# Properties

 Label 2240.4.a.bn.1.2 Level $2240$ Weight $4$ Character 2240.1 Self dual yes Analytic conductor $132.164$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2240,4,Mod(1,2240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2240, 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("2240.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2240 = 2^{6} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2240.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: $$132.164278413$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 35) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 2240.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+4.65685 q^{3} +5.00000 q^{5} -7.00000 q^{7} -5.31371 q^{9} +O(q^{10})$$ $$q+4.65685 q^{3} +5.00000 q^{5} -7.00000 q^{7} -5.31371 q^{9} +52.2548 q^{11} -30.6569 q^{13} +23.2843 q^{15} +37.2254 q^{17} -80.2254 q^{19} -32.5980 q^{21} +25.8335 q^{23} +25.0000 q^{25} -150.480 q^{27} -20.9411 q^{29} -314.558 q^{31} +243.343 q^{33} -35.0000 q^{35} -197.147 q^{37} -142.765 q^{39} +11.3625 q^{41} +33.8335 q^{43} -26.5685 q^{45} -361.676 q^{47} +49.0000 q^{49} +173.353 q^{51} -153.019 q^{53} +261.274 q^{55} -373.598 q^{57} +616.000 q^{59} -15.2649 q^{61} +37.1960 q^{63} -153.284 q^{65} +166.510 q^{67} +120.303 q^{69} -952.000 q^{71} -148.489 q^{73} +116.421 q^{75} -365.784 q^{77} +857.725 q^{79} -557.294 q^{81} -660.528 q^{83} +186.127 q^{85} -97.5198 q^{87} -45.7746 q^{89} +214.598 q^{91} -1464.85 q^{93} -401.127 q^{95} +1682.13 q^{97} -277.667 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + 10 q^{5} - 14 q^{7} + 12 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 + 10 * q^5 - 14 * q^7 + 12 * q^9 $$2 q - 2 q^{3} + 10 q^{5} - 14 q^{7} + 12 q^{9} + 14 q^{11} - 50 q^{13} - 10 q^{15} - 50 q^{17} - 36 q^{19} + 14 q^{21} + 244 q^{23} + 50 q^{25} - 86 q^{27} + 26 q^{29} - 120 q^{31} + 498 q^{33} - 70 q^{35} - 564 q^{37} - 14 q^{39} - 328 q^{41} + 260 q^{43} + 60 q^{45} - 350 q^{47} + 98 q^{49} + 754 q^{51} + 56 q^{53} + 70 q^{55} - 668 q^{57} + 1232 q^{59} - 336 q^{61} - 84 q^{63} - 250 q^{65} + 152 q^{67} - 1332 q^{69} - 1904 q^{71} + 676 q^{73} - 50 q^{75} - 98 q^{77} + 1014 q^{79} - 1454 q^{81} + 376 q^{83} - 250 q^{85} - 410 q^{87} - 216 q^{89} + 350 q^{91} - 2760 q^{93} - 180 q^{95} + 2742 q^{97} - 940 q^{99}+O(q^{100})$$ 2 * q - 2 * q^3 + 10 * q^5 - 14 * q^7 + 12 * q^9 + 14 * q^11 - 50 * q^13 - 10 * q^15 - 50 * q^17 - 36 * q^19 + 14 * q^21 + 244 * q^23 + 50 * q^25 - 86 * q^27 + 26 * q^29 - 120 * q^31 + 498 * q^33 - 70 * q^35 - 564 * q^37 - 14 * q^39 - 328 * q^41 + 260 * q^43 + 60 * q^45 - 350 * q^47 + 98 * q^49 + 754 * q^51 + 56 * q^53 + 70 * q^55 - 668 * q^57 + 1232 * q^59 - 336 * q^61 - 84 * q^63 - 250 * q^65 + 152 * q^67 - 1332 * q^69 - 1904 * q^71 + 676 * q^73 - 50 * q^75 - 98 * q^77 + 1014 * q^79 - 1454 * q^81 + 376 * q^83 - 250 * q^85 - 410 * q^87 - 216 * q^89 + 350 * q^91 - 2760 * q^93 - 180 * q^95 + 2742 * q^97 - 940 * 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
$$3$$ 4.65685 0.896212 0.448106 0.893980i $$-0.352099\pi$$
0.448106 + 0.893980i $$0.352099\pi$$
$$4$$ 0 0
$$5$$ 5.00000 0.447214
$$6$$ 0 0
$$7$$ −7.00000 −0.377964
$$8$$ 0 0
$$9$$ −5.31371 −0.196804
$$10$$ 0 0
$$11$$ 52.2548 1.43231 0.716156 0.697941i $$-0.245900\pi$$
0.716156 + 0.697941i $$0.245900\pi$$
$$12$$ 0 0
$$13$$ −30.6569 −0.654052 −0.327026 0.945015i $$-0.606047\pi$$
−0.327026 + 0.945015i $$0.606047\pi$$
$$14$$ 0 0
$$15$$ 23.2843 0.400798
$$16$$ 0 0
$$17$$ 37.2254 0.531087 0.265544 0.964099i $$-0.414449\pi$$
0.265544 + 0.964099i $$0.414449\pi$$
$$18$$ 0 0
$$19$$ −80.2254 −0.968683 −0.484341 0.874879i $$-0.660941\pi$$
−0.484341 + 0.874879i $$0.660941\pi$$
$$20$$ 0 0
$$21$$ −32.5980 −0.338736
$$22$$ 0 0
$$23$$ 25.8335 0.234202 0.117101 0.993120i $$-0.462640\pi$$
0.117101 + 0.993120i $$0.462640\pi$$
$$24$$ 0 0
$$25$$ 25.0000 0.200000
$$26$$ 0 0
$$27$$ −150.480 −1.07259
$$28$$ 0 0
$$29$$ −20.9411 −0.134092 −0.0670460 0.997750i $$-0.521357\pi$$
−0.0670460 + 0.997750i $$0.521357\pi$$
$$30$$ 0 0
$$31$$ −314.558 −1.82246 −0.911232 0.411894i $$-0.864867\pi$$
−0.911232 + 0.411894i $$0.864867\pi$$
$$32$$ 0 0
$$33$$ 243.343 1.28365
$$34$$ 0 0
$$35$$ −35.0000 −0.169031
$$36$$ 0 0
$$37$$ −197.147 −0.875968 −0.437984 0.898983i $$-0.644307\pi$$
−0.437984 + 0.898983i $$0.644307\pi$$
$$38$$ 0 0
$$39$$ −142.765 −0.586170
$$40$$ 0 0
$$41$$ 11.3625 0.0432810 0.0216405 0.999766i $$-0.493111\pi$$
