Properties

 Label 2496.4.a.be.1.2 Level $2496$ Weight $4$ Character 2496.1 Self dual yes Analytic conductor $147.269$ 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] = [2496,4,Mod(1,2496)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2496, 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("2496.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2496 = 2^{6} \cdot 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2496.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: $$147.268767374$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{113})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 28$$ x^2 - x - 28 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 312) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.2 Root $$-4.81507$$ of defining polynomial Character $$\chi$$ $$=$$ 2496.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.00000 q^{3} +7.63015 q^{5} +5.63015 q^{7} +9.00000 q^{9} +O(q^{10})$$ $$q+3.00000 q^{3} +7.63015 q^{5} +5.63015 q^{7} +9.00000 q^{9} -34.5206 q^{11} -13.0000 q^{13} +22.8904 q^{15} +2.00000 q^{17} +88.1507 q^{19} +16.8904 q^{21} -64.0000 q^{23} -66.7809 q^{25} +27.0000 q^{27} -23.7809 q^{29} -284.452 q^{31} -103.562 q^{33} +42.9588 q^{35} -115.343 q^{37} -39.0000 q^{39} +1.41102 q^{41} +337.041 q^{43} +68.6713 q^{45} -198.219 q^{47} -311.301 q^{49} +6.00000 q^{51} -59.0412 q^{53} -263.397 q^{55} +264.452 q^{57} +188.301 q^{59} -336.987 q^{61} +50.6713 q^{63} -99.1919 q^{65} +531.411 q^{67} -192.000 q^{69} -510.247 q^{71} -164.219 q^{73} -200.343 q^{75} -194.356 q^{77} -29.3148 q^{79} +81.0000 q^{81} +117.507 q^{83} +15.2603 q^{85} -71.3426 q^{87} +508.671 q^{89} -73.1919 q^{91} -853.357 q^{93} +672.603 q^{95} -1020.88 q^{97} -310.685 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{3} - 6 q^{5} - 10 q^{7} + 18 q^{9}+O(q^{10})$$ 2 * q + 6 * q^3 - 6 * q^5 - 10 * q^7 + 18 * q^9 $$2 q + 6 q^{3} - 6 q^{5} - 10 q^{7} + 18 q^{9} + 16 q^{11} - 26 q^{13} - 18 q^{15} + 4 q^{17} + 70 q^{19} - 30 q^{21} - 128 q^{23} - 6 q^{25} + 54 q^{27} + 80 q^{29} - 250 q^{31} + 48 q^{33} + 256 q^{35} + 152 q^{37} - 78 q^{39} - 146 q^{41} + 504 q^{43} - 54 q^{45} - 524 q^{47} - 410 q^{49} + 12 q^{51} + 52 q^{53} - 952 q^{55} + 210 q^{57} + 164 q^{59} + 304 q^{61} - 90 q^{63} + 78 q^{65} + 914 q^{67} - 384 q^{69} - 456 q^{73} - 18 q^{75} - 984 q^{77} - 824 q^{79} + 162 q^{81} - 828 q^{83} - 12 q^{85} + 240 q^{87} + 826 q^{89} + 130 q^{91} - 750 q^{93} + 920 q^{95} + 552 q^{97} + 144 q^{99}+O(q^{100})$$ 2 * q + 6 * q^3 - 6 * q^5 - 10 * q^7 + 18 * q^9 + 16 * q^11 - 26 * q^13 - 18 * q^15 + 4 * q^17 + 70 * q^19 - 30 * q^21 - 128 * q^23 - 6 * q^25 + 54 * q^27 + 80 * q^29 - 250 * q^31 + 48 * q^33 + 256 * q^35 + 152 * q^37 - 78 * q^39 - 146 * q^41 + 504 * q^43 - 54 * q^45 - 524 * q^47 - 410 * q^49 + 12 * q^51 + 52 * q^53 - 952 * q^55 + 210 * q^57 + 164 * q^59 + 304 * q^61 - 90 * q^63 + 78 * q^65 + 914 * q^67 - 384 * q^69 - 456 * q^73 - 18 * q^75 - 984 * q^77 - 824 * q^79 + 162 * q^81 - 828 * q^83 - 12 * q^85 + 240 * q^87 + 826 * q^89 + 130 * q^91 - 750 * q^93 + 920 * q^95 + 552 * q^97 + 144 * 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
$$5$$ 7.63015 0.682461 0.341230 0.939980i $$-0.389156\pi$$
0.341230 + 0.939980i $$0.389156\pi$$
$$6$$ 0 0
$$7$$ 5.63015 0.303999 0.152000 0.988381i $$-0.451429\pi$$
0.152000 + 0.988381i $$0.451429\pi$$
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ −34.5206 −0.946213 −0.473107 0.881005i $$-0.656867\pi$$
−0.473107 + 0.881005i $$0.656867\pi$$
$$12$$ 0 0
$$13$$ −13.0000 −0.277350
$$14$$ 0 0
$$15$$ 22.8904 0.394019
$$16$$ 0 0
$$17$$ 2.00000 0.0285336 0.0142668 0.999898i $$-0.495459\pi$$
0.0142668 + 0.999898i $$0.495459\pi$$
$$18$$ 0 0
$$19$$ 88.1507 1.06438 0.532189 0.846626i $$-0.321370\pi$$
0.532189 + 0.846626i $$0.321370\pi$$
$$20$$ 0 0
$$21$$ 16.8904 0.175514
$$22$$ 0 0
$$23$$ −64.0000 −0.580214 −0.290107 0.956994i $$-0.593691\pi$$
−0.290107 + 0.956994i $$0.593691\pi$$
$$24$$ 0 0
$$25$$ −66.7809 −0.534247
$$26$$ 0 0
$$27$$ 27.0000 0.192450
$$28$$ 0 0
$$29$$ −23.7809 −0.152276 −0.0761379 0.997097i $$-0.524259\pi$$
−0.0761379 + 0.997097i $$0.524259\pi$$
$$30$$ 0 0
$$31$$ −284.452 −1.64804 −0.824018 0.566564i $$-0.808273\pi$$
−0.824018 + 0.566564i $$0.808273\pi$$
$$32$$ 0 0
$$33$$ −103.562 −0.546297
$$34$$ 0 0
$$35$$ 42.9588 0.207468
$$36$$ 0 0
$$37$$ −115.343 −0.512492 −0.256246 0.966612i $$-0.582486\pi$$
