# Properties

 Label 1521.4.a.bi.1.6 Level $1521$ Weight $4$ Character 1521.1 Self dual yes Analytic conductor $89.742$ Analytic rank $1$ Dimension $9$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1521,4,Mod(1,1521)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1521, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1521.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1521 = 3^{2} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1521.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: $$89.7419051187$$ Analytic rank: $$1$$ Dimension: $$9$$ Coefficient field: $$\mathbb{Q}[x]/(x^{9} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{9} - 56x^{7} - 27x^{6} + 945x^{5} + 763x^{4} - 4139x^{3} - 2478x^{2} + 63x + 27$$ x^9 - 56*x^7 - 27*x^6 + 945*x^5 + 763*x^4 - 4139*x^3 - 2478*x^2 + 63*x + 27 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$13^{2}$$ Twist minimal: no (minimal twist has level 507) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.6 Root $$-0.100291$$ of defining polynomial Character $$\chi$$ $$=$$ 1521.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.34727 q^{2} -2.49032 q^{4} +15.3991 q^{5} +10.1317 q^{7} -24.6236 q^{8} +O(q^{10})$$ $$q+2.34727 q^{2} -2.49032 q^{4} +15.3991 q^{5} +10.1317 q^{7} -24.6236 q^{8} +36.1458 q^{10} +15.0669 q^{11} +23.7819 q^{14} -37.8757 q^{16} -90.8352 q^{17} -114.640 q^{19} -38.3486 q^{20} +35.3661 q^{22} -75.7635 q^{23} +112.132 q^{25} -25.2313 q^{28} -214.817 q^{29} +284.476 q^{31} +108.084 q^{32} -213.215 q^{34} +156.019 q^{35} -358.878 q^{37} -269.091 q^{38} -379.181 q^{40} +313.154 q^{41} -296.702 q^{43} -37.5214 q^{44} -177.837 q^{46} -316.691 q^{47} -240.348 q^{49} +263.203 q^{50} -163.911 q^{53} +232.016 q^{55} -249.480 q^{56} -504.233 q^{58} +254.149 q^{59} -935.247 q^{61} +667.742 q^{62} +556.709 q^{64} +240.494 q^{67} +226.209 q^{68} +366.220 q^{70} +947.455 q^{71} +430.712 q^{73} -842.384 q^{74} +285.490 q^{76} +152.654 q^{77} -496.620 q^{79} -583.251 q^{80} +735.058 q^{82} +392.527 q^{83} -1398.78 q^{85} -696.439 q^{86} -371.002 q^{88} +979.895 q^{89} +188.675 q^{92} -743.360 q^{94} -1765.35 q^{95} +553.356 q^{97} -564.162 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$9 q + 6 q^{2} + 44 q^{4} + 33 q^{5} - 83 q^{7} + 87 q^{8}+O(q^{10})$$ 9 * q + 6 * q^2 + 44 * q^4 + 33 * q^5 - 83 * q^7 + 87 * q^8 $$9 q + 6 q^{2} + 44 q^{4} + 33 q^{5} - 83 q^{7} + 87 q^{8} - 54 q^{10} + 85 q^{11} - 158 q^{14} + 216 q^{16} - 178 q^{17} - 352 q^{19} + 402 q^{20} - 630 q^{22} - 150 q^{23} - 20 q^{25} - 940 q^{28} + 97 q^{29} - 717 q^{31} + 707 q^{32} - 632 q^{34} + 418 q^{35} - 1108 q^{37} + 660 q^{38} - 1506 q^{40} + 334 q^{41} + 242 q^{43} - 307 q^{44} - 979 q^{46} - 184 q^{47} - 38 q^{49} - 2031 q^{50} + 151 q^{53} + 2064 q^{55} - 2276 q^{56} - 1161 q^{58} + 537 q^{59} - 1340 q^{61} - 347 q^{62} + 893 q^{64} - 2308 q^{67} - 2785 q^{68} + 1420 q^{70} + 96 q^{71} - 2505 q^{73} + 1191 q^{74} - 2409 q^{76} + 2142 q^{77} - 1591 q^{79} - 2671 q^{80} + 1517 q^{82} + 1539 q^{83} - 4296 q^{85} - 3763 q^{86} - 3716 q^{88} - 592 q^{89} - 515 q^{92} - 692 q^{94} - 4158 q^{95} - 1445 q^{97} + 1457 q^{98}+O(q^{100})$$ 9 * q + 6 * q^2 + 44 * q^4 + 33 * q^5 - 83 * q^7 + 87 * q^8 - 54 * q^10 + 85 * q^11 - 158 * q^14 + 216 * q^16 - 178 * q^17 - 352 * q^19 + 402 * q^20 - 630 * q^22 - 150 * q^23 - 20 * q^25 - 940 * q^28 + 97 * q^29 - 717 * q^31 + 707 * q^32 - 632 * q^34 + 418 * q^35 - 1108 * q^37 + 660 * q^38 - 1506 * q^40 + 334 * q^41 + 242 * q^43 - 307 * q^44 - 979 * q^46 - 184 * q^47 - 38 * q^49 - 2031 * q^50 + 151 * q^53 + 2064 * q^55 - 2276 * q^56 - 1161 * q^58 + 537 * q^59 - 1340 * q^61 - 347 * q^62 + 893 * q^64 - 2308 * q^67 - 2785 * q^68 + 1420 * q^70 + 96 * q^71 - 2505 * q^73 + 1191 * q^74 - 2409 * q^76 + 2142 * q^77 - 1591 * q^79 - 2671 * q^80 + 1517 * q^82 + 1539 * q^83 - 4296 * q^85 - 3763 * q^86 - 3716 * q^88 - 592 * q^89 - 515 * q^92 - 692 * q^94 - 4158 * q^95 - 1445 * q^97 + 1457 * q^98

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.34727 0.829886 0.414943 0.909848i $$-0.363802\pi$$
0.414943 + 0.909848i $$0.363802\pi$$
$$3$$ 0 0
$$4$$ −2.49032 −0.311290
$$5$$ 15.3991 1.37734 0.688668 0.725077i $$-0.258196\pi$$
0.688668 + 0.725077i $$0.258196\pi$$
$$6$$ 0 0
$$7$$ 10.1317 0.547062 0.273531 0.961863i $$-0.411808\pi$$
0.273531 + 0.961863i $$0.411808\pi$$
$$8$$ −24.6236 −1.08822
$$9$$ 0 0
$$10$$ 36.1458 1.14303
$$11$$ 15.0669 0.412985 0.206493 0.978448i $$-0.433795\pi$$
0.206493 + 0.978448i $$0.433795\pi$$
$$12$$ 0 0
$$13$$ 0 0
$$14$$ 23.7819 0.453999
$$15$$ 0 0
$$16$$ −37.8757 −0.591809
$$17$$ −90.8352 −1.29593 −0.647964 0.761671i $$-0.724379\pi$$
−0.647964 + 0.761671i $$0.724379\pi$$
$$18$$ 0 0
$$19$$ −114.640 −1.38422 −0.692110 0.721792i $$-0.743319\pi$$