0.0216405 + 0.999766i $$0.493111\pi$$
$$42$$ 0 0
$$43$$ 33.8335 0.119990 0.0599948 0.998199i $$-0.480892\pi$$
0.0599948 + 0.998199i $$0.480892\pi$$
$$44$$ 0 0
$$45$$ −26.5685 −0.0880134
$$46$$ 0 0
$$47$$ −361.676 −1.12247 −0.561233 0.827658i $$-0.689673\pi$$
−0.561233 + 0.827658i $$0.689673\pi$$
$$48$$ 0 0
$$49$$ 49.0000 0.142857
$$50$$ 0 0
$$51$$ 173.353 0.475967
$$52$$ 0 0
$$53$$ −153.019 −0.396582 −0.198291 0.980143i $$-0.563539\pi$$
−0.198291 + 0.980143i $$0.563539\pi$$
$$54$$ 0 0
$$55$$ 261.274 0.640549
$$56$$ 0 0
$$57$$ −373.598 −0.868145
$$58$$ 0 0
$$59$$ 616.000 1.35926 0.679630 0.733555i $$-0.262140\pi$$
0.679630 + 0.733555i $$0.262140\pi$$
$$60$$ 0 0
$$61$$ −15.2649 −0.0320406 −0.0160203 0.999872i $$-0.505100\pi$$
−0.0160203 + 0.999872i $$0.505100\pi$$
$$62$$ 0 0
$$63$$ 37.1960 0.0743849
$$64$$ 0 0
$$65$$ −153.284 −0.292501
$$66$$ 0 0
$$67$$ 166.510 0.303618 0.151809 0.988410i $$-0.451490\pi$$
0.151809 + 0.988410i $$0.451490\pi$$
$$68$$ 0 0
$$69$$ 120.303 0.209895
$$70$$ 0 0
$$71$$ −952.000 −1.59129 −0.795645 0.605763i $$-0.792868\pi$$
−0.795645 + 0.605763i $$0.792868\pi$$
$$72$$ 0 0
$$73$$ −148.489 −0.238074 −0.119037 0.992890i $$-0.537981\pi$$
−0.119037 + 0.992890i $$0.537981\pi$$
$$74$$ 0 0
$$75$$ 116.421 0.179242
$$76$$ 0 0
$$77$$ −365.784 −0.541363
$$78$$ 0 0
$$79$$ 857.725 1.22154 0.610770 0.791808i $$-0.290860\pi$$
0.610770 + 0.791808i $$0.290860\pi$$
$$80$$ 0 0
$$81$$ −557.294 −0.764464
$$82$$ 0 0
$$83$$ −660.528 −0.873523 −0.436761 0.899577i $$-0.643875\pi$$
−0.436761 + 0.899577i $$0.643875\pi$$
$$84$$ 0 0
$$85$$ 186.127 0.237509
$$86$$ 0 0
$$87$$ −97.5198 −0.120175
$$88$$ 0 0
$$89$$ −45.7746 −0.0545180 −0.0272590 0.999628i $$-0.508678\pi$$
−0.0272590 + 0.999628i $$0.508678\pi$$
$$90$$ 0 0
$$91$$ 214.598 0.247209
$$92$$ 0 0
$$93$$ −1464.85 −1.63331
$$94$$ 0 0
$$95$$ −401.127 −0.433208
$$96$$ 0 0
$$97$$ 1682.13 1.76076 0.880382 0.474265i $$-0.157286\pi$$
0.880382 + 0.474265i $$0.157286\pi$$
$$98$$ 0 0
$$99$$ −277.667 −0.281885
$$100$$ 0 0
$$101$$ 434.167 0.427734 0.213867 0.976863i $$-0.431394\pi$$
0.213867 + 0.976863i $$0.431394\pi$$
$$102$$ 0 0
$$103$$ 345.577 0.330589 0.165295 0.986244i $$-0.447142\pi$$
0.165295 + 0.986244i $$0.447142\pi$$
$$104$$ 0 0
$$105$$ −162.990 −0.151487
$$106$$ 0 0
$$107$$ −217.119 −0.196165 −0.0980825 0.995178i $$-0.531271\pi$$
−0.0980825 + 0.995178i $$0.531271\pi$$
$$108$$ 0 0
$$109$$ −1734.41 −1.52409 −0.762047 0.647521i $$-0.775806\pi$$
−0.762047 + 0.647521i $$0.775806\pi$$
$$110$$ 0 0
$$111$$ −918.086 −0.785053
$$112$$ 0 0
$$113$$ −1854.20 −1.54362 −0.771809 0.635855i $$-0.780648\pi$$
−0.771809 + 0.635855i $$0.780648\pi$$
$$114$$ 0 0
$$115$$ 129.167 0.104738
$$116$$ 0 0
$$117$$ 162.902 0.128720
$$118$$ 0 0
$$119$$ −260.578 −0.200732
$$120$$ 0 0
$$121$$ 1399.57 1.05152
$$122$$ 0 0
$$123$$ 52.9134 0.0387890
$$124$$ 0 0
$$125$$ 125.000 0.0894427
$$126$$ 0 0
$$127$$ 1394.51 0.974352 0.487176 0.873304i $$-0.338027\pi$$
0.487176 + 0.873304i $$0.338027\pi$$
$$128$$ 0 0
$$129$$ 157.558 0.107536
$$130$$ 0 0
$$131$$ −1762.42 −1.17544 −0.587722 0.809063i $$-0.699975\pi$$
−0.587722 + 0.809063i $$0.699975\pi$$
$$132$$ 0 0
$$133$$ 561.578 0.366128
$$134$$ 0 0
$$135$$ −752.401 −0.479677
$$136$$ 0 0
$$137$$ −922.949 −0.575568 −0.287784 0.957695i $$-0.592919\pi$$
−0.287784 + 0.957695i $$0.592919\pi$$
$$138$$ 0 0
$$139$$ 196.039 0.119624 0.0598122 0.998210i $$-0.480950\pi$$
0.0598122 + 0.998210i $$0.480950\pi$$
$$140$$ 0 0
$$141$$ −1684.27 −1.00597
$$142$$ 0 0
$$143$$ −1601.97 −0.936807
$$144$$ 0 0
$$145$$ −104.706 −0.0599678
$$146$$ 0 0
$$147$$ 228.186 0.128030
$$148$$ 0 0
$$149$$ −780.372 −0.429064 −0.214532 0.976717i $$-0.568823\pi$$
−0.214532 + 0.976717i $$0.568823\pi$$
$$150$$ 0 0
$$151$$ −2319.43 −1.25002 −0.625008 0.780618i $$-0.714904\pi$$
−0.625008 + 0.780618i $$0.714904\pi$$
$$152$$ 0 0
$$153$$ −197.805 −0.104520
$$154$$ 0 0
$$155$$ −1572.79 −0.815030
$$156$$ 0 0
$$157$$ −1022.90 −0.519977 −0.259989 0.965612i $$-0.583719\pi$$
−0.259989 + 0.965612i $$0.583719\pi$$
$$158$$ 0 0
$$159$$ −712.589 −0.355421
$$160$$ 0 0
$$161$$ −180.834 −0.0885201
$$162$$ 0 0
$$163$$ 1350.63 0.649013 0.324507 0.945883i $$-0.394802\pi$$
0.324507 + 0.945883i $$0.394802\pi$$
$$164$$ 0 0
$$165$$ 1216.72 0.574068
$$166$$ 0 0
$$167$$ −1230.58 −0.570209 −0.285105 0.958496i $$-0.592028\pi$$
−0.285105 + 0.958496i $$0.592028\pi$$
$$168$$ 0 0
$$169$$ −1257.16 −0.572215
$$170$$ 0 0
$$171$$ 426.294 0.190641
