# Properties

 Label 304.4.a.e.1.1 Level $304$ Weight $4$ Character 304.1 Self dual yes Analytic conductor $17.937$ 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] = [304,4,Mod(1,304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(304, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("304.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$304 = 2^{4} \cdot 19$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 304.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: $$17.9365806417$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{57})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 14$$ x^2 - x - 14 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 152) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-3.27492$$ of defining polynomial Character $$\chi$$ $$=$$ 304.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-3.27492 q^{3} -6.27492 q^{5} +18.0997 q^{7} -16.2749 q^{9} +O(q^{10})$$ $$q-3.27492 q^{3} -6.27492 q^{5} +18.0997 q^{7} -16.2749 q^{9} +33.9244 q^{11} -3.07558 q^{13} +20.5498 q^{15} -14.2508 q^{17} +19.0000 q^{19} -59.2749 q^{21} -114.072 q^{23} -85.6254 q^{25} +141.722 q^{27} -34.5257 q^{29} -107.698 q^{31} -111.100 q^{33} -113.574 q^{35} -181.698 q^{37} +10.0723 q^{39} -444.743 q^{41} -120.323 q^{43} +102.124 q^{45} -306.371 q^{47} -15.4020 q^{49} +46.6703 q^{51} +115.825 q^{53} -212.873 q^{55} -62.2234 q^{57} -161.680 q^{59} +274.571 q^{61} -294.571 q^{63} +19.2990 q^{65} -81.6254 q^{67} +373.577 q^{69} -773.492 q^{71} +148.794 q^{73} +280.416 q^{75} +614.021 q^{77} +557.341 q^{79} -24.7043 q^{81} -768.337 q^{83} +89.4228 q^{85} +113.069 q^{87} +457.884 q^{89} -55.6670 q^{91} +352.701 q^{93} -119.223 q^{95} -1500.82 q^{97} -552.117 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} - 5 q^{5} + 6 q^{7} - 25 q^{9}+O(q^{10})$$ 2 * q + q^3 - 5 * q^5 + 6 * q^7 - 25 * q^9 $$2 q + q^{3} - 5 q^{5} + 6 q^{7} - 25 q^{9} + 15 q^{11} - 59 q^{13} + 26 q^{15} - 104 q^{17} + 38 q^{19} - 111 q^{21} + 21 q^{23} - 209 q^{25} - 11 q^{27} - 137 q^{29} - 4 q^{31} - 192 q^{33} - 129 q^{35} - 152 q^{37} - 229 q^{39} - 210 q^{41} - 67 q^{43} + 91 q^{45} - 273 q^{47} - 212 q^{49} - 337 q^{51} + 209 q^{53} - 237 q^{55} + 19 q^{57} - 799 q^{59} + 149 q^{61} - 189 q^{63} - 52 q^{65} - 201 q^{67} + 951 q^{69} - 792 q^{71} - 246 q^{73} - 247 q^{75} + 