−0.692110 + 0.721792i $$0.743319\pi$$
$$20$$ −38.3486 −0.428751
$$21$$ 0 0
$$22$$ 35.3661 0.342731
$$23$$ −75.7635 −0.686860 −0.343430 0.939178i $$-0.611589\pi$$
−0.343430 + 0.939178i $$0.611589\pi$$
$$24$$ 0 0
$$25$$ 112.132 0.897052
$$26$$ 0 0
$$27$$ 0 0
$$28$$ −25.2313 −0.170295
$$29$$ −214.817 −1.37553 −0.687767 0.725931i $$-0.741409\pi$$
−0.687767 + 0.725931i $$0.741409\pi$$
$$30$$ 0 0
$$31$$ 284.476 1.64817 0.824087 0.566463i $$-0.191689\pi$$
0.824087 + 0.566463i $$0.191689\pi$$
$$32$$ 108.084 0.597087
$$33$$ 0 0
$$34$$ −213.215 −1.07547
$$35$$ 156.019 0.753488
$$36$$ 0 0
$$37$$ −358.878 −1.59457 −0.797286 0.603602i $$-0.793732\pi$$
−0.797286 + 0.603602i $$0.793732\pi$$
$$38$$ −269.091 −1.14874
$$39$$ 0 0
$$40$$ −379.181 −1.49884
$$41$$ 313.154 1.19284 0.596421 0.802672i $$-0.296589\pi$$
0.596421 + 0.802672i $$0.296589\pi$$
$$42$$ 0 0
$$43$$ −296.702 −1.05225 −0.526123 0.850409i $$-0.676355\pi$$
−0.526123 + 0.850409i $$0.676355\pi$$
$$44$$ −37.5214 −0.128558
$$45$$ 0 0
$$46$$ −177.837 −0.570015
$$47$$ −316.691 −0.982854 −0.491427 0.870919i $$-0.663525\pi$$
−0.491427 + 0.870919i $$0.663525\pi$$
$$48$$ 0 0
$$49$$ −240.348 −0.700723
$$50$$ 263.203 0.744451
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −163.911 −0.424810 −0.212405 0.977182i $$-0.568130\pi$$
−0.212405 + 0.977182i $$0.568130\pi$$
$$54$$ 0 0
$$55$$ 232.016 0.568819
$$56$$ −249.480 −0.595324
$$57$$ 0 0
$$58$$ −504.233 −1.14154
$$59$$ 254.149 0.560803 0.280401 0.959883i $$-0.409532\pi$$
0.280401 + 0.959883i $$0.409532\pi$$
$$60$$ 0 0
$$61$$ −935.247 −1.96305 −0.981526 0.191330i $$-0.938720\pi$$
−0.981526 + 0.191330i $$0.938720\pi$$
$$62$$ 667.742 1.36780
$$63$$ 0 0
$$64$$ 556.709 1.08732
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 240.494 0.438522 0.219261 0.975666i $$-0.429635\pi$$
0.219261 + 0.975666i $$0.429635\pi$$
$$68$$ 226.209 0.403409
$$69$$ 0 0
$$70$$ 366.220 0.625309
$$71$$ 947.455 1.58369 0.791847 0.610720i $$-0.209120\pi$$
0.791847 + 0.610720i $$0.209120\pi$$
$$72$$ 0 0
$$73$$ 430.712 0.690562 0.345281 0.938499i $$-0.387784\pi$$
0.345281 + 0.938499i $$0.387784\pi$$
$$74$$ −842.384 −1.32331
$$75$$ 0 0
$$76$$ 285.490 0.430894
$$77$$ 152.654 0.225929
$$78$$ 0 0
$$79$$ −496.620 −0.707268 −0.353634 0.935384i $$-0.615054\pi$$
−0.353634 + 0.935384i $$0.615054\pi$$
$$80$$ −583.251 −0.815119
$$81$$ 0 0
$$82$$ 735.058 0.989922
$$83$$ 392.527 0.519102 0.259551 0.965729i $$-0.416425\pi$$
0.259551 + 0.965729i $$0.416425\pi$$
$$84$$ 0 0
$$85$$ −1398.78 −1.78493
$$86$$ −696.439 −0.873244
$$87$$ 0 0
$$88$$ −371.002 −0.449419
$$89$$ 979.895 1.16706 0.583532 0.812090i $$-0.301670\pi$$
0.583532 + 0.812090i $$0.301670\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 188.675 0.213813
$$93$$ 0 0
$$94$$ −743.360 −0.815656
$$95$$ −1765.35 −1.90654
$$96$$ 0 0
$$97$$ 553.356 0.579225 0.289613 0.957144i $$-0.406474\pi$$
0.289613 + 0.957144i $$0.406474\pi$$
$$98$$ −564.162 −0.581520
$$99$$ 0 0
$$100$$ −279.243 −0.279243
$$101$$ 763.951 0.752633 0.376317 0.926491i $$-0.377190\pi$$
0.376317 + 0.926491i $$0.377190\pi$$
$$102$$ 0 0
$$103$$ 182.518 0.174602 0.0873010 0.996182i $$-0.472176\pi$$
0.0873010 + 0.996182i $$0.472176\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ −384.744 −0.352544
$$107$$ −183.699 −0.165971 −0.0829855 0.996551i $$-0.526446\pi$$
−0.0829855 + 0.996551i $$0.526446\pi$$
$$108$$ 0 0
$$109$$ −1774.42 −1.55925 −0.779627 0.626244i $$-0.784592\pi$$
−0.779627 + 0.626244i $$0.784592\pi$$
$$110$$ 544.605 0.472055
$$111$$ 0 0
$$112$$ −383.747 −0.323756
$$113$$ −417.288 −0.347391 −0.173695 0.984799i $$-0.555571\pi$$
−0.173695 + 0.984799i $$0.555571\pi$$
$$114$$ 0 0
$$115$$ −1166.69 −0.946037
$$116$$ 534.963 0.428190
$$117$$ 0 0
$$118$$ 596.556 0.465402
$$119$$ −920.318 −0.708953
$$120$$ 0 0
$$121$$ −1103.99 −0.829443
$$122$$ −2195.28 −1.62911
$$123$$ 0 0
$$124$$ −708.436 −0.513060
$$125$$ −198.162 −0.141793
$$126$$ 0 0
$$127$$ −1951.69 −1.36366 −0.681828 0.731513i $$-0.738815\pi$$
−0.681828 + 0.731513i $$0.738815\pi$$
$$128$$ 442.072 0.305266
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 1475.61 0.984159 0.492080 0.870550i $$-0.336237\pi$$
0.492080 + 0.870550i $$0.336237\pi$$
$$132$$ 0 0
$$133$$ −1161.50 −0.757255
$$134$$ 564.503 0.363923
$$135$$ 0 0
$$136$$ 2236.69 1.41026
$$137$$ −1900.71 −1.18532 −0.592660 0.805453i $$-0.701922\pi$$
−0.592660 + 0.805453i $$0.701922\pi$$
$$138$$ 0 0
$$139$$ −2326.76 −1.41981 −0.709903 0.704299i $$-0.751261\pi$$
−0.709903 + 0.704299i $$0.751261\pi$$
$$140$$ −388.538 −0.234553
$$141$$ 0 0
$$142$$ 2223.93 1.31428
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −3307.98 −1.89457
$$146$$ 1011.00 0.573088
$$147$$ 0 0
$$148$$ 893.721 0.496374
$$149$$ −1370.92 −0.753761 −0.376881 0.926262i $$-0.623003\pi$$