$$172$$ 0 0
$$173$$ 2487.65 1.09325 0.546626 0.837377i $$-0.315912\pi$$
0.546626 + 0.837377i $$0.315912\pi$$
$$174$$ 0 0
$$175$$ −175.000 −0.0755929
$$176$$ 0 0
$$177$$ 2868.62 1.21819
$$178$$ 0 0
$$179$$ −1621.18 −0.676941 −0.338471 0.940977i $$-0.609910\pi$$
−0.338471 + 0.940977i $$0.609910\pi$$
$$180$$ 0 0
$$181$$ −2593.69 −1.06512 −0.532561 0.846392i $$-0.678770\pi$$
−0.532561 + 0.846392i $$0.678770\pi$$
$$182$$ 0 0
$$183$$ −71.0866 −0.0287151
$$184$$ 0 0
$$185$$ −985.736 −0.391745
$$186$$ 0 0
$$187$$ 1945.21 0.760682
$$188$$ 0 0
$$189$$ 1053.36 0.405401
$$190$$ 0 0
$$191$$ −1823.08 −0.690645 −0.345323 0.938484i $$-0.612231\pi$$
−0.345323 + 0.938484i $$0.612231\pi$$
$$192$$ 0 0
$$193$$ −1541.03 −0.574744 −0.287372 0.957819i $$-0.592782\pi$$
−0.287372 + 0.957819i $$0.592782\pi$$
$$194$$ 0 0
$$195$$ −713.823 −0.262143
$$196$$ 0 0
$$197$$ −701.243 −0.253612 −0.126806 0.991928i $$-0.540473\pi$$
−0.126806 + 0.991928i $$0.540473\pi$$
$$198$$ 0 0
$$199$$ 3294.96 1.17374 0.586868 0.809682i $$-0.300361\pi$$
0.586868 + 0.809682i $$0.300361\pi$$
$$200$$ 0 0
$$201$$ 775.411 0.272106
$$202$$ 0 0
$$203$$ 146.588 0.0506820
$$204$$ 0 0
$$205$$ 56.8124 0.0193559
$$206$$ 0 0
$$207$$ −137.272 −0.0460920
$$208$$ 0 0
$$209$$ −4192.16 −1.38746
$$210$$ 0 0
$$211$$ −4082.35 −1.33195 −0.665974 0.745975i $$-0.731984\pi$$
−0.665974 + 0.745975i $$0.731984\pi$$
$$212$$ 0 0
$$213$$ −4433.33 −1.42613
$$214$$ 0 0
$$215$$ 169.167 0.0536610
$$216$$ 0 0
$$217$$ 2201.91 0.688826
$$218$$ 0 0
$$219$$ −691.494 −0.213364
$$220$$ 0 0
$$221$$ −1141.21 −0.347359
$$222$$ 0 0
$$223$$ 747.161 0.224366 0.112183 0.993688i $$-0.464216\pi$$
0.112183 + 0.993688i $$0.464216\pi$$
$$224$$ 0 0
$$225$$ −132.843 −0.0393608
$$226$$ 0 0
$$227$$ −1665.67 −0.487025 −0.243513 0.969898i $$-0.578300\pi$$
−0.243513 + 0.969898i $$0.578300\pi$$
$$228$$ 0 0
$$229$$ 6628.35 1.91272 0.956362 0.292183i $$-0.0943816\pi$$
0.956362 + 0.292183i $$0.0943816\pi$$
$$230$$ 0 0
$$231$$ −1703.40 −0.485176
$$232$$ 0 0
$$233$$ −432.431 −0.121586 −0.0607929 0.998150i $$-0.519363\pi$$
−0.0607929 + 0.998150i $$0.519363\pi$$
$$234$$ 0 0
$$235$$ −1808.38 −0.501982
$$236$$ 0 0
$$237$$ 3994.30 1.09476
$$238$$ 0 0
$$239$$ 5580.44 1.51033 0.755165 0.655535i $$-0.227557\pi$$
0.755165 + 0.655535i $$0.227557\pi$$
$$240$$ 0 0
$$241$$ −6296.87 −1.68306 −0.841529 0.540212i $$-0.818344\pi$$
−0.841529 + 0.540212i $$0.818344\pi$$
$$242$$ 0 0
$$243$$ 1467.73 0.387468
$$244$$ 0 0
$$245$$ 245.000 0.0638877
$$246$$ 0 0
$$247$$ 2459.46 0.633569
$$248$$ 0 0
$$249$$ −3075.98 −0.782862
$$250$$ 0 0
$$251$$ −311.921 −0.0784393 −0.0392197 0.999231i $$-0.512487\pi$$
−0.0392197 + 0.999231i $$0.512487\pi$$
$$252$$ 0 0
$$253$$ 1349.92 0.335451
$$254$$ 0 0
$$255$$ 866.766 0.212859
$$256$$ 0 0
$$257$$ −7861.39 −1.90809 −0.954046 0.299659i $$-0.903127\pi$$
−0.954046 + 0.299659i $$0.903127\pi$$
$$258$$ 0 0
$$259$$ 1380.03 0.331085
$$260$$ 0 0
$$261$$ 111.275 0.0263899
$$262$$ 0 0
$$263$$ 5227.09 1.22554 0.612769 0.790262i $$-0.290056\pi$$
0.612769 + 0.790262i $$0.290056\pi$$
$$264$$ 0 0
$$265$$ −765.097 −0.177357
$$266$$ 0 0
$$267$$ −213.166 −0.0488596
$$268$$ 0 0
$$269$$ −1281.71 −0.290510 −0.145255 0.989394i $$-0.546400\pi$$
−0.145255 + 0.989394i $$0.546400\pi$$
$$270$$ 0 0
$$271$$ 4704.14 1.05445 0.527226 0.849725i $$-0.323232\pi$$
0.527226 + 0.849725i $$0.323232\pi$$
$$272$$ 0 0
$$273$$ 999.352 0.221551
$$274$$ 0 0
$$275$$ 1306.37 0.286462
$$276$$ 0 0
$$277$$ −8958.56 −1.94321 −0.971603 0.236619i $$-0.923961\pi$$
−0.971603 + 0.236619i $$0.923961\pi$$
$$278$$ 0 0
$$279$$ 1671.47 0.358668
$$280$$ 0 0
$$281$$ −370.904 −0.0787412 −0.0393706 0.999225i $$-0.512535\pi$$
−0.0393706 + 0.999225i $$0.512535\pi$$
$$282$$ 0 0
$$283$$ 5822.26 1.22296 0.611479 0.791261i $$-0.290575\pi$$
0.611479 + 0.791261i $$0.290575\pi$$
$$284$$ 0 0
$$285$$ −1867.99 −0.388246
$$286$$ 0 0
$$287$$ −79.5374 −0.0163587
$$288$$ 0 0
$$289$$ −3527.27 −0.717946
$$290$$ 0 0
$$291$$ 7833.42 1.57802
$$292$$ 0 0
$$293$$ −7443.79 −1.48420 −0.742100 0.670289i $$-0.766170\pi$$
−0.742100 + 0.670289i $$0.766170\pi$$
$$294$$ 0 0
$$295$$ 3080.00 0.607880
$$296$$ 0 0
$$297$$ −7863.32 −1.53628
$$298$$ 0 0
$$299$$ −791.973 −0.153181
$$300$$ 0 0
$$301$$ −236.834 −0.0453518
$$302$$ 0 0
$$303$$ 2021.85 0.383341
$$304$$ 0 0
$$305$$ −76.3247 −0.0143290
$$306$$ 0 0
$$307$$ 761.674 0.141600 0.0707998 0.997491i $$-0.477445\pi$$
0.0707998 + 0.997491i $$0.477445\pi$$
$$308$$ 0 0
