 + 0.984146i $$0.556755\pi$$
$$174$$ 0 0
$$175$$ −535.310 −0.231232
$$176$$ 0 0
$$177$$ 588.000 0.249699
$$178$$ 0 0
$$179$$ −1950.39 −0.814408 −0.407204 0.913337i $$-0.633496\pi$$
−0.407204 + 0.913337i $$0.633496\pi$$
$$180$$ 0 0
$$181$$ −1061.61 −0.435959 −0.217980 0.975953i $$-0.569947\pi$$
−0.217980 + 0.975953i $$0.569947\pi$$
$$182$$ 0 0
$$183$$ 2104.08 0.849933
$$184$$ 0 0
$$185$$ −357.218 −0.141963
$$186$$ 0 0
$$187$$ 804.853 0.314742
$$188$$ 0 0
$$189$$ 128.103 0.0493023
$$190$$ 0 0
$$191$$ −2136.41 −0.809348 −0.404674 0.914461i $$-0.632615\pi$$
−0.404674 + 0.914461i $$0.632615\pi$$
$$192$$ 0 0
$$193$$ 3947.76 1.47236 0.736181 0.676784i $$-0.236627\pi$$
0.736181 + 0.676784i $$0.236627\pi$$
$$194$$ 0 0
$$195$$ 157.238 0.0577439
$$196$$ 0 0
$$197$$ −923.886 −0.334133 −0.167066 0.985946i $$-0.553429\pi$$
−0.167066 + 0.985946i $$0.553429\pi$$
$$198$$ 0 0
$$199$$ 476.152 0.169616 0.0848078 0.996397i $$-0.472972\pi$$
0.0848078 + 0.996397i $$0.472972\pi$$
$$200$$ 0 0
$$201$$ 2701.76 0.948097
$$202$$ 0 0
$$203$$ −1153.52 −0.398824
$$204$$ 0 0
$$205$$ −841.770 −0.286789
$$206$$ 0 0
$$207$$ −1008.00 −0.338458
$$208$$ 0 0
$$209$$ −865.712 −0.286519
$$210$$ 0 0
$$211$$ −4918.24 −1.60467 −0.802336 0.596872i $$-0.796410\pi$$
−0.802336 + 0.596872i $$0.796410\pi$$
$$212$$ 0 0
$$213$$ −2269.50 −0.730064
$$214$$ 0 0
$$215$$ −977.012 −0.309915
$$216$$ 0 0
$$217$$ −1322.39 −0.413686
$$218$$ 0 0
$$219$$ −3059.44 −0.944010
$$220$$ 0 0
$$221$$ 1099.12 0.334546
$$222$$ 0 0
$$223$$ −2100.29 −0.630700 −0.315350 0.948975i $$-0.602122\pi$$
−0.315350 + 0.948975i $$0.602122\pi$$
$$224$$ 0 0
$$225$$ −1015.43 −0.300869
$$226$$ 0 0
$$227$$ −2257.16 −0.659970 −0.329985 0.943986i $$-0.607044\pi$$
−0.329985 + 0.943986i $$0.607044\pi$$
$$228$$ 0 0
$$229$$ 5311.07 1.53260 0.766301 0.642482i $$-0.222095\pi$$
0.766301 + 0.642482i $$0.222095\pi$$
$$230$$ 0 0
$$231$$ 156.571 0.0445956
$$232$$ 0 0
$$233$$ 2466.27 0.693435 0.346718 0.937970i $$-0.387296\pi$$
0.346718 + 0.937970i $$0.387296\pi$$
$$234$$ 0 0
$$235$$ 592.696 0.164524
$$236$$ 0 0
$$237$$ 982.646 0.269324
$$238$$ 0 0
$$239$$ −1429.40 −0.386863 −0.193432 0.981114i $$-0.561962\pi$$
−0.193432 + 0.981114i $$0.561962\pi$$
$$240$$ 0 0
$$241$$ −978.989 −0.261669 −0.130835 0.991404i $$-0.541766\pi$$
−0.130835 + 0.991404i $$0.541766\pi$$
$$242$$ 0 0
$$243$$ 243.000 0.0641500
$$244$$ 0 0
$$245$$ −1118.23 −0.291595
$$246$$ 0 0
$$247$$ −1182.23 −0.304548
$$248$$ 0 0
$$249$$ −2269.86 −0.577696
$$250$$ 0 0
$$251$$ −6530.63 −1.64227 −0.821135 0.570734i $$-0.806659\pi$$
−0.821135 + 0.570734i $$0.806659\pi$$
$$252$$ 0 0
$$253$$ −1232.00 −0.306147
$$254$$ 0 0
$$255$$ 765.882 0.188084
$$256$$ 0 0
$$257$$ 8130.26 1.97335 0.986676 0.162696i $$-0.0520188\pi$$
0.986676 + 0.162696i $$0.0520188\pi$$
$$258$$ 0 0
$$259$$ −485.750 −0.116537
$$260$$ 0 0
$$261$$ −2188.12 −0.518933
$$262$$ 0 0
$$263$$ 4549.42 1.06665 0.533326 0.845910i $$-0.320942\pi$$
0.533326 + 0.845910i $$0.320942\pi$$
$$264$$ 0 0
$$265$$ 1429.33 0.331332
$$266$$ 0 0
$$267$$ 1526.93 0.349988
$$268$$ 0 0
$$269$$ 29.1522 0.00660760 0.00330380 0.999995i $$-0.498948\pi$$
0.00330380 + 0.999995i $$0.498948\pi$$
$$270$$ 0 0
$$271$$ −7711.22 −1.72850 −0.864250 0.503063i $$-0.832206\pi$$
−0.864250 + 0.503063i $$0.832206\pi$$
$$272$$ 0 0
