# Properties

 Label 2112.4.a.bg.1.1 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: $$\Q(\sqrt{97})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 24$$ x^2 - x - 24 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 33) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$5.42443$$ of defining polynomial Character $$\chi$$ $$=$$ 2112.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-3.00000 q^{3} -2.84886 q^{5} -31.6977 q^{7} +9.00000 q^{9} +O(q^{10})$$ $$q-3.00000 q^{3} -2.84886 q^{5} -31.6977 q^{7} +9.00000 q^{9} -11.0000 q^{11} -5.15114 q^{13} +8.54657 q^{15} +121.942 q^{17} +34.8489 q^{19} +95.0931 q^{21} -116.244 q^{23} -116.884 q^{25} -27.0000 q^{27} +69.4534 q^{29} -140.605 q^{31} +33.0000 q^{33} +90.3023 q^{35} +420.070 q^{37} +15.4534 q^{39} -322.058 q^{41} +321.035 q^{43} -25.6397 q^{45} +231.408 q^{47} +661.745 q^{49} -365.826 q^{51} -4.91916 q^{53} +31.3374 q^{55} -104.547 q^{57} +406.443 q^{59} +556.431 q^{61} -285.279 q^{63} +14.6749 q^{65} +84.7452 q^{67} +348.733 q^{69} -49.0808 q^{71} +785.884 q^{73} +350.652 q^{75} +348.675 q^{77} +383.118 q^{79} +81.0000 q^{81} -930.211 q^{83} -347.395 q^{85} -208.360 q^{87} -732.559 q^{89} +163.279 q^{91} +421.814 q^{93} -99.2794 q^{95} -1171.49 q^{97} -99.0000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 6 q^{3} + 14 q^{5} - 24 q^{7} + 18 q^{9}+O(q^{10})$$ 2 * q - 6 * q^3 + 14 * q^5 - 24 * q^7 + 18 * q^9 $$2 q - 6 q^{3} + 14 q^{5} - 24 q^{7} + 18 q^{9} - 22 q^{11} - 30 q^{13} - 42 q^{15} + 106 q^{17} + 50 q^{19} + 72 q^{21} - 134 q^{23} + 42 q^{25} - 54 q^{27} + 198 q^{29} - 360 q^{31} + 66 q^{33} + 220 q^{35} + 328 q^{37} + 90 q^{39} - 782 q^{41} + 386 q^{43} + 126 q^{45} - 266 q^{47} + 378 q^{49} - 318 q^{51} + 522 q^{53} - 154 q^{55} - 150 q^{57} - 172 q^{59} + 778 q^{61} - 216 q^{63} - 404 q^{65} - 776 q^{67} + 402 q^{69} - 630 q^{71} + 1296 q^{73} - 126 q^{75} + 264 q^{77} - 652 q^{79} + 162 q^{81} - 324 q^{83} - 616 q^{85} - 594 q^{87} - 756 q^{89} - 28 q^{91} + 1080 q^{93} + 156 q^{95} - 452 q^{97} - 198 q^{99}+O(q^{100})$$ 2 * q - 6 * q^3 + 14 * q^5 - 24 * q^7 + 18 * q^9 - 22 * q^11 - 30 * q^13 - 42 * q^15 + 106 * q^17 + 50 * q^19 + 72 * q^21 - 134 * q^23 + 42 * q^25 - 54 * q^27 + 198 * q^29 - 360 * q^31 + 66 * q^33 + 220 * q^35 + 328 * q^37 + 90 * q^39 - 782 * q^41 + 386 * q^43 + 126 * q^45 - 266 * q^47 + 378 * q^49 - 318 * q^51 + 522 * q^53 - 154 * q^55 - 150 * q^57 - 172 * q^59 + 778 * q^61 - 216 * q^63 - 404 * q^65 - 776 * q^67 + 402 * q^69 - 630 * q^71 + 1296 * q^73 - 126 * q^75 + 264 * q^77 - 652 * q^79 + 162 * q^81 - 324 * q^83 - 616 * q^85 - 594 * q^87 - 756 * q^89 - 28 * q^91 + 1080 * q^93 + 156 * q^95 - 452 * q^97 - 198 * q^99

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −3.00000 −0.577350
$$4$$ 0 0
$$5$$ −2.84886 −0.254810 −0.127405 0.991851i $$-0.540665\pi$$
−0.127405 + 0.991851i $$0.540665\pi$$
$$6$$ 0 0
$$7$$ −31.6977 −1.71152 −0.855758 0.517377i $$-0.826909\pi$$
−0.855758 + 0.517377i $$0.826909\pi$$
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ −11.0000 −0.301511
$$12$$ 0 0
$$13$$ −5.15114 −0.109898 −0.0549488 0.998489i $$-0.517500\pi$$
−0.0549488 + 0.998489i $$0.517500\pi$$
$$14$$ 0 0
$$15$$ 8.54657 0.147114
$$16$$ 0 0
$$17$$ 121.942 1.73972 0.869861 0.493297i $$-0.164208\pi$$
0.869861 + 0.493297i $$0.164208\pi$$
$$18$$ 0 0
$$19$$ 34.8489 0.420783 0.210391 0.977617i $$-0.432526\pi$$
0.210391 + 0.977617i $$0.432526\pi$$
$$20$$ 0 0
$$21$$ 95.0931 0.988144
$$22$$ 0 0
$$23$$ −116.244 −1.05385 −0.526926 0.849911i $$-0.676656\pi$$
−0.526926 + 0.849911i $$0.676656\pi$$
$$24$$ 0 0
$$25$$ −116.884 −0.935072
$$26$$ 0 0
$$27$$ −27.0000 −0.192450
$$28$$ 0 0
$$29$$ 69.4534 0.444730 0.222365 0.974963i $$-0.428622\pi$$
0.222365 + 0.974963i $$0.428622\pi$$
$$30$$ 0 0
$$31$$ −140.605 −0.814623 −0.407312 0.913289i $$-0.633534\pi$$
−0.407312 + 0.913289i $$0.633534\pi$$
$$32$$ 0 0
$$33$$ 33.0000 0.174078
$$34$$ 0 0
$$35$$ 90.3023 0.436111
$$36$$ 0 0
$$37$$ 420.070 1.86646 0.933232 0.359276i $$-0.116976\pi$$
0.933232 + 0.359276i $$0.116976\pi$$
$$38$$ 0 0
$$39$$ 15.4534 0.0634495
$$40$$ 0 0
$$41$$ −322.058 −1.22676 −0.613378 0.789789i $$-0.710190\pi$$
−0.613378 + 0.789789i $$0.710190\pi$$
$$42$$ 0 0
$$43$$ 321.035 1.13854 0.569272 0.822149i $$-0.307225\pi$$
0.569272 + 0.822149i $$0.307225\pi$$
$$44$$ 0 0
$$45$$ −25.6397 −0.0849365
$$46$$ 0 0