$$309$$ 1609.30 0.296278
$$310$$ 0 0
$$311$$ 7718.69 1.40735 0.703677 0.710520i $$-0.251540\pi$$
0.703677 + 0.710520i $$0.251540\pi$$
$$312$$ 0 0
$$313$$ 8556.00 1.54509 0.772546 0.634959i $$-0.218983\pi$$
0.772546 + 0.634959i $$0.218983\pi$$
$$314$$ 0 0
$$315$$ 185.980 0.0332660
$$316$$ 0 0
$$317$$ 7780.95 1.37862 0.689309 0.724468i $$-0.257914\pi$$
0.689309 + 0.724468i $$0.257914\pi$$
$$318$$ 0 0
$$319$$ −1094.28 −0.192062
$$320$$ 0 0
$$321$$ −1011.09 −0.175805
$$322$$ 0 0
$$323$$ −2986.42 −0.514455
$$324$$ 0 0
$$325$$ −766.421 −0.130810
$$326$$ 0 0
$$327$$ −8076.89 −1.36591
$$328$$ 0 0
$$329$$ 2531.73 0.424252
$$330$$ 0 0
$$331$$ 4932.12 0.819015 0.409507 0.912307i $$-0.365701\pi$$
0.409507 + 0.912307i $$0.365701\pi$$
$$332$$ 0 0
$$333$$ 1047.58 0.172394
$$334$$ 0 0
$$335$$ 832.548 0.135782
$$336$$ 0 0
$$337$$ −7121.13 −1.15108 −0.575538 0.817775i $$-0.695207\pi$$
−0.575538 + 0.817775i $$0.695207\pi$$
$$338$$ 0 0
$$339$$ −8634.76 −1.38341
$$340$$ 0 0
$$341$$ −16437.2 −2.61034
$$342$$ 0 0
$$343$$ −343.000 −0.0539949
$$344$$ 0 0
$$345$$ 601.514 0.0938679
$$346$$ 0 0
$$347$$ −9540.58 −1.47598 −0.737991 0.674811i $$-0.764225\pi$$
−0.737991 + 0.674811i $$0.764225\pi$$
$$348$$ 0 0
$$349$$ −1281.65 −0.196576 −0.0982880 0.995158i $$-0.531337\pi$$
−0.0982880 + 0.995158i $$0.531337\pi$$
$$350$$ 0 0
$$351$$ 4613.25 0.701530
$$352$$ 0 0
$$353$$ 5798.07 0.874221 0.437110 0.899408i $$-0.356002\pi$$
0.437110 + 0.899408i $$0.356002\pi$$
$$354$$ 0 0
$$355$$ −4760.00 −0.711647
$$356$$ 0 0
$$357$$ −1213.47 −0.179899
$$358$$ 0 0
$$359$$ 2267.29 0.333323 0.166662 0.986014i $$-0.446701\pi$$
0.166662 + 0.986014i $$0.446701\pi$$
$$360$$ 0 0
$$361$$ −422.886 −0.0616541
$$362$$ 0 0
$$363$$ 6517.58 0.942381
$$364$$ 0 0
$$365$$ −742.447 −0.106470
$$366$$ 0 0
$$367$$ −7372.85 −1.04866 −0.524332 0.851514i $$-0.675685\pi$$
−0.524332 + 0.851514i $$0.675685\pi$$
$$368$$ 0 0
$$369$$ −60.3769 −0.00851788
$$370$$ 0 0
$$371$$ 1071.14 0.149894
$$372$$ 0 0
$$373$$ −6447.14 −0.894961 −0.447480 0.894294i $$-0.647679\pi$$
−0.447480 + 0.894294i $$0.647679\pi$$
$$374$$ 0 0
$$375$$ 582.107 0.0801596
$$376$$ 0 0
$$377$$ 641.989 0.0877032
$$378$$ 0 0
$$379$$ 4247.57 0.575680 0.287840 0.957678i $$-0.407063\pi$$
0.287840 + 0.957678i $$0.407063\pi$$
$$380$$ 0 0
$$381$$ 6494.03 0.873226
$$382$$ 0 0
$$383$$ −6681.86 −0.891454 −0.445727 0.895169i $$-0.647055\pi$$
−0.445727 + 0.895169i $$0.647055\pi$$
$$384$$ 0 0
$$385$$ −1828.92 −0.242105
$$386$$ 0 0
$$387$$ −179.781 −0.0236145
$$388$$ 0 0
$$389$$ 6371.78 0.830494 0.415247 0.909709i $$-0.363695\pi$$
0.415247 + 0.909709i $$0.363695\pi$$
$$390$$ 0 0
$$391$$ 961.661 0.124382
$$392$$ 0 0
$$393$$ −8207.33 −1.05345
$$394$$ 0 0
$$395$$ 4288.62 0.546289
$$396$$ 0 0
$$397$$ −4247.93 −0.537021 −0.268510 0.963277i $$-0.586531\pi$$
−0.268510 + 0.963277i $$0.586531\pi$$
$$398$$ 0 0
$$399$$ 2615.19 0.328128
$$400$$ 0 0
$$401$$ −8833.62 −1.10008 −0.550038 0.835140i $$-0.685387\pi$$
−0.550038 + 0.835140i $$0.685387\pi$$
$$402$$ 0 0
$$403$$ 9643.37 1.19199
$$404$$ 0 0
$$405$$ −2786.47 −0.341879
$$406$$ 0 0
$$407$$ −10301.9 −1.25466
$$408$$ 0 0
$$409$$ −319.205 −0.0385908 −0.0192954 0.999814i $$-0.506142\pi$$
−0.0192954 + 0.999814i $$0.506142\pi$$
$$410$$ 0 0
$$411$$ −4298.04 −0.515831
$$412$$ 0 0
$$413$$ −4312.00 −0.513752
$$414$$ 0 0
$$415$$ −3302.64 −0.390651
$$416$$ 0 0
$$417$$ 912.924 0.107209
$$418$$ 0 0
$$419$$ 12789.2 1.49115 0.745577 0.666420i $$-0.232174\pi$$
0.745577 + 0.666420i $$0.232174\pi$$
$$420$$ 0 0
$$421$$ 6747.40 0.781112 0.390556 0.920579i $$-0.372283\pi$$
0.390556 + 0.920579i $$0.372283\pi$$
$$422$$ 0 0
$$423$$ 1921.84 0.220906
$$424$$ 0 0
$$425$$ 930.635 0.106217
$$426$$ 0 0
$$427$$ 106.855 0.0121102
$$428$$ 0 0
$$429$$ −7460.14 −0.839577
$$430$$ 0 0
$$431$$ −5184.75 −0.579444 −0.289722 0.957111i $$-0.593563\pi$$
−0.289722 + 0.957111i $$0.593563\pi$$
$$432$$ 0 0
$$433$$ −4242.03 −0.470806 −0.235403 0.971898i $$-0.575641\pi$$
−0.235403 + 0.971898i $$0.575641\pi$$
$$434$$ 0 0
$$435$$ −487.599 −0.0537439
$$436$$ 0 0
$$437$$ −2072.50 −0.226868
$$438$$ 0 0
$$439$$ −5434.12 −0.590789 −0.295394 0.955375i $$-0.595451\pi$$
−0.295394 + 0.955375i $$0.595451\pi$$
$$440$$ 0 0
$$441$$ −260.372 −0.0281149
$$442$$ 0 0
$$443$$ 11493.8 1.23270 0.616350 0.787472i $$-0.288611\pi$$
0.616350 + 0.787472i $$0.288611\pi$$
$$444$$ 0 0
$$445$$ −228.873 −0.0243812
$$446$$ 0 0
$$447$$ −3634.08 −0.384532
$$448$$ 0 0