−0.256246 + 0.966612i $$0.582486\pi$$
$$38$$ 0 0
$$39$$ −39.0000 −0.160128
$$40$$ 0 0
$$41$$ 1.41102 0.00537474 0.00268737 0.999996i $$-0.499145\pi$$
0.00268737 + 0.999996i $$0.499145\pi$$
$$42$$ 0 0
$$43$$ 337.041 1.19531 0.597655 0.801754i $$-0.296099\pi$$
0.597655 + 0.801754i $$0.296099\pi$$
$$44$$ 0 0
$$45$$ 68.6713 0.227487
$$46$$ 0 0
$$47$$ −198.219 −0.615175 −0.307588 0.951520i $$-0.599522\pi$$
−0.307588 + 0.951520i $$0.599522\pi$$
$$48$$ 0 0
$$49$$ −311.301 −0.907584
$$50$$ 0 0
$$51$$ 6.00000 0.0164739
$$52$$ 0 0
$$53$$ −59.0412 −0.153018 −0.0765088 0.997069i $$-0.524377\pi$$
−0.0765088 + 0.997069i $$0.524377\pi$$
$$54$$ 0 0
$$55$$ −263.397 −0.645754
$$56$$ 0 0
$$57$$ 264.452 0.614518
$$58$$ 0 0
$$59$$ 188.301 0.415504 0.207752 0.978181i $$-0.433385\pi$$
0.207752 + 0.978181i $$0.433385\pi$$
$$60$$ 0 0
$$61$$ −336.987 −0.707323 −0.353662 0.935373i $$-0.615064\pi$$
−0.353662 + 0.935373i $$0.615064\pi$$
$$62$$ 0 0
$$63$$ 50.6713 0.101333
$$64$$ 0 0
$$65$$ −99.1919 −0.189281
$$66$$ 0 0
$$67$$ 531.411 0.968988 0.484494 0.874795i $$-0.339004\pi$$
0.484494 + 0.874795i $$0.339004\pi$$
$$68$$ 0 0
$$69$$ −192.000 −0.334987
$$70$$ 0 0
$$71$$ −510.247 −0.852890 −0.426445 0.904514i $$-0.640234\pi$$
−0.426445 + 0.904514i $$0.640234\pi$$
$$72$$ 0 0
$$73$$ −164.219 −0.263293 −0.131647 0.991297i $$-0.542026\pi$$
−0.131647 + 0.991297i $$0.542026\pi$$
$$74$$ 0 0
$$75$$ −200.343 −0.308448
$$76$$ 0 0
$$77$$ −194.356 −0.287648
$$78$$ 0 0
$$79$$ −29.3148 −0.0417490 −0.0208745 0.999782i $$-0.506645\pi$$
−0.0208745 + 0.999782i $$0.506645\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ 117.507 0.155399 0.0776994 0.996977i $$-0.475243\pi$$
0.0776994 + 0.996977i $$0.475243\pi$$
$$84$$ 0 0
$$85$$ 15.2603 0.0194731
$$86$$ 0 0
$$87$$ −71.3426 −0.0879165
$$88$$ 0 0
$$89$$ 508.671 0.605832 0.302916 0.953017i $$-0.402040\pi$$
0.302916 + 0.953017i $$0.402040\pi$$
$$90$$ 0 0
$$91$$ −73.1919 −0.0843142
$$92$$ 0 0
$$93$$ −853.357 −0.951494
$$94$$ 0 0
$$95$$ 672.603 0.726396
$$96$$ 0 0
$$97$$ −1020.88 −1.06860 −0.534301 0.845294i $$-0.679425\pi$$
−0.534301 + 0.845294i $$0.679425\pi$$
$$98$$ 0 0
$$99$$ −310.685 −0.315404
$$100$$ 0 0
$$101$$ 1028.44 1.01320 0.506602 0.862180i $$-0.330901\pi$$
0.506602 + 0.862180i $$0.330901\pi$$
$$102$$ 0 0
$$103$$ −1267.01 −1.21206 −0.606032 0.795440i $$-0.707240\pi$$
−0.606032 + 0.795440i $$0.707240\pi$$
$$104$$ 0 0
$$105$$ 128.877 0.119782
$$106$$ 0 0
$$107$$ −1934.74 −1.74802 −0.874012 0.485905i $$-0.838490\pi$$
−0.874012 + 0.485905i $$0.838490\pi$$
$$108$$ 0 0
$$109$$ −2038.66 −1.79145 −0.895725 0.444608i $$-0.853343\pi$$
−0.895725 + 0.444608i $$0.853343\pi$$
$$110$$ 0 0
$$111$$ −346.028 −0.295887
$$112$$ 0 0
$$113$$ −35.9455 −0.0299245 −0.0149623 0.999888i $$-0.504763\pi$$
−0.0149623 + 0.999888i $$0.504763\pi$$
$$114$$ 0 0
$$115$$ −488.329 −0.395973
$$116$$ 0 0
$$117$$ −117.000 −0.0924500
$$118$$ 0 0
$$119$$ 11.2603 0.00867420
$$120$$ 0 0
$$121$$ −139.329 −0.104680
$$122$$ 0 0
$$123$$ 4.23306 0.00310311
$$124$$ 0 0
$$125$$ −1463.32 −1.04706
$$126$$ 0 0
$$127$$ −1205.01 −0.841950 −0.420975 0.907072i $$-0.638312\pi$$
−0.420975 + 0.907072i $$0.638312\pi$$
$$128$$ 0 0
$$129$$ 1011.12 0.690112
$$130$$ 0 0
$$131$$ −1222.08 −0.815067 −0.407534 0.913190i $$-0.633611\pi$$
−0.407534 + 0.913190i $$0.633611\pi$$
$$132$$ 0 0
$$133$$ 496.301 0.323570
$$134$$ 0 0
$$135$$ 206.014 0.131340
$$136$$ 0 0
$$137$$ 1466.40 0.914474 0.457237 0.889345i $$-0.348839\pi$$
0.457237 + 0.889345i $$0.348839\pi$$
$$138$$ 0 0
$$139$$ 1225.40 0.747748 0.373874 0.927480i $$-0.378029\pi$$
0.373874 + 0.927480i $$0.378029\pi$$
$$140$$ 0 0
$$141$$ −594.657 −0.355172
$$142$$ 0 0
$$143$$ 448.768 0.262432
$$144$$ 0 0
$$145$$ −181.452 −0.103922
$$146$$ 0 0
$$147$$ −933.904 −0.523994
$$148$$ 0 0
$$149$$ −1601.88 −0.880744 −0.440372 0.897815i $$-0.645153\pi$$
−0.440372 + 0.897815i $$0.645153\pi$$
$$150$$ 0 0
$$151$$ 588.234 0.317019 0.158509 0.987357i $$-0.449331\pi$$
0.158509 + 0.987357i $$0.449331\pi$$
$$152$$ 0 0
$$153$$ 18.0000 0.00951120
$$154$$ 0 0
$$155$$ −2170.41 −1.12472
$$156$$ 0 0
$$157$$ 925.919 0.470678 0.235339 0.971913i $$-0.424380\pi$$
0.235339 + 0.971913i $$0.424380\pi$$
$$158$$ 0 0
$$159$$ −177.123 −0.0883447
$$160$$ 0 0
$$161$$ −360.329 −0.176385
$$162$$ 0 0
$$163$$ 2947.52 1.41637 0.708183 0.706029i $$-0.249515\pi$$
0.708183 + 0.706029i $$0.249515\pi$$