−0.376881 + 0.926262i $$0.623003\pi$$
$$150$$ 0 0
$$151$$ −1177.57 −0.634628 −0.317314 0.948320i $$-0.602781\pi$$
−0.317314 + 0.948320i $$0.602781\pi$$
$$152$$ 2822.85 1.50634
$$153$$ 0 0
$$154$$ 358.320 0.187495
$$155$$ 4380.67 2.27009
$$156$$ 0 0
$$157$$ 1621.57 0.824302 0.412151 0.911116i $$-0.364778\pi$$
0.412151 + 0.911116i $$0.364778\pi$$
$$158$$ −1165.70 −0.586951
$$159$$ 0 0
$$160$$ 1664.40 0.822389
$$161$$ −767.616 −0.375755
$$162$$ 0 0
$$163$$ 133.130 0.0639727 0.0319864 0.999488i $$-0.489817\pi$$
0.0319864 + 0.999488i $$0.489817\pi$$
$$164$$ −779.854 −0.371320
$$165$$ 0 0
$$166$$ 921.368 0.430795
$$167$$ −490.724 −0.227385 −0.113693 0.993516i $$-0.536268\pi$$
−0.113693 + 0.993516i $$0.536268\pi$$
$$168$$ 0 0
$$169$$ 0 0
$$170$$ −3283.31 −1.48129
$$171$$ 0 0
$$172$$ 738.882 0.327554
$$173$$ −2008.13 −0.882518 −0.441259 0.897380i $$-0.645468\pi$$
−0.441259 + 0.897380i $$0.645468\pi$$
$$174$$ 0 0
$$175$$ 1136.09 0.490743
$$176$$ −570.670 −0.244408
$$177$$ 0 0
$$178$$ 2300.08 0.968529
$$179$$ 2152.60 0.898843 0.449421 0.893320i $$-0.351630\pi$$
0.449421 + 0.893320i $$0.351630\pi$$
$$180$$ 0 0
$$181$$ 834.690 0.342774 0.171387 0.985204i $$-0.445175\pi$$
0.171387 + 0.985204i $$0.445175\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 1865.57 0.747455
$$185$$ −5526.39 −2.19626
$$186$$ 0 0
$$187$$ −1368.60 −0.535200
$$188$$ 788.662 0.305953
$$189$$ 0 0
$$190$$ −4143.75 −1.58221
$$191$$ −4464.71 −1.69139 −0.845695 0.533667i $$-0.820814\pi$$
−0.845695 + 0.533667i $$0.820814\pi$$
$$192$$ 0 0
$$193$$ 3299.84 1.23071 0.615357 0.788248i $$-0.289012\pi$$
0.615357 + 0.788248i $$0.289012\pi$$
$$194$$ 1298.88 0.480691
$$195$$ 0 0
$$196$$ 598.543 0.218128
$$197$$ 2973.59 1.07543 0.537714 0.843127i $$-0.319288\pi$$
0.537714 + 0.843127i $$0.319288\pi$$
$$198$$ 0 0
$$199$$ −5430.82 −1.93457 −0.967287 0.253684i $$-0.918358\pi$$
−0.967287 + 0.253684i $$0.918358\pi$$
$$200$$ −2761.08 −0.976191
$$201$$ 0 0
$$202$$ 1793.20 0.624600
$$203$$ −2176.47 −0.752503
$$204$$ 0 0
$$205$$ 4822.29 1.64294
$$206$$ 428.419 0.144900
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −1727.27 −0.571663
$$210$$ 0 0
$$211$$ 1228.25 0.400742 0.200371 0.979720i $$-0.435785\pi$$
0.200371 + 0.979720i $$0.435785\pi$$
$$212$$ 408.192 0.132239
$$213$$ 0 0
$$214$$ −431.192 −0.137737
$$215$$ −4568.93 −1.44930
$$216$$ 0 0
$$217$$ 2882.24 0.901654
$$218$$ −4165.05 −1.29400
$$219$$ 0 0
$$220$$ −577.795 −0.177068
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 685.256 0.205776 0.102888 0.994693i $$-0.467192\pi$$
0.102888 + 0.994693i $$0.467192\pi$$
$$224$$ 1095.08 0.326644
$$225$$ 0 0
$$226$$ −979.487 −0.288294
$$227$$ 287.067 0.0839354 0.0419677 0.999119i $$-0.486637\pi$$
0.0419677 + 0.999119i $$0.486637\pi$$
$$228$$ 0 0
$$229$$ 2302.23 0.664347 0.332174 0.943218i $$-0.392218\pi$$
0.332174 + 0.943218i $$0.392218\pi$$
$$230$$ −2738.53 −0.785102
$$231$$ 0 0
$$232$$ 5289.57 1.49688
$$233$$ 970.620 0.272908 0.136454 0.990646i $$-0.456429\pi$$
0.136454 + 0.990646i $$0.456429\pi$$
$$234$$ 0 0
$$235$$ −4876.75 −1.35372
$$236$$ −632.912 −0.174572
$$237$$ 0 0
$$238$$ −2160.24 −0.588350
$$239$$ −5007.40 −1.35524 −0.677619 0.735413i $$-0.736988\pi$$
−0.677619 + 0.735413i $$0.736988\pi$$
$$240$$ 0 0
$$241$$ 540.092 0.144358 0.0721792 0.997392i $$-0.477005\pi$$
0.0721792 + 0.997392i $$0.477005\pi$$
$$242$$ −2591.36 −0.688343
$$243$$ 0 0
$$244$$ 2329.07 0.611078
$$245$$ −3701.14 −0.965130
$$246$$ 0 0
$$247$$ 0 0
$$248$$ −7004.83 −1.79358
$$249$$ 0 0
$$250$$ −465.141 −0.117672
$$251$$ 6087.80 1.53091 0.765455 0.643489i $$-0.222514\pi$$
0.765455 + 0.643489i $$0.222514\pi$$
$$252$$ 0 0
$$253$$ −1141.52 −0.283663
$$254$$ −4581.14 −1.13168
$$255$$ 0 0
$$256$$ −3416.01 −0.833987
$$257$$ 5096.34 1.23697 0.618484 0.785797i $$-0.287747\pi$$
0.618484 + 0.785797i $$0.287747\pi$$
$$258$$ 0 0
$$259$$ −3636.06 −0.872330
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 3463.66 0.816739
$$263$$ −3405.50 −0.798450 −0.399225 0.916853i $$-0.630721\pi$$
−0.399225 + 0.916853i $$0.630721\pi$$
$$264$$ 0 0
$$265$$ −2524.08 −0.585106
$$266$$ −2726.36 −0.628435
$$267$$ 0 0
$$268$$ −598.906 −0.136507
$$269$$ 2720.44 0.616611 0.308306 0.951287i $$-0.400238\pi$$
0.308306 + 0.951287i $$0.400238\pi$$
$$270$$ 0 0
$$271$$ −6954.27 −1.55883 −0.779413 0.626511i $$-0.784482\pi$$
−0.779413 + 0.626511i $$0.784482\pi$$
$$272$$ 3440.45 0.766941
$$273$$ 0 0
$$274$$ −4461.49 −0.983680
$$275$$ 1689.47 0.370470
$$276$$ 0 0
$$277$$ −6563.96 −1.42379 −0.711895 0.702286i $$-0.752163\pi$$
−0.711895 + 0.702286i $$0.752163\pi$$
$$278$$ −5461.53 −1.17828
$$279$$ 0 0
$$280$$ −3841.76 −0.819961
$$281$$ −652.800 −0.138586 −0.0692932 0.997596i $$-0.522074\pi$$
−0.0692932 + 0.997596i $$0.522074\pi$$
$$282$$ 0 0
$$283$$ 6010.62 1.26252 0.631262 0.775570i $$-0.282537\pi$$