843 q^{77} + 254 q^{79} - 442 q^{81} - 374 q^{83} - 25 q^{85} - 325 q^{87} - 564 q^{89} + 621 q^{91} + 796 q^{93} - 95 q^{95} - 178 q^{97} - 387 q^{99}+O(q^{100})$$ 2 * q + q^3 - 5 * q^5 + 6 * q^7 - 25 * q^9 + 15 * q^11 - 59 * q^13 + 26 * q^15 - 104 * q^17 + 38 * q^19 - 111 * q^21 + 21 * q^23 - 209 * q^25 - 11 * q^27 - 137 * q^29 - 4 * q^31 - 192 * q^33 - 129 * q^35 - 152 * q^37 - 229 * q^39 - 210 * q^41 - 67 * q^43 + 91 * q^45 - 273 * q^47 - 212 * q^49 - 337 * q^51 + 209 * q^53 - 237 * q^55 + 19 * q^57 - 799 * q^59 + 149 * q^61 - 189 * q^63 - 52 * q^65 - 201 * q^67 + 951 * q^69 - 792 * q^71 - 246 * q^73 - 247 * q^75 + 843 * q^77 + 254 * q^79 - 442 * q^81 - 374 * q^83 - 25 * q^85 - 325 * q^87 - 564 * q^89 + 621 * q^91 + 796 * q^93 - 95 * q^95 - 178 * q^97 - 387 * 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
$$3$$ −3.27492 −0.630258 −0.315129 0.949049i $$-0.602048\pi$$
−0.315129 + 0.949049i $$0.602048\pi$$
$$4$$ 0 0
$$5$$ −6.27492 −0.561246 −0.280623 0.959818i $$-0.590541\pi$$
−0.280623 + 0.959818i $$0.590541\pi$$
$$6$$ 0 0
$$7$$ 18.0997 0.977290 0.488645 0.872483i $$-0.337491\pi$$
0.488645 + 0.872483i $$0.337491\pi$$
$$8$$ 0 0
$$9$$ −16.2749 −0.602775
$$10$$ 0 0
$$11$$ 33.9244 0.929873 0.464936 0.885344i $$-0.346077\pi$$
0.464936 + 0.885344i $$0.346077\pi$$
$$12$$ 0 0
$$13$$ −3.07558 −0.0656163 −0.0328082 0.999462i $$-0.510445\pi$$
−0.0328082 + 0.999462i $$0.510445\pi$$
$$14$$ 0 0
$$15$$ 20.5498 0.353730
$$16$$ 0 0
$$17$$ −14.2508 −0.203314 −0.101657 0.994820i $$-0.532414\pi$$
−0.101657 + 0.994820i $$0.532414\pi$$
$$18$$ 0 0
$$19$$ 19.0000 0.229416
$$20$$ 0 0
$$21$$ −59.2749 −0.615945
$$22$$ 0 0
$$23$$ −114.072 −1.03416 −0.517081 0.855937i $$-0.672981\pi$$
−0.517081 + 0.855937i $$0.672981\pi$$
$$24$$ 0 0
$$25$$ −85.6254 −0.685003
$$26$$ 0 0
$$27$$ 141.722 1.01016
$$28$$ 0 0
$$29$$ −34.5257 −0.221078 −0.110539 0.993872i $$-0.535258\pi$$
−0.110539 + 0.993872i $$0.535258\pi$$
$$30$$ 0 0
$$31$$ −107.698 −0.623970 −0.311985 0.950087i $$-0.600994\pi$$
−0.311985 + 0.950087i $$0.600994\pi$$
$$32$$ 0 0
$$33$$ −111.100 −0.586060
$$34$$ 0 0
$$35$$ −113.574 −0.548500
$$36$$ 0 0
$$37$$ −181.698 −0.807322 −0.403661 0.914909i $$-0.632262\pi$$
−0.403661 + 0.914909i $$0.632262\pi$$
$$38$$ 0 0
$$39$$ 10.0723 0.0413552
$$40$$ 0 0
$$41$$ −444.743 −1.69408 −0.847038 0.531532i $$-0.821616\pi$$