$$164$$ 0 0
$$165$$ −790.191 −0.372826
$$166$$ 0 0
$$167$$ −715.372 −0.331480 −0.165740 0.986169i $$-0.553001\pi$$
−0.165740 + 0.986169i $$0.553001\pi$$
$$168$$ 0 0
$$169$$ 169.000 0.0769231
$$170$$ 0 0
$$171$$ 793.357 0.354792
$$172$$ 0 0
$$173$$ −335.396 −0.147397 −0.0736985 0.997281i $$-0.523480\pi$$
−0.0736985 + 0.997281i $$0.523480\pi$$
$$174$$ 0 0
$$175$$ −375.986 −0.162411
$$176$$ 0 0
$$177$$ 564.904 0.239892
$$178$$ 0 0
$$179$$ 3549.04 1.48194 0.740972 0.671536i $$-0.234365\pi$$
0.740972 + 0.671536i $$0.234365\pi$$
$$180$$ 0 0
$$181$$ −1169.26 −0.480168 −0.240084 0.970752i $$-0.577175\pi$$
−0.240084 + 0.970752i $$0.577175\pi$$
$$182$$ 0 0
$$183$$ −1010.96 −0.408373
$$184$$ 0 0
$$185$$ −880.081 −0.349756
$$186$$ 0 0
$$187$$ −69.0412 −0.0269989
$$188$$ 0 0
$$189$$ 152.014 0.0585047
$$190$$ 0 0
$$191$$ 4233.86 1.60394 0.801968 0.597367i $$-0.203787\pi$$
0.801968 + 0.597367i $$0.203787\pi$$
$$192$$ 0 0
$$193$$ 750.494 0.279905 0.139953 0.990158i $$-0.455305\pi$$
0.139953 + 0.990158i $$0.455305\pi$$
$$194$$ 0 0
$$195$$ −297.576 −0.109281
$$196$$ 0 0
$$197$$ 745.986 0.269793 0.134897 0.990860i $$-0.456930\pi$$
0.134897 + 0.990860i $$0.456930\pi$$
$$198$$ 0 0
$$199$$ 2347.97 0.836400 0.418200 0.908355i $$-0.362661\pi$$
0.418200 + 0.908355i $$0.362661\pi$$
$$200$$ 0 0
$$201$$ 1594.23 0.559445
$$202$$ 0 0
$$203$$ −133.890 −0.0462917
$$204$$ 0 0
$$205$$ 10.7663 0.00366805
$$206$$ 0 0
$$207$$ −576.000 −0.193405
$$208$$ 0 0
$$209$$ −3043.01 −1.00713
$$210$$ 0 0
$$211$$ 684.056 0.223186 0.111593 0.993754i $$-0.464405\pi$$
0.111593 + 0.993754i $$0.464405\pi$$
$$212$$ 0 0
$$213$$ −1530.74 −0.492416
$$214$$ 0 0
$$215$$ 2571.67 0.815752
$$216$$ 0 0
$$217$$ −1601.51 −0.501002
$$218$$ 0 0
$$219$$ −492.657 −0.152012
$$220$$ 0 0
$$221$$ −26.0000 −0.00791380
$$222$$ 0 0
$$223$$ −1583.90 −0.475632 −0.237816 0.971310i $$-0.576432\pi$$
−0.237816 + 0.971310i $$0.576432\pi$$
$$224$$ 0 0
$$225$$ −601.028 −0.178082
$$226$$ 0 0
$$227$$ −3675.23 −1.07460 −0.537299 0.843392i $$-0.680555\pi$$
−0.537299 + 0.843392i $$0.680555\pi$$
$$228$$ 0 0
$$229$$ 2742.33 0.791347 0.395673 0.918391i $$-0.370511\pi$$
0.395673 + 0.918391i $$0.370511\pi$$
$$230$$ 0 0
$$231$$ −583.068 −0.166074
$$232$$ 0 0
$$233$$ −2459.83 −0.691627 −0.345813 0.938303i $$-0.612397\pi$$
−0.345813 + 0.938303i $$0.612397\pi$$
$$234$$ 0 0
$$235$$ −1512.44 −0.419833
$$236$$ 0 0
$$237$$ −87.9443 −0.0241038
$$238$$ 0 0
$$239$$ −334.688 −0.0905822 −0.0452911 0.998974i $$-0.514422\pi$$
−0.0452911 + 0.998974i $$0.514422\pi$$
$$240$$ 0 0
$$241$$ −2697.81 −0.721083 −0.360542 0.932743i $$-0.617408\pi$$
−0.360542 + 0.932743i $$0.617408\pi$$
$$242$$ 0 0
$$243$$ 243.000 0.0641500
$$244$$ 0 0
$$245$$ −2375.28 −0.619391
$$246$$ 0 0
$$247$$ −1145.96 −0.295205
$$248$$ 0 0
$$249$$ 352.522 0.0897195
$$250$$ 0 0
$$251$$ −2665.64 −0.670335 −0.335167 0.942159i $$-0.608793\pi$$
−0.335167 + 0.942159i $$0.608793\pi$$
$$252$$ 0 0
$$253$$ 2209.32 0.549006
$$254$$ 0 0
$$255$$ 45.7809 0.0112428
$$256$$ 0 0
$$257$$ −6877.42 −1.66927 −0.834634 0.550805i $$-0.814321\pi$$
−0.834634 + 0.550805i $$0.814321\pi$$
$$258$$ 0 0
$$259$$ −649.396 −0.155797
$$260$$ 0 0
$$261$$ −214.028 −0.0507586
$$262$$ 0 0
$$263$$ −55.5086 −0.0130145 −0.00650724 0.999979i $$-0.502071\pi$$
−0.00650724 + 0.999979i $$0.502071\pi$$
$$264$$ 0 0
$$265$$ −450.493 −0.104428
$$266$$ 0 0
$$267$$ 1526.01 0.349777
$$268$$ 0 0
$$269$$ 3726.85 0.844722 0.422361 0.906428i $$-0.361202\pi$$
0.422361 + 0.906428i $$0.361202\pi$$
$$270$$ 0 0
$$271$$ 1536.59 0.344432 0.172216 0.985059i $$-0.444907\pi$$
0.172216 + 0.985059i $$0.444907\pi$$
$$272$$ 0 0
$$273$$ −219.576 −0.0486788
$$274$$ 0 0
$$275$$ 2305.31 0.505512
$$276$$ 0 0
$$277$$ −4917.23 −1.06660 −0.533300 0.845926i $$-0.679048\pi$$
−0.533300 + 0.845926i $$0.679048\pi$$
$$278$$ 0 0
$$279$$ −2560.07 −0.549345
$$280$$ 0 0
$$281$$ −5281.66 −1.12127 −0.560636 0.828062i $$-0.689443\pi$$
−0.560636 + 0.828062i $$0.689443\pi$$
$$282$$ 0 0
$$283$$ 5091.37 1.06944 0.534718 0.845030i $$-0.320418\pi$$
0.534718 + 0.845030i $$0.320418\pi$$
$$284$$ 0 0
$$285$$ 2017.81 0.419385
$$286$$ 0 0
$$287$$ 7.94425 0.00163392
$$288$$ 0 0
$$289$$ −4909.00 −0.999186
$$290$$ 0 0
$$291$$ −3062.63 −0.616958
$$292$$ 0 0
$$293$$ 4275.98 0.852579 0.426290 0.904587i $$-0.359820\pi$$
0.426290 + 0.904587i $$0.359820\pi$$
$$294$$ 0 0
$$295$$ 1436.77 0.283566
$$296$$ 0 0
