# Properties

 Label 528.4.a.r.1.2 Level $528$ Weight $4$ Character 528.1 Self dual yes Analytic conductor $31.153$ 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] = [528,4,Mod(1,528)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(528, 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("528.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$528 = 2^{4} \cdot 3 \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 528.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: $$31.1530084830$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 264) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-1.56155$$ of defining polynomial Character $$\chi$$ $$=$$ 528.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.00000 q^{3} +1.12311 q^{5} -0.876894 q^{7} +9.00000 q^{9} +O(q^{10})$$ $$q+3.00000 q^{3} +1.12311 q^{5} -0.876894 q^{7} +9.00000 q^{9} -11.0000 q^{11} -71.0928 q^{13} +3.36932 q^{15} +22.2159 q^{17} -125.693 q^{19} -2.63068 q^{21} +104.600 q^{23} -123.739 q^{25} +27.0000 q^{27} -303.201 q^{29} +196.924 q^{31} -33.0000 q^{33} -0.984845 q^{35} +247.848 q^{37} -213.278 q^{39} -448.462 q^{41} +196.371 q^{43} +10.1080 q^{45} -28.5398 q^{47} -342.231 q^{49} +66.6477 q^{51} -38.4451 q^{53} -12.3542 q^{55} -377.080 q^{57} -14.8599 q^{59} -625.589 q^{61} -7.89205 q^{63} -79.8447 q^{65} -668.742 q^{67} +313.801 q^{69} +179.244 q^{71} +1053.58 q^{73} -371.216 q^{75} +9.64584 q^{77} -458.941 q^{79} +81.0000 q^{81} +626.098 q^{83} +24.9508 q^{85} -909.602 q^{87} -1570.68 q^{89} +62.3409 q^{91} +590.773 q^{93} -141.167 q^{95} +827.322 q^{97} -99.0000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{3} - 6 q^{5} - 10 q^{7} + 18 q^{9}+O(q^{10})$$ 2 * q + 6 * q^3 - 6 * q^5 - 10 * q^7 + 18 * q^9 $$2 q + 6 q^{3} - 6 q^{5} - 10 q^{7} + 18 q^{9} - 22 q^{11} - 2 q^{13} - 18 q^{15} - 104 q^{17} - 4 q^{19} - 30 q^{21} + 102 q^{23} - 198 q^{25} + 54 q^{27} - 392 q^{29} + 64 q^{31} - 66 q^{33} + 64 q^{35} - 164 q^{37} - 6 q^{39} - 732 q^{41} - 168 q^{43} - 54 q^{45} + 314 q^{47} - 602 q^{49} - 312 q^{51} - 382 q^{53} + 66 q^{55} - 12 q^{57} - 508 q^{59} - 6 q^{61} - 90 q^{63} - 572 q^{65} - 216 q^{67} + 306 q^{69} + 878 q^{71} + 260 q^{73} - 594 q^{75} + 110 q^{77} - 118 q^{79} + 162 q^{81} - 496 q^{83} + 924 q^{85} - 1176 q^{87} - 1756 q^{89} - 568 q^{91} + 192 q^{93} - 1008 q^{95} + 1968 q^{97} - 198 q^{99}+O(q^{100})$$ 2 * q + 6 * q^3 - 6 * q^5 - 10 * q^7 + 18 * q^9 - 22 * q^11 - 2 * q^13 - 18 * q^15 - 104 * q^17 - 4 * q^19 - 30 * q^21 + 102 * q^23 - 198 * q^25 + 54 * q^27 - 392 * q^29 + 64 * q^31 - 66 * q^33 + 64 * q^35 - 164 * q^37 - 6 * q^39 - 732 * q^41 - 168 * q^43 - 54 * q^45 + 314 * q^47 - 602 * q^49 - 312 * q^51 - 382 * q^53 + 66 * q^55 - 12 * q^57 - 508 * q^59 - 6 * q^61 - 90 * q^63 - 572 * q^65 - 216 * q^67 + 306 * q^69 + 878 * q^71 + 260 * q^73 - 594 * q^75 + 110 * q^77 - 118 * q^79 + 162 * q^81 - 496 * q^83 + 924 * q^85 - 1176 * q^87 - 1756 * q^89 - 568 * q^91 + 192 * q^93 - 1008 * q^95 + 1968 * 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
$$5$$ 1.12311 0.100454 0.0502268 0.998738i $$-0.484006\pi$$
0.0502268 + 0.998738i $$0.484006\pi$$
$$6$$ 0 0
$$7$$ −0.876894 −0.0473478 −0.0236739 0.999720i $$-0.507536\pi$$
−0.0236739 + 0.999720i $$0.507536\pi$$
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ −11.0000 −0.301511
$$12$$ 0 0
$$13$$ −71.0928 −1.51674 −0.758369 0.651825i $$-0.774003\pi$$
−0.758369 + 0.651825i $$0.774003\pi$$
$$14$$ 0 0
$$15$$ 3.36932 0.0579969
$$16$$ 0 0
$$17$$ 22.2159 0.316950 0.158475 0.987363i $$-0.449342\pi$$
0.158475 + 0.987363i $$0.449342\pi$$
$$18$$ 0 0
$$19$$ −125.693 −1.51768 −0.758842 0.651275i $$-0.774234\pi$$
−0.758842 + 0.651275i $$0.774234\pi$$
$$20$$ 0 0
$$21$$ −2.63068 −0.0273363
$$22$$ 0 0
$$23$$ 104.600 0.948291 0.474145 0.880447i $$-0.342757\pi$$
0.474145 + 0.880447i $$0.342757\pi$$
$$24$$ 0 0
$$25$$ −123.739 −0.989909
$$26$$ 0 0
$$27$$ 27.0000 0.192450
$$28$$ 0 0
$$29$$ −303.201 −1.94148 −0.970741 0.240130i $$-0.922810\pi$$
−0.970741 + 0.240130i $$0.922810\pi$$
$$30$$ 0 0
$$31$$ 196.924 1.14092 0.570462 0.821324i $$-0.306764\pi$$
0.570462 + 0.821324i $$0.306764\pi$$
$$32$$ 0 0
$$33$$ −33.0000 −0.174078
$$34$$ 0 0
$$35$$ −0.984845 −0.00475626
$$36$$ 0 0
$$37$$ 247.848 1.10124 0.550622 0.834755i $$-0.314391\pi$$
0.550622 + 0.834755i $$0.314391\pi$$
$$38$$ 0 0
$$39$$ −213.278 −0.875689