−0.847038 + 0.531532i $$0.821616\pi$$
$$42$$ 0 0
$$43$$ −120.323 −0.426723 −0.213362 0.976973i $$-0.568441\pi$$
−0.213362 + 0.976973i $$0.568441\pi$$
$$44$$ 0 0
$$45$$ 102.124 0.338305
$$46$$ 0 0
$$47$$ −306.371 −0.950826 −0.475413 0.879763i $$-0.657701\pi$$
−0.475413 + 0.879763i $$0.657701\pi$$
$$48$$ 0 0
$$49$$ −15.4020 −0.0449038
$$50$$ 0 0
$$51$$ 46.6703 0.128140
$$52$$ 0 0
$$53$$ 115.825 0.300184 0.150092 0.988672i $$-0.452043\pi$$
0.150092 + 0.988672i $$0.452043\pi$$
$$54$$ 0 0
$$55$$ −212.873 −0.521887
$$56$$ 0 0
$$57$$ −62.2234 −0.144591
$$58$$ 0 0
$$59$$ −161.680 −0.356762 −0.178381 0.983961i $$-0.557086\pi$$
−0.178381 + 0.983961i $$0.557086\pi$$
$$60$$ 0 0
$$61$$ 274.571 0.576314 0.288157 0.957583i $$-0.406957\pi$$
0.288157 + 0.957583i $$0.406957\pi$$
$$62$$ 0 0
$$63$$ −294.571 −0.589086
$$64$$ 0 0
$$65$$ 19.2990 0.0368269
$$66$$ 0 0
$$67$$ −81.6254 −0.148838 −0.0744189 0.997227i $$-0.523710\pi$$
−0.0744189 + 0.997227i $$0.523710\pi$$
$$68$$ 0 0
$$69$$ 373.577 0.651789
$$70$$ 0 0
$$71$$ −773.492 −1.29291 −0.646455 0.762952i $$-0.723749\pi$$
−0.646455 + 0.762952i $$0.723749\pi$$
$$72$$ 0 0
$$73$$ 148.794 0.238562 0.119281 0.992861i $$-0.461941\pi$$
0.119281 + 0.992861i $$0.461941\pi$$
$$74$$ 0 0
$$75$$ 280.416 0.431729
$$76$$ 0 0
$$77$$ 614.021 0.908755
$$78$$ 0 0
$$79$$ 557.341 0.793743 0.396872 0.917874i $$-0.370096\pi$$
0.396872 + 0.917874i $$0.370096\pi$$
$$80$$ 0 0
$$81$$ −24.7043 −0.0338879
$$82$$ 0 0
$$83$$ −768.337 −1.01610 −0.508048 0.861329i $$-0.669633\pi$$
−0.508048 + 0.861329i $$0.669633\pi$$
$$84$$ 0 0
$$85$$ 89.4228 0.114109
$$86$$ 0 0
$$87$$ 113.069 0.139336
$$88$$ 0 0
$$89$$ 457.884 0.545344 0.272672 0.962107i $$-0.412093\pi$$
0.272672 + 0.962107i $$0.412093\pi$$
$$90$$ 0 0
$$91$$ −55.6670 −0.0641262
$$92$$ 0 0
$$93$$ 352.701 0.393262
$$94$$ 0 0
$$95$$ −119.223 −0.128759
$$96$$ 0 0
$$97$$ −1500.82 −1.57098 −0.785490 0.618874i $$-0.787589\pi$$
−0.785490 + 0.618874i $$0.787589\pi$$
$$98$$ 0 0
$$99$$ −552.117 −0.560504
$$100$$ 0 0
$$101$$ −1731.06 −1.70541 −0.852707 0.522389i $$-0.825041\pi$$
−0.852707 + 0.522389i $$0.825041\pi$$
$$102$$ 0 0
$$103$$ 1887.77 1.80590 0.902951 0.429743i $$-0.141396\pi$$
0.902951 + 0.429743i $$0.141396\pi$$
$$104$$ 0 0
$$105$$ 371.945 0.345697
$$106$$ 0 0
$$107$$ 411.468 0.371758 0.185879 0.982573i $$-0.440487\pi$$