$$47$$ 231.408 0.718176 0.359088 0.933304i $$-0.383088\pi$$
0.359088 + 0.933304i $$0.383088\pi$$
$$48$$ 0 0
$$49$$ 661.745 1.92929
$$50$$ 0 0
$$51$$ −365.826 −1.00443
$$52$$ 0 0
$$53$$ −4.91916 −0.0127490 −0.00637452 0.999980i $$-0.502029\pi$$
−0.00637452 + 0.999980i $$0.502029\pi$$
$$54$$ 0 0
$$55$$ 31.3374 0.0768280
$$56$$ 0 0
$$57$$ −104.547 −0.242939
$$58$$ 0 0
$$59$$ 406.443 0.896854 0.448427 0.893820i $$-0.351984\pi$$
0.448427 + 0.893820i $$0.351984\pi$$
$$60$$ 0 0
$$61$$ 556.431 1.16793 0.583964 0.811779i $$-0.301501\pi$$
0.583964 + 0.811779i $$0.301501\pi$$
$$62$$ 0 0
$$63$$ −285.279 −0.570505
$$64$$ 0 0
$$65$$ 14.6749 0.0280030
$$66$$ 0 0
$$67$$ 84.7452 0.154526 0.0772632 0.997011i $$-0.475382\pi$$
0.0772632 + 0.997011i $$0.475382\pi$$
$$68$$ 0 0
$$69$$ 348.733 0.608442
$$70$$ 0 0
$$71$$ −49.0808 −0.0820398 −0.0410199 0.999158i $$-0.513061\pi$$
−0.0410199 + 0.999158i $$0.513061\pi$$
$$72$$ 0 0
$$73$$ 785.884 1.26001 0.630005 0.776591i $$-0.283053\pi$$
0.630005 + 0.776591i $$0.283053\pi$$
$$74$$ 0 0
$$75$$ 350.652 0.539864
$$76$$ 0 0
$$77$$ 348.675 0.516041
$$78$$ 0 0
$$79$$ 383.118 0.545622 0.272811 0.962068i $$-0.412047\pi$$
0.272811 + 0.962068i $$0.412047\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ −930.211 −1.23017 −0.615084 0.788462i $$-0.710878\pi$$
−0.615084 + 0.788462i $$0.710878\pi$$
$$84$$ 0 0
$$85$$ −347.395 −0.443298
$$86$$ 0 0
$$87$$ −208.360 −0.256765
$$88$$ 0 0
$$89$$ −732.559 −0.872484 −0.436242 0.899829i $$-0.643691\pi$$
−0.436242 + 0.899829i $$0.643691\pi$$
$$90$$ 0 0
$$91$$ 163.279 0.188092
$$92$$ 0 0
$$93$$ 421.814 0.470323
$$94$$ 0 0
$$95$$ −99.2794 −0.107220
$$96$$ 0 0
$$97$$ −1171.49 −1.22626 −0.613128 0.789984i $$-0.710089\pi$$
−0.613128 + 0.789984i $$0.710089\pi$$
$$98$$ 0 0
$$99$$ −99.0000 −0.100504
$$100$$ 0 0
$$101$$ 1221.27 1.20318 0.601589 0.798806i $$-0.294535\pi$$
0.601589 + 0.798806i $$0.294535\pi$$
$$102$$ 0 0
$$103$$ −516.745 −0.494334 −0.247167 0.968973i $$-0.579500\pi$$
−0.247167 + 0.968973i $$0.579500\pi$$
$$104$$ 0 0
$$105$$ −270.907 −0.251789
$$106$$ 0 0
$$107$$ −152.025 −0.137353 −0.0686765 0.997639i $$-0.521878\pi$$
−0.0686765 + 0.997639i $$0.521878\pi$$
$$108$$ 0 0
$$109$$ −2170.32 −1.90714 −0.953572 0.301164i $$-0.902625\pi$$
−0.953572 + 0.301164i $$0.902625\pi$$
$$110$$ 0 0
$$111$$ −1260.21 −1.07760
$$112$$ 0 0
$$113$$ −646.397 −0.538123 −0.269062 0.963123i $$-0.586714\pi$$
−0.269062 + 0.963123i $$0.586714\pi$$
$$114$$ 0 0
$$115$$ 331.163 0.268532
$$116$$ 0 0
$$117$$ −46.3603 −0.0366326
$$118$$ 0 0
$$119$$ −3865.28 −2.97756
$$120$$ 0 0
$$121$$ 121.000 0.0909091
$$122$$ 0 0
$$123$$ 966.174 0.708268
$$124$$ 0 0
$$125$$ 689.093 0.493075
$$126$$ 0 0
$$127$$ 993.304 0.694027 0.347014 0.937860i $$-0.387196\pi$$
0.347014 + 0.937860i $$0.387196\pi$$
$$128$$ 0 0
$$129$$ −963.105 −0.657339
$$130$$ 0 0
$$131$$ 385.814 0.257318 0.128659 0.991689i $$-0.458933\pi$$
0.128659 + 0.991689i $$0.458933\pi$$
$$132$$ 0 0
$$133$$ −1104.63 −0.720177
$$134$$ 0 0
$$135$$ 76.9192 0.0490381
$$136$$ 0 0
$$137$$ 884.840 0.551803 0.275901 0.961186i $$-0.411024\pi$$
0.275901 + 0.961186i $$0.411024\pi$$
$$138$$ 0 0
$$139$$ −1091.94 −0.666312 −0.333156 0.942872i $$-0.608114\pi$$
−0.333156 + 0.942872i $$0.608114\pi$$
$$140$$ 0 0
$$141$$ −694.223 −0.414639
$$142$$ 0 0
$$143$$ 56.6626 0.0331354
$$144$$ 0 0
$$145$$ −197.863 −0.113322
$$146$$ 0 0
$$147$$ −1985.24 −1.11387
$$148$$ 0 0
$$149$$ −297.014 −0.163304 −0.0816522 0.996661i $$-0.526020\pi$$
−0.0816522 + 0.996661i $$0.526020\pi$$
$$150$$ 0 0
$$151$$ 1887.86 1.01743 0.508716 0.860935i $$-0.330120\pi$$
0.508716 + 0.860935i $$0.330120\pi$$
$$152$$ 0 0
$$153$$ 1097.48 0.579907
$$154$$ 0 0
$$155$$ 400.562 0.207574
$$156$$ 0 0
$$157$$ 56.5343 0.0287384 0.0143692 0.999897i $$-0.495426\pi$$
0.0143692 + 0.999897i $$0.495426\pi$$
$$158$$ 0 0
$$159$$ 14.7575 0.00736066
$$160$$ 0 0
$$161$$ 3684.68 1.80369
$$162$$ 0 0
$$163$$ −49.2338 −0.0236582 −0.0118291 0.999930i $$-0.503765\pi$$
−0.0118291 + 0.999930i $$0.503765\pi$$
$$164$$ 0 0
$$165$$ −94.0123 −0.0443567
$$166$$ 0 0
$$167$$ −2068.75 −0.958589 −0.479294 0.877654i $$-0.659107\pi$$
−0.479294 + 0.877654i $$0.659107\pi$$
$$168$$ 0 0
$$169$$ −2170.47 −0.987923
$$170$$ 0 0
$$171$$ 313.640 0.140261
$$172$$ 0 0
$$173$$ 604.012 0.265446 0.132723 0.991153i $$-0.457628\pi$$
0.132723 + 0.991153i $$0.457628\pi$$
$$174$$ 0 0
$$175$$ 3704.96 1.60039
$$176$$ 0 0