$$40$$ 0 0
$$41$$ −448.462 −1.70824 −0.854122 0.520072i $$-0.825905\pi$$
−0.854122 + 0.520072i $$0.825905\pi$$
$$42$$ 0 0
$$43$$ 196.371 0.696426 0.348213 0.937415i $$-0.386789\pi$$
0.348213 + 0.937415i $$0.386789\pi$$
$$44$$ 0 0
$$45$$ 10.1080 0.0334845
$$46$$ 0 0
$$47$$ −28.5398 −0.0885734 −0.0442867 0.999019i $$-0.514102\pi$$
−0.0442867 + 0.999019i $$0.514102\pi$$
$$48$$ 0 0
$$49$$ −342.231 −0.997758
$$50$$ 0 0
$$51$$ 66.6477 0.182991
$$52$$ 0 0
$$53$$ −38.4451 −0.0996385 −0.0498192 0.998758i $$-0.515865\pi$$
−0.0498192 + 0.998758i $$0.515865\pi$$
$$54$$ 0 0
$$55$$ −12.3542 −0.0302879
$$56$$ 0 0
$$57$$ −377.080 −0.876235
$$58$$ 0 0
$$59$$ −14.8599 −0.0327897 −0.0163948 0.999866i $$-0.505219\pi$$
−0.0163948 + 0.999866i $$0.505219\pi$$
$$60$$ 0 0
$$61$$ −625.589 −1.31309 −0.656545 0.754287i $$-0.727983\pi$$
−0.656545 + 0.754287i $$0.727983\pi$$
$$62$$ 0 0
$$63$$ −7.89205 −0.0157826
$$64$$ 0 0
$$65$$ −79.8447 −0.152362
$$66$$ 0 0
$$67$$ −668.742 −1.21940 −0.609701 0.792632i $$-0.708710\pi$$
−0.609701 + 0.792632i $$0.708710\pi$$
$$68$$ 0 0
$$69$$ 313.801 0.547496
$$70$$ 0 0
$$71$$ 179.244 0.299611 0.149806 0.988715i $$-0.452135\pi$$
0.149806 + 0.988715i $$0.452135\pi$$
$$72$$ 0 0
$$73$$ 1053.58 1.68920 0.844601 0.535397i $$-0.179838\pi$$
0.844601 + 0.535397i $$0.179838\pi$$
$$74$$ 0 0
$$75$$ −371.216 −0.571524
$$76$$ 0 0
$$77$$ 9.64584 0.0142759
$$78$$ 0 0
$$79$$ −458.941 −0.653607 −0.326803 0.945092i $$-0.605971\pi$$
−0.326803 + 0.945092i $$0.605971\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ 626.098 0.827991 0.413995 0.910279i $$-0.364133\pi$$
0.413995 + 0.910279i $$0.364133\pi$$
$$84$$ 0 0
$$85$$ 24.9508 0.0318388
$$86$$ 0 0
$$87$$ −909.602 −1.12091
$$88$$ 0 0
$$89$$ −1570.68 −1.87070 −0.935348 0.353729i $$-0.884913\pi$$
−0.935348 + 0.353729i $$0.884913\pi$$
$$90$$ 0 0
$$91$$ 62.3409 0.0718143
$$92$$ 0 0
$$93$$ 590.773 0.658713
$$94$$ 0 0
$$95$$ −141.167 −0.152457
$$96$$ 0 0
$$97$$ 827.322 0.865998 0.432999 0.901394i $$-0.357455\pi$$
0.432999 + 0.901394i $$0.357455\pi$$
$$98$$ 0 0
$$99$$ −99.0000 −0.100504
$$100$$ 0 0
$$101$$ 137.852 0.135810 0.0679050 0.997692i $$-0.478369\pi$$
0.0679050 + 0.997692i $$0.478369\pi$$
$$102$$ 0 0
$$103$$ −623.352 −0.596318 −0.298159 0.954516i $$-0.596373\pi$$
−0.298159 + 0.954516i $$0.596373\pi$$
$$104$$ 0 0
$$105$$ −2.95454 −0.00274603
$$106$$ 0 0
$$107$$ −996.068 −0.899940 −0.449970 0.893044i $$-0.648565\pi$$
−0.449970 + 0.893044i $$0.648565\pi$$
$$108$$ 0 0
$$109$$ 567.180 0.498404 0.249202 0.968452i $$-0.419832\pi$$
0.249202 + 0.968452i $$0.419832\pi$$
$$110$$ 0 0
$$111$$ 743.545 0.635804
$$112$$ 0 0
$$113$$ −440.644 −0.366834 −0.183417 0.983035i $$-0.558716\pi$$
−0.183417 + 0.983035i $$0.558716\pi$$
$$114$$ 0 0
$$115$$ 117.477 0.0952592
$$116$$ 0 0
$$117$$ −639.835 −0.505579
$$118$$ 0 0
$$119$$ −19.4810 −0.0150069
$$120$$ 0 0
$$121$$ 121.000 0.0909091
$$122$$ 0 0
$$123$$ −1345.39 −0.986255
$$124$$ 0 0
$$125$$ −279.360 −0.199894
$$126$$ 0 0
$$127$$ −533.032 −0.372433 −0.186216 0.982509i $$-0.559623\pi$$
−0.186216 + 0.982509i $$0.559623\pi$$
$$128$$ 0 0
$$129$$ 589.114 0.402082
$$130$$ 0 0
$$131$$ −1983.45 −1.32286 −0.661432 0.750005i $$-0.730051\pi$$
−0.661432 + 0.750005i $$0.730051\pi$$
$$132$$ 0 0
$$133$$ 110.220 0.0718591
$$134$$ 0 0
$$135$$ 30.3239 0.0193323
$$136$$ 0 0
$$137$$ −1756.75 −1.09554 −0.547770 0.836629i $$-0.684523\pi$$
−0.547770 + 0.836629i $$0.684523\pi$$
$$138$$ 0 0
$$139$$ −1453.38 −0.886865 −0.443433 0.896308i $$-0.646239\pi$$
−0.443433 + 0.896308i $$0.646239\pi$$
$$140$$ 0 0
$$141$$ −85.6193 −0.0511379
$$142$$ 0 0
$$143$$ 782.021 0.457314
$$144$$ 0 0
$$145$$ −340.526 −0.195029
$$146$$ 0 0
$$147$$ −1026.69 −0.576056
$$148$$ 0 0
$$149$$ 745.151 0.409699 0.204850 0.978793i $$-0.434329\pi$$
0.204850 + 0.978793i $$0.434329\pi$$
$$150$$ 0 0
$$151$$ 2647.53 1.42684 0.713420 0.700737i $$-0.247145\pi$$
0.713420 + 0.700737i $$0.247145\pi$$
$$152$$ 0 0
$$153$$ 199.943 0.105650
$$154$$ 0 0
$$155$$ 221.167 0.114610
$$156$$ 0 0
$$157$$ 2537.55 1.28993 0.644963 0.764214i $$-0.276873\pi$$
0.644963 + 0.764214i $$0.276873\pi$$
$$158$$ 0 0
$$159$$ −115.335 −0.0575263
$$160$$ 0 0
$$161$$ −91.7235 −0.0448995
$$162$$ 0 0
$$163$$ 84.8636 0.0407793 0.0203897 0.999792i $$-0.493509\pi$$
0.0203897 + 0.999792i $$0.493509\pi$$
$$164$$ 0 0
$$165$$ −37.0625 −0.0174867
$$166$$ 0 0