$$273$$ 213.815 0.0474017
$$274$$ 0 0
$$275$$ −1241.09 −0.272147
$$276$$ 0 0
$$277$$ −1127.52 −0.244571 −0.122286 0.992495i $$-0.539022\pi$$
−0.122286 + 0.992495i $$0.539022\pi$$
$$278$$ 0 0
$$279$$ −2508.46 −0.538270
$$280$$ 0 0
$$281$$ −1872.47 −0.397517 −0.198758 0.980049i $$-0.563691\pi$$
−0.198758 + 0.980049i $$0.563691\pi$$
$$282$$ 0 0
$$283$$ 2124.48 0.446245 0.223123 0.974790i $$-0.428375\pi$$
0.223123 + 0.974790i $$0.428375\pi$$
$$284$$ 0 0
$$285$$ −823.794 −0.171219
$$286$$ 0 0
$$287$$ −1144.65 −0.235424
$$288$$ 0 0
$$289$$ 440.621 0.0896846
$$290$$ 0 0
$$291$$ 1843.07 0.371282
$$292$$ 0 0
$$293$$ 3324.19 0.662802 0.331401 0.943490i $$-0.392479\pi$$
0.331401 + 0.943490i $$0.392479\pi$$
$$294$$ 0 0
$$295$$ 683.869 0.134971
$$296$$ 0 0
$$297$$ 297.000 0.0580259
$$298$$ 0 0
$$299$$ −1682.44 −0.325411
$$300$$ 0 0
$$301$$ −1328.55 −0.254407
$$302$$ 0 0
$$303$$ 3047.75 0.577851
$$304$$ 0 0
$$305$$ 2447.13 0.459417
$$306$$ 0 0
$$307$$ −1698.94 −0.315843 −0.157921 0.987452i $$-0.550479\pi$$
−0.157921 + 0.987452i $$0.550479\pi$$
$$308$$ 0 0
$$309$$ −3306.49 −0.608736
$$310$$ 0 0
$$311$$ −6928.83 −1.26334 −0.631668 0.775239i $$-0.717630\pi$$
−0.631668 + 0.775239i $$0.717630\pi$$
$$312$$ 0 0
$$313$$ −3560.75 −0.643020 −0.321510 0.946906i $$-0.604190\pi$$
−0.321510 + 0.946906i $$0.604190\pi$$
$$314$$ 0 0
$$315$$ 148.989 0.0266495
$$316$$ 0 0
$$317$$ −332.750 −0.0589561 −0.0294780 0.999565i $$-0.509385\pi$$
−0.0294780 + 0.999565i $$0.509385\pi$$
$$318$$ 0 0
$$319$$ −2674.37 −0.469393
$$320$$ 0 0
$$321$$ 4132.73 0.718587
$$322$$ 0 0
$$323$$ −5758.43 −0.991975
$$324$$ 0 0
$$325$$ −1694.84 −0.289271
$$326$$ 0 0
$$327$$ −960.652 −0.162459
$$328$$ 0 0
$$329$$ 805.957 0.135057
$$330$$ 0 0
$$331$$ −541.445 −0.0899108 −0.0449554 0.998989i $$-0.514315\pi$$
−0.0449554 + 0.998989i $$0.514315\pi$$
$$332$$ 0 0
$$333$$ −921.423 −0.151633
$$334$$ 0 0
$$335$$ 3142.26 0.512478
$$336$$ 0 0
$$337$$ 816.531 0.131986 0.0659930 0.997820i $$-0.478978\pi$$
0.0659930 + 0.997820i $$0.478978\pi$$
$$338$$ 0 0
$$339$$ −4888.34 −0.783180
$$340$$ 0 0
$$341$$ −3065.89 −0.486883
$$342$$ 0 0
$$343$$ −3147.97 −0.495552
$$344$$ 0 0
$$345$$ −1172.35 −0.182948
$$346$$ 0 0
$$347$$ 6260.53 0.968539 0.484269 0.874919i $$-0.339086\pi$$
0.484269 + 0.874919i $$0.339086\pi$$
$$348$$ 0 0
$$349$$ 12768.5 1.95840 0.979198 0.202906i $$-0.0650386\pi$$
0.979198 + 0.202906i $$0.0650386\pi$$
$$350$$ 0 0
$$351$$ 405.587 0.0616771
$$352$$ 0 0
$$353$$ −2649.28 −0.399453 −0.199727 0.979852i $$-0.564005\pi$$
−0.199727 + 0.979852i $$0.564005\pi$$
$$354$$ 0 0
$$355$$ −2639.52 −0.394623
$$356$$ 0 0
$$357$$ 1041.46 0.154397
$$358$$ 0 0
$$359$$ 3203.91 0.471020 0.235510 0.971872i $$-0.424324\pi$$
0.235510 + 0.971872i $$0.424324\pi$$
$$360$$ 0 0
$$361$$ −665.143 −0.0969737
$$362$$ 0 0
$$363$$ 363.000 0.0524864
$$364$$ 0 0
$$365$$ −3558.26 −0.510268
$$366$$ 0 0
$$367$$ 8429.40 1.19894 0.599470 0.800397i $$-0.295378\pi$$
0.599470 + 0.800397i $$0.295378\pi$$
$$368$$ 0 0
$$369$$ −2171.30 −0.306323
$$370$$ 0 0
$$371$$ 1943.62 0.271988
$$372$$ 0 0
$$373$$ 9388.53 1.30327 0.651635 0.758533i $$-0.274083\pi$$
0.651635 + 0.758533i $$0.274083\pi$$
$$374$$ 0 0
$$375$$ −2489.41 −0.342807
$$376$$ 0 0
$$377$$ −3652.16 −0.498928
$$378$$ 0 0
$$379$$ −14264.5 −1.93329 −0.966647 0.256112i $$-0.917558\pi$$