0.631262 + 0.775570i $$0.282537\pi$$
$$284$$ −2359.47 −0.492988
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 3172.80 0.652558
$$288$$ 0 0
$$289$$ 3338.04 0.679430
$$290$$ −7764.73 −1.57228
$$291$$ 0 0
$$292$$ −1072.61 −0.214965
$$293$$ −1912.72 −0.381372 −0.190686 0.981651i $$-0.561071\pi$$
−0.190686 + 0.981651i $$0.561071\pi$$
$$294$$ 0 0
$$295$$ 3913.66 0.772413
$$296$$ 8836.87 1.73525
$$297$$ 0 0
$$298$$ −3217.93 −0.625535
$$299$$ 0 0
$$300$$ 0 0
$$301$$ −3006.10 −0.575644
$$302$$ −2764.06 −0.526669
$$303$$ 0 0
$$304$$ 4342.07 0.819194
$$305$$ −14401.9 −2.70378
$$306$$ 0 0
$$307$$ 1983.07 0.368665 0.184332 0.982864i $$-0.440988\pi$$
0.184332 + 0.982864i $$0.440988\pi$$
$$308$$ −380.157 −0.0703294
$$309$$ 0 0
$$310$$ 10282.6 1.88391
$$311$$ −1893.76 −0.345291 −0.172645 0.984984i $$-0.555231\pi$$
−0.172645 + 0.984984i $$0.555231\pi$$
$$312$$ 0 0
$$313$$ 4574.19 0.826034 0.413017 0.910723i $$-0.364475\pi$$
0.413017 + 0.910723i $$0.364475\pi$$
$$314$$ 3806.26 0.684076
$$315$$ 0 0
$$316$$ 1236.74 0.220165
$$317$$ 7594.53 1.34559 0.672794 0.739830i $$-0.265094\pi$$
0.672794 + 0.739830i $$0.265094\pi$$
$$318$$ 0 0
$$319$$ −3236.62 −0.568076
$$320$$ 8572.81 1.49761
$$321$$ 0 0
$$322$$ −1801.80 −0.311834
$$323$$ 10413.3 1.79385
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 312.492 0.0530901
$$327$$ 0 0
$$328$$ −7710.99 −1.29807
$$329$$ −3208.63 −0.537682
$$330$$ 0 0
$$331$$ 1738.58 0.288705 0.144352 0.989526i $$-0.453890\pi$$
0.144352 + 0.989526i $$0.453890\pi$$
$$332$$ −977.519 −0.161591
$$333$$ 0 0
$$334$$ −1151.86 −0.188704
$$335$$ 3703.38 0.603992
$$336$$ 0 0
$$337$$ −2710.61 −0.438149 −0.219074 0.975708i $$-0.570304\pi$$
−0.219074 + 0.975708i $$0.570304\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 3483.41 0.555630
$$341$$ 4286.17 0.680672
$$342$$ 0 0
$$343$$ −5910.33 −0.930401
$$344$$ 7305.87 1.14508
$$345$$ 0 0
$$346$$ −4713.63 −0.732389
$$347$$ 2506.81 0.387817 0.193908 0.981020i $$-0.437884\pi$$
0.193908 + 0.981020i $$0.437884\pi$$
$$348$$ 0 0
$$349$$ −7536.48 −1.15593 −0.577963 0.816063i $$-0.696152\pi$$
−0.577963 + 0.816063i $$0.696152\pi$$
$$350$$ 2666.70 0.407261
$$351$$ 0 0
$$352$$ 1628.50 0.246588
$$353$$ −8992.88 −1.35593 −0.677964 0.735095i $$-0.737138\pi$$
−0.677964 + 0.735095i $$0.737138\pi$$
$$354$$ 0 0
$$355$$ 14589.9 2.18128
$$356$$ −2440.25 −0.363295
$$357$$ 0 0
$$358$$ 5052.74 0.745937
$$359$$ −4566.64 −0.671359 −0.335679 0.941976i $$-0.608966\pi$$
−0.335679 + 0.941976i $$0.608966\pi$$
$$360$$ 0 0
$$361$$ 6283.31 0.916068
$$362$$ 1959.24 0.284463
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 6632.57 0.951136
$$366$$ 0 0
$$367$$ −6449.50 −0.917332 −0.458666 0.888609i $$-0.651673\pi$$
−0.458666 + 0.888609i $$0.651673\pi$$
$$368$$ 2869.60 0.406490
$$369$$ 0 0
$$370$$ −12971.9 −1.82264
$$371$$ −1660.71 −0.232398
$$372$$ 0 0
$$373$$ 7648.89 1.06178 0.530891 0.847440i $$-0.321857\pi$$
0.530891 + 0.847440i $$0.321857\pi$$
$$374$$ −3212.49 −0.444154
$$375$$ 0 0
$$376$$ 7798.08 1.06956
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 10297.8 1.39567 0.697837 0.716256i $$-0.254146\pi$$
0.697837 + 0.716256i $$0.254146\pi$$
$$380$$ 4396.28 0.593486
$$381$$ 0 0
$$382$$ −10479.9 −1.40366
$$383$$ 10258.0 1.36856 0.684282 0.729217i $$-0.260116\pi$$
0.684282 + 0.729217i $$0.260116\pi$$
$$384$$ 0 0
$$385$$ 2350.73 0.311180
$$386$$ 7745.63 1.02135
$$387$$ 0 0
$$388$$ −1378.03 −0.180307
$$389$$ 4771.57 0.621924 0.310962 0.950422i $$-0.399349\pi$$
0.310962 + 0.950422i $$0.399349\pi$$
$$390$$ 0 0
$$391$$ 6882.00 0.890122
$$392$$ 5918.24 0.762541
$$393$$ 0 0
$$394$$ 6979.82 0.892483
$$395$$ −7647.50 −0.974145
$$396$$ 0 0
$$397$$ −2291.22 −0.289655 −0.144827 0.989457i $$-0.546263\pi$$
−0.144827 + 0.989457i $$0.546263\pi$$
$$398$$ −12747.6 −1.60548
$$399$$ 0 0
$$400$$ −4247.07 −0.530883
$$401$$ 7534.63 0.938308 0.469154 0.883116i $$-0.344559\pi$$
0.469154 + 0.883116i $$0.344559\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ −1902.48 −0.234287
$$405$$ 0 0
$$406$$ −5108.76 −0.624491
$$407$$ −5407.18 −0.658535
$$408$$ 0 0
$$409$$ 1517.97 0.183517 0.0917587 0.995781i $$-0.470751\pi$$
0.0917587 + 0.995781i $$0.470751\pi$$
$$410$$ 11319.2 1.36345
$$411$$ 0 0
$$412$$ −454.528 −0.0543519
$$413$$ 2574.97 0.306794
$$414$$ 0 0
$$415$$ 6044.56 0.714978
$$416$$ 0 0
$$417$$ 0 0
$$418$$ −4054.36 −0.474415
$$419$$ −8080.97 −0.942199 −0.471100 0.882080i $$-0.656143\pi$$
−0.471100 + 0.882080i $$0.656143\pi$$
$$420$$ 0 0
$$421$$ −8073.13 −0.934585 −0.467293 0.884103i $$-0.654771\pi$$
−0.467293 + 0.884103i $$0.654771\pi$$
$$422$$ 2883.05 0.332570
$$423$$ 0 0
$$424$$ 4036.09 0.462287
$$425$$ −10185.5 −1.16252
$$426$$ 0 0
$$427$$ −9475.68 −1.07391
$$428$$ 457.470 0.0516651
$$429$$ 0 0