0.185879 + 0.982573i $$0.440487\pi$$
$$108$$ 0 0
$$109$$ 1186.86 1.04294 0.521469 0.853270i $$-0.325384\pi$$
0.521469 + 0.853270i $$0.325384\pi$$
$$110$$ 0 0
$$111$$ 595.045 0.508821
$$112$$ 0 0
$$113$$ 509.478 0.424139 0.212069 0.977255i $$-0.431980\pi$$
0.212069 + 0.977255i $$0.431980\pi$$
$$114$$ 0 0
$$115$$ 715.794 0.580419
$$116$$ 0 0
$$117$$ 50.0548 0.0395519
$$118$$ 0 0
$$119$$ −257.935 −0.198697
$$120$$ 0 0
$$121$$ −180.134 −0.135337
$$122$$ 0 0
$$123$$ 1456.50 1.06771
$$124$$ 0 0
$$125$$ 1321.66 0.945701
$$126$$ 0 0
$$127$$ 1169.71 0.817284 0.408642 0.912695i $$-0.366002\pi$$
0.408642 + 0.912695i $$0.366002\pi$$
$$128$$ 0 0
$$129$$ 394.048 0.268946
$$130$$ 0 0
$$131$$ 1052.31 0.701838 0.350919 0.936406i $$-0.385869\pi$$
0.350919 + 0.936406i $$0.385869\pi$$
$$132$$ 0 0
$$133$$ 343.894 0.224206
$$134$$ 0 0
$$135$$ −889.292 −0.566949
$$136$$ 0 0
$$137$$ −2116.92 −1.32015 −0.660075 0.751200i $$-0.729476\pi$$
−0.660075 + 0.751200i $$0.729476\pi$$
$$138$$ 0 0
$$139$$ 1115.90 0.680929 0.340464 0.940257i $$-0.389416\pi$$
0.340464 + 0.940257i $$0.389416\pi$$
$$140$$ 0 0
$$141$$ 1003.34 0.599266
$$142$$ 0 0
$$143$$ −104.337 −0.0610148
$$144$$ 0 0
$$145$$ 216.646 0.124079
$$146$$ 0 0
$$147$$ 50.4402 0.0283010
$$148$$ 0 0
$$149$$ 472.693 0.259896 0.129948 0.991521i $$-0.458519\pi$$
0.129948 + 0.991521i $$0.458519\pi$$
$$150$$ 0 0
$$151$$ 1635.40 0.881372 0.440686 0.897661i $$-0.354735\pi$$
0.440686 + 0.897661i $$0.354735\pi$$
$$152$$ 0 0
$$153$$ 231.931 0.122552
$$154$$ 0 0
$$155$$ 675.794 0.350201
$$156$$ 0 0
$$157$$ −785.904 −0.399503 −0.199751 0.979847i $$-0.564013\pi$$
−0.199751 + 0.979847i $$0.564013\pi$$
$$158$$ 0 0
$$159$$ −379.316 −0.189193
$$160$$ 0 0
$$161$$ −2064.67 −1.01068
$$162$$ 0 0
$$163$$ −1224.95 −0.588623 −0.294311 0.955710i $$-0.595090\pi$$
−0.294311 + 0.955710i $$0.595090\pi$$
$$164$$ 0 0
$$165$$ 697.141 0.328923
$$166$$ 0 0
$$167$$ −1071.26 −0.496386 −0.248193 0.968711i $$-0.579837\pi$$
−0.248193 + 0.968711i $$0.579837\pi$$
$$168$$ 0 0
$$169$$ −2187.54 −0.995694
$$170$$ 0 0
$$171$$ −309.223 −0.138286
$$172$$ 0 0
$$173$$ 825.821 0.362925 0.181462 0.983398i $$-0.441917\pi$$
0.181462 + 0.983398i $$0.441917\pi$$
$$174$$ 0 0
$$175$$ −1549.79 −0.669447
$$176$$ 0 0
$$177$$ 529.489 0.224852
$$178$$ 0 0
$$179$$ 465.802 0.194501 0.0972506 0.995260i $$-0.468995\pi$$
0.0972506 + 0.995260i $$0.468995\pi$$