−0.966647 + 0.256112i $$0.917558\pi$$
$$380$$ 0 0
$$381$$ −6873.77 −0.924288
$$382$$ 0 0
$$383$$ −13462.2 −1.79605 −0.898026 0.439942i $$-0.854999\pi$$
−0.898026 + 0.439942i $$0.854999\pi$$
$$384$$ 0 0
$$385$$ 182.098 0.0241054
$$386$$ 0 0
$$387$$ −2520.15 −0.331024
$$388$$ 0 0
$$389$$ 941.881 0.122764 0.0613821 0.998114i $$-0.480449\pi$$
0.0613821 + 0.998114i $$0.480449\pi$$
$$390$$ 0 0
$$391$$ −8194.87 −1.05993
$$392$$ 0 0
$$393$$ −3442.24 −0.441827
$$394$$ 0 0
$$395$$ 1142.86 0.145578
$$396$$ 0 0
$$397$$ 847.839 0.107183 0.0535917 0.998563i $$-0.482933\pi$$
0.0535917 + 0.998563i $$0.482933\pi$$
$$398$$ 0 0
$$399$$ −1120.21 −0.140553
$$400$$ 0 0
$$401$$ 12203.6 1.51975 0.759875 0.650069i $$-0.225260\pi$$
0.759875 + 0.650069i $$0.225260\pi$$
$$402$$ 0 0
$$403$$ −4186.82 −0.517520
$$404$$ 0 0
$$405$$ 282.619 0.0346752
$$406$$ 0 0
$$407$$ −1126.18 −0.137157
$$408$$ 0 0
$$409$$ 8759.53 1.05900 0.529500 0.848310i $$-0.322380\pi$$
0.529500 + 0.848310i $$0.322380\pi$$
$$410$$ 0 0
$$411$$ 3805.79 0.456754
$$412$$ 0 0
$$413$$ 929.934 0.110797
$$414$$ 0 0
$$415$$ −2639.94 −0.312264
$$416$$ 0 0
$$417$$ −1458.86 −0.171321
$$418$$ 0 0
$$419$$ −11188.4 −1.30451 −0.652256 0.757999i $$-0.726177\pi$$
−0.652256 + 0.757999i $$0.726177\pi$$
$$420$$ 0 0
$$421$$ 14082.3 1.63023 0.815116 0.579298i $$-0.196673\pi$$
0.815116 + 0.579298i $$0.196673\pi$$
$$422$$ 0 0
$$423$$ 1528.83 0.175731
$$424$$ 0 0
$$425$$ −8255.30 −0.942214
$$426$$ 0 0
$$427$$ 3327.64 0.377133
$$428$$ 0 0
$$429$$ 495.718 0.0557890
$$430$$ 0 0
$$431$$ 5616.05 0.627647 0.313823 0.949481i $$-0.398390\pi$$
0.313823 + 0.949481i $$0.398390\pi$$
$$432$$ 0 0
$$433$$ 7195.75 0.798627 0.399314 0.916814i $$-0.369248\pi$$
0.399314 + 0.916814i $$0.369248\pi$$
$$434$$ 0 0
$$435$$ −2544.88 −0.280500
$$436$$ 0 0
$$437$$ 8814.52 0.964887
$$438$$ 0 0
$$439$$ −101.959 −0.0110848 −0.00554240 0.999985i $$-0.501764\pi$$
−0.00554240 + 0.999985i $$0.501764\pi$$
$$440$$ 0 0
$$441$$ −2884.40 −0.311457
$$442$$ 0 0
$$443$$ 4953.74 0.531285 0.265642 0.964072i $$-0.414416\pi$$
0.265642 + 0.964072i $$0.414416\pi$$
$$444$$ 0 0
$$445$$ 1775.89 0.189180
$$446$$ 0 0
$$447$$ −7062.34 −0.747287
$$448$$ 0 0
$$449$$ −11602.0 −1.21945 −0.609723 0.792615i $$-0.708719\pi$$
−0.609723 + 0.792615i $$0.708719\pi$$
$$450$$ 0 0
$$451$$ −2653.81 −0.277080
$$452$$ 0 0
$$453$$ 1710.21 0.177379
$$454$$ 0 0
$$455$$ 248.676 0.0256222
$$456$$ 0 0
$$457$$ −3530.68 −0.361397 −0.180698 0.983539i $$-0.557836\pi$$
−0.180698 + 0.983539i $$0.557836\pi$$
$$458$$ 0 0
$$459$$ 1975.55 0.200895
$$460$$ 0 0
$$461$$ −11566.3 −1.16854 −0.584271 0.811559i $$-0.698619\pi$$
−0.584271 + 0.811559i $$0.698619\pi$$
$$462$$ 0 0
$$463$$ −10888.5 −1.09294 −0.546470 0.837479i $$-0.684029\pi$$
−0.546470 + 0.837479i $$0.684029\pi$$
$$464$$ 0 0
$$465$$ −2917.44 −0.290953
$$466$$ 0 0
$$467$$ 10688.0 1.05906 0.529529 0.848292i $$-0.322369\pi$$
0.529529 + 0.848292i $$0.322369\pi$$
$$468$$ 0 0
$$469$$ 4272.89 0.420690
$$470$$ 0 0
$$471$$ 6218.02 0.608304
$$472$$ 0 0
$$473$$ −3080.18 −0.299422
$$474$$ 0 0
$$475$$ 8879.53 0.857728
$$476$$ 0 0
$$477$$ 3686.87 0.353900
$$478$$ 0 0
$$479$$ −2341.90 −0.223391 −0.111696 0.993742i $$-0.535628\pi$$
−0.111696 + 0.993742i $$0.535628\pi$$
$$480$$ 0 0
$$481$$ −1537.93 −0.145787
$$482$$ 0 0