$$177$$ −1219.33 −0.517799
$$178$$ 0 0
$$179$$ −2132.02 −0.890251 −0.445126 0.895468i $$-0.646841\pi$$
−0.445126 + 0.895468i $$0.646841\pi$$
$$180$$ 0 0
$$181$$ 589.371 0.242031 0.121015 0.992651i $$-0.461385\pi$$
0.121015 + 0.992651i $$0.461385\pi$$
$$182$$ 0 0
$$183$$ −1669.29 −0.674304
$$184$$ 0 0
$$185$$ −1196.72 −0.475593
$$186$$ 0 0
$$187$$ −1341.36 −0.524546
$$188$$ 0 0
$$189$$ 855.838 0.329381
$$190$$ 0 0
$$191$$ 2160.90 0.818624 0.409312 0.912395i $$-0.365769\pi$$
0.409312 + 0.912395i $$0.365769\pi$$
$$192$$ 0 0
$$193$$ −1490.91 −0.556052 −0.278026 0.960574i $$-0.589680\pi$$
−0.278026 + 0.960574i $$0.589680\pi$$
$$194$$ 0 0
$$195$$ −44.0246 −0.0161675
$$196$$ 0 0
$$197$$ 230.529 0.0833732 0.0416866 0.999131i $$-0.486727\pi$$
0.0416866 + 0.999131i $$0.486727\pi$$
$$198$$ 0 0
$$199$$ −22.4007 −0.00797963 −0.00398982 0.999992i $$-0.501270\pi$$
−0.00398982 + 0.999992i $$0.501270\pi$$
$$200$$ 0 0
$$201$$ −254.236 −0.0892159
$$202$$ 0 0
$$203$$ −2201.51 −0.761163
$$204$$ 0 0
$$205$$ 917.497 0.312589
$$206$$ 0 0
$$207$$ −1046.20 −0.351284
$$208$$ 0 0
$$209$$ −383.337 −0.126871
$$210$$ 0 0
$$211$$ −1051.64 −0.343117 −0.171558 0.985174i $$-0.554880\pi$$
−0.171558 + 0.985174i $$0.554880\pi$$
$$212$$ 0 0
$$213$$ 147.243 0.0473657
$$214$$ 0 0
$$215$$ −914.583 −0.290112
$$216$$ 0 0
$$217$$ 4456.84 1.39424
$$218$$ 0 0
$$219$$ −2357.65 −0.727467
$$220$$ 0 0
$$221$$ −628.141 −0.191191
$$222$$ 0 0
$$223$$ −3861.80 −1.15966 −0.579832 0.814736i $$-0.696882\pi$$
−0.579832 + 0.814736i $$0.696882\pi$$
$$224$$ 0 0
$$225$$ −1051.96 −0.311691
$$226$$ 0 0
$$227$$ −872.721 −0.255174 −0.127587 0.991827i $$-0.540723\pi$$
−0.127587 + 0.991827i $$0.540723\pi$$
$$228$$ 0 0
$$229$$ −1841.72 −0.531459 −0.265730 0.964048i $$-0.585613\pi$$
−0.265730 + 0.964048i $$0.585613\pi$$
$$230$$ 0 0
$$231$$ −1046.02 −0.297937
$$232$$ 0 0
$$233$$ 3932.14 1.10559 0.552796 0.833317i $$-0.313561\pi$$
0.552796 + 0.833317i $$0.313561\pi$$
$$234$$ 0 0
$$235$$ −659.248 −0.182998
$$236$$ 0 0
$$237$$ −1149.35 −0.315015
$$238$$ 0 0
$$239$$ −4772.10 −1.29155 −0.645777 0.763526i $$-0.723466\pi$$
−0.645777 + 0.763526i $$0.723466\pi$$
$$240$$ 0 0
$$241$$ 3988.84 1.06616 0.533078 0.846066i $$-0.321035\pi$$
0.533078 + 0.846066i $$0.321035\pi$$
$$242$$ 0 0
$$243$$ −243.000 −0.0641500
$$244$$ 0 0
$$245$$ −1885.22 −0.491601
$$246$$ 0 0
$$247$$ −179.511 −0.0462431
$$248$$ 0 0
$$249$$ 2790.63 0.710238
$$250$$ 0 0
$$251$$ −5474.22 −1.37661 −0.688306 0.725421i $$-0.741645\pi$$
−0.688306 + 0.725421i $$0.741645\pi$$
$$252$$ 0 0
$$253$$ 1278.69 0.317749
$$254$$ 0 0
$$255$$ 1042.19 0.255938
$$256$$ 0 0
$$257$$ −6434.01 −1.56164 −0.780822 0.624754i $$-0.785199\pi$$
−0.780822 + 0.624754i $$0.785199\pi$$
$$258$$ 0 0
$$259$$ −13315.3 −3.19448
$$260$$ 0 0
$$261$$ 625.081 0.148243
$$262$$ 0 0
$$263$$ −7589.00 −1.77931 −0.889654 0.456636i $$-0.849054\pi$$
−0.889654 + 0.456636i $$0.849054\pi$$
$$264$$ 0 0
$$265$$ 14.0140 0.00324858
$$266$$ 0 0
$$267$$ 2197.68 0.503729
$$268$$ 0 0
$$269$$ −478.178 −0.108383 −0.0541914 0.998531i $$-0.517258\pi$$
−0.0541914 + 0.998531i $$0.517258\pi$$
$$270$$ 0 0
$$271$$ 122.323 0.0274192 0.0137096 0.999906i $$-0.495636\pi$$
0.0137096 + 0.999906i $$0.495636\pi$$
$$272$$ 0 0
$$273$$ −489.838 −0.108595
$$274$$ 0 0
$$275$$ 1285.72 0.281935
$$276$$ 0 0
$$277$$ −8199.41 −1.77854 −0.889269 0.457385i $$-0.848786\pi$$
−0.889269 + 0.457385i $$0.848786\pi$$
$$278$$ 0 0
$$279$$ −1265.44 −0.271541
$$280$$ 0 0
$$281$$ 6943.79 1.47413 0.737067 0.675820i $$-0.236210\pi$$
0.737067 + 0.675820i $$0.236210\pi$$
$$282$$ 0 0
$$283$$ 1035.14 0.217429 0.108715 0.994073i $$-0.465327\pi$$
0.108715 + 0.994073i $$0.465327\pi$$
$$284$$ 0 0
$$285$$ 297.838 0.0619032
$$286$$ 0 0
$$287$$ 10208.5 2.09961
$$288$$ 0 0
$$289$$ 9956.85 2.02663
$$290$$ 0 0
$$291$$ 3514.47 0.707979
$$292$$ 0 0
$$293$$ 6144.81 1.22520 0.612600 0.790393i $$-0.290124\pi$$
0.612600 + 0.790393i $$0.290124\pi$$
$$294$$ 0 0
$$295$$ −1157.90 −0.228527
$$296$$ 0 0
$$297$$ 297.000 0.0580259
$$298$$ 0 0
$$299$$ 598.791 0.115816
$$300$$ 0 0
$$301$$ −10176.1 −1.94864
$$302$$ 0 0
$$303$$ −3663.81 −0.694655
$$304$$ 0 0
$$305$$ −1585.19 −0.297599
$$306$$ 0 0
$$307$$ −2186.09 −0.406406 −0.203203 0.979137i $$-0.565135\pi$$
−0.203203 + 0.979137i $$0.565135\pi$$
$$308$$ 0 0
$$309$$ 1550.24 0.285404
$$310$$ 0 0
$$311$$ 7484.83 1.36471 0.682357 0.731019i $$-0.260955\pi$$