$$297$$ −932.056 −0.182099
$$298$$ 0 0
$$299$$ 832.000 0.160922
$$300$$ 0 0
$$301$$ 1897.59 0.363373
$$302$$ 0 0
$$303$$ 3085.32 0.584973
$$304$$ 0 0
$$305$$ −2571.26 −0.482721
$$306$$ 0 0
$$307$$ −306.312 −0.0569450 −0.0284725 0.999595i $$-0.509064\pi$$
−0.0284725 + 0.999595i $$0.509064\pi$$
$$308$$ 0 0
$$309$$ −3801.04 −0.699786
$$310$$ 0 0
$$311$$ −3267.24 −0.595717 −0.297859 0.954610i $$-0.596272\pi$$
−0.297859 + 0.954610i $$0.596272\pi$$
$$312$$ 0 0
$$313$$ −2456.19 −0.443553 −0.221776 0.975098i $$-0.571185\pi$$
−0.221776 + 0.975098i $$0.571185\pi$$
$$314$$ 0 0
$$315$$ 386.630 0.0691559
$$316$$ 0 0
$$317$$ 1034.48 0.183288 0.0916441 0.995792i $$-0.470788\pi$$
0.0916441 + 0.995792i $$0.470788\pi$$
$$318$$ 0 0
$$319$$ 820.930 0.144085
$$320$$ 0 0
$$321$$ −5804.22 −1.00922
$$322$$ 0 0
$$323$$ 176.301 0.0303705
$$324$$ 0 0
$$325$$ 868.151 0.148173
$$326$$ 0 0
$$327$$ −6115.98 −1.03429
$$328$$ 0 0
$$329$$ −1116.00 −0.187013
$$330$$ 0 0
$$331$$ −4221.63 −0.701033 −0.350516 0.936557i $$-0.613994\pi$$
−0.350516 + 0.936557i $$0.613994\pi$$
$$332$$ 0 0
$$333$$ −1038.08 −0.170831
$$334$$ 0 0
$$335$$ 4054.74 0.661296
$$336$$ 0 0
$$337$$ 5043.43 0.815232 0.407616 0.913154i $$-0.366360\pi$$
0.407616 + 0.913154i $$0.366360\pi$$
$$338$$ 0 0
$$339$$ −107.837 −0.0172769
$$340$$ 0 0
$$341$$ 9819.46 1.55939
$$342$$ 0 0
$$343$$ −3683.81 −0.579904
$$344$$ 0 0
$$345$$ −1464.99 −0.228615
$$346$$ 0 0
$$347$$ −6689.76 −1.03494 −0.517471 0.855700i $$-0.673127\pi$$
−0.517471 + 0.855700i $$0.673127\pi$$
$$348$$ 0 0
$$349$$ −12252.0 −1.87918 −0.939591 0.342299i $$-0.888794\pi$$
−0.939591 + 0.342299i $$0.888794\pi$$
$$350$$ 0 0
$$351$$ −351.000 −0.0533761
$$352$$ 0 0
$$353$$ −11099.6 −1.67358 −0.836790 0.547523i $$-0.815571\pi$$
−0.836790 + 0.547523i $$0.815571\pi$$
$$354$$ 0 0
$$355$$ −3893.26 −0.582064
$$356$$ 0 0
$$357$$ 33.7809 0.00500805
$$358$$ 0 0
$$359$$ 9354.11 1.37518 0.687592 0.726097i $$-0.258668\pi$$
0.687592 + 0.726097i $$0.258668\pi$$
$$360$$ 0 0
$$361$$ 911.551 0.132899
$$362$$ 0 0
$$363$$ −417.988 −0.0604371
$$364$$ 0 0
$$365$$ −1253.02 −0.179687
$$366$$ 0 0
$$367$$ 1468.14 0.208818 0.104409 0.994534i $$-0.466705\pi$$
0.104409 + 0.994534i $$0.466705\pi$$
$$368$$ 0 0
$$369$$ 12.6992 0.00179158
$$370$$ 0 0
$$371$$ −332.410 −0.0465172
$$372$$ 0 0
$$373$$ −6216.36 −0.862925 −0.431462 0.902131i $$-0.642002\pi$$
−0.431462 + 0.902131i $$0.642002\pi$$
$$374$$ 0 0
$$375$$ −4389.95 −0.604523
$$376$$ 0 0
$$377$$ 309.151 0.0422337
$$378$$ 0 0
$$379$$ 1365.52 0.185072 0.0925358 0.995709i $$-0.470503\pi$$
0.0925358 + 0.995709i $$0.470503\pi$$
$$380$$ 0 0
$$381$$ −3615.04 −0.486100
$$382$$ 0 0
$$383$$ 8311.16 1.10883 0.554413 0.832242i $$-0.312943\pi$$
0.554413 + 0.832242i $$0.312943\pi$$
$$384$$ 0 0
$$385$$ −1482.96 −0.196309
$$386$$ 0 0
$$387$$ 3033.37 0.398436
$$388$$ 0 0
$$389$$ −6160.09 −0.802902 −0.401451 0.915881i $$-0.631494\pi$$
−0.401451 + 0.915881i $$0.631494\pi$$
$$390$$ 0 0
$$391$$ −128.000 −0.0165556
$$392$$ 0 0
$$393$$ −3666.25 −0.470579
$$394$$ 0 0
$$395$$ −223.676 −0.0284920
$$396$$ 0 0
$$397$$ −12658.6 −1.60029 −0.800146 0.599805i $$-0.795245\pi$$
−0.800146 + 0.599805i $$0.795245\pi$$
$$398$$ 0 0
$$399$$ 1488.90 0.186813
$$400$$ 0 0
$$401$$ 13609.8 1.69487 0.847434 0.530900i $$-0.178146\pi$$
0.847434 + 0.530900i $$0.178146\pi$$
$$402$$ 0 0
$$403$$ 3697.88 0.457083
$$404$$ 0 0
$$405$$ 618.042 0.0758290
$$406$$ 0 0
$$407$$ 3981.69 0.484927
$$408$$ 0 0
$$409$$ 5286.90 0.639170 0.319585 0.947558i $$-0.396457\pi$$
0.319585 + 0.947558i $$0.396457\pi$$
$$410$$ 0 0
$$411$$ 4399.20 0.527972
$$412$$ 0 0
$$413$$ 1060.16 0.126313
$$414$$ 0 0
$$415$$ 896.598 0.106054
$$416$$ 0 0
$$417$$ 3676.20 0.431712
$$418$$ 0 0
$$419$$ 4058.86 0.473241 0.236621 0.971602i $$-0.423960\pi$$
0.236621 + 0.971602i $$0.423960\pi$$
$$420$$ 0 0
$$421$$ −11383.0 −1.31776 −0.658878 0.752249i $$-0.728969\pi$$
−0.658878 + 0.752249i $$0.728969\pi$$
$$422$$ 0 0
$$423$$ −1783.97 −0.205058
$$424$$ 0 0
$$425$$ −133.562 −0.0152440
$$426$$ 0 0
$$427$$ −1897.28 −0.215026
$$428$$ 0 0
$$429$$ 1346.30 0.151515
$$430$$ 0 0
$$431$$ −4896.19 −0.547195 −0.273598 0.961844i $$-0.588214\pi$$
−0.273598 + 0.961844i $$0.588214\pi$$
$$432$$ 0 0
$$433$$ 6775.12 0.751944 0.375972 0.926631i $$-0.377309\pi$$
0.375972 + 0.926631i $$0.377309\pi$$
$$434$$ 0 0
$$435$$ −544.355 −0.0599996
$$436$$ 0 0
$$437$$ −5641.65 −0.617566
$$438$$ 0 0