$$167$$ 4133.03 1.91511 0.957556 0.288247i $$-0.0930723\pi$$
0.957556 + 0.288247i $$0.0930723\pi$$
$$168$$ 0 0
$$169$$ 2857.19 1.30049
$$170$$ 0 0
$$171$$ −1131.24 −0.505895
$$172$$ 0 0
$$173$$ −1177.29 −0.517386 −0.258693 0.965960i $$-0.583292\pi$$
−0.258693 + 0.965960i $$0.583292\pi$$
$$174$$ 0 0
$$175$$ 108.506 0.0468701
$$176$$ 0 0
$$177$$ −44.5796 −0.0189311
$$178$$ 0 0
$$179$$ 241.087 0.100669 0.0503343 0.998732i $$-0.483971\pi$$
0.0503343 + 0.998732i $$0.483971\pi$$
$$180$$ 0 0
$$181$$ 2125.85 0.873002 0.436501 0.899704i $$-0.356218\pi$$
0.436501 + 0.899704i $$0.356218\pi$$
$$182$$ 0 0
$$183$$ −1876.77 −0.758113
$$184$$ 0 0
$$185$$ 278.360 0.110624
$$186$$ 0 0
$$187$$ −244.375 −0.0955640
$$188$$ 0 0
$$189$$ −23.6761 −0.00911210
$$190$$ 0 0
$$191$$ −3348.26 −1.26844 −0.634219 0.773154i $$-0.718678\pi$$
−0.634219 + 0.773154i $$0.718678\pi$$
$$192$$ 0 0
$$193$$ −1437.61 −0.536173 −0.268087 0.963395i $$-0.586391\pi$$
−0.268087 + 0.963395i $$0.586391\pi$$
$$194$$ 0 0
$$195$$ −239.534 −0.0879661
$$196$$ 0 0
$$197$$ −3276.09 −1.18483 −0.592415 0.805633i $$-0.701825\pi$$
−0.592415 + 0.805633i $$0.701825\pi$$
$$198$$ 0 0
$$199$$ 4479.61 1.59573 0.797867 0.602833i $$-0.205962\pi$$
0.797867 + 0.602833i $$0.205962\pi$$
$$200$$ 0 0
$$201$$ −2006.23 −0.704022
$$202$$ 0 0
$$203$$ 265.875 0.0919250
$$204$$ 0 0
$$205$$ −503.670 −0.171599
$$206$$ 0 0
$$207$$ 941.403 0.316097
$$208$$ 0 0
$$209$$ 1382.62 0.457599
$$210$$ 0 0
$$211$$ 4326.58 1.41163 0.705816 0.708395i $$-0.250581\pi$$
0.705816 + 0.708395i $$0.250581\pi$$
$$212$$ 0 0
$$213$$ 537.733 0.172981
$$214$$ 0 0
$$215$$ 220.546 0.0699585
$$216$$ 0 0
$$217$$ −172.682 −0.0540203
$$218$$ 0 0
$$219$$ 3160.73 0.975261
$$220$$ 0 0
$$221$$ −1579.39 −0.480730
$$222$$ 0 0
$$223$$ −898.969 −0.269953 −0.134976 0.990849i $$-0.543096\pi$$
−0.134976 + 0.990849i $$0.543096\pi$$
$$224$$ 0 0
$$225$$ −1113.65 −0.329970
$$226$$ 0 0
$$227$$ −601.530 −0.175881 −0.0879405 0.996126i $$-0.528029\pi$$
−0.0879405 + 0.996126i $$0.528029\pi$$
$$228$$ 0 0
$$229$$ 3143.94 0.907238 0.453619 0.891196i $$-0.350133\pi$$
0.453619 + 0.891196i $$0.350133\pi$$
$$230$$ 0 0
$$231$$ 28.9375 0.00824220
$$232$$ 0 0
$$233$$ 1699.18 0.477756 0.238878 0.971050i $$-0.423220\pi$$
0.238878 + 0.971050i $$0.423220\pi$$
$$234$$ 0 0
$$235$$ −32.0532 −0.00889752
$$236$$ 0 0
$$237$$ −1376.82 −0.377360
$$238$$ 0 0
$$239$$ 7204.05 1.94975 0.974877 0.222744i $$-0.0715014\pi$$
0.974877 + 0.222744i $$0.0715014\pi$$
$$240$$ 0 0
$$241$$ 3347.54 0.894747 0.447374 0.894347i $$-0.352359\pi$$
0.447374 + 0.894347i $$0.352359\pi$$
$$242$$ 0 0
$$243$$ 243.000 0.0641500
$$244$$ 0 0
$$245$$ −384.362 −0.100228
$$246$$ 0 0
$$247$$ 8935.88 2.30193
$$248$$ 0 0
$$249$$ 1878.30 0.478041
$$250$$ 0 0
$$251$$ −50.8673 −0.0127917 −0.00639585 0.999980i $$-0.502036\pi$$
−0.00639585 + 0.999980i $$0.502036\pi$$
$$252$$ 0 0
$$253$$ −1150.60 −0.285920
$$254$$ 0 0
$$255$$ 74.8524 0.0183821
$$256$$ 0 0
$$257$$ 4216.53 1.02342 0.511712 0.859157i $$-0.329012\pi$$
0.511712 + 0.859157i $$0.329012\pi$$
$$258$$ 0 0
$$259$$ −217.337 −0.0521415
$$260$$ 0 0
$$261$$ −2728.81 −0.647161
$$262$$ 0 0
$$263$$ 2000.38 0.469006 0.234503 0.972115i $$-0.424654\pi$$
0.234503 + 0.972115i $$0.424654\pi$$
$$264$$ 0 0
$$265$$ −43.1779 −0.0100090
$$266$$ 0 0
$$267$$ −4712.05 −1.08005
$$268$$ 0 0
$$269$$ 2305.18 0.522487 0.261244 0.965273i $$-0.415867\pi$$
0.261244 + 0.965273i $$0.415867\pi$$
$$270$$ 0 0
$$271$$ 4668.18 1.04639 0.523195 0.852213i $$-0.324740\pi$$
0.523195 + 0.852213i $$0.324740\pi$$
$$272$$ 0 0
$$273$$ 187.023 0.0414620
$$274$$ 0 0
$$275$$ 1361.12 0.298469
$$276$$ 0 0
$$277$$ −2933.96 −0.636406 −0.318203 0.948023i $$-0.603079\pi$$
−0.318203 + 0.948023i $$0.603079\pi$$
$$278$$ 0 0
$$279$$ 1772.32 0.380308
$$280$$ 0 0
$$281$$ −4395.40 −0.933124 −0.466562 0.884488i $$-0.654508\pi$$
−0.466562 + 0.884488i $$0.654508\pi$$
$$282$$ 0 0
$$283$$ 1527.07 0.320759 0.160379 0.987055i $$-0.448728\pi$$
0.160379 + 0.987055i $$0.448728\pi$$
$$284$$ 0 0
$$285$$ −423.500 −0.0880210
$$286$$ 0 0
$$287$$ 393.254 0.0808817
$$288$$ 0 0
$$289$$ −4419.45 −0.899543
$$290$$ 0 0
$$291$$ 2481.97 0.499984
$$292$$ 0 0
$$293$$ 4641.14 0.925386 0.462693 0.886519i $$-0.346883\pi$$
0.462693 + 0.886519i $$0.346883\pi$$
$$294$$ 0 0
$$295$$ −16.6892 −0.00329384
$$296$$ 0 0
$$297$$ −297.000 −0.0580259
$$298$$ 0 0
$$299$$ −7436.33 −1.43831
$$300$$ 0 0
$$301$$ −172.197 −0.0329743