$$483$$ −1594.17 −0.150181
$$484$$ 0 0
$$485$$ 2143.57 0.200690
$$486$$ 0 0
$$487$$ −6748.91 −0.627972 −0.313986 0.949428i $$-0.601665\pi$$
−0.313986 + 0.949428i $$0.601665\pi$$
$$488$$ 0 0
$$489$$ 8029.53 0.742552
$$490$$ 0 0
$$491$$ 7361.40 0.676609 0.338305 0.941037i $$-0.390147\pi$$
0.338305 + 0.941037i $$0.390147\pi$$
$$492$$ 0 0
$$493$$ −17789.1 −1.62511
$$494$$ 0 0
$$495$$ 345.423 0.0313649
$$496$$ 0 0
$$497$$ −3589.26 −0.323944
$$498$$ 0 0
$$499$$ 10381.7 0.931359 0.465680 0.884953i $$-0.345810\pi$$
0.465680 + 0.884953i $$0.345810\pi$$
$$500$$ 0 0
$$501$$ 3564.36 0.317852
$$502$$ 0 0
$$503$$ −19149.0 −1.69744 −0.848721 0.528840i $$-0.822627\pi$$
−0.848721 + 0.528840i $$0.822627\pi$$
$$504$$ 0 0
$$505$$ 3544.67 0.312348
$$506$$ 0 0
$$507$$ −5914.04 −0.518051
$$508$$ 0 0
$$509$$ −16073.2 −1.39967 −0.699836 0.714303i $$-0.746744\pi$$
−0.699836 + 0.714303i $$0.746744\pi$$
$$510$$ 0 0
$$511$$ −4838.58 −0.418877
$$512$$ 0 0
$$513$$ −2124.93 −0.182881
$$514$$ 0 0
$$515$$ −3845.58 −0.329042
$$516$$ 0 0
$$517$$ 1868.56 0.158954
$$518$$ 0 0
$$519$$ −2421.44 −0.204797
$$520$$ 0 0
$$521$$ −18955.3 −1.59395 −0.796975 0.604012i $$-0.793568\pi$$
−0.796975 + 0.604012i $$0.793568\pi$$
$$522$$ 0 0
$$523$$ −4442.19 −0.371402 −0.185701 0.982606i $$-0.559456\pi$$
−0.185701 + 0.982606i $$0.559456\pi$$
$$524$$ 0 0
$$525$$ −1605.93 −0.133502
$$526$$ 0 0
$$527$$ −20393.3 −1.68567
$$528$$ 0 0
$$529$$ 377.000 0.0309855
$$530$$ 0 0
$$531$$ 1764.00 0.144164
$$532$$ 0 0
$$533$$ −3624.08 −0.294515
$$534$$ 0 0
$$535$$ 4806.54 0.388420
$$536$$ 0 0
$$537$$ −5851.17 −0.470199
$$538$$ 0 0
$$539$$ −3525.38 −0.281723
$$540$$ 0 0
$$541$$ −2180.90 −0.173316 −0.0866580 0.996238i $$-0.527619\pi$$
−0.0866580 + 0.996238i $$0.527619\pi$$
$$542$$ 0 0
$$543$$ −3184.82 −0.251701
$$544$$ 0 0
$$545$$ −1117.28 −0.0878146
$$546$$ 0 0
$$547$$ 8225.04 0.642920 0.321460 0.946923i $$-0.395826\pi$$
0.321460 + 0.946923i $$0.395826\pi$$
$$548$$ 0 0
$$549$$ 6312.23 0.490709
$$550$$ 0 0
$$551$$ 19134.2 1.47939
$$552$$ 0 0
$$553$$ 1554.08 0.119505
$$554$$ 0 0
$$555$$ −1071.65 −0.0819625
$$556$$ 0 0
$$557$$ 25181.9 1.91561 0.957804 0.287423i $$-0.0927986\pi$$
0.957804 + 0.287423i $$0.0927986\pi$$
$$558$$ 0 0
$$559$$ −4206.33 −0.318263
$$560$$ 0 0
$$561$$ 2414.56 0.181716
$$562$$ 0 0
$$563$$ −4504.50 −0.337197 −0.168599 0.985685i $$-0.553924\pi$$
−0.168599 + 0.985685i $$0.553924\pi$$
$$564$$ 0 0
$$565$$ −5685.34 −0.423335
$$566$$ 0 0
$$567$$ 384.310 0.0284647
$$568$$ 0 0
$$569$$ −13447.0 −0.990732 −0.495366 0.868684i $$-0.664966\pi$$
−0.495366 + 0.868684i $$0.664966\pi$$
$$570$$ 0 0
$$571$$ −2605.52 −0.190959 −0.0954795 0.995431i $$-0.530438\pi$$
−0.0954795 + 0.995431i $$0.530438\pi$$
$$572$$ 0 0
$$573$$ −6409.24 −0.467277
$$574$$ 0 0
$$575$$ 12636.5 0.916485
$$576$$ 0 0
$$577$$ 6339.65 0.457406 0.228703 0.973496i $$-0.426552\pi$$
0.228703 + 0.973496i $$0.426552\pi$$
$$578$$ 0 0
$$579$$ 11843.3 0.850069
$$580$$ 0 0
$$581$$ −3589.83 −0.256336
$$582$$ 0 0
$$583$$ 4506.17 0.320114
$$584$$ 0 0
$$585$$ 471.715 0.0333385
$$586$$ 0 0
$$587$$ −13370.6 −0.940140 −0.470070 0.882629i $$-0.655771\pi$$
−0.470070 + 0.882629i $$0.655771\pi$$
$$588$$ 0 0
$$589$$ 21935.3 1.53452
$$590$$ 0 0
$$591$$ −2771.66 −0.192912
$$592$$ 0 0
$$593$$ 14319.3 0.991608 0.495804 0.868434i $$-0.334873\pi$$