$$180$$ 0 0
$$181$$ −1214.74 −0.498847 −0.249423 0.968395i $$-0.580241\pi$$
−0.249423 + 0.968395i $$0.580241\pi$$
$$182$$ 0 0
$$183$$ −899.196 −0.363227
$$184$$ 0 0
$$185$$ 1140.14 0.453106
$$186$$ 0 0
$$187$$ −483.451 −0.189056
$$188$$ 0 0
$$189$$ 2565.12 0.987221
$$190$$ 0 0
$$191$$ −154.943 −0.0586980 −0.0293490 0.999569i $$-0.509343\pi$$
−0.0293490 + 0.999569i $$0.509343\pi$$
$$192$$ 0 0
$$193$$ −2776.64 −1.03558 −0.517790 0.855508i $$-0.673245\pi$$
−0.517790 + 0.855508i $$0.673245\pi$$
$$194$$ 0 0
$$195$$ −63.2026 −0.0232104
$$196$$ 0 0
$$197$$ 4865.09 1.75951 0.879754 0.475429i $$-0.157707\pi$$
0.879754 + 0.475429i $$0.157707\pi$$
$$198$$ 0 0
$$199$$ 3014.43 1.07381 0.536903 0.843644i $$-0.319594\pi$$
0.536903 + 0.843644i $$0.319594\pi$$
$$200$$ 0 0
$$201$$ 267.316 0.0938062
$$202$$ 0 0
$$203$$ −624.905 −0.216058
$$204$$ 0 0
$$205$$ 2790.72 0.950793
$$206$$ 0 0
$$207$$ 1856.52 0.623366
$$208$$ 0 0
$$209$$ 644.564 0.213327
$$210$$ 0 0
$$211$$ −3434.46 −1.12056 −0.560280 0.828303i $$-0.689307\pi$$
−0.560280 + 0.828303i $$0.689307\pi$$
$$212$$ 0 0
$$213$$ 2533.12 0.814867
$$214$$ 0 0
$$215$$ 755.017 0.239497
$$216$$ 0 0
$$217$$ −1949.29 −0.609800
$$218$$ 0 0
$$219$$ −487.288 −0.150356
$$220$$ 0 0
$$221$$ 43.8296 0.0133407
$$222$$ 0 0
$$223$$ 2247.73 0.674975 0.337488 0.941330i $$-0.390423\pi$$
0.337488 + 0.941330i $$0.390423\pi$$
$$224$$ 0 0
$$225$$ 1393.55 0.412903
$$226$$ 0 0
$$227$$ −1883.81 −0.550806 −0.275403 0.961329i $$-0.588811\pi$$
−0.275403 + 0.961329i $$0.588811\pi$$
$$228$$ 0 0
$$229$$ 3594.49 1.03725 0.518626 0.855001i $$-0.326444\pi$$
0.518626 + 0.855001i $$0.326444\pi$$
$$230$$ 0 0
$$231$$ −2010.87 −0.572750
$$232$$ 0 0
$$233$$ −5530.01 −1.55486 −0.777431 0.628968i $$-0.783478\pi$$
−0.777431 + 0.628968i $$0.783478\pi$$
$$234$$ 0 0
$$235$$ 1922.45 0.533647
$$236$$ 0 0
$$237$$ −1825.24 −0.500263
$$238$$ 0 0
$$239$$ −2587.82 −0.700387 −0.350193 0.936677i $$-0.613884\pi$$
−0.350193 + 0.936677i $$0.613884\pi$$
$$240$$ 0 0
$$241$$ −3576.45 −0.955932 −0.477966 0.878378i $$-0.658626\pi$$
−0.477966 + 0.878378i $$0.658626\pi$$
$$242$$ 0 0
$$243$$ −3745.58 −0.988804
$$244$$ 0 0
$$245$$ 96.6462 0.0252020
$$246$$ 0 0
$$247$$ −58.4360 −0.0150534
$$248$$ 0 0
$$249$$ 2516.24 0.640403
$$250$$ 0 0
$$251$$ −5966.87 −1.50050 −0.750251 0.661154i $$-0.770067\pi$$