0.682357 + 0.731019i $$0.260955\pi$$
$$312$$ 0 0
$$313$$ −6833.33 −1.23400 −0.617001 0.786962i $$-0.711653\pi$$
−0.617001 + 0.786962i $$0.711653\pi$$
$$314$$ 0 0
$$315$$ 812.721 0.145370
$$316$$ 0 0
$$317$$ −924.265 −0.163760 −0.0818800 0.996642i $$-0.526092\pi$$
−0.0818800 + 0.996642i $$0.526092\pi$$
$$318$$ 0 0
$$319$$ −763.988 −0.134091
$$320$$ 0 0
$$321$$ 456.074 0.0793008
$$322$$ 0 0
$$323$$ 4249.54 0.732046
$$324$$ 0 0
$$325$$ 602.086 0.102762
$$326$$ 0 0
$$327$$ 6510.95 1.10109
$$328$$ 0 0
$$329$$ −7335.10 −1.22917
$$330$$ 0 0
$$331$$ −9820.46 −1.63076 −0.815380 0.578927i $$-0.803472\pi$$
−0.815380 + 0.578927i $$0.803472\pi$$
$$332$$ 0 0
$$333$$ 3780.63 0.622154
$$334$$ 0 0
$$335$$ −241.427 −0.0393748
$$336$$ 0 0
$$337$$ 600.808 0.0971161 0.0485580 0.998820i $$-0.484537\pi$$
0.0485580 + 0.998820i $$0.484537\pi$$
$$338$$ 0 0
$$339$$ 1939.19 0.310686
$$340$$ 0 0
$$341$$ 1546.65 0.245618
$$342$$ 0 0
$$343$$ −10103.5 −1.59049
$$344$$ 0 0
$$345$$ −993.490 −0.155037
$$346$$ 0 0
$$347$$ −3143.41 −0.486303 −0.243152 0.969988i $$-0.578181\pi$$
−0.243152 + 0.969988i $$0.578181\pi$$
$$348$$ 0 0
$$349$$ −720.663 −0.110533 −0.0552667 0.998472i $$-0.517601\pi$$
−0.0552667 + 0.998472i $$0.517601\pi$$
$$350$$ 0 0
$$351$$ 139.081 0.0211498
$$352$$ 0 0
$$353$$ 1207.12 0.182007 0.0910034 0.995851i $$-0.470993\pi$$
0.0910034 + 0.995851i $$0.470993\pi$$
$$354$$ 0 0
$$355$$ 139.824 0.0209045
$$356$$ 0 0
$$357$$ 11595.8 1.71910
$$358$$ 0 0
$$359$$ −8748.31 −1.28612 −0.643062 0.765814i $$-0.722336\pi$$
−0.643062 + 0.765814i $$0.722336\pi$$
$$360$$ 0 0
$$361$$ −5644.56 −0.822942
$$362$$ 0 0
$$363$$ −363.000 −0.0524864
$$364$$ 0 0
$$365$$ −2238.87 −0.321063
$$366$$ 0 0
$$367$$ 6730.45 0.957293 0.478647 0.878008i $$-0.341128\pi$$
0.478647 + 0.878008i $$0.341128\pi$$
$$368$$ 0 0
$$369$$ −2898.52 −0.408919
$$370$$ 0 0
$$371$$ 155.926 0.0218202
$$372$$ 0 0
$$373$$ 227.394 0.0315657 0.0157828 0.999875i $$-0.494976\pi$$
0.0157828 + 0.999875i $$0.494976\pi$$
$$374$$ 0 0
$$375$$ −2067.28 −0.284677
$$376$$ 0 0
$$377$$ −357.764 −0.0488748
$$378$$ 0 0
$$379$$ 11356.2 1.53913 0.769565 0.638568i $$-0.220473\pi$$
0.769565 + 0.638568i $$0.220473\pi$$
$$380$$ 0 0
$$381$$ −2979.91 −0.400697
$$382$$ 0 0
$$383$$ −10753.6 −1.43468 −0.717338 0.696725i $$-0.754640\pi$$
−0.717338 + 0.696725i $$0.754640\pi$$
$$384$$ 0 0
$$385$$ −993.325 −0.131492
$$386$$ 0 0
$$387$$ 2889.32 0.379515
$$388$$ 0 0
$$389$$ 11727.1 1.52850 0.764252 0.644918i $$-0.223109\pi$$
0.764252 + 0.644918i $$0.223109\pi$$
$$390$$ 0 0
$$391$$ −14175.1 −1.83341
$$392$$ 0 0
$$393$$ −1157.44 −0.148563
$$394$$ 0 0
$$395$$ −1091.45 −0.139030
$$396$$ 0 0
$$397$$ 359.905 0.0454990 0.0227495 0.999741i $$-0.492758\pi$$
0.0227495 + 0.999741i $$0.492758\pi$$
$$398$$ 0 0
$$399$$ 3313.89 0.415794
$$400$$ 0 0
$$401$$ −4066.71 −0.506438 −0.253219 0.967409i $$-0.581489\pi$$
−0.253219 + 0.967409i $$0.581489\pi$$
$$402$$ 0 0
$$403$$ 724.274 0.0895252
$$404$$ 0 0
$$405$$ −230.757 −0.0283122
$$406$$ 0 0
$$407$$ −4620.77 −0.562760
$$408$$ 0 0
$$409$$ −13488.8 −1.63076 −0.815379 0.578927i $$-0.803472\pi$$
−0.815379 + 0.578927i $$0.803472\pi$$
$$410$$ 0 0
$$411$$ −2654.52 −0.318584
$$412$$ 0 0
$$413$$ −12883.3 −1.53498
$$414$$ 0 0
$$415$$ 2650.04 0.313459
$$416$$ 0 0
$$417$$ 3275.83 0.384695
$$418$$ 0 0
$$419$$ −7040.12 −0.820841 −0.410420 0.911896i $$-0.634618\pi$$
−0.410420 + 0.911896i $$0.634618\pi$$
$$420$$ 0 0
$$421$$ −9171.74 −1.06177 −0.530883 0.847445i $$-0.678140\pi$$
−0.530883 + 0.847445i $$0.678140\pi$$
$$422$$ 0 0
$$423$$ 2082.67 0.239392
$$424$$ 0 0
$$425$$ −14253.1 −1.62677
$$426$$ 0 0
$$427$$ −17637.6 −1.99893
$$428$$ 0 0
$$429$$ −169.988 −0.0191307
$$430$$ 0 0
$$431$$ −992.995 −0.110976 −0.0554882 0.998459i $$-0.517672\pi$$
−0.0554882 + 0.998459i $$0.517672\pi$$
$$432$$ 0 0
$$433$$ 3790.21 0.420660 0.210330 0.977630i $$-0.432546\pi$$
0.210330 + 0.977630i $$0.432546\pi$$
$$434$$ 0 0
$$435$$ 593.589 0.0654262
$$436$$ 0 0
$$437$$ −4050.98 −0.443443
$$438$$ 0 0
$$439$$ 5136.97 0.558483 0.279242 0.960221i $$-0.409917\pi$$
0.279242 + 0.960221i $$0.409917\pi$$
$$440$$ 0 0
$$441$$ 5955.71 0.643095
$$442$$ 0 0
$$443$$ 10676.8 1.14508 0.572541 0.819876i $$-0.305958\pi$$
0.572541 + 0.819876i $$0.305958\pi$$
$$444$$ 0 0
$$445$$ 2086.96 0.222317
$$446$$ 0 0
$$447$$ 891.042 0.0942838
$$448$$ 0 0
$$449$$ 10529.9 1.10676 0.553379 0.832929i $$-0.313338\pi$$