$$449$$ −16849.3 −1.77098 −0.885489 0.464661i $$-0.846176\pi$$
−0.885489 + 0.464661i $$0.846176\pi$$
$$450$$ 0 0
$$451$$ 593.745 0.0619919
$$452$$ 0 0
$$453$$ −10801.2 −1.12028
$$454$$ 0 0
$$455$$ 1072.99 0.110555
$$456$$ 0 0
$$457$$ 15348.5 1.57106 0.785528 0.618826i $$-0.212391\pi$$
0.785528 + 0.618826i $$0.212391\pi$$
$$458$$ 0 0
$$459$$ −5601.69 −0.569639
$$460$$ 0 0
$$461$$ −14038.4 −1.41830 −0.709148 0.705059i $$-0.750920\pi$$
−0.709148 + 0.705059i $$0.750920\pi$$
$$462$$ 0 0
$$463$$ −8661.23 −0.869377 −0.434689 0.900581i $$-0.643142\pi$$
−0.434689 + 0.900581i $$0.643142\pi$$
$$464$$ 0 0
$$465$$ −7324.26 −0.730440
$$466$$ 0 0
$$467$$ −7014.71 −0.695079 −0.347539 0.937665i $$-0.612983\pi$$
−0.347539 + 0.937665i $$0.612983\pi$$
$$468$$ 0 0
$$469$$ −1165.57 −0.114757
$$470$$ 0 0
$$471$$ −4763.50 −0.466010
$$472$$ 0 0
$$473$$ 1767.96 0.171863
$$474$$ 0 0
$$475$$ −2005.63 −0.193737
$$476$$ 0 0
$$477$$ 813.100 0.0780488
$$478$$ 0 0
$$479$$ 18134.7 1.72984 0.864922 0.501907i $$-0.167368\pi$$
0.864922 + 0.501907i $$0.167368\pi$$
$$480$$ 0 0
$$481$$ 6043.91 0.572929
$$482$$ 0 0
$$483$$ −842.119 −0.0793328
$$484$$ 0 0
$$485$$ 8410.63 0.787438
$$486$$ 0 0
$$487$$ 16537.8 1.53881 0.769405 0.638761i $$-0.220553\pi$$
0.769405 + 0.638761i $$0.220553\pi$$
$$488$$ 0 0
$$489$$ 6289.67 0.581654
$$490$$ 0 0
$$491$$ −220.608 −0.0202768 −0.0101384 0.999949i $$-0.503227\pi$$
−0.0101384 + 0.999949i $$0.503227\pi$$
$$492$$ 0 0
$$493$$ −779.542 −0.0712146
$$494$$ 0 0
$$495$$ −1388.33 −0.126063
$$496$$ 0 0
$$497$$ 6664.00 0.601451
$$498$$ 0 0
$$499$$ −5939.04 −0.532801 −0.266401 0.963862i $$-0.585834\pi$$
−0.266401 + 0.963862i $$0.585834\pi$$
$$500$$ 0 0
$$501$$ −5730.62 −0.511029
$$502$$ 0 0
$$503$$ −11604.8 −1.02869 −0.514345 0.857584i $$-0.671965\pi$$
−0.514345 + 0.857584i $$0.671965\pi$$
$$504$$ 0 0
$$505$$ 2170.83 0.191289
$$506$$ 0 0
$$507$$ −5854.40 −0.512826
$$508$$ 0 0
$$509$$ 1867.67 0.162639 0.0813193 0.996688i $$-0.474087\pi$$
0.0813193 + 0.996688i $$0.474087\pi$$
$$510$$ 0 0
$$511$$ 1039.43 0.0899834
$$512$$ 0 0
$$513$$ 12072.3 1.03900
$$514$$ 0 0
$$515$$ 1727.88 0.147844
$$516$$ 0 0
$$517$$ −18899.3 −1.60772
$$518$$ 0 0
$$519$$ 11584.6 0.979786
$$520$$ 0 0
$$521$$ 6117.21 0.514395 0.257197 0.966359i $$-0.417201\pi$$
0.257197 + 0.966359i $$0.417201\pi$$
$$522$$ 0 0
$$523$$ 16685.6 1.39505 0.697524 0.716561i $$-0.254285\pi$$
0.697524 + 0.716561i $$0.254285\pi$$
$$524$$ 0 0
$$525$$ −814.949 −0.0677473
$$526$$ 0 0
$$527$$ −11709.6 −0.967887
$$528$$ 0 0
$$529$$ −11499.6 −0.945149
$$530$$ 0 0
$$531$$ −3273.24 −0.267508
$$532$$ 0 0
$$533$$ −348.338 −0.0283081
$$534$$ 0 0
$$535$$ −1085.59 −0.0877276
$$536$$ 0 0
$$537$$ −7549.59 −0.606683
$$538$$ 0 0
$$539$$ 2560.49 0.204616
$$540$$ 0 0
$$541$$ −9309.03 −0.739790 −0.369895 0.929074i $$-0.620606\pi$$
−0.369895 + 0.929074i $$0.620606\pi$$
$$542$$ 0 0
$$543$$ −12078.4 −0.954575
$$544$$ 0 0
$$545$$ −8672.05 −0.681596
$$546$$ 0 0
$$547$$ −10894.7 −0.851598 −0.425799 0.904818i $$-0.640007\pi$$
−0.425799 + 0.904818i $$0.640007\pi$$
$$548$$ 0 0
$$549$$ 81.1134 0.00630571
$$550$$ 0 0
$$551$$ 1680.01 0.129893
$$552$$ 0 0
$$553$$ −6004.07 −0.461698
$$554$$ 0 0
$$555$$ −4590.43 −0.351086
$$556$$ 0 0
$$557$$ 7873.90 0.598973 0.299486 0.954101i $$-0.403185\pi$$
0.299486 + 0.954101i $$0.403185\pi$$
$$558$$ 0 0
$$559$$ −1037.23 −0.0784796
$$560$$ 0 0
$$561$$ 9058.55 0.681733
$$562$$ 0 0
$$563$$ −21770.7 −1.62971 −0.814854 0.579666i $$-0.803183\pi$$
−0.814854 + 0.579666i $$0.803183\pi$$
$$564$$ 0 0
$$565$$ −9271.02 −0.690327
$$566$$ 0 0
$$567$$ 3901.06 0.288940
$$568$$ 0 0
$$569$$ −12381.3 −0.912213 −0.456106 0.889925i $$-0.650756\pi$$
−0.456106 + 0.889925i $$0.650756\pi$$
$$570$$ 0 0
$$571$$ 5768.38 0.422765 0.211383 0.977403i $$-0.432203\pi$$
0.211383 + 0.977403i $$0.432203\pi$$
$$572$$ 0 0
$$573$$ −8489.81 −0.618965
$$574$$ 0 0
$$575$$ 645.837 0.0468405
$$576$$ 0 0
$$577$$ 4733.38 0.341513 0.170757 0.985313i $$-0.445379\pi$$
0.170757 + 0.985313i $$0.445379\pi$$
$$578$$ 0 0
$$579$$ −7176.34 −0.515093
$$580$$ 0 0
$$581$$ 4623.70 0.330161
$$582$$ 0 0
$$583$$ −7996.00 −0.568028
$$584$$ 0 0
$$585$$ 814.508 0.0575654
$$586$$ 0 0
$$587$$ 8441.67 0.593569 0.296785 0.954944i $$-0.404086\pi$$
0.296785 + 0.954944i $$0.404086\pi$$
$$588$$ 0 0
$$589$$ 25235.6 1.76539
$$590$$ 0 0
$$591$$ −3265.59 −0.227290
$$592$$ 0 0
$$593$$ 18939.9 1.31158 0.655791 0.754943i $$-0.272335\pi$$
0.655791 + 0.754943i $$0.272335\pi$$
$$594$$ 0 0