$$430$$ −10724.5 −1.20275
$$431$$ 2241.39 0.250496 0.125248 0.992125i $$-0.460027\pi$$
0.125248 + 0.992125i $$0.460027\pi$$
$$432$$ 0 0
$$433$$ 10237.9 1.13626 0.568130 0.822939i $$-0.307667\pi$$
0.568130 + 0.822939i $$0.307667\pi$$
$$434$$ 6765.39 0.748270
$$435$$ 0 0
$$436$$ 4418.88 0.485380
$$437$$ 8685.52 0.950766
$$438$$ 0 0
$$439$$ −5416.69 −0.588894 −0.294447 0.955668i $$-0.595135\pi$$
−0.294447 + 0.955668i $$0.595135\pi$$
$$440$$ −5713.08 −0.619001
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 2537.44 0.272138 0.136069 0.990699i $$-0.456553\pi$$
0.136069 + 0.990699i $$0.456553\pi$$
$$444$$ 0 0
$$445$$ 15089.5 1.60744
$$446$$ 1608.48 0.170771
$$447$$ 0 0
$$448$$ 5640.43 0.594833
$$449$$ 7790.61 0.818846 0.409423 0.912345i $$-0.365730\pi$$
0.409423 + 0.912345i $$0.365730\pi$$
$$450$$ 0 0
$$451$$ 4718.26 0.492626
$$452$$ 1039.18 0.108139
$$453$$ 0 0
$$454$$ 673.825 0.0696568
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −6599.71 −0.675539 −0.337770 0.941229i $$-0.609672\pi$$
−0.337770 + 0.941229i $$0.609672\pi$$
$$458$$ 5403.95 0.551332
$$459$$ 0 0
$$460$$ 2905.43 0.294492
$$461$$ 4482.57 0.452872 0.226436 0.974026i $$-0.427293\pi$$
0.226436 + 0.974026i $$0.427293\pi$$
$$462$$ 0 0
$$463$$ 3805.86 0.382016 0.191008 0.981589i $$-0.438824\pi$$
0.191008 + 0.981589i $$0.438824\pi$$
$$464$$ 8136.35 0.814053
$$465$$ 0 0
$$466$$ 2278.31 0.226482
$$467$$ −14778.8 −1.46441 −0.732207 0.681082i $$-0.761510\pi$$
−0.732207 + 0.681082i $$0.761510\pi$$
$$468$$ 0 0
$$469$$ 2436.62 0.239899
$$470$$ −11447.0 −1.12343
$$471$$ 0 0
$$472$$ −6258.06 −0.610277
$$473$$ −4470.37 −0.434562
$$474$$ 0 0
$$475$$ −12854.8 −1.24172
$$476$$ 2291.89 0.220690
$$477$$ 0 0
$$478$$ −11753.7 −1.12469
$$479$$ 11171.2 1.06560 0.532801 0.846240i $$-0.321139\pi$$
0.532801 + 0.846240i $$0.321139\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 1267.74 0.119801
$$483$$ 0 0
$$484$$ 2749.29 0.258197
$$485$$ 8521.18 0.797787
$$486$$ 0 0
$$487$$ −6046.57 −0.562621 −0.281310 0.959617i $$-0.590769\pi$$
−0.281310 + 0.959617i $$0.590769\pi$$
$$488$$ 23029.2 2.13623
$$489$$ 0 0
$$490$$ −8687.57 −0.800948
$$491$$ 1035.04 0.0951338 0.0475669 0.998868i $$-0.484853\pi$$
0.0475669 + 0.998868i $$0.484853\pi$$
$$492$$ 0 0
$$493$$ 19512.9 1.78259
$$494$$ 0 0
$$495$$ 0 0
$$496$$ −10774.7 −0.975404
$$497$$ 9599.37 0.866379
$$498$$ 0 0
$$499$$ 11698.3 1.04947 0.524736 0.851265i $$-0.324164\pi$$
0.524736 + 0.851265i $$0.324164\pi$$
$$500$$ 493.488 0.0441389
$$501$$ 0 0
$$502$$ 14289.7 1.27048
$$503$$ 13552.0 1.20130 0.600651 0.799511i $$-0.294908\pi$$
0.600651 + 0.799511i $$0.294908\pi$$
$$504$$ 0 0
$$505$$ 11764.1 1.03663
$$506$$ −2679.46 −0.235408
$$507$$ 0 0
$$508$$ 4860.33 0.424492
$$509$$ 5076.46 0.442064 0.221032 0.975267i $$-0.429058\pi$$
0.221032 + 0.975267i $$0.429058\pi$$
$$510$$ 0 0
$$511$$ 4363.86 0.377781
$$512$$ −11554.9 −0.997380
$$513$$ 0 0
$$514$$ 11962.5 1.02654
$$515$$ 2810.60 0.240486
$$516$$ 0 0
$$517$$ −4771.55 −0.405904
$$518$$ −8534.81 −0.723934
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 8493.73 0.714236 0.357118 0.934059i $$-0.383759\pi$$
0.357118 + 0.934059i $$0.383759\pi$$
$$522$$ 0 0
$$523$$ −1384.53 −0.115758 −0.0578789 0.998324i $$-0.518434\pi$$
−0.0578789 + 0.998324i $$0.518434\pi$$
$$524$$ −3674.75 −0.306359
$$525$$ 0 0
$$526$$ −7993.64 −0.662622
$$527$$ −25840.4 −2.13592
$$528$$ 0 0
$$529$$ −6426.89 −0.528223
$$530$$ −5924.71 −0.485571
$$531$$ 0 0
$$532$$ 2892.51 0.235726
$$533$$ 0 0
$$534$$ 0 0
$$535$$ −2828.80 −0.228598
$$536$$ −5921.82 −0.477208
$$537$$ 0 0
$$538$$ 6385.62 0.511717
$$539$$ −3621.30 −0.289388
$$540$$ 0 0
$$541$$ −5026.83 −0.399483 −0.199742 0.979849i $$-0.564010\pi$$
−0.199742 + 0.979849i $$0.564010\pi$$
$$542$$ −16323.6 −1.29365
$$543$$ 0 0
$$544$$ −9817.87 −0.773782
$$545$$ −27324.5 −2.14762
$$546$$ 0 0
$$547$$ 1540.36 0.120404 0.0602021 0.998186i $$-0.480825\pi$$
0.0602021 + 0.998186i $$0.480825\pi$$
$$548$$ 4733.38 0.368978
$$549$$ 0 0
$$550$$ 3965.65 0.307447
$$551$$ 24626.6 1.90404
$$552$$ 0 0
$$553$$ −5031.63 −0.386920
$$554$$ −15407.4 −1.18158
$$555$$ 0 0
$$556$$ 5794.38 0.441972
$$557$$ −8550.51 −0.650443 −0.325221 0.945638i $$-0.605439\pi$$
−0.325221 + 0.945638i $$0.605439\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ −5909.35 −0.445921
$$561$$ 0 0
$$562$$ −1532.30 −0.115011
$$563$$ 6569.35 0.491768 0.245884 0.969299i $$-0.420922\pi$$
0.245884 + 0.969299i $$0.420922\pi$$
$$564$$ 0 0
$$565$$ −6425.85 −0.478473
$$566$$ 14108.5 1.04775
$$567$$ 0 0
$$568$$ −23329.8 −1.72341
$$569$$ −23766.1 −1.75102 −0.875508 0.483204i $$-0.839473\pi$$
−0.875508 + 0.483204i $$0.839473\pi$$
$$570$$ 0 0
$$571$$ −24971.7 −1.83018 −0.915091 0.403248i $$-0.867881\pi$$
−0.915091 + 0.403248i $$0.867881\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 7447.41 0.541549