0.553379 + 0.832929i $$0.313338\pi$$
$$450$$ 0 0
$$451$$ 3542.64 0.369881
$$452$$ 0 0
$$453$$ −5663.59 −0.587414
$$454$$ 0 0
$$455$$ −465.160 −0.0479275
$$456$$ 0 0
$$457$$ −14072.5 −1.44045 −0.720225 0.693741i $$-0.755961\pi$$
−0.720225 + 0.693741i $$0.755961\pi$$
$$458$$ 0 0
$$459$$ −3292.43 −0.334810
$$460$$ 0 0
$$461$$ 30.8173 0.00311346 0.00155673 0.999999i $$-0.499504\pi$$
0.00155673 + 0.999999i $$0.499504\pi$$
$$462$$ 0 0
$$463$$ −17591.3 −1.76573 −0.882867 0.469622i $$-0.844390\pi$$
−0.882867 + 0.469622i $$0.844390\pi$$
$$464$$ 0 0
$$465$$ −1201.69 −0.119843
$$466$$ 0 0
$$467$$ 13273.1 1.31522 0.657609 0.753360i $$-0.271568\pi$$
0.657609 + 0.753360i $$0.271568\pi$$
$$468$$ 0 0
$$469$$ −2686.23 −0.264474
$$470$$ 0 0
$$471$$ −169.603 −0.0165921
$$472$$ 0 0
$$473$$ −3531.39 −0.343284
$$474$$ 0 0
$$475$$ −4073.27 −0.393462
$$476$$ 0 0
$$477$$ −44.2724 −0.00424968
$$478$$ 0 0
$$479$$ −2496.68 −0.238155 −0.119077 0.992885i $$-0.537994\pi$$
−0.119077 + 0.992885i $$0.537994\pi$$
$$480$$ 0 0
$$481$$ −2163.84 −0.205120
$$482$$ 0 0
$$483$$ −11054.0 −1.04136
$$484$$ 0 0
$$485$$ 3337.41 0.312462
$$486$$ 0 0
$$487$$ 3464.42 0.322357 0.161178 0.986925i $$-0.448471\pi$$
0.161178 + 0.986925i $$0.448471\pi$$
$$488$$ 0 0
$$489$$ 147.701 0.0136591
$$490$$ 0 0
$$491$$ −16224.6 −1.49125 −0.745625 0.666366i $$-0.767849\pi$$
−0.745625 + 0.666366i $$0.767849\pi$$
$$492$$ 0 0
$$493$$ 8469.29 0.773707
$$494$$ 0 0
$$495$$ 282.037 0.0256093
$$496$$ 0 0
$$497$$ 1555.75 0.140412
$$498$$ 0 0
$$499$$ 9993.81 0.896562 0.448281 0.893893i $$-0.352036\pi$$
0.448281 + 0.893893i $$0.352036\pi$$
$$500$$ 0 0
$$501$$ 6206.24 0.553441
$$502$$ 0 0
$$503$$ 15334.8 1.35933 0.679667 0.733520i $$-0.262124\pi$$
0.679667 + 0.733520i $$0.262124\pi$$
$$504$$ 0 0
$$505$$ −3479.23 −0.306581
$$506$$ 0 0
$$507$$ 6511.40 0.570377
$$508$$ 0 0
$$509$$ 7291.23 0.634927 0.317464 0.948270i $$-0.397169\pi$$
0.317464 + 0.948270i $$0.397169\pi$$
$$510$$ 0 0
$$511$$ −24910.7 −2.15653
$$512$$ 0 0
$$513$$ −940.919 −0.0809797
$$514$$ 0 0
$$515$$ 1472.13 0.125961
$$516$$ 0 0
$$517$$ −2545.49 −0.216538
$$518$$ 0 0
$$519$$ −1812.04 −0.153255
$$520$$ 0 0
$$521$$ 16794.3 1.41223 0.706114 0.708098i $$-0.250447\pi$$
0.706114 + 0.708098i $$0.250447\pi$$
$$522$$ 0 0
$$523$$ −21009.4 −1.75655 −0.878275 0.478157i $$-0.841305\pi$$
−0.878275 + 0.478157i $$0.841305\pi$$
$$524$$ 0 0
$$525$$ −11114.9 −0.923986
$$526$$ 0 0
$$527$$ −17145.6 −1.41722
$$528$$ 0 0
$$529$$ 1345.73 0.110605
$$530$$ 0 0
$$531$$ 3657.99 0.298951
$$532$$ 0 0
$$533$$ 1658.97 0.134818
$$534$$ 0 0
$$535$$ 433.097 0.0349989
$$536$$ 0 0
$$537$$ 6396.07 0.513987
$$538$$ 0 0
$$539$$ −7279.20 −0.581702
$$540$$ 0 0
$$541$$ 16802.8 1.33532 0.667662 0.744464i $$-0.267295\pi$$
0.667662 + 0.744464i $$0.267295\pi$$
$$542$$ 0 0
$$543$$ −1768.11 −0.139737
$$544$$ 0 0
$$545$$ 6182.93 0.485959
$$546$$ 0 0
$$547$$ 16784.5 1.31198 0.655990 0.754770i $$-0.272251\pi$$
0.655990 + 0.754770i $$0.272251\pi$$
$$548$$ 0 0
$$549$$ 5007.88 0.389309
$$550$$ 0 0
$$551$$ 2420.37 0.187135
$$552$$ 0 0
$$553$$ −12144.0 −0.933840
$$554$$ 0 0
$$555$$ 3590.16 0.274584
$$556$$ 0 0
$$557$$ −18127.0 −1.37893 −0.689467 0.724317i $$-0.742155\pi$$
−0.689467 + 0.724317i $$0.742155\pi$$
$$558$$ 0 0
$$559$$ −1653.70 −0.125123
$$560$$ 0 0
$$561$$ 4024.09 0.302847
$$562$$ 0 0
$$563$$ 2090.88 0.156518 0.0782592 0.996933i $$-0.475064\pi$$
0.0782592 + 0.996933i $$0.475064\pi$$
$$564$$ 0 0
$$565$$ 1841.49 0.137119
$$566$$ 0 0
$$567$$ −2567.51 −0.190168
$$568$$ 0 0
$$569$$ 6249.23 0.460424 0.230212 0.973140i $$-0.426058\pi$$
0.230212 + 0.973140i $$0.426058\pi$$
$$570$$ 0 0
$$571$$ 6048.79 0.443317 0.221659 0.975124i $$-0.428853\pi$$
0.221659 + 0.975124i $$0.428853\pi$$
$$572$$ 0 0
$$573$$ −6482.69 −0.472633
$$574$$ 0 0
$$575$$ 13587.1 0.985428
$$576$$ 0 0
$$577$$ −15729.1 −1.13486 −0.567429 0.823423i $$-0.692062\pi$$
−0.567429 + 0.823423i $$0.692062\pi$$
$$578$$ 0 0
$$579$$ 4472.73 0.321037
$$580$$ 0 0
$$581$$ 29485.6 2.10545
$$582$$ 0 0
$$583$$ 54.1108 0.00384398
$$584$$ 0 0
$$585$$ 132.074 0.00933433
$$586$$ 0 0
$$587$$ 15620.5 1.09835 0.549173 0.835709i $$-0.314943\pi$$
0.549173 + 0.835709i $$0.314943\pi$$
$$588$$ 0 0
$$589$$ −4899.91 −0.342780
$$590$$ 0 0
$$591$$ −691.587 −0.0481355
$$592$$ 0 0
$$593$$ −493.541 −0.0341776 −0.0170888 0.999854i $$-0.505440\pi$$
−0.0170888 + 0.999854i $$0.505440\pi$$
$$594$$ 0 0