−0.750251 + 0.661154i $$0.770067\pi$$
$$252$$ 0 0
$$253$$ −3869.84 −0.961638
$$254$$ 0 0
$$255$$ −292.852 −0.0719181
$$256$$ 0 0
$$257$$ 3841.01 0.932278 0.466139 0.884712i $$-0.345645\pi$$
0.466139 + 0.884712i $$0.345645\pi$$
$$258$$ 0 0
$$259$$ −3288.67 −0.788988
$$260$$ 0 0
$$261$$ 561.904 0.133260
$$262$$ 0 0
$$263$$ −3956.41 −0.927614 −0.463807 0.885936i $$-0.653517\pi$$
−0.463807 + 0.885936i $$0.653517\pi$$
$$264$$ 0 0
$$265$$ −726.791 −0.168477
$$266$$ 0 0
$$267$$ −1499.53 −0.343707
$$268$$ 0 0
$$269$$ 2158.66 0.489279 0.244639 0.969614i $$-0.421330\pi$$
0.244639 + 0.969614i $$0.421330\pi$$
$$270$$ 0 0
$$271$$ 2321.77 0.520435 0.260217 0.965550i $$-0.416206\pi$$
0.260217 + 0.965550i $$0.416206\pi$$
$$272$$ 0 0
$$273$$ 182.305 0.0404161
$$274$$ 0 0
$$275$$ −2904.79 −0.636966
$$276$$ 0 0
$$277$$ −6184.24 −1.34143 −0.670713 0.741717i $$-0.734012\pi$$
−0.670713 + 0.741717i $$0.734012\pi$$
$$278$$ 0 0
$$279$$ 1752.77 0.376113
$$280$$ 0 0
$$281$$ −7981.21 −1.69437 −0.847187 0.531294i $$-0.821706\pi$$
−0.847187 + 0.531294i $$0.821706\pi$$
$$282$$ 0 0
$$283$$ 4851.71 1.01910 0.509548 0.860442i $$-0.329813\pi$$
0.509548 + 0.860442i $$0.329813\pi$$
$$284$$ 0 0
$$285$$ 390.447 0.0811511
$$286$$ 0 0
$$287$$ −8049.69 −1.65560
$$288$$ 0 0
$$289$$ −4709.91 −0.958664
$$290$$ 0 0
$$291$$ 4915.06 0.990123
$$292$$ 0 0
$$293$$ −1269.75 −0.253172 −0.126586 0.991956i $$-0.540402\pi$$
−0.126586 + 0.991956i $$0.540402\pi$$
$$294$$ 0 0
$$295$$ 1014.53 0.200231
$$296$$ 0 0
$$297$$ 4807.83 0.939322
$$298$$ 0 0
$$299$$ 350.838 0.0678579
$$300$$ 0 0
$$301$$ −2177.81 −0.417032
$$302$$ 0 0
$$303$$ 5669.08 1.07485
$$304$$ 0 0
$$305$$ −1722.91 −0.323454
$$306$$ 0 0
$$307$$ 9699.35 1.80316 0.901581 0.432610i $$-0.142407\pi$$
0.901581 + 0.432610i $$0.142407\pi$$
$$308$$ 0 0
$$309$$ −6182.30 −1.13818
$$310$$ 0 0
$$311$$ 323.284 0.0589446 0.0294723 0.999566i $$-0.490617\pi$$
0.0294723 + 0.999566i $$0.490617\pi$$
$$312$$ 0 0
$$313$$ −3251.29 −0.587137 −0.293569 0.955938i $$-0.594843\pi$$
−0.293569 + 0.955938i $$0.594843\pi$$
$$314$$ 0 0
$$315$$ 1848.41 0.330622
$$316$$ 0 0
$$317$$ −7512.25 −1.33101 −0.665505 0.746393i $$-0.731784\pi$$
−0.665505 + 0.746393i $$0.731784\pi$$
$$318$$ 0 0
$$319$$ −1171.27 −0.205575
$$320$$ 0 0
$$321$$ −1347.52 −0.234303
$$322$$ 0 0
$$323$$ −270.766 −0.0466434
$$324$$ 0 0
$$325$$ 263.348 0.0449474