$$302$$ 0 0
$$303$$ 413.557 0.0784099
$$304$$ 0 0
$$305$$ −702.602 −0.131905
$$306$$ 0 0
$$307$$ −5918.64 −1.10031 −0.550154 0.835063i $$-0.685431\pi$$
−0.550154 + 0.835063i $$0.685431\pi$$
$$308$$ 0 0
$$309$$ −1870.06 −0.344284
$$310$$ 0 0
$$311$$ −4622.47 −0.842817 −0.421409 0.906871i $$-0.638464\pi$$
−0.421409 + 0.906871i $$0.638464\pi$$
$$312$$ 0 0
$$313$$ 9862.20 1.78097 0.890486 0.455010i $$-0.150364\pi$$
0.890486 + 0.455010i $$0.150364\pi$$
$$314$$ 0 0
$$315$$ −8.86361 −0.00158542
$$316$$ 0 0
$$317$$ 6797.67 1.20440 0.602201 0.798345i $$-0.294291\pi$$
0.602201 + 0.798345i $$0.294291\pi$$
$$318$$ 0 0
$$319$$ 3335.21 0.585379
$$320$$ 0 0
$$321$$ −2988.20 −0.519580
$$322$$ 0 0
$$323$$ −2792.39 −0.481030
$$324$$ 0 0
$$325$$ 8796.93 1.50143
$$326$$ 0 0
$$327$$ 1701.54 0.287753
$$328$$ 0 0
$$329$$ 25.0263 0.00419376
$$330$$ 0 0
$$331$$ 5847.55 0.971029 0.485514 0.874229i $$-0.338632\pi$$
0.485514 + 0.874229i $$0.338632\pi$$
$$332$$ 0 0
$$333$$ 2230.64 0.367081
$$334$$ 0 0
$$335$$ −751.068 −0.122493
$$336$$ 0 0
$$337$$ 3797.52 0.613840 0.306920 0.951735i $$-0.400702\pi$$
0.306920 + 0.951735i $$0.400702\pi$$
$$338$$ 0 0
$$339$$ −1321.93 −0.211792
$$340$$ 0 0
$$341$$ −2166.17 −0.344001
$$342$$ 0 0
$$343$$ 600.875 0.0945895
$$344$$ 0 0
$$345$$ 352.432 0.0549979
$$346$$ 0 0
$$347$$ −11221.2 −1.73599 −0.867993 0.496577i $$-0.834590\pi$$
−0.867993 + 0.496577i $$0.834590\pi$$
$$348$$ 0 0
$$349$$ −8130.46 −1.24703 −0.623515 0.781811i $$-0.714296\pi$$
−0.623515 + 0.781811i $$0.714296\pi$$
$$350$$ 0 0
$$351$$ −1919.51 −0.291896
$$352$$ 0 0
$$353$$ −11060.0 −1.66761 −0.833804 0.552061i $$-0.813842\pi$$
−0.833804 + 0.552061i $$0.813842\pi$$
$$354$$ 0 0
$$355$$ 201.310 0.0300970
$$356$$ 0 0
$$357$$ −58.4430 −0.00866423
$$358$$ 0 0
$$359$$ −9040.68 −1.32911 −0.664553 0.747242i $$-0.731378\pi$$
−0.664553 + 0.747242i $$0.731378\pi$$
$$360$$ 0 0
$$361$$ 8939.77 1.30336
$$362$$ 0 0
$$363$$ 363.000 0.0524864
$$364$$ 0 0
$$365$$ 1183.28 0.169686
$$366$$ 0 0
$$367$$ 2896.46 0.411973 0.205986 0.978555i $$-0.433960\pi$$
0.205986 + 0.978555i $$0.433960\pi$$
$$368$$ 0 0
$$369$$ −4036.16 −0.569415
$$370$$ 0 0
$$371$$ 33.7123 0.00471767
$$372$$ 0 0
$$373$$ −12226.8 −1.69726 −0.848631 0.528985i $$-0.822573\pi$$
−0.848631 + 0.528985i $$0.822573\pi$$
$$374$$ 0 0
$$375$$ −838.079 −0.115409
$$376$$ 0 0
$$377$$ 21555.4 2.94472
$$378$$ 0 0
$$379$$ −10363.3 −1.40455 −0.702277 0.711903i $$-0.747833\pi$$
−0.702277 + 0.711903i $$0.747833\pi$$
$$380$$ 0 0
$$381$$ −1599.10 −0.215024
$$382$$ 0 0
$$383$$ −4571.03 −0.609840 −0.304920 0.952378i $$-0.598630\pi$$
−0.304920 + 0.952378i $$0.598630\pi$$
$$384$$ 0 0
$$385$$ 10.8333 0.00143407
$$386$$ 0 0
$$387$$ 1767.34 0.232142
$$388$$ 0 0
$$389$$ 7931.49 1.03378 0.516892 0.856050i $$-0.327089\pi$$
0.516892 + 0.856050i $$0.327089\pi$$
$$390$$ 0 0
$$391$$ 2323.79 0.300561
$$392$$ 0 0
$$393$$ −5950.36 −0.763756
$$394$$ 0 0
$$395$$ −515.439 −0.0656572
$$396$$ 0 0
$$397$$ −7087.28 −0.895970 −0.447985 0.894041i $$-0.647858\pi$$
−0.447985 + 0.894041i $$0.647858\pi$$
$$398$$ 0 0
$$399$$ 330.659 0.0414878
$$400$$ 0 0
$$401$$ −2459.84 −0.306331 −0.153166 0.988201i $$-0.548947\pi$$
−0.153166 + 0.988201i $$0.548947\pi$$
$$402$$ 0 0
$$403$$ −13999.9 −1.73048
$$404$$ 0 0
$$405$$ 90.9716 0.0111615
$$406$$ 0 0
$$407$$ −2726.33 −0.332038
$$408$$ 0 0
$$409$$ −1755.37 −0.212218 −0.106109 0.994354i $$-0.533839\pi$$
−0.106109 + 0.994354i $$0.533839\pi$$
$$410$$ 0 0
$$411$$ −5270.24 −0.632510
$$412$$ 0 0
$$413$$ 13.0305 0.00155252
$$414$$ 0 0
$$415$$ 703.175 0.0831747
$$416$$ 0 0
$$417$$ −4360.15 −0.512032
$$418$$ 0 0
$$419$$ −10632.1 −1.23965 −0.619823 0.784742i $$-0.712796\pi$$
−0.619823 + 0.784742i $$0.712796\pi$$
$$420$$ 0 0
$$421$$ −4067.83 −0.470912 −0.235456 0.971885i $$-0.575658\pi$$
−0.235456 + 0.971885i $$0.575658\pi$$
$$422$$ 0 0
$$423$$ −256.858 −0.0295245
$$424$$ 0 0
$$425$$ −2748.97 −0.313752
$$426$$ 0 0
$$427$$ 548.575 0.0621720
$$428$$ 0 0
$$429$$ 2346.06 0.264030
$$430$$ 0 0
$$431$$ 13789.1 1.54106 0.770532 0.637401i $$-0.219990\pi$$
0.770532 + 0.637401i $$0.219990\pi$$
$$432$$ 0 0
$$433$$ 15964.2 1.77180 0.885901 0.463873i $$-0.153541\pi$$
0.885901 + 0.463873i $$0.153541\pi$$
$$434$$ 0 0
$$435$$ −1021.58 −0.112600
$$436$$ 0 0
$$437$$ −13147.6 −1.43921
$$438$$ 0 0
$$439$$ 7233.47 0.786412 0.393206 0.919450i $$-0.371366\pi$$