$$595$$ 11011.6 0.758711
$$596$$ 0 0
$$597$$ 67.2022 0.00460704
$$598$$ 0 0
$$599$$ 12455.1 0.849585 0.424793 0.905291i $$-0.360347\pi$$
0.424793 + 0.905291i $$0.360347\pi$$
$$600$$ 0 0
$$601$$ 12454.8 0.845329 0.422664 0.906286i $$-0.361095\pi$$
0.422664 + 0.906286i $$0.361095\pi$$
$$602$$ 0 0
$$603$$ 762.707 0.0515088
$$604$$ 0 0
$$605$$ −344.712 −0.0231645
$$606$$ 0 0
$$607$$ 4243.19 0.283733 0.141867 0.989886i $$-0.454690\pi$$
0.141867 + 0.989886i $$0.454690\pi$$
$$608$$ 0 0
$$609$$ 6604.54 0.439458
$$610$$ 0 0
$$611$$ −1192.01 −0.0789259
$$612$$ 0 0
$$613$$ −5733.14 −0.377748 −0.188874 0.982001i $$-0.560484\pi$$
−0.188874 + 0.982001i $$0.560484\pi$$
$$614$$ 0 0
$$615$$ −2752.49 −0.180473
$$616$$ 0 0
$$617$$ 15642.1 1.02063 0.510314 0.859988i $$-0.329529\pi$$
0.510314 + 0.859988i $$0.329529\pi$$
$$618$$ 0 0
$$619$$ −7467.40 −0.484879 −0.242440 0.970167i $$-0.577948\pi$$
−0.242440 + 0.970167i $$0.577948\pi$$
$$620$$ 0 0
$$621$$ 3138.60 0.202814
$$622$$ 0 0
$$623$$ 23220.4 1.49327
$$624$$ 0 0
$$625$$ 12647.4 0.809432
$$626$$ 0 0
$$627$$ 1150.01 0.0732489
$$628$$ 0 0
$$629$$ 51224.2 3.24713
$$630$$ 0 0
$$631$$ 1486.38 0.0937745 0.0468872 0.998900i $$-0.485070\pi$$
0.0468872 + 0.998900i $$0.485070\pi$$
$$632$$ 0 0
$$633$$ 3154.91 0.198099
$$634$$ 0 0
$$635$$ −2829.78 −0.176845
$$636$$ 0 0
$$637$$ −3408.74 −0.212024
$$638$$ 0 0
$$639$$ −441.728 −0.0273466
$$640$$ 0 0
$$641$$ 12386.0 0.763211 0.381606 0.924325i $$-0.375371\pi$$
0.381606 + 0.924325i $$0.375371\pi$$
$$642$$ 0 0
$$643$$ −14458.1 −0.886737 −0.443369 0.896339i $$-0.646217\pi$$
−0.443369 + 0.896339i $$0.646217\pi$$
$$644$$ 0 0
$$645$$ 2743.75 0.167496
$$646$$ 0 0
$$647$$ −15792.8 −0.959625 −0.479813 0.877371i $$-0.659295\pi$$
−0.479813 + 0.877371i $$0.659295\pi$$
$$648$$ 0 0
$$649$$ −4470.87 −0.270412
$$650$$ 0 0
$$651$$ −13370.5 −0.804965
$$652$$ 0 0
$$653$$ 3179.93 0.190567 0.0952837 0.995450i $$-0.469624\pi$$
0.0952837 + 0.995450i $$0.469624\pi$$
$$654$$ 0 0
$$655$$ −1099.13 −0.0655672
$$656$$ 0 0
$$657$$ 7072.96 0.420003
$$658$$ 0 0
$$659$$ 11593.5 0.685308 0.342654 0.939462i $$-0.388674\pi$$
0.342654 + 0.939462i $$0.388674\pi$$
$$660$$ 0 0
$$661$$ −3233.88 −0.190293 −0.0951464 0.995463i $$-0.530332\pi$$
−0.0951464 + 0.995463i $$0.530332\pi$$
$$662$$ 0 0
$$663$$ 1884.42 0.110384
$$664$$ 0 0
$$665$$ 3146.93 0.183508
$$666$$ 0 0
$$667$$ −8073.56 −0.468680
$$668$$ 0 0
$$669$$ 11585.4 0.669532
$$670$$ 0 0
$$671$$ −6120.74 −0.352144
$$672$$ 0 0
$$673$$ −5495.72 −0.314776 −0.157388 0.987537i $$-0.550307\pi$$
−0.157388 + 0.987537i $$0.550307\pi$$
$$674$$ 0 0
$$675$$ 3155.87 0.179955
$$676$$ 0 0
$$677$$ −33836.7 −1.92090 −0.960451 0.278448i $$-0.910180\pi$$
−0.960451 + 0.278448i $$0.910180\pi$$
$$678$$ 0 0
$$679$$ 37133.6 2.09876
$$680$$ 0 0
$$681$$ 2618.16 0.147325
$$682$$ 0 0
$$683$$ −21080.3 −1.18099 −0.590493 0.807043i $$-0.701067\pi$$
−0.590493 + 0.807043i $$0.701067\pi$$
$$684$$ 0 0
$$685$$ −2520.78 −0.140605
$$686$$ 0 0
$$687$$ 5525.16 0.306838
$$688$$ 0 0
$$689$$ 25.3393 0.00140109
$$690$$ 0 0
$$691$$ 11811.3 0.650253 0.325127 0.945671i $$-0.394593\pi$$
0.325127 + 0.945671i $$0.394593\pi$$
$$692$$ 0 0
$$693$$ 3138.07 0.172014
$$694$$ 0 0
$$695$$ 3110.79 0.169783
$$696$$ 0 0
$$697$$ −39272.4 −2.13422
$$698$$ 0 0
$$699$$ −11796.4 −0.638313
$$700$$ 0 0
$$701$$ −4244.99 −0.228718 −0.114359 0.993440i $$-0.536481\pi$$
−0.114359 + 0.993440i $$0.536481\pi$$
$$702$$ 0 0
$$703$$ 14639.0 0.785376
$$704$$ 0 0
$$705$$ 1977.74 0.105654
$$706$$ 0 0
$$707$$ −38711.5 −2.05926
$$708$$ 0 0
$$709$$ 898.822 0.0476107 0.0238053 0.999717i $$-0.492422\pi$$
0.0238053 + 0.999717i $$0.492422\pi$$
$$710$$ 0 0
$$711$$ 3448.06 0.181874
$$712$$ 0 0
$$713$$ 16344.5 0.858493
$$714$$ 0 0
$$715$$ −161.424 −0.00844322
$$716$$ 0 0
$$717$$ 14316.3 0.745679
$$718$$ 0 0
$$719$$ −10741.8 −0.557165 −0.278582 0.960412i $$-0.589865\pi$$
−0.278582 + 0.960412i $$0.589865\pi$$
$$720$$ 0 0
$$721$$ 16379.6 0.846061
$$722$$ 0 0
$$723$$ −11966.5 −0.615546
$$724$$ 0 0
$$725$$ −8117.99 −0.415855
$$726$$ 0 0
$$727$$ −16794.2 −0.856758 −0.428379 0.903599i $$-0.640915\pi$$
−0.428379 + 0.903599i $$0.640915\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ 39147.7 1.98075
$$732$$ 0 0
$$733$$ −8659.40 −0.436347 −0.218173 0.975910i $$-0.570010\pi$$
−0.218173 + 0.975910i $$0.570010\pi$$
$$734$$ 0 0
$$735$$ 5655.65 0.283826
$$736$$ 0 0
$$737$$ −932.197 −0.0465915