0.495804 + 0.868434i $$0.334873\pi$$
$$594$$ 0 0
$$595$$ 1211.26 0.0834567
$$596$$ 0 0
$$597$$ 1428.46 0.0979276
$$598$$ 0 0
$$599$$ 5788.63 0.394853 0.197427 0.980318i $$-0.436742\pi$$
0.197427 + 0.980318i $$0.436742\pi$$
$$600$$ 0 0
$$601$$ 23968.1 1.62675 0.813375 0.581739i $$-0.197628\pi$$
0.813375 + 0.581739i $$0.197628\pi$$
$$602$$ 0 0
$$603$$ 8105.28 0.547384
$$604$$ 0 0
$$605$$ 422.184 0.0283706
$$606$$ 0 0
$$607$$ 23526.6 1.57317 0.786585 0.617482i $$-0.211847\pi$$
0.786585 + 0.617482i $$0.211847\pi$$
$$608$$ 0 0
$$609$$ −3460.56 −0.230261
$$610$$ 0 0
$$611$$ 2551.74 0.168956
$$612$$ 0 0
$$613$$ −1228.07 −0.0809159 −0.0404579 0.999181i $$-0.512882\pi$$
−0.0404579 + 0.999181i $$0.512882\pi$$
$$614$$ 0 0
$$615$$ −2525.31 −0.165578
$$616$$ 0 0
$$617$$ −9844.90 −0.642368 −0.321184 0.947017i $$-0.604081\pi$$
−0.321184 + 0.947017i $$0.604081\pi$$
$$618$$ 0 0
$$619$$ −6551.68 −0.425419 −0.212709 0.977115i $$-0.568229\pi$$
−0.212709 + 0.977115i $$0.568229\pi$$
$$620$$ 0 0
$$621$$ −3024.00 −0.195409
$$622$$ 0 0
$$623$$ 2414.88 0.155297
$$624$$ 0 0
$$625$$ 11208.0 0.717309
$$626$$ 0 0
$$627$$ −2597.14 −0.165422
$$628$$ 0 0
$$629$$ −7491.01 −0.474859
$$630$$ 0 0
$$631$$ 26440.5 1.66812 0.834058 0.551677i $$-0.186012\pi$$
0.834058 + 0.551677i $$0.186012\pi$$
$$632$$ 0 0
$$633$$ −14754.7 −0.926458
$$634$$ 0 0
$$635$$ −7994.48 −0.499608
$$636$$ 0 0
$$637$$ −4814.31 −0.299450
$$638$$ 0 0
$$639$$ −6808.50 −0.421502
$$640$$ 0 0
$$641$$ −27927.2 −1.72084 −0.860421 0.509584i $$-0.829799\pi$$
−0.860421 + 0.509584i $$0.829799\pi$$
$$642$$ 0 0
$$643$$ −16737.7 −1.02655 −0.513274 0.858225i $$-0.671568\pi$$
−0.513274 + 0.858225i $$0.671568\pi$$
$$644$$ 0 0
$$645$$ −2931.03 −0.178929
$$646$$ 0 0
$$647$$ −7818.70 −0.475092 −0.237546 0.971376i $$-0.576343\pi$$
−0.237546 + 0.971376i $$0.576343\pi$$
$$648$$ 0 0
$$649$$ 2156.00 0.130401
$$650$$ 0 0
$$651$$ −3967.17 −0.238842
$$652$$ 0 0
$$653$$ −19747.6 −1.18344 −0.591719 0.806144i $$-0.701550\pi$$
−0.591719 + 0.806144i $$0.701550\pi$$
$$654$$ 0 0
$$655$$ −4003.47 −0.238822
$$656$$ 0 0
$$657$$ −9178.33 −0.545024
$$658$$ 0 0
$$659$$ 7867.72 0.465072 0.232536 0.972588i $$-0.425298\pi$$
0.232536 + 0.972588i $$0.425298\pi$$
$$660$$ 0 0
$$661$$ −4227.41 −0.248755 −0.124378 0.992235i $$-0.539693\pi$$
−0.124378 + 0.992235i $$0.539693\pi$$
$$662$$ 0 0
$$663$$ 3297.35 0.193150
$$664$$ 0 0
$$665$$ −1302.85 −0.0759733
$$666$$ 0 0
$$667$$ 27230.0 1.58073
$$668$$ 0 0
$$669$$ −6300.88 −0.364135
$$670$$ 0 0
$$671$$ 7714.94 0.443863
$$672$$ 0 0
$$673$$ 29397.6 1.68379 0.841897 0.539638i $$-0.181439\pi$$
0.841897 + 0.539638i $$0.181439\pi$$
$$674$$ 0 0
$$675$$ −3046.30 −0.173707
$$676$$ 0 0
$$677$$ −5737.14 −0.325696 −0.162848 0.986651i $$-0.552068\pi$$
−0.162848 + 0.986651i $$0.552068\pi$$
$$678$$ 0 0
$$679$$ 2914.86 0.164745
$$680$$ 0 0
$$681$$ −6771.49 −0.381034
$$682$$ 0 0
$$683$$ 32097.6 1.79821 0.899107 0.437729i $$-0.144217\pi$$
0.899107 + 0.437729i $$0.144217\pi$$
$$684$$ 0 0
$$685$$ 4426.30 0.246891
$$686$$ 0 0
$$687$$ 15933.2 0.884848
$$688$$ 0 0
$$689$$ 6153.69 0.340257
$$690$$ 0 0
$$691$$ −16456.2 −0.905965 −0.452983 0.891519i $$-0.649640\pi$$
−0.452983 + 0.891519i $$0.649640\pi$$
$$692$$ 0 0
$$693$$ 469.712 0.0257473
$$694$$ 0 0
$$695$$ −1696.72 −0.0926047
$$696$$ 0 0