$$439$$ 4163.40 0.452639 0.226319 0.974053i $$-0.427331\pi$$
0.226319 + 0.974053i $$0.427331\pi$$
$$440$$ 0 0
$$441$$ −2801.71 −0.302528
$$442$$ 0 0
$$443$$ 11825.0 1.26823 0.634114 0.773240i $$-0.281365\pi$$
0.634114 + 0.773240i $$0.281365\pi$$
$$444$$ 0 0
$$445$$ 3881.24 0.413457
$$446$$ 0 0
$$447$$ −4805.63 −0.508498
$$448$$ 0 0
$$449$$ −5455.22 −0.573381 −0.286690 0.958023i $$-0.592555\pi$$
−0.286690 + 0.958023i $$0.592555\pi$$
$$450$$ 0 0
$$451$$ −48.7093 −0.00508565
$$452$$ 0 0
$$453$$ 1764.70 0.183031
$$454$$ 0 0
$$455$$ −558.465 −0.0575412
$$456$$ 0 0
$$457$$ −4052.72 −0.414832 −0.207416 0.978253i $$-0.566505\pi$$
−0.207416 + 0.978253i $$0.566505\pi$$
$$458$$ 0 0
$$459$$ 54.0000 0.00549129
$$460$$ 0 0
$$461$$ −17679.6 −1.78616 −0.893079 0.449900i $$-0.851460\pi$$
−0.893079 + 0.449900i $$0.851460\pi$$
$$462$$ 0 0
$$463$$ 19202.9 1.92750 0.963752 0.266799i $$-0.0859661\pi$$
0.963752 + 0.266799i $$0.0859661\pi$$
$$464$$ 0 0
$$465$$ −6511.23 −0.649358
$$466$$ 0 0
$$467$$ 489.310 0.0484851 0.0242426 0.999706i $$-0.492283\pi$$
0.0242426 + 0.999706i $$0.492283\pi$$
$$468$$ 0 0
$$469$$ 2991.92 0.294572
$$470$$ 0 0
$$471$$ 2777.76 0.271746
$$472$$ 0 0
$$473$$ −11634.9 −1.13102
$$474$$ 0 0
$$475$$ −5886.78 −0.568640
$$476$$ 0 0
$$477$$ −531.370 −0.0510058
$$478$$ 0 0
$$479$$ 8629.64 0.823170 0.411585 0.911371i $$-0.364975\pi$$
0.411585 + 0.911371i $$0.364975\pi$$
$$480$$ 0 0
$$481$$ 1499.45 0.142140
$$482$$ 0 0
$$483$$ −1080.99 −0.101836
$$484$$ 0 0
$$485$$ −7789.45 −0.729279
$$486$$ 0 0
$$487$$ −2511.57 −0.233697 −0.116848 0.993150i $$-0.537279\pi$$
−0.116848 + 0.993150i $$0.537279\pi$$
$$488$$ 0 0
$$489$$ 8842.57 0.817740
$$490$$ 0 0
$$491$$ 19977.9 1.83623 0.918115 0.396314i $$-0.129711\pi$$
0.918115 + 0.396314i $$0.129711\pi$$
$$492$$ 0 0
$$493$$ −47.5617 −0.00434498
$$494$$ 0 0
$$495$$ −2370.57 −0.215251
$$496$$ 0 0
$$497$$ −2872.77 −0.259278
$$498$$ 0 0
$$499$$ −8946.90 −0.802642 −0.401321 0.915938i $$-0.631449\pi$$
−0.401321 + 0.915938i $$0.631449\pi$$
$$500$$ 0 0
$$501$$ −2146.12 −0.191380
$$502$$ 0 0
$$503$$ −4061.20 −0.360000 −0.180000 0.983667i $$-0.557610\pi$$
−0.180000 + 0.983667i $$0.557610\pi$$
$$504$$ 0 0
$$505$$ 7847.14 0.691472
$$506$$ 0 0
$$507$$ 507.000 0.0444116
$$508$$ 0 0
$$509$$ −16439.6 −1.43157 −0.715787 0.698319i $$-0.753932\pi$$
−0.715787 + 0.698319i $$0.753932\pi$$
$$510$$ 0 0
$$511$$ −924.578 −0.0800409
$$512$$ 0 0
$$513$$ 2380.07 0.204839
$$514$$ 0 0
$$515$$ −9667.51 −0.827187
$$516$$ 0 0
$$517$$ 6842.64 0.582087
$$518$$ 0 0
$$519$$ −1006.19 −0.0850997
$$520$$ 0 0
$$521$$ −19585.2 −1.64692 −0.823459 0.567376i $$-0.807958\pi$$
−0.823459 + 0.567376i $$0.807958\pi$$
$$522$$ 0 0
$$523$$ −1682.34 −0.140657 −0.0703283 0.997524i $$-0.522405\pi$$
−0.0703283 + 0.997524i $$0.522405\pi$$
$$524$$ 0 0
$$525$$ −1127.96 −0.0937679
$$526$$ 0 0
$$527$$ −568.904 −0.0470244
$$528$$ 0 0
$$529$$ −8071.00 −0.663352
$$530$$ 0 0
$$531$$ 1694.71 0.138501
$$532$$ 0 0
$$533$$ −18.3433 −0.00149069
$$534$$ 0 0
$$535$$ −14762.4 −1.19296
$$536$$ 0 0
$$537$$ 10647.1 0.855601
$$538$$ 0 0
$$539$$ 10746.3 0.858769
$$540$$ 0 0
$$541$$ −13519.3 −1.07438 −0.537189 0.843462i $$-0.680514\pi$$
−0.537189 + 0.843462i $$0.680514\pi$$
$$542$$ 0 0
$$543$$ −3507.78 −0.277225
$$544$$ 0 0
$$545$$ −15555.3 −1.22259
$$546$$ 0 0
$$547$$ 747.313 0.0584147 0.0292073 0.999573i $$-0.490702\pi$$
0.0292073 + 0.999573i $$0.490702\pi$$
$$548$$ 0 0
$$549$$ −3032.88 −0.235774
$$550$$ 0 0
$$551$$ −2096.30 −0.162079
$$552$$ 0 0
$$553$$ −165.046 −0.0126917
$$554$$ 0 0
$$555$$ −2640.24 −0.201932
$$556$$ 0 0
$$557$$ 779.939 0.0593305 0.0296653 0.999560i $$-0.490556\pi$$
0.0296653 + 0.999560i $$0.490556\pi$$
$$558$$ 0 0
$$559$$ −4381.54 −0.331519
$$560$$ 0 0
$$561$$ −207.123 −0.0155878
$$562$$ 0 0
$$563$$ 4041.76 0.302558 0.151279 0.988491i $$-0.451661\pi$$
0.151279 + 0.988491i $$0.451661\pi$$
$$564$$ 0 0
$$565$$ −274.270 −0.0204223
$$566$$ 0 0
$$567$$ 456.042 0.0337777
$$568$$ 0 0
$$569$$ 17848.8 1.31505 0.657524 0.753434i $$-0.271604\pi$$
0.657524 + 0.753434i $$0.271604\pi$$
$$570$$ 0 0
$$571$$ −10136.5 −0.742908 −0.371454 0.928451i $$-0.621141\pi$$
−0.371454 + 0.928451i $$0.621141\pi$$
$$572$$ 0 0
$$573$$ 12701.6 0.926033
$$574$$ 0 0
$$575$$ 4273.98 0.309978
$$576$$ 0 0
$$577$$ 5111.16 0.368770 0.184385 0.982854i $$-0.440971\pi$$
0.184385 + 0.982854i $$0.440971\pi$$
$$578$$ 0 0
$$579$$ 2251.48 0.161603