$$575$$ −8495.48 −0.616150
$$576$$ 0 0
$$577$$ 16643.4 1.20082 0.600409 0.799693i $$-0.295004\pi$$
0.600409 + 0.799693i $$0.295004\pi$$
$$578$$ 7835.28 0.563849
$$579$$ 0 0
$$580$$ 8237.93 0.589761
$$581$$ 3976.98 0.283981
$$582$$ 0 0
$$583$$ −2469.64 −0.175441
$$584$$ −10605.7 −0.751484
$$585$$ 0 0
$$586$$ −4489.66 −0.316495
$$587$$ −6720.67 −0.472558 −0.236279 0.971685i $$-0.575928\pi$$
−0.236279 + 0.971685i $$0.575928\pi$$
$$588$$ 0 0
$$589$$ −32612.3 −2.28144
$$590$$ 9186.41 0.641014
$$591$$ 0 0
$$592$$ 13592.8 0.943681
$$593$$ 26349.8 1.82471 0.912357 0.409395i $$-0.134260\pi$$
0.912357 + 0.409395i $$0.134260\pi$$
$$594$$ 0 0
$$595$$ −14172.1 −0.976466
$$596$$ 3414.04 0.234638
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 6136.81 0.418603 0.209302 0.977851i $$-0.432881\pi$$
0.209302 + 0.977851i $$0.432881\pi$$
$$600$$ 0 0
$$601$$ 12493.7 0.847966 0.423983 0.905670i $$-0.360632\pi$$
0.423983 + 0.905670i $$0.360632\pi$$
$$602$$ −7056.14 −0.477719
$$603$$ 0 0
$$604$$ 2932.51 0.197553
$$605$$ −17000.4 −1.14242
$$606$$ 0 0
$$607$$ 18696.6 1.25020 0.625099 0.780545i $$-0.285059\pi$$
0.625099 + 0.780545i $$0.285059\pi$$
$$608$$ −12390.8 −0.826501
$$609$$ 0 0
$$610$$ −33805.3 −2.24383
$$611$$ 0 0
$$612$$ 0 0
$$613$$ −1238.64 −0.0816123 −0.0408062 0.999167i $$-0.512993\pi$$
−0.0408062 + 0.999167i $$0.512993\pi$$
$$614$$ 4654.81 0.305949
$$615$$ 0 0
$$616$$ −3758.89 −0.245860
$$617$$ 16549.5 1.07983 0.539917 0.841718i $$-0.318456\pi$$
0.539917 + 0.841718i $$0.318456\pi$$
$$618$$ 0 0
$$619$$ −13945.4 −0.905513 −0.452756 0.891634i $$-0.649559\pi$$
−0.452756 + 0.891634i $$0.649559\pi$$
$$620$$ −10909.3 −0.706656
$$621$$ 0 0
$$622$$ −4445.17 −0.286552
$$623$$ 9928.03 0.638456
$$624$$ 0 0
$$625$$ −17068.0 −1.09235
$$626$$ 10736.9 0.685513
$$627$$ 0 0
$$628$$ −4038.23 −0.256597
$$629$$ 32598.8 2.06645
$$630$$ 0 0
$$631$$ −15343.1 −0.967987 −0.483994 0.875072i $$-0.660814\pi$$
−0.483994 + 0.875072i $$0.660814\pi$$
$$632$$ 12228.6 0.769663
$$633$$ 0 0
$$634$$ 17826.4 1.11668
$$635$$ −30054.2 −1.87821
$$636$$ 0 0
$$637$$ 0 0
$$638$$ −7597.23 −0.471438
$$639$$ 0 0
$$640$$ 6807.51 0.420454
$$641$$ −11067.7 −0.681979 −0.340989 0.940067i $$-0.610762\pi$$
−0.340989 + 0.940067i $$0.610762\pi$$
$$642$$ 0 0
$$643$$ 25118.7 1.54057 0.770284 0.637701i $$-0.220115\pi$$
0.770284 + 0.637701i $$0.220115\pi$$
$$644$$ 1911.61 0.116969
$$645$$ 0 0
$$646$$ 24442.9 1.48869
$$647$$ 4447.03 0.270217 0.135109 0.990831i $$-0.456862\pi$$
0.135109 + 0.990831i $$0.456862\pi$$
$$648$$ 0 0
$$649$$ 3829.23 0.231603
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −331.537 −0.0199141
$$653$$ 19426.2 1.16418 0.582088 0.813126i $$-0.302236\pi$$
0.582088 + 0.813126i $$0.302236\pi$$
$$654$$ 0 0
$$655$$ 22723.1 1.35552
$$656$$ −11861.0 −0.705934
$$657$$ 0 0
$$658$$ −7531.52 −0.446215
$$659$$ 14099.4 0.833438 0.416719 0.909035i $$-0.363180\pi$$
0.416719 + 0.909035i $$0.363180\pi$$
$$660$$ 0 0
$$661$$ −2754.25 −0.162070 −0.0810348 0.996711i $$-0.525823\pi$$
−0.0810348 + 0.996711i $$0.525823\pi$$
$$662$$ 4080.93 0.239592
$$663$$ 0 0
$$664$$ −9665.44 −0.564898
$$665$$ −17886.0 −1.04299
$$666$$ 0 0
$$667$$ 16275.3 0.944800
$$668$$ 1222.06 0.0707828
$$669$$ 0 0
$$670$$ 8692.83 0.501244
$$671$$ −14091.3 −0.810712
$$672$$ 0 0
$$673$$ 11936.8 0.683700 0.341850 0.939754i $$-0.388946\pi$$
0.341850 + 0.939754i $$0.388946\pi$$
$$674$$ −6362.53 −0.363613
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 10255.2 0.582186 0.291093 0.956695i $$-0.405981\pi$$
0.291093 + 0.956695i $$0.405981\pi$$
$$678$$ 0 0
$$679$$ 5606.46 0.316872
$$680$$ 34443.0 1.94239
$$681$$ 0 0
$$682$$ 10060.8 0.564880
$$683$$ 10605.7 0.594169 0.297085 0.954851i $$-0.403986\pi$$
0.297085 + 0.954851i $$0.403986\pi$$
$$684$$ 0 0
$$685$$ −29269.2 −1.63258
$$686$$ −13873.1 −0.772127
$$687$$ 0 0
$$688$$ 11237.8 0.622728
$$689$$ 0 0
$$690$$ 0 0
$$691$$ −4660.14 −0.256556 −0.128278 0.991738i $$-0.540945\pi$$
−0.128278 + 0.991738i $$0.540945\pi$$
$$692$$ 5000.90 0.274719
$$693$$ 0 0
$$694$$ 5884.15 0.321844
$$695$$ −35829.9 −1.95555
$$696$$ 0 0
$$697$$ −28445.4 −1.54584
$$698$$ −17690.2 −0.959287
$$699$$ 0 0
$$700$$ −2829.22 −0.152764
$$701$$ 24016.9 1.29402 0.647008 0.762483i $$-0.276020\pi$$
0.647008 + 0.762483i $$0.276020\pi$$
$$702$$ 0 0
$$703$$ 41141.7 2.20724
$$704$$ 8387.88 0.449048
$$705$$ 0 0
$$706$$ −21108.7 −1.12527
$$707$$ 7740.15 0.411737
$$708$$ 0 0
$$709$$ 7252.56 0.384169 0.192084 0.981378i $$-0.438475\pi$$
0.192084 + 0.981378i $$0.438475\pi$$
$$710$$ 34246.5 1.81021
$$711$$ 0 0
$$712$$ −24128.6 −1.27002
$$713$$ −21552.9 −1.13207
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −5360.66 −0.279801
$$717$$ 0 0
$$718$$ −10719.1 −0.557151
$$719$$ −10951.7 −0.568050 −0.284025 0.958817i $$-0.591670\pi$$