0.393206 + 0.919450i $$0.371366\pi$$
$$440$$ 0 0
$$441$$ −3080.08 −0.332586
$$442$$ 0 0
$$443$$ −8411.08 −0.902082 −0.451041 0.892503i $$-0.648947\pi$$
−0.451041 + 0.892503i $$0.648947\pi$$
$$444$$ 0 0
$$445$$ −1764.04 −0.187918
$$446$$ 0 0
$$447$$ 2235.45 0.236540
$$448$$ 0 0
$$449$$ −1821.18 −0.191418 −0.0957090 0.995409i $$-0.530512\pi$$
−0.0957090 + 0.995409i $$0.530512\pi$$
$$450$$ 0 0
$$451$$ 4933.08 0.515055
$$452$$ 0 0
$$453$$ 7942.58 0.823786
$$454$$ 0 0
$$455$$ 70.0154 0.00721400
$$456$$ 0 0
$$457$$ −8637.94 −0.884170 −0.442085 0.896973i $$-0.645761\pi$$
−0.442085 + 0.896973i $$0.645761\pi$$
$$458$$ 0 0
$$459$$ 599.829 0.0609970
$$460$$ 0 0
$$461$$ 1775.51 0.179379 0.0896894 0.995970i $$-0.471413\pi$$
0.0896894 + 0.995970i $$0.471413\pi$$
$$462$$ 0 0
$$463$$ −14687.5 −1.47427 −0.737134 0.675746i $$-0.763822\pi$$
−0.737134 + 0.675746i $$0.763822\pi$$
$$464$$ 0 0
$$465$$ 663.500 0.0661701
$$466$$ 0 0
$$467$$ 12523.4 1.24093 0.620463 0.784236i $$-0.286945\pi$$
0.620463 + 0.784236i $$0.286945\pi$$
$$468$$ 0 0
$$469$$ 586.416 0.0577360
$$470$$ 0 0
$$471$$ 7612.65 0.744739
$$472$$ 0 0
$$473$$ −2160.08 −0.209980
$$474$$ 0 0
$$475$$ 15553.1 1.50237
$$476$$ 0 0
$$477$$ −346.006 −0.0332128
$$478$$ 0 0
$$479$$ −8938.18 −0.852601 −0.426301 0.904582i $$-0.640183\pi$$
−0.426301 + 0.904582i $$0.640183\pi$$
$$480$$ 0 0
$$481$$ −17620.2 −1.67030
$$482$$ 0 0
$$483$$ −275.170 −0.0259228
$$484$$ 0 0
$$485$$ 929.170 0.0869926
$$486$$ 0 0
$$487$$ −7493.41 −0.697246 −0.348623 0.937263i $$-0.613351\pi$$
−0.348623 + 0.937263i $$0.613351\pi$$
$$488$$ 0 0
$$489$$ 254.591 0.0235440
$$490$$ 0 0
$$491$$ −2890.76 −0.265699 −0.132849 0.991136i $$-0.542413\pi$$
−0.132849 + 0.991136i $$0.542413\pi$$
$$492$$ 0 0
$$493$$ −6735.88 −0.615352
$$494$$ 0 0
$$495$$ −111.187 −0.0100960
$$496$$ 0 0
$$497$$ −157.178 −0.0141859
$$498$$ 0 0
$$499$$ −14757.4 −1.32391 −0.661957 0.749542i $$-0.730274\pi$$
−0.661957 + 0.749542i $$0.730274\pi$$
$$500$$ 0 0
$$501$$ 12399.1 1.10569
$$502$$ 0 0
$$503$$ −4214.22 −0.373564 −0.186782 0.982401i $$-0.559806\pi$$
−0.186782 + 0.982401i $$0.559806\pi$$
$$504$$ 0 0
$$505$$ 154.823 0.0136426
$$506$$ 0 0
$$507$$ 8571.56 0.750841
$$508$$ 0 0
$$509$$ −15335.5 −1.33543 −0.667716 0.744416i $$-0.732728\pi$$
−0.667716 + 0.744416i $$0.732728\pi$$
$$510$$ 0 0
$$511$$ −923.875 −0.0799800
$$512$$ 0 0
$$513$$ −3393.72 −0.292078
$$514$$ 0 0
$$515$$ −700.090 −0.0599023
$$516$$ 0 0
$$517$$ 313.937 0.0267059
$$518$$ 0 0
$$519$$ −3531.88 −0.298713
$$520$$ 0 0
$$521$$ −6502.03 −0.546755 −0.273377 0.961907i $$-0.588141\pi$$
−0.273377 + 0.961907i $$0.588141\pi$$
$$522$$ 0 0
$$523$$ −6816.32 −0.569898 −0.284949 0.958543i $$-0.591977\pi$$
−0.284949 + 0.958543i $$0.591977\pi$$
$$524$$ 0 0
$$525$$ 325.517 0.0270604
$$526$$ 0 0
$$527$$ 4374.85 0.361616
$$528$$ 0 0
$$529$$ −1225.76 −0.100745
$$530$$ 0 0
$$531$$ −133.739 −0.0109299
$$532$$ 0 0
$$533$$ 31882.4 2.59096
$$534$$ 0 0
$$535$$ −1118.69 −0.0904022
$$536$$ 0 0
$$537$$ 723.261 0.0581211
$$538$$ 0 0
$$539$$ 3764.54 0.300835
$$540$$ 0 0
$$541$$ −6746.57 −0.536151 −0.268076 0.963398i $$-0.586388\pi$$
−0.268076 + 0.963398i $$0.586388\pi$$
$$542$$ 0 0
$$543$$ 6377.56 0.504028
$$544$$ 0 0
$$545$$ 637.003 0.0500664
$$546$$ 0 0
$$547$$ 14961.8 1.16951 0.584754 0.811210i $$-0.301191\pi$$
0.584754 + 0.811210i $$0.301191\pi$$
$$548$$ 0 0
$$549$$ −5630.30 −0.437696
$$550$$ 0 0
$$551$$ 38110.3 2.94655
$$552$$ 0 0
$$553$$ 402.443 0.0309469
$$554$$ 0 0
$$555$$ 835.080 0.0638688
$$556$$ 0 0
$$557$$ −3962.80 −0.301453 −0.150727 0.988575i $$-0.548161\pi$$
−0.150727 + 0.988575i $$0.548161\pi$$
$$558$$ 0 0
$$559$$ −13960.6 −1.05630
$$560$$ 0 0
$$561$$ −733.125 −0.0551739
$$562$$ 0 0
$$563$$ −3231.60 −0.241910 −0.120955 0.992658i $$-0.538596\pi$$
−0.120955 + 0.992658i $$0.538596\pi$$
$$564$$ 0 0
$$565$$ −494.890 −0.0368499
$$566$$ 0 0
$$567$$ −71.0284 −0.00526087
$$568$$ 0 0
$$569$$ −18512.3 −1.36393 −0.681965 0.731385i $$-0.738874\pi$$
−0.681965 + 0.731385i $$0.738874\pi$$
$$570$$ 0 0
$$571$$ 1859.16 0.136258 0.0681292 0.997677i $$-0.478297\pi$$
0.0681292 + 0.997677i $$0.478297\pi$$
$$572$$ 0 0
$$573$$ −10044.8 −0.732333
$$574$$ 0 0
$$575$$ −12943.1 −0.938722
$$576$$ 0 0
$$577$$ 5922.28 0.427292 0.213646 0.976911i $$-0.431466\pi$$
0.213646 + 0.976911i $$0.431466\pi$$
$$578$$ 0 0
$$579$$ −4312.83 −0.309560
$$580$$ 0 0
$$581$$ −549.022 −0.0392036
$$582$$ 0 0