$$738$$ 0 0
$$739$$ 16705.7 0.831567 0.415783 0.909464i $$-0.363507\pi$$
0.415783 + 0.909464i $$0.363507\pi$$
$$740$$ 0 0
$$741$$ 538.534 0.0266984
$$742$$ 0 0
$$743$$ −1292.12 −0.0637996 −0.0318998 0.999491i $$-0.510156\pi$$
−0.0318998 + 0.999491i $$0.510156\pi$$
$$744$$ 0 0
$$745$$ 846.151 0.0416115
$$746$$ 0 0
$$747$$ −8371.90 −0.410056
$$748$$ 0 0
$$749$$ 4818.83 0.235082
$$750$$ 0 0
$$751$$ 14980.4 0.727886 0.363943 0.931421i $$-0.381430\pi$$
0.363943 + 0.931421i $$0.381430\pi$$
$$752$$ 0 0
$$753$$ 16422.7 0.794787
$$754$$ 0 0
$$755$$ −5378.25 −0.259251
$$756$$ 0 0
$$757$$ −3003.41 −0.144202 −0.0721010 0.997397i $$-0.522970\pi$$
−0.0721010 + 0.997397i $$0.522970\pi$$
$$758$$ 0 0
$$759$$ −3836.06 −0.183452
$$760$$ 0 0
$$761$$ −20375.0 −0.970555 −0.485277 0.874360i $$-0.661281\pi$$
−0.485277 + 0.874360i $$0.661281\pi$$
$$762$$ 0 0
$$763$$ 68794.1 3.26411
$$764$$ 0 0
$$765$$ −3126.56 −0.147766
$$766$$ 0 0
$$767$$ −2093.65 −0.0985621
$$768$$ 0 0
$$769$$ −12372.4 −0.580184 −0.290092 0.956999i $$-0.593686\pi$$
−0.290092 + 0.956999i $$0.593686\pi$$
$$770$$ 0 0
$$771$$ 19302.0 0.901615
$$772$$ 0 0
$$773$$ −21023.6 −0.978225 −0.489113 0.872221i $$-0.662679\pi$$
−0.489113 + 0.872221i $$0.662679\pi$$
$$774$$ 0 0
$$775$$ 16434.4 0.761732
$$776$$ 0 0
$$777$$ 39945.8 1.84433
$$778$$ 0 0
$$779$$ −11223.4 −0.516198
$$780$$ 0 0
$$781$$ 539.889 0.0247359
$$782$$ 0 0
$$783$$ −1875.24 −0.0855884
$$784$$ 0 0
$$785$$ −161.058 −0.00732281
$$786$$ 0 0
$$787$$ 30286.2 1.37177 0.685886 0.727709i $$-0.259415\pi$$
0.685886 + 0.727709i $$0.259415\pi$$
$$788$$ 0 0
$$789$$ 22767.0 1.02728
$$790$$ 0 0
$$791$$ 20489.3 0.921007
$$792$$ 0 0
$$793$$ −2866.25 −0.128353
$$794$$ 0 0
$$795$$ −42.0420 −0.00187557
$$796$$ 0 0
$$797$$ −32337.8 −1.43722 −0.718610 0.695413i $$-0.755221\pi$$
−0.718610 + 0.695413i $$0.755221\pi$$
$$798$$ 0 0
$$799$$ 28218.3 1.24943
$$800$$ 0 0
$$801$$ −6593.03 −0.290828
$$802$$ 0 0
$$803$$ −8644.72 −0.379907
$$804$$ 0 0
$$805$$ −10497.1 −0.459596
$$806$$ 0 0
$$807$$ 1434.53 0.0625749
$$808$$ 0 0
$$809$$ −891.707 −0.0387525 −0.0193762 0.999812i $$-0.506168\pi$$
−0.0193762 + 0.999812i $$0.506168\pi$$
$$810$$ 0 0
$$811$$ −10114.9 −0.437957 −0.218978 0.975730i $$-0.570272\pi$$
−0.218978 + 0.975730i $$0.570272\pi$$
$$812$$ 0 0
$$813$$ −366.970 −0.0158305
$$814$$ 0 0
$$815$$ 140.260 0.00602833
$$816$$ 0 0
$$817$$ 11187.7 0.479080
$$818$$ 0 0
$$819$$ 1469.51 0.0626972
$$820$$ 0 0
$$821$$ −10833.5 −0.460525 −0.230262 0.973129i $$-0.573958\pi$$
−0.230262 + 0.973129i $$0.573958\pi$$
$$822$$ 0 0
$$823$$ −31958.5 −1.35359 −0.676794 0.736173i $$-0.736631\pi$$
−0.676794 + 0.736173i $$0.736631\pi$$
$$824$$ 0 0
$$825$$ −3857.17 −0.162775
$$826$$ 0 0
$$827$$ 34847.3 1.46525 0.732624 0.680634i $$-0.238296\pi$$
0.732624 + 0.680634i $$0.238296\pi$$
$$828$$ 0 0
$$829$$ −6537.91 −0.273910 −0.136955 0.990577i $$-0.543732\pi$$
−0.136955 + 0.990577i $$0.543732\pi$$
$$830$$ 0 0
$$831$$ 24598.2 1.02684
$$832$$ 0 0
$$833$$ 80694.5 3.35642
$$834$$ 0 0
$$835$$ 5893.56 0.244258
$$836$$ 0 0
$$837$$ 3796.32 0.156774
$$838$$ 0 0
$$839$$ −2710.34 −0.111527 −0.0557635 0.998444i $$-0.517759\pi$$
−0.0557635 + 0.998444i $$0.517759\pi$$
$$840$$ 0 0
$$841$$ −19565.2 −0.802215
$$842$$ 0 0
$$843$$ −20831.4 −0.851092
$$844$$ 0 0
$$845$$ 6183.35 0.251732
$$846$$ 0 0
$$847$$ −3835.42 −0.155592
$$848$$ 0 0
$$849$$ −3105.41 −0.125533
$$850$$ 0 0
$$851$$ −48830.8 −1.96698
$$852$$ 0 0
$$853$$ −9759.32 −0.391738 −0.195869 0.980630i $$-0.562753\pi$$
−0.195869 + 0.980630i $$0.562753\pi$$
$$854$$ 0 0
$$855$$ −893.515 −0.0357398
$$856$$ 0 0
$$857$$ −13649.8 −0.544072 −0.272036 0.962287i $$-0.587697\pi$$
−0.272036 + 0.962287i $$0.587697\pi$$
$$858$$ 0 0
$$859$$ 7796.42 0.309674 0.154837 0.987940i $$-0.450515\pi$$
0.154837 + 0.987940i $$0.450515\pi$$
$$860$$ 0 0
$$861$$ −30625.5 −1.21221
$$862$$ 0 0
$$863$$ −7183.57 −0.283350 −0.141675 0.989913i $$-0.545249\pi$$
−0.141675 + 0.989913i $$0.545249\pi$$
$$864$$ 0 0
$$865$$ −1720.75 −0.0676383
$$866$$ 0 0
$$867$$ −29870.6 −1.17008
$$868$$ 0 0
$$869$$ −4214.30 −0.164511
$$870$$ 0 0
$$871$$ −436.534 −0.0169821
$$872$$ 0 0
$$873$$ −10543.4 −0.408752
$$874$$ 0 0
$$875$$ −21842.7 −0.843905
$$876$$ 0 0
$$877$$ −17063.1 −0.656991 −0.328495 0.944506i $$-0.606542\pi$$
−0.328495 + 0.944506i $$0.606542\pi$$
$$878$$ 0 0
$$879$$ −18434.4 −0.707369
$$880$$ 0 0