−0.284025 + 0.958817i $$0.591670\pi$$
$$720$$ 0 0
$$721$$ 1849.22 0.0955182
$$722$$ 14748.6 0.760231
$$723$$ 0 0
$$724$$ −2078.65 −0.106702
$$725$$ −24087.7 −1.23393
$$726$$ 0 0
$$727$$ −27856.0 −1.42108 −0.710538 0.703658i $$-0.751549\pi$$
−0.710538 + 0.703658i $$0.751549\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 15568.4 0.789334
$$731$$ 26951.0 1.36364
$$732$$ 0 0
$$733$$ 31213.8 1.57286 0.786431 0.617678i $$-0.211927\pi$$
0.786431 + 0.617678i $$0.211927\pi$$
$$734$$ −15138.7 −0.761281
$$735$$ 0 0
$$736$$ −8188.85 −0.410116
$$737$$ 3623.49 0.181103
$$738$$ 0 0
$$739$$ −14423.1 −0.717946 −0.358973 0.933348i $$-0.616873\pi$$
−0.358973 + 0.933348i $$0.616873\pi$$
$$740$$ 13762.5 0.683674
$$741$$ 0 0
$$742$$ −3898.13 −0.192864
$$743$$ 13469.7 0.665079 0.332539 0.943089i $$-0.392095\pi$$
0.332539 + 0.943089i $$0.392095\pi$$
$$744$$ 0 0
$$745$$ −21111.0 −1.03818
$$746$$ 17954.0 0.881158
$$747$$ 0 0
$$748$$ 3408.26 0.166602
$$749$$ −1861.19 −0.0907965
$$750$$ 0 0
$$751$$ −32033.6 −1.55649 −0.778245 0.627961i $$-0.783890\pi$$
−0.778245 + 0.627961i $$0.783890\pi$$
$$752$$ 11994.9 0.581661
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −18133.4 −0.874096
$$756$$ 0 0
$$757$$ −26097.2 −1.25300 −0.626498 0.779423i $$-0.715512\pi$$
−0.626498 + 0.779423i $$0.715512\pi$$
$$758$$ 24171.6 1.15825
$$759$$ 0 0
$$760$$ 43469.3 2.07473
$$761$$ 18238.2 0.868772 0.434386 0.900727i $$-0.356965\pi$$
0.434386 + 0.900727i $$0.356965\pi$$
$$762$$ 0 0
$$763$$ −17978.0 −0.853009
$$764$$ 11118.6 0.526513
$$765$$ 0 0
$$766$$ 24078.4 1.13575
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 18817.0 0.882390 0.441195 0.897411i $$-0.354555\pi$$
0.441195 + 0.897411i $$0.354555\pi$$
$$770$$ 5517.79 0.258243
$$771$$ 0 0
$$772$$ −8217.67 −0.383109
$$773$$ −13795.6 −0.641904 −0.320952 0.947095i $$-0.604003\pi$$
−0.320952 + 0.947095i $$0.604003\pi$$
$$774$$ 0 0
$$775$$ 31898.7 1.47850
$$776$$ −13625.6 −0.630325
$$777$$ 0 0
$$778$$ 11200.2 0.516126
$$779$$ −35900.0 −1.65116
$$780$$ 0 0
$$781$$ 14275.2 0.654043
$$782$$ 16153.9 0.738699
$$783$$ 0 0
$$784$$ 9103.36 0.414694
$$785$$ 24970.7 1.13534
$$786$$ 0 0
$$787$$ 40545.4 1.83645 0.918227 0.396055i $$-0.129621\pi$$
0.918227 + 0.396055i $$0.129621\pi$$
$$788$$ −7405.19 −0.334770
$$789$$ 0 0
$$790$$ −17950.7 −0.808429
$$791$$ −4227.85 −0.190044
$$792$$ 0 0
$$793$$ 0 0
$$794$$ −5378.11 −0.240380
$$795$$ 0 0
$$796$$ 13524.5 0.602214
$$797$$ 31576.4 1.40338 0.701690 0.712482i $$-0.252429\pi$$
0.701690 + 0.712482i $$0.252429\pi$$
$$798$$ 0 0
$$799$$ 28766.7 1.27371
$$800$$ 12119.7 0.535619
$$801$$ 0 0
$$802$$ 17685.8 0.778688
$$803$$ 6489.50 0.285192
$$804$$ 0 0
$$805$$ −11820.6 −0.517541
$$806$$ 0 0
$$807$$ 0 0
$$808$$ −18811.2 −0.819031
$$809$$ −31130.1 −1.35287 −0.676437 0.736501i $$-0.736477\pi$$
−0.676437 + 0.736501i $$0.736477\pi$$
$$810$$ 0 0
$$811$$ 8733.71 0.378153 0.189076 0.981962i $$-0.439451\pi$$
0.189076 + 0.981962i $$0.439451\pi$$
$$812$$ 5420.10 0.234247
$$813$$ 0 0
$$814$$ −12692.1 −0.546509
$$815$$ 2050.08 0.0881119
$$816$$ 0 0
$$817$$ 34013.8 1.45654
$$818$$ 3563.08 0.152298
$$819$$ 0 0
$$820$$ −12009.0 −0.511431
$$821$$ −36960.1 −1.57115 −0.785576 0.618765i $$-0.787633\pi$$
−0.785576 + 0.618765i $$0.787633\pi$$
$$822$$ 0 0
$$823$$ 20509.7 0.868679 0.434340 0.900749i $$-0.356982\pi$$
0.434340 + 0.900749i $$0.356982\pi$$
$$824$$ −4494.25 −0.190006
$$825$$ 0 0
$$826$$ 6044.15 0.254604
$$827$$ −11533.4 −0.484952 −0.242476 0.970157i $$-0.577960\pi$$
−0.242476 + 0.970157i $$0.577960\pi$$
$$828$$ 0 0
$$829$$ 34096.4 1.42849 0.714244 0.699897i $$-0.246771\pi$$
0.714244 + 0.699897i $$0.246771\pi$$
$$830$$ 14188.2 0.593350
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 21832.1 0.908087
$$834$$ 0 0
$$835$$ −7556.69 −0.313186
$$836$$ 4301.45 0.177953
$$837$$ 0 0
$$838$$ −18968.2 −0.781917
$$839$$ −1855.42 −0.0763484 −0.0381742 0.999271i $$-0.512154\pi$$
−0.0381742 + 0.999271i $$0.512154\pi$$
$$840$$ 0 0
$$841$$ 21757.3 0.892094
$$842$$ −18949.8 −0.775599
$$843$$ 0 0
$$844$$ −3058.75 −0.124747
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −11185.3 −0.453757
$$848$$ 6208.26 0.251406
$$849$$ 0 0
$$850$$ −23908.1 −0.964755
$$851$$ 27189.9 1.09525
$$852$$ 0 0
$$853$$ 28668.7 1.15076 0.575380 0.817886i $$-0.304854\pi$$
0.575380 + 0.817886i $$0.304854\pi$$
$$854$$ −22242.0 −0.891224
$$855$$ 0 0
$$856$$ 4523.35 0.180613
$$857$$ −449.310 −0.0179091 −0.00895457 0.999960i $$-0.502850\pi$$
−0.00895457 + 0.999960i $$0.502850\pi$$
$$858$$ 0 0
$$859$$ 33466.9 1.32931 0.664654 0.747151i $$-0.268579\pi$$
0.664654 + 0.747151i $$0.268579\pi$$
$$860$$ 11378.1 0.451151
$$861$$ 0 0
$$862$$ 5261.14 0.207883
$$863$$ −25097.5 −0.989953 −0.494976 0.868906i $$-0.664823\pi$$