$$583$$ 422.896 0.0300421
$$584$$ 0 0
$$585$$ −718.602 −0.0507873
$$586$$ 0 0
$$587$$ −2908.32 −0.204496 −0.102248 0.994759i $$-0.532604\pi$$
−0.102248 + 0.994759i $$0.532604\pi$$
$$588$$ 0 0
$$589$$ −24752.0 −1.73156
$$590$$ 0 0
$$591$$ −9828.26 −0.684062
$$592$$ 0 0
$$593$$ −10853.0 −0.751565 −0.375782 0.926708i $$-0.622626\pi$$
−0.375782 + 0.926708i $$0.622626\pi$$
$$594$$ 0 0
$$595$$ −21.8792 −0.00150750
$$596$$ 0 0
$$597$$ 13438.8 0.921298
$$598$$ 0 0
$$599$$ 15577.7 1.06258 0.531290 0.847190i $$-0.321707\pi$$
0.531290 + 0.847190i $$0.321707\pi$$
$$600$$ 0 0
$$601$$ −1353.97 −0.0918961 −0.0459481 0.998944i $$-0.514631\pi$$
−0.0459481 + 0.998944i $$0.514631\pi$$
$$602$$ 0 0
$$603$$ −6018.68 −0.406467
$$604$$ 0 0
$$605$$ 135.896 0.00913215
$$606$$ 0 0
$$607$$ 16849.6 1.12670 0.563348 0.826220i $$-0.309513\pi$$
0.563348 + 0.826220i $$0.309513\pi$$
$$608$$ 0 0
$$609$$ 797.625 0.0530729
$$610$$ 0 0
$$611$$ 2028.97 0.134343
$$612$$ 0 0
$$613$$ −6436.00 −0.424058 −0.212029 0.977263i $$-0.568007\pi$$
−0.212029 + 0.977263i $$0.568007\pi$$
$$614$$ 0 0
$$615$$ −1511.01 −0.0990729
$$616$$ 0 0
$$617$$ −4556.21 −0.297287 −0.148644 0.988891i $$-0.547491\pi$$
−0.148644 + 0.988891i $$0.547491\pi$$
$$618$$ 0 0
$$619$$ −15419.6 −1.00124 −0.500620 0.865667i $$-0.666895\pi$$
−0.500620 + 0.865667i $$0.666895\pi$$
$$620$$ 0 0
$$621$$ 2824.21 0.182499
$$622$$ 0 0
$$623$$ 1377.32 0.0885734
$$624$$ 0 0
$$625$$ 15153.6 0.969829
$$626$$ 0 0
$$627$$ 4147.87 0.264195
$$628$$ 0 0
$$629$$ 5506.18 0.349039
$$630$$ 0 0
$$631$$ 10148.5 0.640265 0.320132 0.947373i $$-0.396273\pi$$
0.320132 + 0.947373i $$0.396273\pi$$
$$632$$ 0 0
$$633$$ 12979.8 0.815006
$$634$$ 0 0
$$635$$ −598.651 −0.0374122
$$636$$ 0 0
$$637$$ 24330.2 1.51334
$$638$$ 0 0
$$639$$ 1613.20 0.0998704
$$640$$ 0 0
$$641$$ −24451.5 −1.50667 −0.753336 0.657635i $$-0.771557\pi$$
−0.753336 + 0.657635i $$0.771557\pi$$
$$642$$ 0 0
$$643$$ 24688.2 1.51416 0.757082 0.653320i $$-0.226624\pi$$
0.757082 + 0.653320i $$0.226624\pi$$
$$644$$ 0 0
$$645$$ 661.637 0.0403906
$$646$$ 0 0
$$647$$ 7484.57 0.454789 0.227395 0.973803i $$-0.426979\pi$$
0.227395 + 0.973803i $$0.426979\pi$$
$$648$$ 0 0
$$649$$ 163.459 0.00988646
$$650$$ 0 0
$$651$$ −518.045 −0.0311886
$$652$$ 0 0
$$653$$ 12122.3 0.726465 0.363233 0.931699i $$-0.381673\pi$$
0.363233 + 0.931699i $$0.381673\pi$$
$$654$$ 0 0
$$655$$ −2227.63 −0.132887
$$656$$ 0 0
$$657$$ 9482.18 0.563067
$$658$$ 0 0
$$659$$ 31197.8 1.84415 0.922074 0.387013i $$-0.126493\pi$$
0.922074 + 0.387013i $$0.126493\pi$$
$$660$$ 0 0
$$661$$ 3983.93 0.234428 0.117214 0.993107i $$-0.462604\pi$$
0.117214 + 0.993107i $$0.462604\pi$$
$$662$$ 0 0
$$663$$ −4738.17 −0.277550
$$664$$ 0 0
$$665$$ 123.788 0.00721850
$$666$$ 0 0
$$667$$ −31714.9 −1.84109
$$668$$ 0 0
$$669$$ −2696.91 −0.155857
$$670$$ 0 0
$$671$$ 6881.48 0.395911
$$672$$ 0 0
$$673$$ 159.587 0.00914060 0.00457030 0.999990i $$-0.498545\pi$$
0.00457030 + 0.999990i $$0.498545\pi$$
$$674$$ 0 0
$$675$$ −3340.94 −0.190508
$$676$$ 0 0
$$677$$ −23774.6 −1.34968 −0.674838 0.737965i $$-0.735787\pi$$
−0.674838 + 0.737965i $$0.735787\pi$$
$$678$$ 0 0
$$679$$ −725.474 −0.0410031
$$680$$ 0 0
$$681$$ −1804.59 −0.101545
$$682$$ 0 0
$$683$$ 17460.9 0.978215 0.489108 0.872223i $$-0.337323\pi$$
0.489108 + 0.872223i $$0.337323\pi$$
$$684$$ 0 0
$$685$$ −1973.01 −0.110051
$$686$$ 0 0
$$687$$ 9431.83 0.523794
$$688$$ 0 0
$$689$$ 2733.17 0.151125
$$690$$ 0 0
$$691$$ −25778.2 −1.41917 −0.709587 0.704617i $$-0.751119\pi$$
−0.709587 + 0.704617i $$0.751119\pi$$
$$692$$ 0 0
$$693$$ 86.8125 0.00475864
$$694$$ 0 0
$$695$$ −1632.30 −0.0890889
$$696$$ 0 0
$$697$$ −9962.99 −0.541428
$$698$$ 0 0
$$699$$ 5097.54 0.275832
$$700$$ 0 0
$$701$$ −5531.04 −0.298009 −0.149005 0.988837i $$-0.547607\pi$$
−0.149005 + 0.988837i $$0.547607\pi$$
$$702$$ 0 0
$$703$$ −31152.9 −1.67134
$$704$$ 0 0
$$705$$ −96.1595 −0.00513699
$$706$$ 0 0
$$707$$ −120.882 −0.00643031
$$708$$ 0 0
$$709$$ −5591.04 −0.296158 −0.148079 0.988976i $$-0.547309\pi$$
−0.148079 + 0.988976i $$0.547309\pi$$
$$710$$ 0 0
$$711$$ −4130.47 −0.217869
$$712$$ 0 0
$$713$$ 20598.3 1.08193
$$714$$ 0 0
$$715$$ 878.292 0.0459388
$$716$$ 0 0
$$717$$ 21612.1 1.12569
$$718$$ 0 0
$$719$$ −16219.3 −0.841276 −0.420638 0.907228i $$-0.638194\pi$$
−0.420638 + 0.907228i $$0.638194\pi$$
$$720$$ 0 0
$$721$$ 546.614 0.0282344
$$722$$ 0 0
$$723$$ 10042.6 0.516583