$$595$$ −1302.89 −0.0897701
$$596$$ 0 0
$$597$$ 15344.2 1.05192
$$598$$ 0 0
$$599$$ 22655.3 1.54536 0.772681 0.634794i $$-0.218915\pi$$
0.772681 + 0.634794i $$0.218915\pi$$
$$600$$ 0 0
$$601$$ −15947.4 −1.08237 −0.541187 0.840902i $$-0.682025\pi$$
−0.541187 + 0.840902i $$0.682025\pi$$
$$602$$ 0 0
$$603$$ −884.784 −0.0597532
$$604$$ 0 0
$$605$$ 6997.84 0.470252
$$606$$ 0 0
$$607$$ −25993.2 −1.73811 −0.869053 0.494719i $$-0.835271\pi$$
−0.869053 + 0.494719i $$0.835271\pi$$
$$608$$ 0 0
$$609$$ 682.638 0.0454218
$$610$$ 0 0
$$611$$ 11087.9 0.734152
$$612$$ 0 0
$$613$$ −665.408 −0.0438427 −0.0219213 0.999760i $$-0.506978\pi$$
−0.0219213 + 0.999760i $$0.506978\pi$$
$$614$$ 0 0
$$615$$ 264.567 0.0173470
$$616$$ 0 0
$$617$$ 18401.3 1.20066 0.600330 0.799752i $$-0.295036\pi$$
0.600330 + 0.799752i $$0.295036\pi$$
$$618$$ 0 0
$$619$$ 11150.6 0.724040 0.362020 0.932170i $$-0.382087\pi$$
0.362020 + 0.932170i $$0.382087\pi$$
$$620$$ 0 0
$$621$$ −3887.43 −0.251203
$$622$$ 0 0
$$623$$ 320.422 0.0206059
$$624$$ 0 0
$$625$$ 625.000 0.0400000
$$626$$ 0 0
$$627$$ −19522.3 −1.24345
$$628$$ 0 0
$$629$$ −7338.88 −0.465215
$$630$$ 0 0
$$631$$ 5381.79 0.339534 0.169767 0.985484i $$-0.445699\pi$$
0.169767 + 0.985484i $$0.445699\pi$$
$$632$$ 0 0
$$633$$ −19010.9 −1.19371
$$634$$ 0 0
$$635$$ 6972.55 0.435744
$$636$$ 0 0
$$637$$ −1502.19 −0.0934361
$$638$$ 0 0
$$639$$ 5058.65 0.313172
$$640$$ 0 0
$$641$$ −19455.1 −1.19880 −0.599398 0.800451i $$-0.704593\pi$$
−0.599398 + 0.800451i $$0.704593\pi$$
$$642$$ 0 0
$$643$$ 14695.8 0.901317 0.450658 0.892696i $$-0.351189\pi$$
0.450658 + 0.892696i $$0.351189\pi$$
$$644$$ 0 0
$$645$$ 787.788 0.0480917
$$646$$ 0 0
$$647$$ −12694.8 −0.771383 −0.385691 0.922628i $$-0.626037\pi$$
−0.385691 + 0.922628i $$0.626037\pi$$
$$648$$ 0 0
$$649$$ 32189.0 1.94688
$$650$$ 0 0
$$651$$ 10254.0 0.617334
$$652$$ 0 0
$$653$$ 12385.6 0.742247 0.371124 0.928583i $$-0.378973\pi$$
0.371124 + 0.928583i $$0.378973\pi$$
$$654$$ 0 0
$$655$$ −8812.09 −0.525675
$$656$$ 0 0
$$657$$ 789.030 0.0468539
$$658$$ 0 0
$$659$$ 2072.18 0.122489 0.0612447 0.998123i $$-0.480493\pi$$
0.0612447 + 0.998123i $$0.480493\pi$$
$$660$$ 0 0
$$661$$ −1074.36 −0.0632193 −0.0316096 0.999500i $$-0.510063\pi$$
−0.0316096 + 0.999500i $$0.510063\pi$$
$$662$$ 0 0
$$663$$ −5314.47 −0.311307
$$664$$ 0 0
$$665$$ 2807.89 0.163737
$$666$$ 0 0
$$667$$ −540.982 −0.0314047
$$668$$ 0 0
$$669$$ 3479.42 0.201080
$$670$$ 0 0
$$671$$ −797.667 −0.0458921
$$672$$ 0 0
$$673$$ 26195.2 1.50037 0.750186 0.661226i $$-0.229964\pi$$
0.750186 + 0.661226i $$0.229964\pi$$
$$674$$ 0 0
$$675$$ −3762.01 −0.214518
$$676$$ 0 0
$$677$$ 4228.44 0.240047 0.120024 0.992771i $$-0.461703\pi$$
0.120024 + 0.992771i $$0.461703\pi$$
$$678$$ 0 0
$$679$$ −11774.9 −0.665506
$$680$$ 0 0
$$681$$ −7756.80 −0.436478
$$682$$ 0 0
$$683$$ −27525.5 −1.54207 −0.771036 0.636792i $$-0.780261\pi$$
−0.771036 + 0.636792i $$0.780261\pi$$
$$684$$ 0 0
$$685$$ −4614.74 −0.257402
$$686$$ 0 0
$$687$$ 30867.3 1.71421
$$688$$ 0 0
$$689$$ 4691.09 0.259385
$$690$$ 0 0
$$691$$ 33324.4 1.83462 0.917309 0.398177i $$-0.130357\pi$$
0.917309 + 0.398177i $$0.130357\pi$$
$$692$$ 0 0
$$693$$ 1943.67 0.106542
$$694$$ 0 0
$$695$$ 980.193 0.0534976
$$696$$ 0 0
$$697$$ 422.973 0.0229860
$$698$$ 0 0
$$699$$ −2013.77 −0.108967
$$700$$ 0 0
$$701$$ 33262.9 1.79219 0.896094 0.443864i $$-0.146393\pi$$
0.896094 + 0.443864i $$0.146393\pi$$
$$702$$ 0 0
$$703$$ 15816.2 0.848534
$$704$$ 0 0
$$705$$ −8421.37 −0.449882
$$706$$ 0 0
$$707$$ −3039.17 −0.161668
$$708$$ 0 0
$$709$$ −13703.0 −0.725851 −0.362926 0.931818i $$-0.618222\pi$$
−0.362926 + 0.931818i $$0.618222\pi$$
$$710$$ 0 0
$$711$$ −4557.70 −0.240404
$$712$$ 0 0
$$713$$ −8126.14 −0.426825
$$714$$ 0 0
$$715$$ −8009.84 −0.418953
$$716$$ 0 0
$$717$$ 25987.3 1.35358
$$718$$ 0 0
$$719$$ −8074.93 −0.418838 −0.209419 0.977826i $$-0.567157\pi$$
−0.209419 + 0.977826i $$0.567157\pi$$
$$720$$ 0 0
$$721$$ −2419.04 −0.124951
$$722$$ 0 0
$$723$$ −29323.6 −1.50838
$$724$$ 0 0
$$725$$ −523.528 −0.0268184
$$726$$ 0 0
$$727$$ −3668.70 −0.187159 −0.0935794 0.995612i $$-0.529831\pi$$
−0.0935794 + 0.995612i $$0.529831\pi$$
$$728$$ 0 0
$$729$$ 21881.9 1.11172
$$730$$ 0 0
$$731$$ 1259.46 0.0637250
$$732$$ 0 0
$$733$$ 14980.3 0.754857 0.377428 0.926039i $$-0.376808\pi$$
0.377428 + 0.926039i $$0.376808\pi$$
$$734$$ 0 0
$$735$$ 1140.93 0.0572569
$$736$$ 0 0
$$737$$ 8700.94 0.434875
$$738$$ 0 0