$$580$$ 0 0
$$581$$ 661.583 0.0472411
$$582$$ 0 0
$$583$$ 2038.14 0.144787
$$584$$ 0 0
$$585$$ −892.727 −0.0630935
$$586$$ 0 0
$$587$$ 20129.9 1.41542 0.707708 0.706505i $$-0.249729\pi$$
0.707708 + 0.706505i $$0.249729\pi$$
$$588$$ 0 0
$$589$$ −25074.7 −1.75413
$$590$$ 0 0
$$591$$ 2237.96 0.155765
$$592$$ 0 0
$$593$$ 3531.09 0.244527 0.122264 0.992498i $$-0.460985\pi$$
0.122264 + 0.992498i $$0.460985\pi$$
$$594$$ 0 0
$$595$$ 85.9177 0.00591980
$$596$$ 0 0
$$597$$ 7043.92 0.482896
$$598$$ 0 0
$$599$$ −14765.7 −1.00720 −0.503598 0.863938i $$-0.667991\pi$$
−0.503598 + 0.863938i $$0.667991\pi$$
$$600$$ 0 0
$$601$$ −14004.6 −0.950513 −0.475257 0.879847i $$-0.657645\pi$$
−0.475257 + 0.879847i $$0.657645\pi$$
$$602$$ 0 0
$$603$$ 4782.70 0.322996
$$604$$ 0 0
$$605$$ −1063.10 −0.0714401
$$606$$ 0 0
$$607$$ −24883.8 −1.66393 −0.831963 0.554831i $$-0.812783\pi$$
−0.831963 + 0.554831i $$0.812783\pi$$
$$608$$ 0 0
$$609$$ −401.669 −0.0267265
$$610$$ 0 0
$$611$$ 2576.85 0.170619
$$612$$ 0 0
$$613$$ 9656.87 0.636276 0.318138 0.948044i $$-0.396942\pi$$
0.318138 + 0.948044i $$0.396942\pi$$
$$614$$ 0 0
$$615$$ 32.2989 0.00211775
$$616$$ 0 0
$$617$$ 10779.4 0.703345 0.351672 0.936123i $$-0.385613\pi$$
0.351672 + 0.936123i $$0.385613\pi$$
$$618$$ 0 0
$$619$$ −11588.6 −0.752481 −0.376241 0.926522i $$-0.622783\pi$$
−0.376241 + 0.926522i $$0.622783\pi$$
$$620$$ 0 0
$$621$$ −1728.00 −0.111662
$$622$$ 0 0
$$623$$ 2863.89 0.184173
$$624$$ 0 0
$$625$$ −2817.71 −0.180333
$$626$$ 0 0
$$627$$ −9129.04 −0.581466
$$628$$ 0 0
$$629$$ −230.685 −0.0146232
$$630$$ 0 0
$$631$$ −69.9543 −0.00441337 −0.00220669 0.999998i $$-0.500702\pi$$
−0.00220669 + 0.999998i $$0.500702\pi$$
$$632$$ 0 0
$$633$$ 2052.17 0.128857
$$634$$ 0 0
$$635$$ −9194.43 −0.574598
$$636$$ 0 0
$$637$$ 4046.92 0.251719
$$638$$ 0 0
$$639$$ −4592.22 −0.284297
$$640$$ 0 0
$$641$$ 22360.1 1.37780 0.688900 0.724856i $$-0.258094\pi$$
0.688900 + 0.724856i $$0.258094\pi$$
$$642$$ 0 0
$$643$$ 18216.6 1.11725 0.558626 0.829419i $$-0.311329\pi$$
0.558626 + 0.829419i $$0.311329\pi$$
$$644$$ 0 0
$$645$$ 7715.02 0.470975
$$646$$ 0 0
$$647$$ −16594.2 −1.00832 −0.504162 0.863609i $$-0.668198\pi$$
−0.504162 + 0.863609i $$0.668198\pi$$
$$648$$ 0 0
$$649$$ −6500.28 −0.393156
$$650$$ 0 0
$$651$$ −4804.52 −0.289254
$$652$$ 0 0
$$653$$ 14687.0 0.880161 0.440081 0.897958i $$-0.354950\pi$$
0.440081 + 0.897958i $$0.354950\pi$$
$$654$$ 0 0
$$655$$ −9324.67 −0.556252
$$656$$ 0 0
$$657$$ −1477.97 −0.0877644
$$658$$ 0 0
$$659$$ −12906.0 −0.762891 −0.381446 0.924391i $$-0.624574\pi$$
−0.381446 + 0.924391i $$0.624574\pi$$
$$660$$ 0 0
$$661$$ 6640.83 0.390769 0.195385 0.980727i $$-0.437404\pi$$
0.195385 + 0.980727i $$0.437404\pi$$
$$662$$ 0 0
$$663$$ −78.0000 −0.00456903
$$664$$ 0 0
$$665$$ 3786.85 0.220824
$$666$$ 0 0
$$667$$ 1521.98 0.0883525
$$668$$ 0 0
$$669$$ −4751.71 −0.274606
$$670$$ 0 0
$$671$$ 11633.0 0.669279
$$672$$ 0 0
$$673$$ −26318.2 −1.50742 −0.753709 0.657209i $$-0.771737\pi$$
−0.753709 + 0.657209i $$0.771737\pi$$
$$674$$ 0 0
$$675$$ −1803.08 −0.102816
$$676$$ 0 0
$$677$$ −12702.6 −0.721121 −0.360561 0.932736i $$-0.617415\pi$$
−0.360561 + 0.932736i $$0.617415\pi$$
$$678$$ 0 0
$$679$$ −5747.69 −0.324854
$$680$$ 0 0
$$681$$ −11025.7 −0.620420
$$682$$ 0 0
$$683$$ 32848.6 1.84029 0.920144 0.391580i $$-0.128071\pi$$
0.920144 + 0.391580i $$0.128071\pi$$
$$684$$ 0 0
$$685$$ 11188.8 0.624093
$$686$$ 0 0
$$687$$ 8227.00 0.456884
$$688$$ 0 0
$$689$$ 767.535 0.0424394
$$690$$ 0 0
$$691$$ 7183.85 0.395494 0.197747 0.980253i $$-0.436638\pi$$
0.197747 + 0.980253i $$0.436638\pi$$
$$692$$ 0 0
$$693$$ −1749.20 −0.0958827
$$694$$ 0 0
$$695$$ 9349.97 0.510309
$$696$$ 0 0
$$697$$ 2.82204 0.000153361 0
$$698$$ 0 0
$$699$$ −7379.50 −0.399311
$$700$$ 0 0
$$701$$ 34532.1 1.86057 0.930285 0.366838i $$-0.119560\pi$$
0.930285 + 0.366838i $$0.119560\pi$$
$$702$$ 0 0
$$703$$ −10167.5 −0.545485
$$704$$ 0 0
$$705$$ −4537.32 −0.242391
$$706$$ 0 0
$$707$$ 5790.26 0.308013
$$708$$ 0 0
$$709$$ −6369.28 −0.337381 −0.168691 0.985669i $$-0.553954\pi$$
−0.168691 + 0.985669i $$0.553954\pi$$
$$710$$ 0 0
$$711$$ −263.833 −0.0139163
$$712$$ 0 0
$$713$$ 18204.9 0.956214
$$714$$ 0 0
$$715$$ 3424.16 0.179100
$$716$$ 0 0
$$717$$ −1004.06 −0.0522977
$$718$$ 0 0
$$719$$ 18776.2 0.973899 0.486949 0.873430i $$-0.338110\pi$$
0.486949 + 0.873430i $$0.338110\pi$$
$$720$$ 0 0
$$721$$ −7133.48 −0.368467