$$724$$ 0 0
$$725$$ 37517.6 1.92189
$$726$$ 0 0
$$727$$ −34227.8 −1.74613 −0.873067 0.487601i $$-0.837872\pi$$
−0.873067 + 0.487601i $$0.837872\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ 4362.56 0.220732
$$732$$ 0 0
$$733$$ −12844.2 −0.647220 −0.323610 0.946191i $$-0.604896\pi$$
−0.323610 + 0.946191i $$0.604896\pi$$
$$734$$ 0 0
$$735$$ −1153.08 −0.0578669
$$736$$ 0 0
$$737$$ 7356.17 0.367663
$$738$$ 0 0
$$739$$ 11145.6 0.554800 0.277400 0.960755i $$-0.410527\pi$$
0.277400 + 0.960755i $$0.410527\pi$$
$$740$$ 0 0
$$741$$ 26807.6 1.32902
$$742$$ 0 0
$$743$$ −15771.0 −0.778711 −0.389356 0.921088i $$-0.627302\pi$$
−0.389356 + 0.921088i $$0.627302\pi$$
$$744$$ 0 0
$$745$$ 836.884 0.0411558
$$746$$ 0 0
$$747$$ 5634.89 0.275997
$$748$$ 0 0
$$749$$ 873.446 0.0426102
$$750$$ 0 0
$$751$$ 18340.4 0.891146 0.445573 0.895246i $$-0.353000\pi$$
0.445573 + 0.895246i $$0.353000\pi$$
$$752$$ 0 0
$$753$$ −152.602 −0.00738529
$$754$$ 0 0
$$755$$ 2973.45 0.143331
$$756$$ 0 0
$$757$$ −3727.76 −0.178980 −0.0894900 0.995988i $$-0.528524\pi$$
−0.0894900 + 0.995988i $$0.528524\pi$$
$$758$$ 0 0
$$759$$ −3451.81 −0.165076
$$760$$ 0 0
$$761$$ 36537.0 1.74043 0.870214 0.492674i $$-0.163980\pi$$
0.870214 + 0.492674i $$0.163980\pi$$
$$762$$ 0 0
$$763$$ −497.357 −0.0235983
$$764$$ 0 0
$$765$$ 224.557 0.0106129
$$766$$ 0 0
$$767$$ 1056.43 0.0497333
$$768$$ 0 0
$$769$$ −4108.97 −0.192683 −0.0963416 0.995348i $$-0.530714\pi$$
−0.0963416 + 0.995348i $$0.530714\pi$$
$$770$$ 0 0
$$771$$ 12649.6 0.590874
$$772$$ 0 0
$$773$$ 22430.1 1.04367 0.521833 0.853048i $$-0.325248\pi$$
0.521833 + 0.853048i $$0.325248\pi$$
$$774$$ 0 0
$$775$$ −24367.1 −1.12941
$$776$$ 0 0
$$777$$ −652.011 −0.0301039
$$778$$ 0 0
$$779$$ 56368.6 2.59257
$$780$$ 0 0
$$781$$ −1971.69 −0.0903362
$$782$$ 0 0
$$783$$ −8186.42 −0.373638
$$784$$ 0 0
$$785$$ 2849.94 0.129578
$$786$$ 0 0
$$787$$ 12213.2 0.553181 0.276590 0.960988i $$-0.410795\pi$$
0.276590 + 0.960988i $$0.410795\pi$$
$$788$$ 0 0
$$789$$ 6001.14 0.270781
$$790$$ 0 0
$$791$$ 386.398 0.0173688
$$792$$ 0 0
$$793$$ 44474.9 1.99161
$$794$$ 0 0
$$795$$ −129.534 −0.00577873
$$796$$ 0 0
$$797$$ −30552.4 −1.35787 −0.678934 0.734199i $$-0.737558\pi$$
−0.678934 + 0.734199i $$0.737558\pi$$
$$798$$ 0 0
$$799$$ −634.036 −0.0280733
$$800$$ 0 0
$$801$$ −14136.1 −0.623565
$$802$$ 0 0
$$803$$ −11589.3 −0.509313
$$804$$ 0 0
$$805$$ −103.015 −0.00451032
$$806$$ 0 0
$$807$$ 6915.53 0.301658
$$808$$ 0 0
$$809$$ 17537.6 0.762163 0.381081 0.924541i $$-0.375552\pi$$
0.381081 + 0.924541i $$0.375552\pi$$
$$810$$ 0 0
$$811$$ −42391.0 −1.83545 −0.917725 0.397217i $$-0.869976\pi$$
−0.917725 + 0.397217i $$0.869976\pi$$
$$812$$ 0 0
$$813$$ 14004.6 0.604134
$$814$$ 0 0
$$815$$ 95.3108 0.00409643
$$816$$ 0 0
$$817$$ −24682.5 −1.05695
$$818$$ 0 0
$$819$$ 561.068 0.0239381
$$820$$ 0 0
$$821$$ 3922.86 0.166759 0.0833793 0.996518i $$-0.473429\pi$$
0.0833793 + 0.996518i $$0.473429\pi$$
$$822$$ 0 0
$$823$$ −10990.0 −0.465477 −0.232739 0.972539i $$-0.574769\pi$$
−0.232739 + 0.972539i $$0.574769\pi$$
$$824$$ 0 0
$$825$$ 4083.37 0.172321
$$826$$ 0 0
$$827$$ 41723.5 1.75438 0.877188 0.480146i $$-0.159416\pi$$
0.877188 + 0.480146i $$0.159416\pi$$
$$828$$ 0 0
$$829$$ 37425.8 1.56798 0.783989 0.620775i $$-0.213182\pi$$
0.783989 + 0.620775i $$0.213182\pi$$
$$830$$ 0 0
$$831$$ −8801.88 −0.367429
$$832$$ 0 0
$$833$$ −7602.97 −0.316239
$$834$$ 0 0
$$835$$ 4641.83 0.192380
$$836$$ 0 0
$$837$$ 5316.95 0.219571
$$838$$ 0 0
$$839$$ −43490.7 −1.78959 −0.894794 0.446479i $$-0.852678\pi$$
−0.894794 + 0.446479i $$0.852678\pi$$
$$840$$ 0 0
$$841$$ 67541.7 2.76935
$$842$$ 0 0
$$843$$ −13186.2 −0.538739
$$844$$ 0 0
$$845$$ 3208.92 0.130639
$$846$$ 0 0
$$847$$ −106.104 −0.00430435
$$848$$ 0 0
$$849$$ 4581.20 0.185190
$$850$$ 0 0
$$851$$ 25925.0 1.04430
$$852$$ 0 0
$$853$$ 6522.01 0.261793 0.130896 0.991396i $$-0.458214\pi$$
0.130896 + 0.991396i $$0.458214\pi$$
$$854$$ 0 0
$$855$$ −1270.50 −0.0508189
$$856$$ 0 0
$$857$$ −29468.6 −1.17459 −0.587297 0.809371i $$-0.699808\pi$$
−0.587297 + 0.809371i $$0.699808\pi$$
$$858$$ 0 0
$$859$$ 16576.4 0.658416 0.329208 0.944257i $$-0.393218\pi$$
0.329208 + 0.944257i $$0.393218\pi$$
$$860$$ 0 0
$$861$$ 1179.76 0.0466971
$$862$$ 0 0
$$863$$ 36972.2 1.45834 0.729170 0.684332i $$-0.239906\pi$$
0.729170 + 0.684332i $$0.239906\pi$$
$$864$$ 0 0
$$865$$ −1322.22 −0.0519733