$$739$$ −6530.59 −0.325077 −0.162538 0.986702i $$-0.551968\pi$$
−0.162538 + 0.986702i $$0.551968\pi$$
$$740$$ 0 0
$$741$$ 11453.3 0.567812
$$742$$ 0 0
$$743$$ 25952.0 1.28141 0.640704 0.767788i $$-0.278643\pi$$
0.640704 + 0.767788i $$0.278643\pi$$
$$744$$ 0 0
$$745$$ −3901.86 −0.191883
$$746$$ 0 0
$$747$$ 3509.85 0.171913
$$748$$ 0 0
$$749$$ 1519.83 0.0741434
$$750$$ 0 0
$$751$$ −14093.9 −0.684813 −0.342407 0.939552i $$-0.611242\pi$$
−0.342407 + 0.939552i $$0.611242\pi$$
$$752$$ 0 0
$$753$$ −1452.57 −0.0702983
$$754$$ 0 0
$$755$$ −11597.1 −0.559024
$$756$$ 0 0
$$757$$ 2554.41 0.122644 0.0613220 0.998118i $$-0.480468\pi$$
0.0613220 + 0.998118i $$0.480468\pi$$
$$758$$ 0 0
$$759$$ 6286.40 0.300635
$$760$$ 0 0
$$761$$ 2219.08 0.105705 0.0528527 0.998602i $$-0.483169\pi$$
0.0528527 + 0.998602i $$0.483169\pi$$
$$762$$ 0 0
$$763$$ 12140.9 0.576054
$$764$$ 0 0
$$765$$ −989.025 −0.0467428
$$766$$ 0 0
$$767$$ −18884.6 −0.889028
$$768$$ 0 0
$$769$$ −22466.2 −1.05352 −0.526758 0.850015i $$-0.676592\pi$$
−0.526758 + 0.850015i $$0.676592\pi$$
$$770$$ 0 0
$$771$$ −36609.3 −1.71006
$$772$$ 0 0
$$773$$ −9674.79 −0.450165 −0.225083 0.974340i $$-0.572265\pi$$
−0.225083 + 0.974340i $$0.572265\pi$$
$$774$$ 0 0
$$775$$ −7863.96 −0.364493
$$776$$ 0 0
$$777$$ 6426.60 0.296722
$$778$$ 0 0
$$779$$ −911.560 −0.0419256
$$780$$ 0 0
$$781$$ −49746.6 −2.27922
$$782$$ 0 0
$$783$$ 3151.23 0.143826
$$784$$ 0 0
$$785$$ −5114.51 −0.232541
$$786$$ 0 0
$$787$$ 20942.8 0.948577 0.474288 0.880370i $$-0.342705\pi$$
0.474288 + 0.880370i $$0.342705\pi$$
$$788$$ 0 0
$$789$$ 24341.8 1.09834
$$790$$ 0 0
$$791$$ 12979.4 0.583433
$$792$$ 0 0
$$793$$ 467.975 0.0209562
$$794$$ 0 0
$$795$$ −3562.94 −0.158949
$$796$$ 0 0
$$797$$ −23526.6 −1.04561 −0.522807 0.852451i $$-0.675115\pi$$
−0.522807 + 0.852451i $$0.675115\pi$$
$$798$$ 0 0
$$799$$ −13463.5 −0.596127
$$800$$ 0 0
$$801$$ 243.233 0.0107294
$$802$$ 0 0
$$803$$ −7759.29 −0.340996
$$804$$ 0 0
$$805$$ −904.172 −0.0395874
$$806$$ 0 0
$$807$$ −5968.72 −0.260358
$$808$$ 0 0
$$809$$ −18202.2 −0.791047 −0.395523 0.918456i $$-0.629437\pi$$
−0.395523 + 0.918456i $$0.629437\pi$$
$$810$$ 0 0
$$811$$ 2510.24 0.108689 0.0543443 0.998522i $$-0.482693\pi$$
0.0543443 + 0.998522i $$0.482693\pi$$
$$812$$ 0 0
$$813$$ 21906.5 0.945012
$$814$$ 0 0
$$815$$ 6753.13 0.290248
$$816$$ 0 0
$$817$$ −2714.30 −0.116232
$$818$$ 0 0
$$819$$ −1140.31 −0.0486516
$$820$$ 0 0
$$821$$ −17899.6 −0.760903 −0.380451 0.924801i $$-0.624231\pi$$
−0.380451 + 0.924801i $$0.624231\pi$$
$$822$$ 0 0
$$823$$ 14039.5 0.594637 0.297318 0.954778i $$-0.403908\pi$$
0.297318 + 0.954778i $$0.403908\pi$$
$$824$$ 0 0
$$825$$ 6083.58 0.256731
$$826$$ 0 0
$$827$$ −15127.4 −0.636073 −0.318036 0.948079i $$-0.603023\pi$$
−0.318036 + 0.948079i $$0.603023\pi$$
$$828$$ 0 0
$$829$$ −21986.5 −0.921136 −0.460568 0.887624i $$-0.652354\pi$$
−0.460568 + 0.887624i $$0.652354\pi$$
$$830$$ 0 0
$$831$$ −41718.7 −1.74152
$$832$$ 0 0
$$833$$ 1824.04 0.0758696
$$834$$ 0 0
$$835$$ −6152.89 −0.255005
$$836$$ 0 0
$$837$$ 47334.8 1.95476
$$838$$ 0 0
$$839$$ −2276.89 −0.0936914 −0.0468457 0.998902i $$-0.514917\pi$$
−0.0468457 + 0.998902i $$0.514917\pi$$
$$840$$ 0 0
$$841$$ −23950.5 −0.982019
$$842$$ 0 0
$$843$$ −1727.25 −0.0705688
$$844$$ 0 0
$$845$$ −6285.79 −0.255903
$$846$$ 0 0
$$847$$ −9796.97 −0.397436
$$848$$ 0 0
$$849$$ 27113.4 1.09603
$$850$$ 0 0
$$851$$ −5093.00 −0.205154
$$852$$ 0 0
$$853$$ −13342.6 −0.535570 −0.267785 0.963479i $$-0.586292\pi$$
−0.267785 + 0.963479i $$0.586292\pi$$
$$854$$ 0 0
$$855$$ 2131.47 0.0852571
$$856$$ 0 0
$$857$$ 18690.9 0.745003 0.372502 0.928032i $$-0.378500\pi$$
0.372502 + 0.928032i $$0.378500\pi$$
$$858$$ 0 0
$$859$$ −18318.9 −0.727628 −0.363814 0.931472i $$-0.618526\pi$$
−0.363814 + 0.931472i $$0.618526\pi$$
$$860$$ 0 0
$$861$$ −370.394 −0.0146609
$$862$$ 0 0
$$863$$ 38133.1 1.50413 0.752067 0.659087i $$-0.229057\pi$$
0.752067 + 0.659087i $$0.229057\pi$$
$$864$$ 0 0
$$865$$ 12438.3 0.488917
$$866$$ 0 0
$$867$$ −16426.0 −0.643432
$$868$$ 0 0
$$869$$ 44820.3 1.74962
$$870$$ 0 0
$$871$$ −5104.66 −0.198582
$$872$$ 0 0
$$873$$ −8938.33 −0.346525
$$874$$ 0 0
$$875$$ −875.000 −0.0338062
$$876$$ 0 0
$$877$$ 19707.5 0.758807 0.379404 0.925231i $$-0.376129\pi$$
0.379404 + 0.925231i $$0.376129\pi$$
$$878$$ 0 0
$$879$$ −34664.6 −1.33016
$$880$$ 0 0
$$881$$ −14091.5 −0.538883 −0.269441 0.963017i $$-0.586839\pi$$
−0.269441 + 0.963017i $$0.586839\pi$$
$$882$$ 0 0