$$722$$ 0 0
$$723$$ −8093.42 −0.416318
$$724$$ 0 0
$$725$$ 1588.11 0.0813529
$$726$$ 0 0
$$727$$ −25307.8 −1.29108 −0.645539 0.763728i $$-0.723367\pi$$
−0.645539 + 0.763728i $$0.723367\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ 674.082 0.0341065
$$732$$ 0 0
$$733$$ −377.025 −0.0189983 −0.00949914 0.999955i $$-0.503024\pi$$
−0.00949914 + 0.999955i $$0.503024\pi$$
$$734$$ 0 0
$$735$$ −7125.83 −0.357606
$$736$$ 0 0
$$737$$ −18344.6 −0.916869
$$738$$ 0 0
$$739$$ −35868.3 −1.78544 −0.892718 0.450616i $$-0.851204\pi$$
−0.892718 + 0.450616i $$0.851204\pi$$
$$740$$ 0 0
$$741$$ −3437.88 −0.170437
$$742$$ 0 0
$$743$$ −10183.8 −0.502835 −0.251417 0.967879i $$-0.580897\pi$$
−0.251417 + 0.967879i $$0.580897\pi$$
$$744$$ 0 0
$$745$$ −12222.6 −0.601074
$$746$$ 0 0
$$747$$ 1057.57 0.0517996
$$748$$ 0 0
$$749$$ −10892.9 −0.531398
$$750$$ 0 0
$$751$$ −13074.3 −0.635271 −0.317635 0.948213i $$-0.602889\pi$$
−0.317635 + 0.948213i $$0.602889\pi$$
$$752$$ 0 0
$$753$$ −7996.93 −0.387018
$$754$$ 0 0
$$755$$ 4488.31 0.216353
$$756$$ 0 0
$$757$$ 26803.7 1.28692 0.643458 0.765481i $$-0.277499\pi$$
0.643458 + 0.765481i $$0.277499\pi$$
$$758$$ 0 0
$$759$$ 6627.95 0.316969
$$760$$ 0 0
$$761$$ 30625.8 1.45885 0.729424 0.684062i $$-0.239788\pi$$
0.729424 + 0.684062i $$0.239788\pi$$
$$762$$ 0 0
$$763$$ −11477.9 −0.544600
$$764$$ 0 0
$$765$$ 137.343 0.00649102
$$766$$ 0 0
$$767$$ −2447.92 −0.115240
$$768$$ 0 0
$$769$$ 16457.3 0.771737 0.385868 0.922554i $$-0.373902\pi$$
0.385868 + 0.922554i $$0.373902\pi$$
$$770$$ 0 0
$$771$$ −20632.3 −0.963753
$$772$$ 0 0
$$773$$ 22499.4 1.04689 0.523446 0.852059i $$-0.324646\pi$$
0.523446 + 0.852059i $$0.324646\pi$$
$$774$$ 0 0
$$775$$ 18996.0 0.880458
$$776$$ 0 0
$$777$$ −1948.19 −0.0899496
$$778$$ 0 0
$$779$$ 124.383 0.00572075
$$780$$ 0 0
$$781$$ 17614.0 0.807016
$$782$$ 0 0
$$783$$ −642.084 −0.0293055
$$784$$ 0 0
$$785$$ 7064.90 0.321219
$$786$$ 0 0
$$787$$ −26500.7 −1.20031 −0.600157 0.799882i $$-0.704895\pi$$
−0.600157 + 0.799882i $$0.704895\pi$$
$$788$$ 0 0
$$789$$ −166.526 −0.00751391
$$790$$ 0 0
$$791$$ −202.379 −0.00909704
$$792$$ 0 0
$$793$$ 4380.83 0.196176
$$794$$ 0 0
$$795$$ −1351.48 −0.0602918
$$796$$ 0 0
$$797$$ −27751.0 −1.23337 −0.616683 0.787212i $$-0.711524\pi$$
−0.616683 + 0.787212i $$0.711524\pi$$
$$798$$ 0 0
$$799$$ −396.438 −0.0175532
$$800$$ 0 0
$$801$$ 4578.04 0.201944
$$802$$ 0 0
$$803$$ 5668.94 0.249131
$$804$$ 0 0
$$805$$ −2749.37 −0.120376
$$806$$ 0 0
$$807$$ 11180.6 0.487700
$$808$$ 0 0
$$809$$ 25964.3 1.12838 0.564189 0.825646i $$-0.309189\pi$$
0.564189 + 0.825646i $$0.309189\pi$$
$$810$$ 0 0
$$811$$ 23042.2 0.997685 0.498843 0.866693i $$-0.333759\pi$$
0.498843 + 0.866693i $$0.333759\pi$$
$$812$$ 0 0
$$813$$ 4609.77 0.198858
$$814$$ 0 0
$$815$$ 22490.0 0.966615
$$816$$ 0 0
$$817$$ 29710.4 1.27226
$$818$$ 0 0
$$819$$ −658.727 −0.0281047
$$820$$ 0 0
$$821$$ −5645.55 −0.239989 −0.119994 0.992775i $$-0.538288\pi$$
−0.119994 + 0.992775i $$0.538288\pi$$
$$822$$ 0 0
$$823$$ −38009.9 −1.60989 −0.804946 0.593348i $$-0.797806\pi$$
−0.804946 + 0.593348i $$0.797806\pi$$
$$824$$ 0 0
$$825$$ 6915.94 0.291857
$$826$$ 0 0
$$827$$ −7195.40 −0.302550 −0.151275 0.988492i $$-0.548338\pi$$
−0.151275 + 0.988492i $$0.548338\pi$$
$$828$$ 0 0
$$829$$ 40503.6 1.69692 0.848461 0.529259i $$-0.177530\pi$$
0.848461 + 0.529259i $$0.177530\pi$$
$$830$$ 0 0
$$831$$ −14751.7 −0.615801
$$832$$ 0 0
$$833$$ −622.603 −0.0258967
$$834$$ 0 0
$$835$$ −5458.39 −0.226222
$$836$$ 0 0
$$837$$ −7680.21 −0.317165
$$838$$ 0 0
$$839$$ 5322.63 0.219020 0.109510 0.993986i $$-0.465072\pi$$
0.109510 + 0.993986i $$0.465072\pi$$
$$840$$ 0 0
$$841$$ −23823.5 −0.976812
$$842$$ 0 0
$$843$$ −15845.0 −0.647367
$$844$$ 0 0
$$845$$ 1289.49 0.0524970
$$846$$ 0 0
$$847$$ −784.444 −0.0318227
$$848$$ 0 0
$$849$$ 15274.1 0.617439
$$850$$ 0 0
$$851$$ 7381.93 0.297355
$$852$$ 0 0
$$853$$ 44680.0 1.79345 0.896726 0.442586i $$-0.145939\pi$$
0.896726 + 0.442586i $$0.145939\pi$$
$$854$$ 0 0
$$855$$ 6053.43 0.242132
$$856$$ 0 0
$$857$$ −9103.80 −0.362870 −0.181435 0.983403i $$-0.558074\pi$$
−0.181435 + 0.983403i $$0.558074\pi$$
$$858$$ 0 0
$$859$$ 41297.8 1.64035 0.820177 0.572110i $$-0.193875\pi$$
0.820177 + 0.572110i $$0.193875\pi$$
$$860$$ 0 0
$$861$$ 23.8328 0.000943343 0
$$862$$ 0 0
$$863$$ 28878.2 1.13908 0.569540 0.821964i $$-0.307121\pi$$
0.569540 + 0.821964i $$0.307121\pi$$