$$866$$ 0 0
$$867$$ −13258.4 −0.519351
$$868$$ 0 0
$$869$$ 5048.35 0.197070
$$870$$ 0 0
$$871$$ 47542.8 1.84951
$$872$$ 0 0
$$873$$ 7445.90 0.288666
$$874$$ 0 0
$$875$$ 244.969 0.00946453
$$876$$ 0 0
$$877$$ −44872.7 −1.72776 −0.863878 0.503701i $$-0.831972\pi$$
−0.863878 + 0.503701i $$0.831972\pi$$
$$878$$ 0 0
$$879$$ 13923.4 0.534272
$$880$$ 0 0
$$881$$ −33961.3 −1.29873 −0.649367 0.760475i $$-0.724966\pi$$
−0.649367 + 0.760475i $$0.724966\pi$$
$$882$$ 0 0
$$883$$ 30892.1 1.17735 0.588676 0.808369i $$-0.299650\pi$$
0.588676 + 0.808369i $$0.299650\pi$$
$$884$$ 0 0
$$885$$ −50.0676 −0.00190170
$$886$$ 0 0
$$887$$ 7879.82 0.298284 0.149142 0.988816i $$-0.452349\pi$$
0.149142 + 0.988816i $$0.452349\pi$$
$$888$$ 0 0
$$889$$ 467.413 0.0176339
$$890$$ 0 0
$$891$$ −891.000 −0.0335013
$$892$$ 0 0
$$893$$ 3587.25 0.134426
$$894$$ 0 0
$$895$$ 270.766 0.0101125
$$896$$ 0 0
$$897$$ −22309.0 −0.830408
$$898$$ 0 0
$$899$$ −59707.6 −2.21508
$$900$$ 0 0
$$901$$ −854.092 −0.0315804
$$902$$ 0 0
$$903$$ −516.590 −0.0190377
$$904$$ 0 0
$$905$$ 2387.56 0.0876962
$$906$$ 0 0
$$907$$ 12109.4 0.443316 0.221658 0.975124i $$-0.428853\pi$$
0.221658 + 0.975124i $$0.428853\pi$$
$$908$$ 0 0
$$909$$ 1240.67 0.0452700
$$910$$ 0 0
$$911$$ −27148.2 −0.987334 −0.493667 0.869651i $$-0.664344\pi$$
−0.493667 + 0.869651i $$0.664344\pi$$
$$912$$ 0 0
$$913$$ −6887.08 −0.249649
$$914$$ 0 0
$$915$$ −2107.81 −0.0761552
$$916$$ 0 0
$$917$$ 1739.28 0.0626348
$$918$$ 0 0
$$919$$ −2030.55 −0.0728854 −0.0364427 0.999336i $$-0.511603\pi$$
−0.0364427 + 0.999336i $$0.511603\pi$$
$$920$$ 0 0
$$921$$ −17755.9 −0.635263
$$922$$ 0 0
$$923$$ −12743.0 −0.454432
$$924$$ 0 0
$$925$$ −30668.4 −1.09013
$$926$$ 0 0
$$927$$ −5610.17 −0.198773
$$928$$ 0 0
$$929$$ 16071.8 0.567598 0.283799 0.958884i $$-0.408405\pi$$
0.283799 + 0.958884i $$0.408405\pi$$
$$930$$ 0 0
$$931$$ 43016.1 1.51428
$$932$$ 0 0
$$933$$ −13867.4 −0.486601
$$934$$ 0 0
$$935$$ −274.459 −0.00959975
$$936$$ 0 0
$$937$$ 771.363 0.0268936 0.0134468 0.999910i $$-0.495720\pi$$
0.0134468 + 0.999910i $$0.495720\pi$$
$$938$$ 0 0
$$939$$ 29586.6 1.02824
$$940$$ 0 0
$$941$$ 16905.2 0.585645 0.292823 0.956167i $$-0.405405\pi$$
0.292823 + 0.956167i $$0.405405\pi$$
$$942$$ 0 0
$$943$$ −46909.3 −1.61991
$$944$$ 0 0
$$945$$ −26.5908 −0.000915343 0
$$946$$ 0 0
$$947$$ −32050.0 −1.09977 −0.549886 0.835240i $$-0.685329\pi$$
−0.549886 + 0.835240i $$0.685329\pi$$
$$948$$ 0 0
$$949$$ −74901.6 −2.56208
$$950$$ 0 0
$$951$$ 20393.0 0.695362
$$952$$ 0 0
$$953$$ 1653.20 0.0561934 0.0280967 0.999605i $$-0.491055\pi$$
0.0280967 + 0.999605i $$0.491055\pi$$
$$954$$ 0 0
$$955$$ −3760.45 −0.127419
$$956$$ 0 0
$$957$$ 10005.6 0.337969
$$958$$ 0 0
$$959$$ 1540.48 0.0518714
$$960$$ 0 0
$$961$$ 8988.15 0.301707
$$962$$ 0 0
$$963$$ −8964.61 −0.299980
$$964$$ 0 0
$$965$$ −1614.59 −0.0538605
$$966$$ 0 0
$$967$$ −47148.3 −1.56793 −0.783964 0.620807i $$-0.786805\pi$$
−0.783964 + 0.620807i $$0.786805\pi$$
$$968$$ 0 0
$$969$$ −8377.16 −0.277723
$$970$$ 0 0
$$971$$ 47880.7 1.58245 0.791227 0.611522i $$-0.209443\pi$$
0.791227 + 0.611522i $$0.209443\pi$$
$$972$$ 0 0
$$973$$ 1274.46 0.0419912
$$974$$ 0 0
$$975$$ 26390.8 0.866853
$$976$$ 0 0
$$977$$ −42111.0 −1.37897 −0.689483 0.724302i $$-0.742162\pi$$
−0.689483 + 0.724302i $$0.742162\pi$$
$$978$$ 0 0
$$979$$ 17277.5 0.564036
$$980$$ 0 0
$$981$$ 5104.62 0.166135
$$982$$ 0 0
$$983$$ 1243.50 0.0403474 0.0201737 0.999796i $$-0.493578\pi$$
0.0201737 + 0.999796i $$0.493578\pi$$
$$984$$ 0 0
$$985$$ −3679.39 −0.119020
$$986$$ 0 0
$$987$$ 75.0790 0.00242127
$$988$$ 0 0
$$989$$ 20540.5 0.660414
$$990$$ 0 0
$$991$$ −3354.93 −0.107541 −0.0537703 0.998553i $$-0.517124\pi$$
−0.0537703 + 0.998553i $$0.517124\pi$$
$$992$$ 0 0
$$993$$ 17542.7 0.560624
$$994$$ 0 0
$$995$$ 5031.07 0.160297
$$996$$ 0 0
$$997$$ −15908.8 −0.505353 −0.252677 0.967551i $$-0.581311\pi$$
−0.252677 + 0.967551i $$0.581311\pi$$
$$998$$ 0 0
$$999$$ 6691.91 0.211935
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 528.4.a.r.1.2 2
3.2 odd 2 1584.4.a.bf.1.1 2
4.3 odd 2 264.4.a.e.1.2 2
8.3 odd 2 2112.4.a.bl.1.1 2
8.5 even 2 2112.4.a.be.1.1 2
12.11 even 2 792.4.a.h.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
264.4.a.e.1.2 2 4.3 odd 2
528.4.a.r.1.2 2 1.1 even 1 trivial
792.4.a.h.1.1 2 12.11 even 2
1584.4.a.bf.1.1 2 3.2 odd 2
2112.4.a.be.1.1 2 8.5 even 2
2112.4.a.bl.1.1 2 8.3 odd 2