$$883$$ −3115.87 −0.118751 −0.0593757 0.998236i $$-0.518911\pi$$
−0.0593757 + 0.998236i $$0.518911\pi$$
$$884$$ 0 0
$$885$$ 14343.1 0.544789
$$886$$ 0 0
$$887$$ 38734.6 1.46627 0.733134 0.680084i $$-0.238057\pi$$
0.733134 + 0.680084i $$0.238057\pi$$
$$888$$ 0 0
$$889$$ −9761.57 −0.368270
$$890$$ 0 0
$$891$$ −29121.3 −1.09495
$$892$$ 0 0
$$893$$ 29015.6 1.08731
$$894$$ 0 0
$$895$$ −8105.89 −0.302737
$$896$$ 0 0
$$897$$ −3688.10 −0.137282
$$898$$ 0 0
$$899$$ 6587.21 0.244378
$$900$$ 0 0
$$901$$ −5696.21 −0.210619
$$902$$ 0 0
$$903$$ −1102.90 −0.0406449
$$904$$ 0 0
$$905$$ −12968.4 −0.476337
$$906$$ 0 0
$$907$$ −19242.9 −0.704464 −0.352232 0.935913i $$-0.614577\pi$$
−0.352232 + 0.935913i $$0.614577\pi$$
$$908$$ 0 0
$$909$$ −2307.03 −0.0841799
$$910$$ 0 0
$$911$$ 34613.3 1.25882 0.629412 0.777072i $$-0.283296\pi$$
0.629412 + 0.777072i $$0.283296\pi$$
$$912$$ 0 0
$$913$$ −34515.8 −1.25116
$$914$$ 0 0
$$915$$ −355.433 −0.0128418
$$916$$ 0 0
$$917$$ 12336.9 0.444276
$$918$$ 0 0
$$919$$ 25826.4 0.927022 0.463511 0.886091i $$-0.346589\pi$$
0.463511 + 0.886091i $$0.346589\pi$$
$$920$$ 0 0
$$921$$ 3547.01 0.126903
$$922$$ 0 0
$$923$$ 29185.3 1.04079
$$924$$ 0 0
$$925$$ −4928.68 −0.175194
$$926$$ 0 0
$$927$$ −1836.29 −0.0650613
$$928$$ 0 0
$$929$$ 19451.6 0.686960 0.343480 0.939160i $$-0.388394\pi$$
0.343480 + 0.939160i $$0.388394\pi$$
$$930$$ 0 0
$$931$$ −3931.04 −0.138383
$$932$$ 0 0
$$933$$ 35944.8 1.26129
$$934$$ 0 0
$$935$$ 9726.03 0.340188
$$936$$ 0 0
$$937$$ 34469.1 1.20177 0.600884 0.799336i $$-0.294815\pi$$
0.600884 + 0.799336i $$0.294815\pi$$
$$938$$ 0 0
$$939$$ 39844.1 1.38473
$$940$$ 0 0
$$941$$ −14156.4 −0.490419 −0.245209 0.969470i $$-0.578857\pi$$
−0.245209 + 0.969470i $$0.578857\pi$$
$$942$$ 0 0
$$943$$ 293.532 0.0101365
$$944$$ 0 0
$$945$$ 5266.81 0.181301
$$946$$ 0 0
$$947$$ 38092.4 1.30711 0.653557 0.756877i $$-0.273276\pi$$
0.653557 + 0.756877i $$0.273276\pi$$
$$948$$ 0 0
$$949$$ 4552.22 0.155713
$$950$$ 0 0
$$951$$ 36234.8 1.23553
$$952$$ 0 0
$$953$$ 5037.40 0.171225 0.0856126 0.996329i $$-0.472715\pi$$
0.0856126 + 0.996329i $$0.472715\pi$$
$$954$$ 0 0
$$955$$ −9115.39 −0.308866
$$956$$ 0 0
$$957$$ −5095.88 −0.172128
$$958$$ 0 0
$$959$$ 6460.64 0.217544
$$960$$ 0 0
$$961$$ 69156.0 2.32137
$$962$$ 0 0
$$963$$ 1153.71 0.0386060
$$964$$ 0 0
$$965$$ −7705.14 −0.257033
$$966$$ 0 0
$$967$$ 11495.3 0.382278 0.191139 0.981563i $$-0.438782\pi$$
0.191139 + 0.981563i $$0.438782\pi$$
$$968$$ 0 0
$$969$$ −13907.3 −0.461061
$$970$$ 0 0
$$971$$ −22352.7 −0.738757 −0.369379 0.929279i $$-0.620429\pi$$
−0.369379 + 0.929279i $$0.620429\pi$$
$$972$$ 0 0
$$973$$ −1372.27 −0.0452138
$$974$$ 0 0
$$975$$ −3569.11 −0.117234
$$976$$ 0 0
$$977$$ 14345.7 0.469765 0.234882 0.972024i $$-0.424530\pi$$
0.234882 + 0.972024i $$0.424530\pi$$
$$978$$ 0 0
$$979$$ −2391.94 −0.0780867
$$980$$ 0 0
$$981$$ 9216.15 0.299948
$$982$$ 0 0
$$983$$ 34460.9 1.11814 0.559070 0.829120i $$-0.311158\pi$$
0.559070 + 0.829120i $$0.311158\pi$$
$$984$$ 0 0
$$985$$ −3506.21 −0.113419
$$986$$ 0 0
$$987$$ 11789.9 0.380220
$$988$$ 0 0
$$989$$ 874.036 0.0281019
$$990$$ 0 0
$$991$$ −35189.6 −1.12799 −0.563993 0.825780i $$-0.690735\pi$$
−0.563993 + 0.825780i $$0.690735\pi$$
$$992$$ 0 0
$$993$$ 22968.2 0.734011
$$994$$ 0 0
$$995$$ 16474.8 0.524911
$$996$$ 0 0
$$997$$ 50730.0 1.61147 0.805734 0.592277i $$-0.201771\pi$$
0.805734 + 0.592277i $$0.201771\pi$$
$$998$$ 0 0
$$999$$ 29666.8 0.939554
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 2240.4.a.bn.1.2 2
4.3 odd 2 2240.4.a.bo.1.1 2
8.3 odd 2 560.4.a.r.1.2 2
8.5 even 2 35.4.a.b.1.2 2
24.5 odd 2 315.4.a.f.1.1 2
40.13 odd 4 175.4.b.c.99.1 4
40.29 even 2 175.4.a.c.1.1 2
40.37 odd 4 175.4.b.c.99.4 4
56.5 odd 6 245.4.e.i.116.1 4
56.13 odd 2 245.4.a.k.1.2 2
56.37 even 6 245.4.e.h.116.1 4
56.45 odd 6 245.4.e.i.226.1 4
56.53 even 6 245.4.e.h.226.1 4
120.29 odd 2 1575.4.a.z.1.2 2
168.125 even 2 2205.4.a.u.1.1 2
280.69 odd 2 1225.4.a.m.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
35.4.a.b.1.2 2 8.5 even 2
175.4.a.c.1.1 2 40.29 even 2
175.4.b.c.99.1 4 40.13 odd 4
175.4.b.c.99.4 4 40.37 odd 4
245.4.a.k.1.2 2 56.13 odd 2
245.4.e.h.116.1 4 56.37 even 6
245.4.e.h.226.1 4 56.53 even 6
245.4.e.i.116.1 4 56.5 odd 6
245.4.e.i.226.1 4 56.45 odd 6
315.4.a.f.1.1 2 24.5 odd 2
560.4.a.r.1.2 2 8.3 odd 2
1225.4.a.m.1.1 2 280.69 odd 2
1575.4.a.z.1.2 2 120.29 odd 2
2205.4.a.u.1.1 2 168.125 even 2
2240.4.a.bn.1.2 2 1.1 even 1 trivial
2240.4.a.bo.1.1 2 4.3 odd 2