$$697$$ −17652.3 −0.959294
$$698$$ 0 0
$$699$$ 7398.80 0.400355
$$700$$ 0 0
$$701$$ −27238.1 −1.46758 −0.733788 0.679379i $$-0.762249\pi$$
−0.733788 + 0.679379i $$0.762249\pi$$
$$702$$ 0 0
$$703$$ 8057.44 0.432279
$$704$$ 0 0
$$705$$ 1778.09 0.0949882
$$706$$ 0 0
$$707$$ 4820.09 0.256405
$$708$$ 0 0
$$709$$ −28761.4 −1.52349 −0.761747 0.647875i $$-0.775658\pi$$
−0.761747 + 0.647875i $$0.775658\pi$$
$$710$$ 0 0
$$711$$ 2947.94 0.155494
$$712$$ 0 0
$$713$$ 31216.3 1.63964
$$714$$ 0 0
$$715$$ 576.540 0.0301558
$$716$$ 0 0
$$717$$ −4288.21 −0.223356
$$718$$ 0 0
$$719$$ 27272.0 1.41456 0.707282 0.706931i $$-0.249921\pi$$
0.707282 + 0.706931i $$0.249921\pi$$
$$720$$ 0 0
$$721$$ −5229.28 −0.270109
$$722$$ 0 0
$$723$$ −2936.97 −0.151075
$$724$$ 0 0
$$725$$ 27430.8 1.40518
$$726$$ 0 0
$$727$$ −3979.75 −0.203027 −0.101514 0.994834i $$-0.532369\pi$$
−0.101514 + 0.994834i $$0.532369\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ −20488.3 −1.03665
$$732$$ 0 0
$$733$$ −9342.48 −0.470767 −0.235384 0.971903i $$-0.575635\pi$$
−0.235384 + 0.971903i $$0.575635\pi$$
$$734$$ 0 0
$$735$$ −3354.68 −0.168353
$$736$$ 0 0
$$737$$ 9906.45 0.495127
$$738$$ 0 0
$$739$$ −28928.0 −1.43997 −0.719983 0.693992i $$-0.755850\pi$$
−0.719983 + 0.693992i $$0.755850\pi$$
$$740$$ 0 0
$$741$$ −3546.68 −0.175831
$$742$$ 0 0
$$743$$ −4857.04 −0.239822 −0.119911 0.992785i $$-0.538261\pi$$
−0.119911 + 0.992785i $$0.538261\pi$$
$$744$$ 0 0
$$745$$ −8213.80 −0.403933
$$746$$ 0 0
$$747$$ −6809.57 −0.333533
$$748$$ 0 0
$$749$$ 6536.00 0.318852
$$750$$ 0 0
$$751$$ −14355.4 −0.697517 −0.348759 0.937213i $$-0.613397\pi$$
−0.348759 + 0.937213i $$0.613397\pi$$
$$752$$ 0 0
$$753$$ −19591.9 −0.948165
$$754$$ 0 0
$$755$$ 1989.05 0.0958792
$$756$$ 0 0
$$757$$ 17714.9 0.850538 0.425269 0.905067i $$-0.360179\pi$$
0.425269 + 0.905067i $$0.360179\pi$$
$$758$$ 0 0
$$759$$ −3696.00 −0.176754
$$760$$ 0 0
$$761$$ −7945.82 −0.378497 −0.189248 0.981929i $$-0.560605\pi$$
−0.189248 + 0.981929i $$0.560605\pi$$
$$762$$ 0 0
$$763$$ −1519.29 −0.0720866
$$764$$ 0 0
$$765$$ 2297.64 0.108590
$$766$$ 0 0
$$767$$ 2944.26 0.138606
$$768$$ 0 0
$$769$$ 27308.1 1.28057 0.640284 0.768139i $$-0.278817\pi$$
0.640284 + 0.768139i $$0.278817\pi$$
$$770$$ 0 0
$$771$$ 24390.8 1.13932
$$772$$ 0 0
$$773$$ 18872.6 0.878136 0.439068 0.898454i $$-0.355309\pi$$
0.439068 + 0.898454i $$0.355309\pi$$
$$774$$ 0 0
$$775$$ 31446.6 1.45754
$$776$$ 0 0
$$777$$ −1457.25 −0.0672826
$$778$$ 0 0
$$779$$ 18987.1 0.873276
$$780$$ 0 0
$$781$$ −8321.50 −0.381263
$$782$$ 0 0
$$783$$ −6564.37 −0.299606
$$784$$ 0 0
$$785$$ 7231.82 0.328808
$$786$$ 0 0
$$787$$ −14512.1 −0.657307 −0.328654 0.944450i $$-0.606595\pi$$
−0.328654 + 0.944450i $$0.606595\pi$$
$$788$$ 0 0
$$789$$ 13648.3 0.615832
$$790$$ 0 0
$$791$$ −7731.01 −0.347513
$$792$$ 0 0
$$793$$ 10535.6 0.471792
$$794$$ 0 0
$$795$$ 4287.98 0.191294
$$796$$ 0 0
$$797$$ −29108.9 −1.29371 −0.646856 0.762612i $$-0.723917\pi$$
−0.646856 + 0.762612i $$0.723917\pi$$
$$798$$ 0 0
$$799$$ 12429.1 0.550325
$$800$$ 0 0
$$801$$ 4580.80 0.202066
$$802$$ 0 0
$$803$$ −11218.0 −0.492993
$$804$$ 0 0
$$805$$ −1854.09 −0.0811777
$$806$$ 0 0
$$807$$ 87.4567 0.00381490
$$808$$ 0 0
$$809$$ −3000.83 −0.130413 −0.0652063 0.997872i $$-0.520771\pi$$
−0.0652063 + 0.997872i $$0.520771\pi$$