$$881$$ −32174.9 −1.23042 −0.615210 0.788363i $$-0.710929\pi$$
−0.615210 + 0.788363i $$0.710929\pi$$
$$882$$ 0 0
$$883$$ 2843.68 0.108378 0.0541889 0.998531i $$-0.482743\pi$$
0.0541889 + 0.998531i $$0.482743\pi$$
$$884$$ 0 0
$$885$$ 3473.69 0.131940
$$886$$ 0 0
$$887$$ 31417.8 1.18930 0.594649 0.803985i $$-0.297291\pi$$
0.594649 + 0.803985i $$0.297291\pi$$
$$888$$ 0 0
$$889$$ −31485.5 −1.18784
$$890$$ 0 0
$$891$$ −891.000 −0.0335013
$$892$$ 0 0
$$893$$ 8064.30 0.302196
$$894$$ 0 0
$$895$$ 6073.83 0.226845
$$896$$ 0 0
$$897$$ −1796.37 −0.0668664
$$898$$ 0 0
$$899$$ −9765.47 −0.362288
$$900$$ 0 0
$$901$$ −599.852 −0.0221798
$$902$$ 0 0
$$903$$ 30528.2 1.12505
$$904$$ 0 0
$$905$$ −1679.03 −0.0616718
$$906$$ 0 0
$$907$$ 12253.1 0.448573 0.224287 0.974523i $$-0.427995\pi$$
0.224287 + 0.974523i $$0.427995\pi$$
$$908$$ 0 0
$$909$$ 10991.4 0.401059
$$910$$ 0 0
$$911$$ 48422.4 1.76104 0.880518 0.474012i $$-0.157195\pi$$
0.880518 + 0.474012i $$0.157195\pi$$
$$912$$ 0 0
$$913$$ 10232.3 0.370909
$$914$$ 0 0
$$915$$ 4755.57 0.171819
$$916$$ 0 0
$$917$$ −12229.4 −0.440404
$$918$$ 0 0
$$919$$ −5546.18 −0.199077 −0.0995385 0.995034i $$-0.531737\pi$$
−0.0995385 + 0.995034i $$0.531737\pi$$
$$920$$ 0 0
$$921$$ 6558.26 0.234638
$$922$$ 0 0
$$923$$ 252.822 0.00901598
$$924$$ 0 0
$$925$$ −49099.5 −1.74528
$$926$$ 0 0
$$927$$ −4650.71 −0.164778
$$928$$ 0 0
$$929$$ −35684.5 −1.26025 −0.630125 0.776494i $$-0.716996\pi$$
−0.630125 + 0.776494i $$0.716996\pi$$
$$930$$ 0 0
$$931$$ 23061.1 0.811811
$$932$$ 0 0
$$933$$ −22454.5 −0.787918
$$934$$ 0 0
$$935$$ 3821.35 0.133659
$$936$$ 0 0
$$937$$ −48903.6 −1.70503 −0.852514 0.522705i $$-0.824923\pi$$
−0.852514 + 0.522705i $$0.824923\pi$$
$$938$$ 0 0
$$939$$ 20500.0 0.712451
$$940$$ 0 0
$$941$$ 23741.9 0.822490 0.411245 0.911525i $$-0.365094\pi$$
0.411245 + 0.911525i $$0.365094\pi$$
$$942$$ 0 0
$$943$$ 37437.4 1.29282
$$944$$ 0 0
$$945$$ −2438.16 −0.0839295
$$946$$ 0 0
$$947$$ 37612.4 1.29064 0.645321 0.763911i $$-0.276724\pi$$
0.645321 + 0.763911i $$0.276724\pi$$
$$948$$ 0 0
$$949$$ −4048.20 −0.138472
$$950$$ 0 0
$$951$$ 2772.80 0.0945469
$$952$$ 0 0
$$953$$ −48294.3 −1.64156 −0.820779 0.571246i $$-0.806460\pi$$
−0.820779 + 0.571246i $$0.806460\pi$$
$$954$$ 0 0
$$955$$ −6156.09 −0.208593
$$956$$ 0 0
$$957$$ 2291.96 0.0774176
$$958$$ 0 0
$$959$$ −28047.4 −0.944419
$$960$$ 0 0
$$961$$ −10021.4 −0.336389
$$962$$ 0 0
$$963$$ −1368.22 −0.0457843
$$964$$ 0 0
$$965$$ 4247.39 0.141687
$$966$$ 0 0
$$967$$ −1840.92 −0.0612204 −0.0306102 0.999531i $$-0.509745\pi$$
−0.0306102 + 0.999531i $$0.509745\pi$$
$$968$$ 0 0
$$969$$ −12748.6 −0.422647
$$970$$ 0 0
$$971$$ 31461.8 1.03981 0.519906 0.854223i $$-0.325967\pi$$
0.519906 + 0.854223i $$0.325967\pi$$
$$972$$ 0 0
$$973$$ 34612.1 1.14040
$$974$$ 0 0
$$975$$ −1806.26 −0.0593298
$$976$$ 0 0
$$977$$ −7040.11 −0.230535 −0.115268 0.993334i $$-0.536773\pi$$
−0.115268 + 0.993334i $$0.536773\pi$$
$$978$$ 0 0
$$979$$ 8058.15 0.263064
$$980$$ 0 0
$$981$$ −19532.9 −0.635715
$$982$$ 0 0
$$983$$ 24610.9 0.798541 0.399270 0.916833i $$-0.369263\pi$$
0.399270 + 0.916833i $$0.369263\pi$$
$$984$$ 0 0
$$985$$ −656.744 −0.0212443
$$986$$ 0 0
$$987$$ 22005.3 0.709662
$$988$$ 0 0
$$989$$ −37318.5 −1.19986
$$990$$ 0 0
$$991$$ 40003.3 1.28229 0.641144 0.767421i $$-0.278460\pi$$
0.641144 + 0.767421i $$0.278460\pi$$
$$992$$ 0 0
$$993$$ 29461.4 0.941519
$$994$$ 0 0
$$995$$ 63.8165 0.00203329
$$996$$ 0 0
$$997$$ 7342.61 0.233242 0.116621 0.993176i $$-0.462794\pi$$
0.116621 + 0.993176i $$0.462794\pi$$
$$998$$ 0 0
$$999$$ −11341.9 −0.359201
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.bg.1.1 2
4.3 odd 2 2112.4.a.bn.1.1 2
8.3 odd 2 33.4.a.c.1.1 2
8.5 even 2 528.4.a.p.1.2 2
24.5 odd 2 1584.4.a.bj.1.1 2
24.11 even 2 99.4.a.f.1.2 2
40.3 even 4 825.4.c.h.199.3 4
40.19 odd 2 825.4.a.l.1.2 2
40.27 even 4 825.4.c.h.199.2 4
56.27 even 2 1617.4.a.k.1.1 2
88.43 even 2 363.4.a.i.1.2 2
120.59 even 2 2475.4.a.p.1.1 2
264.131 odd 2 1089.4.a.u.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
33.4.a.c.1.1 2 8.3 odd 2
99.4.a.f.1.2 2 24.11 even 2
363.4.a.i.1.2 2 88.43 even 2
528.4.a.p.1.2 2 8.5 even 2
825.4.a.l.1.2 2 40.19 odd 2
825.4.c.h.199.2 4 40.27 even 4
825.4.c.h.199.3 4 40.3 even 4
1089.4.a.u.1.1 2 264.131 odd 2
1584.4.a.bj.1.1 2 24.5 odd 2
1617.4.a.k.1.1 2 56.27 even 2
2112.4.a.bg.1.1 2 1.1 even 1 trivial
2112.4.a.bn.1.1 2 4.3 odd 2
2475.4.a.p.1.1 2 120.59 even 2