$$864$$ 0 0
$$865$$ −2559.12 −0.100593
$$866$$ 0 0
$$867$$ −14727.0 −0.576880
$$868$$ 0 0
$$869$$ 1011.96 0.0395034
$$870$$ 0 0
$$871$$ −6908.34 −0.268749
$$872$$ 0 0
$$873$$ −9187.90 −0.356201
$$874$$ 0 0
$$875$$ −8238.68 −0.318307
$$876$$ 0 0
$$877$$ −36361.7 −1.40005 −0.700027 0.714117i $$-0.746829\pi$$
−0.700027 + 0.714117i $$0.746829\pi$$
$$878$$ 0 0
$$879$$ 12828.0 0.492237
$$880$$ 0 0
$$881$$ −14003.7 −0.535523 −0.267761 0.963485i $$-0.586284\pi$$
−0.267761 + 0.963485i $$0.586284\pi$$
$$882$$ 0 0
$$883$$ −45492.4 −1.73379 −0.866897 0.498488i $$-0.833889\pi$$
−0.866897 + 0.498488i $$0.833889\pi$$
$$884$$ 0 0
$$885$$ 4310.30 0.163717
$$886$$ 0 0
$$887$$ −1388.94 −0.0525773 −0.0262886 0.999654i $$-0.508369\pi$$
−0.0262886 + 0.999654i $$0.508369\pi$$
$$888$$ 0 0
$$889$$ −6784.40 −0.255952
$$890$$ 0 0
$$891$$ −2796.17 −0.105135
$$892$$ 0 0
$$893$$ −17473.2 −0.654778
$$894$$ 0 0
$$895$$ 27079.7 1.01137
$$896$$ 0 0
$$897$$ 2496.00 0.0929086
$$898$$ 0 0
$$899$$ 6764.52 0.250956
$$900$$ 0 0
$$901$$ −118.082 −0.00436614
$$902$$ 0 0
$$903$$ 5692.77 0.209794
$$904$$ 0 0
$$905$$ −8921.63 −0.327696
$$906$$ 0 0
$$907$$ −28974.2 −1.06072 −0.530361 0.847772i $$-0.677943\pi$$
−0.530361 + 0.847772i $$0.677943\pi$$
$$908$$ 0 0
$$909$$ 9255.96 0.337735
$$910$$ 0 0
$$911$$ 41047.3 1.49282 0.746408 0.665488i $$-0.231777\pi$$
0.746408 + 0.665488i $$0.231777\pi$$
$$912$$ 0 0
$$913$$ −4056.42 −0.147040
$$914$$ 0 0
$$915$$ −7713.77 −0.278699
$$916$$ 0 0
$$917$$ −6880.50 −0.247780
$$918$$ 0 0
$$919$$ 25401.8 0.911783 0.455891 0.890035i $$-0.349321\pi$$
0.455891 + 0.890035i $$0.349321\pi$$
$$920$$ 0 0
$$921$$ −918.935 −0.0328772
$$922$$ 0 0
$$923$$ 6633.21 0.236549
$$924$$ 0 0
$$925$$ 7702.68 0.273797
$$926$$ 0 0
$$927$$ −11403.1 −0.404022
$$928$$ 0 0
$$929$$ 46342.1 1.63664 0.818319 0.574765i $$-0.194907\pi$$
0.818319 + 0.574765i $$0.194907\pi$$
$$930$$ 0 0
$$931$$ −27441.5 −0.966012
$$932$$ 0 0
$$933$$ −9801.71 −0.343937
$$934$$ 0 0
$$935$$ −526.794 −0.0184257
$$936$$ 0 0
$$937$$ 8386.36 0.292391 0.146196 0.989256i $$-0.453297\pi$$
0.146196 + 0.989256i $$0.453297\pi$$
$$938$$ 0 0
$$939$$ −7368.57 −0.256085
$$940$$ 0 0
$$941$$ 7119.76 0.246650 0.123325 0.992366i $$-0.460644\pi$$
0.123325 + 0.992366i $$0.460644\pi$$
$$942$$ 0 0
$$943$$ −90.3053 −0.00311850
$$944$$ 0 0
$$945$$ 1159.89 0.0399272
$$946$$ 0 0
$$947$$ 19592.2 0.672293 0.336147 0.941810i $$-0.390876\pi$$
0.336147 + 0.941810i $$0.390876\pi$$
$$948$$ 0 0
$$949$$ 2134.85 0.0730244
$$950$$ 0 0
$$951$$ 3103.45 0.105821
$$952$$ 0 0
$$953$$ 48924.0 1.66296 0.831482 0.555552i $$-0.187493\pi$$
0.831482 + 0.555552i $$0.187493\pi$$
$$954$$ 0 0
$$955$$ 32305.0 1.09462
$$956$$ 0 0
$$957$$ 2462.79 0.0831877
$$958$$ 0 0
$$959$$ 8256.04 0.277999
$$960$$ 0 0
$$961$$ 51122.0 1.71602
$$962$$ 0 0
$$963$$ −17412.7 −0.582674
$$964$$ 0 0
$$965$$ 5726.38 0.191025
$$966$$ 0 0
$$967$$ 29095.8 0.967588 0.483794 0.875182i $$-0.339258\pi$$
0.483794 + 0.875182i $$0.339258\pi$$
$$968$$ 0 0
$$969$$ 528.904 0.0175344
$$970$$ 0 0
$$971$$ 41888.1 1.38440 0.692200 0.721706i $$-0.256642\pi$$
0.692200 + 0.721706i $$0.256642\pi$$
$$972$$ 0 0
$$973$$ 6899.17 0.227315
$$974$$ 0 0
$$975$$ 2604.45 0.0855480
$$976$$ 0 0
$$977$$ −37197.0 −1.21805 −0.609027 0.793150i $$-0.708440\pi$$
−0.609027 + 0.793150i $$0.708440\pi$$
$$978$$ 0 0
$$979$$ −17559.6 −0.573246
$$980$$ 0 0
$$981$$ −18347.9 −0.597150
$$982$$ 0 0
$$983$$ 44554.4 1.44564 0.722820 0.691037i $$-0.242846\pi$$
0.722820 + 0.691037i $$0.242846\pi$$
$$984$$ 0 0
$$985$$ 5691.98 0.184123
$$986$$ 0 0
$$987$$ −3348.01 −0.107972
$$988$$ 0 0
$$989$$ −21570.6 −0.693535
$$990$$ 0 0
$$991$$ −23053.2 −0.738961 −0.369480 0.929238i $$-0.620464\pi$$
−0.369480 + 0.929238i $$0.620464\pi$$
$$992$$ 0 0
$$993$$ −12664.9 −0.404742
$$994$$ 0 0
$$995$$ 17915.4 0.570810
$$996$$ 0 0
$$997$$ 36408.5 1.15654 0.578269 0.815846i $$-0.303729\pi$$
0.578269 + 0.815846i $$0.303729\pi$$
$$998$$ 0 0
$$999$$ −3114.25 −0.0986292
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 2496.4.a.be.1.2 2
4.3 odd 2 2496.4.a.v.1.2 2
8.3 odd 2 624.4.a.q.1.1 2
8.5 even 2 312.4.a.c.1.1 2
24.5 odd 2 936.4.a.d.1.2 2
24.11 even 2 1872.4.a.x.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
312.4.a.c.1.1 2 8.5 even 2
624.4.a.q.1.1 2 8.3 odd 2
936.4.a.d.1.2 2 24.5 odd 2
1872.4.a.x.1.2 2 24.11 even 2
2496.4.a.v.1.2 2 4.3 odd 2
2496.4.a.be.1.2 2 1.1 even 1 trivial