$$810$$ 0 0
$$811$$ 6239.39 0.270154 0.135077 0.990835i $$-0.456872\pi$$
0.135077 + 0.990835i $$0.456872\pi$$
$$812$$ 0 0
$$813$$ −23133.7 −0.997950
$$814$$ 0 0
$$815$$ 9338.68 0.401374
$$816$$ 0 0
$$817$$ 22037.6 0.943693
$$818$$ 0 0
$$819$$ 641.445 0.0273674
$$820$$ 0 0
$$821$$ −14922.4 −0.634342 −0.317171 0.948368i $$-0.602733\pi$$
−0.317171 + 0.948368i $$0.602733\pi$$
$$822$$ 0 0
$$823$$ 25737.8 1.09011 0.545057 0.838399i $$-0.316508\pi$$
0.545057 + 0.838399i $$0.316508\pi$$
$$824$$ 0 0
$$825$$ −3723.26 −0.157124
$$826$$ 0 0
$$827$$ 27043.4 1.13711 0.568555 0.822645i $$-0.307503\pi$$
0.568555 + 0.822645i $$0.307503\pi$$
$$828$$ 0 0
$$829$$ 9795.41 0.410384 0.205192 0.978722i $$-0.434218\pi$$
0.205192 + 0.978722i $$0.434218\pi$$
$$830$$ 0 0
$$831$$ −3382.56 −0.141203
$$832$$ 0 0
$$833$$ −23449.7 −0.975370
$$834$$ 0 0
$$835$$ 4145.50 0.171809
$$836$$ 0 0
$$837$$ −7525.37 −0.310770
$$838$$ 0 0
$$839$$ −28875.5 −1.18819 −0.594095 0.804395i $$-0.702490\pi$$
−0.594095 + 0.804395i $$0.702490\pi$$
$$840$$ 0 0
$$841$$ 34720.7 1.42362
$$842$$ 0 0
$$843$$ −5617.41 −0.229506
$$844$$ 0 0
$$845$$ −6878.28 −0.280024
$$846$$ 0 0
$$847$$ 574.092 0.0232893
$$848$$ 0 0
$$849$$ 6373.45 0.257640
$$850$$ 0 0
$$851$$ 11466.6 0.461892
$$852$$ 0 0
$$853$$ 47157.1 1.89288 0.946441 0.322878i $$-0.104650\pi$$
0.946441 + 0.322878i $$0.104650\pi$$
$$854$$ 0 0
$$855$$ −2471.38 −0.0988531
$$856$$ 0 0
$$857$$ 5021.31 0.200145 0.100073 0.994980i $$-0.468092\pi$$
0.100073 + 0.994980i $$0.468092\pi$$
$$858$$ 0 0
$$859$$ −22921.1 −0.910428 −0.455214 0.890382i $$-0.650437\pi$$
−0.455214 + 0.890382i $$0.650437\pi$$
$$860$$ 0 0
$$861$$ −3433.95 −0.135922
$$862$$ 0 0
$$863$$ 19488.1 0.768693 0.384347 0.923189i $$-0.374427\pi$$
0.384347 + 0.923189i $$0.374427\pi$$
$$864$$ 0 0
$$865$$ −2816.24 −0.110699
$$866$$ 0 0
$$867$$ 1321.86 0.0517794
$$868$$ 0 0
$$869$$ 3603.04 0.140650
$$870$$ 0 0
$$871$$ 13528.4 0.526282
$$872$$ 0 0
$$873$$ 5529.22 0.214360
$$874$$ 0 0
$$875$$ −3937.06 −0.152111
$$876$$ 0 0
$$877$$ 8455.67 0.325573 0.162787 0.986661i $$-0.447952\pi$$
0.162787 + 0.986661i $$0.447952\pi$$
$$878$$ 0 0
$$879$$ 9972.56 0.382669
$$880$$ 0 0
$$881$$ −11291.2 −0.431794 −0.215897 0.976416i $$-0.569268\pi$$
−0.215897 + 0.976416i $$0.569268\pi$$
$$882$$ 0 0
$$883$$ 31818.1 1.21264 0.606322 0.795219i $$-0.292644\pi$$
0.606322 + 0.795219i $$0.292644\pi$$
$$884$$ 0 0
$$885$$ 2051.61 0.0779254
$$886$$ 0 0
$$887$$ −17481.1 −0.661732 −0.330866 0.943678i $$-0.607341\pi$$
−0.330866 + 0.943678i $$0.607341\pi$$
$$888$$ 0 0
$$889$$ −10871.0 −0.410126
$$890$$ 0 0
$$891$$ 891.000 0.0335013
$$892$$ 0 0
$$893$$ −13368.9 −0.500978
$$894$$ 0 0
$$895$$ −6805.16 −0.254158
$$896$$ 0 0
$$897$$ −5047.31 −0.187876
$$898$$ 0 0
$$899$$ 67763.1 2.51393
$$900$$ 0 0
$$901$$ 29973.6 1.10829
$$902$$ 0 0
$$903$$ −3985.66 −0.146882
$$904$$ 0 0
$$905$$ −3704.08 −0.136053
$$906$$ 0 0
$$907$$ 10607.4 0.388326 0.194163 0.980969i $$-0.437801\pi$$
0.194163 + 0.980969i $$0.437801\pi$$
$$908$$ 0 0
$$909$$ 9143.26 0.333623
$$910$$ 0 0
$$911$$ 41249.2 1.50016 0.750080 0.661347i $$-0.230015\pi$$
0.750080 + 0.661347i $$0.230015\pi$$
$$912$$ 0 0
$$913$$ −8322.81 −0.301692
$$914$$ 0 0
$$915$$ 7341.38 0.265244
$$916$$ 0 0
$$917$$ −5443.97 −0.196048
$$918$$ 0 0
$$919$$ 13858.1 0.497429 0.248714 0.968577i $$-0.419992\pi$$