−0.494976 + 0.868906i $$0.664823\pi$$
$$864$$ 0 0
$$865$$ −30923.4 −1.21552
$$866$$ 24031.0 0.942965
$$867$$ 0 0
$$868$$ −7177.69 −0.280676
$$869$$ −7482.53 −0.292091
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 43692.7 1.69681
$$873$$ 0 0
$$874$$ 20387.3 0.789027
$$875$$ −2007.73 −0.0775698
$$876$$ 0 0
$$877$$ −27015.2 −1.04018 −0.520090 0.854112i $$-0.674101\pi$$
−0.520090 + 0.854112i $$0.674101\pi$$
$$878$$ −12714.4 −0.488715
$$879$$ 0 0
$$880$$ −8787.79 −0.336632
$$881$$ −48638.7 −1.86002 −0.930010 0.367534i $$-0.880202\pi$$
−0.930010 + 0.367534i $$0.880202\pi$$
$$882$$ 0 0
$$883$$ −25479.0 −0.971048 −0.485524 0.874223i $$-0.661371\pi$$
−0.485524 + 0.874223i $$0.661371\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 5956.05 0.225843
$$887$$ −42359.7 −1.60350 −0.801748 0.597663i $$-0.796096\pi$$
−0.801748 + 0.597663i $$0.796096\pi$$
$$888$$ 0 0
$$889$$ −19774.0 −0.746005
$$890$$ 35419.1 1.33399
$$891$$ 0 0
$$892$$ −1706.51 −0.0640561
$$893$$ 36305.4 1.36049
$$894$$ 0 0
$$895$$ 33148.1 1.23801
$$896$$ 4478.96 0.167000
$$897$$ 0 0
$$898$$ 18286.7 0.679548
$$899$$ −61110.3 −2.26712
$$900$$ 0 0
$$901$$ 14888.9 0.550524
$$902$$ 11075.0 0.408823
$$903$$ 0 0
$$904$$ 10275.1 0.378038
$$905$$ 12853.5 0.472114
$$906$$ 0 0
$$907$$ −5314.91 −0.194574 −0.0972871 0.995256i $$-0.531016\pi$$
−0.0972871 + 0.995256i $$0.531016\pi$$
$$908$$ −714.890 −0.0261283
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −2471.71 −0.0898919 −0.0449459 0.998989i $$-0.514312\pi$$
−0.0449459 + 0.998989i $$0.514312\pi$$
$$912$$ 0 0
$$913$$ 5914.17 0.214382
$$914$$ −15491.3 −0.560620
$$915$$ 0 0
$$916$$ −5733.29 −0.206805
$$917$$ 14950.5 0.538396
$$918$$ 0 0
$$919$$ −9636.13 −0.345883 −0.172942 0.984932i $$-0.555327\pi$$
−0.172942 + 0.984932i $$0.555327\pi$$
$$920$$ 28728.1 1.02950
$$921$$ 0 0
$$922$$ 10521.8 0.375832
$$923$$ 0 0
$$924$$ 0 0
$$925$$ −40241.5 −1.43041
$$926$$ 8933.38 0.317029
$$927$$ 0 0
$$928$$ −23218.3 −0.821314
$$929$$ −53249.1 −1.88057 −0.940283 0.340394i $$-0.889439\pi$$
−0.940283 + 0.340394i $$0.889439\pi$$
$$930$$ 0 0
$$931$$ 27553.5 0.969955
$$932$$ −2417.16 −0.0849534
$$933$$ 0 0
$$934$$ −34689.8 −1.21530
$$935$$ −21075.3 −0.737149
$$936$$ 0 0
$$937$$ −41122.6 −1.43374 −0.716871 0.697205i $$-0.754427\pi$$
−0.716871 + 0.697205i $$0.754427\pi$$
$$938$$ 5719.40 0.199088
$$939$$ 0 0
$$940$$ 12144.7 0.421399
$$941$$ −8005.71 −0.277342 −0.138671 0.990339i $$-0.544283\pi$$
−0.138671 + 0.990339i $$0.544283\pi$$
$$942$$ 0 0
$$943$$ −23725.7 −0.819315
$$944$$ −9626.07 −0.331888
$$945$$ 0 0
$$946$$ −10493.2 −0.360637
$$947$$ −17468.6 −0.599424 −0.299712 0.954030i $$-0.596891\pi$$
−0.299712 + 0.954030i $$0.596891\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ −30173.6 −1.03048
$$951$$ 0 0
$$952$$ 22661.6 0.771498
$$953$$ −30681.4 −1.04288 −0.521442 0.853287i $$-0.674606\pi$$
−0.521442 + 0.853287i $$0.674606\pi$$
$$954$$ 0 0
$$955$$ −68752.5 −2.32961
$$956$$ 12470.0 0.421872
$$957$$ 0 0
$$958$$ 26221.8 0.884328
$$959$$ −19257.5 −0.648444
$$960$$ 0 0
$$961$$ 51135.6 1.71648
$$962$$ 0 0
$$963$$ 0 0
$$964$$ −1345.00 −0.0449373
$$965$$ 50814.5 1.69511
$$966$$ 0 0
$$967$$ −15616.0 −0.519313 −0.259656 0.965701i $$-0.583609\pi$$
−0.259656 + 0.965701i $$0.583609\pi$$
$$968$$ 27184.2 0.902617
$$969$$ 0 0
$$970$$ 20001.5 0.662072
$$971$$ −7185.98 −0.237496 −0.118748 0.992924i $$-0.537888\pi$$
−0.118748 + 0.992924i $$0.537888\pi$$
$$972$$ 0 0
$$973$$ −23574.1 −0.776723
$$974$$ −14192.9 −0.466911
$$975$$ 0 0
$$976$$ 35423.2 1.16175
$$977$$ 33165.3 1.08603 0.543015 0.839723i $$-0.317283\pi$$
0.543015 + 0.839723i $$0.317283\pi$$
$$978$$ 0 0
$$979$$ 14764.0 0.481980
$$980$$ 9217.01 0.300435
$$981$$ 0 0
$$982$$ 2429.52 0.0789502
$$983$$ −11658.7 −0.378287 −0.189143 0.981949i $$-0.560571\pi$$
−0.189143 + 0.981949i $$0.560571\pi$$
$$984$$ 0 0
$$985$$ 45790.5 1.48123
$$986$$ 45802.1 1.47935
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 22479.2 0.722746
$$990$$ 0 0
$$991$$ −1957.32 −0.0627409 −0.0313704 0.999508i $$-0.509987\pi$$
−0.0313704 + 0.999508i $$0.509987\pi$$
$$992$$ 30747.4 0.984104
$$993$$ 0 0
$$994$$ 22532.3 0.718996
$$995$$ −83629.5 −2.66456
$$996$$ 0 0
$$997$$ −11434.8 −0.363233 −0.181617 0.983369i $$-0.558133\pi$$
−0.181617 + 0.983369i $$0.558133\pi$$
$$998$$ 27459.0 0.870942
$$999$$ 0 0
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 1521.4.a.bi.1.6 9
3.2 odd 2 507.4.a.o.1.4 9
13.12 even 2 1521.4.a.bf.1.4 9
39.5 even 4 507.4.b.k.337.12 18
39.8 even 4 507.4.b.k.337.7 18
39.38 odd 2 507.4.a.p.1.6 yes 9

By twisted newform
Twist Min Dim Char Parity Ord Type
507.4.a.o.1.4 9 3.2 odd 2
507.4.a.p.1.6 yes 9 39.38 odd 2
507.4.b.k.337.7 18 39.8 even 4
507.4.b.k.337.12 18 39.5 even 4
1521.4.a.bf.1.4 9 13.12 even 2
1521.4.a.bi.1.6 9 1.1 even 1 trivial