0.248714 + 0.968577i $$0.419992\pi$$
$$920$$ 0 0
$$921$$ −5096.83 −0.182352
$$922$$ 0 0
$$923$$ −11363.9 −0.405253
$$924$$ 0 0
$$925$$ 11551.2 0.410595
$$926$$ 0 0
$$927$$ −9919.47 −0.351454
$$928$$ 0 0
$$929$$ 20893.7 0.737890 0.368945 0.929451i $$-0.379719\pi$$
0.368945 + 0.929451i $$0.379719\pi$$
$$930$$ 0 0
$$931$$ 25222.8 0.887911
$$932$$ 0 0
$$933$$ −20786.5 −0.729388
$$934$$ 0 0
$$935$$ 2808.23 0.0982236
$$936$$ 0 0
$$937$$ 3203.52 0.111691 0.0558454 0.998439i $$-0.482215\pi$$
0.0558454 + 0.998439i $$0.482215\pi$$
$$938$$ 0 0
$$939$$ −10682.2 −0.371248
$$940$$ 0 0
$$941$$ −19951.6 −0.691182 −0.345591 0.938385i $$-0.612322\pi$$
−0.345591 + 0.938385i $$0.612322\pi$$
$$942$$ 0 0
$$943$$ 27020.6 0.933099
$$944$$ 0 0
$$945$$ 446.968 0.0153861
$$946$$ 0 0
$$947$$ −38216.7 −1.31138 −0.655689 0.755031i $$-0.727622\pi$$
−0.655689 + 0.755031i $$0.727622\pi$$
$$948$$ 0 0
$$949$$ −15319.4 −0.524014
$$950$$ 0 0
$$951$$ −998.249 −0.0340383
$$952$$ 0 0
$$953$$ −47661.4 −1.62004 −0.810022 0.586399i $$-0.800545\pi$$
−0.810022 + 0.586399i $$0.800545\pi$$
$$954$$ 0 0
$$955$$ −7454.21 −0.252579
$$956$$ 0 0
$$957$$ −8023.12 −0.271004
$$958$$ 0 0
$$959$$ 6018.94 0.202671
$$960$$ 0 0
$$961$$ 47892.3 1.60761
$$962$$ 0 0
$$963$$ 12398.2 0.414876
$$964$$ 0 0
$$965$$ 13774.2 0.459490
$$966$$ 0 0
$$967$$ −18933.2 −0.629628 −0.314814 0.949153i $$-0.601942\pi$$
−0.314814 + 0.949153i $$0.601942\pi$$
$$968$$ 0 0
$$969$$ −17275.3 −0.572717
$$970$$ 0 0
$$971$$ −40660.3 −1.34382 −0.671911 0.740632i $$-0.734526\pi$$
−0.671911 + 0.740632i $$0.734526\pi$$
$$972$$ 0 0
$$973$$ −2307.23 −0.0760188
$$974$$ 0 0
$$975$$ −5084.53 −0.167011
$$976$$ 0 0
$$977$$ −22502.8 −0.736876 −0.368438 0.929652i $$-0.620107\pi$$
−0.368438 + 0.929652i $$0.620107\pi$$
$$978$$ 0 0
$$979$$ 5598.76 0.182775
$$980$$ 0 0
$$981$$ −2881.96 −0.0937959
$$982$$ 0 0
$$983$$ 4435.20 0.143907 0.0719536 0.997408i $$-0.477077\pi$$
0.0719536 + 0.997408i $$0.477077\pi$$
$$984$$ 0 0
$$985$$ −3223.55 −0.104275
$$986$$ 0 0
$$987$$ 2417.87 0.0779753
$$988$$ 0 0
$$989$$ 31361.8 1.00834
$$990$$ 0 0
$$991$$ −7362.76 −0.236010 −0.118005 0.993013i $$-0.537650\pi$$
−0.118005 + 0.993013i $$0.537650\pi$$
$$992$$ 0 0
$$993$$ −1624.33 −0.0519101
$$994$$ 0 0
$$995$$ 1661.35 0.0529331
$$996$$ 0 0
$$997$$ 53480.1 1.69883 0.849413 0.527728i $$-0.176956\pi$$
0.849413 + 0.527728i $$0.176956\pi$$
$$998$$ 0 0
$$999$$ −2764.27 −0.0875452
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 2112.4.a.bh.1.2 2
4.3 odd 2 2112.4.a.ba.1.2 2
8.3 odd 2 33.4.a.d.1.2 2
8.5 even 2 528.4.a.o.1.1 2
24.5 odd 2 1584.4.a.x.1.2 2
24.11 even 2 99.4.a.e.1.1 2
40.3 even 4 825.4.c.i.199.1 4
40.19 odd 2 825.4.a.k.1.1 2
40.27 even 4 825.4.c.i.199.4 4
56.27 even 2 1617.4.a.j.1.2 2
88.43 even 2 363.4.a.j.1.1 2
120.59 even 2 2475.4.a.o.1.2 2
264.131 odd 2 1089.4.a.t.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
33.4.a.d.1.2 2 8.3 odd 2
99.4.a.e.1.1 2 24.11 even 2
363.4.a.j.1.1 2 88.43 even 2
528.4.a.o.1.1 2 8.5 even 2
825.4.a.k.1.1 2 40.19 odd 2
825.4.c.i.199.1 4 40.3 even 4
825.4.c.i.199.4 4 40.27 even 4
1089.4.a.t.1.2 2 264.131 odd 2
1584.4.a.x.1.2 2 24.5 odd 2
1617.4.a.j.1.2 2 56.27 even 2
2112.4.a.ba.1.2 2 4.3 odd 2
2112.4.a.bh.1.2 2 1.1 even 1 trivial
2475.4.a.o.1.2 2 120.59 even 2