$$326$$ 0 0
$$327$$ −3886.86 −0.657320
$$328$$ 0 0
$$329$$ −5545.22 −0.929233
$$330$$ 0 0
$$331$$ −2857.14 −0.474449 −0.237225 0.971455i $$-0.576238\pi$$
−0.237225 + 0.971455i $$0.576238\pi$$
$$332$$ 0 0
$$333$$ 2957.11 0.486633
$$334$$ 0 0
$$335$$ 512.193 0.0835346
$$336$$ 0 0
$$337$$ 10635.9 1.71921 0.859606 0.510958i $$-0.170709\pi$$
0.859606 + 0.510958i $$0.170709\pi$$
$$338$$ 0 0
$$339$$ −1668.50 −0.267317
$$340$$ 0 0
$$341$$ −3653.58 −0.580213
$$342$$ 0 0
$$343$$ −6486.96 −1.02117
$$344$$ 0 0
$$345$$ −2344.17 −0.365814
$$346$$ 0 0
$$347$$ 8357.08 1.29289 0.646443 0.762962i $$-0.276256\pi$$
0.646443 + 0.762962i $$0.276256\pi$$
$$348$$ 0 0
$$349$$ 7741.28 1.18734 0.593669 0.804709i $$-0.297679\pi$$
0.593669 + 0.804709i $$0.297679\pi$$
$$350$$ 0 0
$$351$$ −435.877 −0.0662831
$$352$$ 0 0
$$353$$ 2480.46 0.373999 0.186999 0.982360i $$-0.440124\pi$$
0.186999 + 0.982360i $$0.440124\pi$$
$$354$$ 0 0
$$355$$ 4853.60 0.725640
$$356$$ 0 0
$$357$$ 844.717 0.125230
$$358$$ 0 0
$$359$$ −4310.17 −0.633655 −0.316827 0.948483i $$-0.602618\pi$$
−0.316827 + 0.948483i $$0.602618\pi$$
$$360$$ 0 0
$$361$$ 361.000 0.0526316
$$362$$ 0 0
$$363$$ 589.923 0.0852973
$$364$$ 0 0
$$365$$ −933.670 −0.133892
$$366$$ 0 0
$$367$$ 8484.71 1.20681 0.603404 0.797436i $$-0.293811\pi$$
0.603404 + 0.797436i $$0.293811\pi$$
$$368$$ 0 0
$$369$$ 7238.15 1.02115
$$370$$ 0 0
$$371$$ 2096.39 0.293367
$$372$$ 0 0
$$373$$ 8039.16 1.11596 0.557978 0.829855i $$-0.311577\pi$$
0.557978 + 0.829855i $$0.311577\pi$$
$$374$$ 0 0
$$375$$ −4328.32 −0.596036
$$376$$ 0 0
$$377$$ 106.187 0.0145063
$$378$$ 0 0
$$379$$ 2976.22 0.403372 0.201686 0.979450i $$-0.435358\pi$$
0.201686 + 0.979450i $$0.435358\pi$$
$$380$$ 0 0
$$381$$ −3830.71 −0.515100
$$382$$ 0 0
$$383$$ 5118.15 0.682834 0.341417 0.939912i $$-0.389093\pi$$
0.341417 + 0.939912i $$0.389093\pi$$
$$384$$ 0 0
$$385$$ −3852.93 −0.510035
$$386$$ 0 0
$$387$$ 1958.25 0.257218
$$388$$ 0 0
$$389$$ −6758.74 −0.880929 −0.440465 0.897770i $$-0.645186\pi$$
−0.440465 + 0.897770i $$0.645186\pi$$
$$390$$ 0 0
$$391$$ 1625.62 0.210259
$$392$$ 0 0
$$393$$ −3446.23 −0.442339
$$394$$ 0 0
$$395$$ −3497.27 −0.445485
$$396$$ 0 0
$$397$$ 14529.0 1.83675 0.918373 0.395716i $$-0.129504\pi$$
0.918373 + 0.395716i $$0.129504\pi$$
$$398$$ 0 0
$$399$$ −1126.22 −0.141308
$$400$$ 0 0
$$401$$ 12416.4 1.54625 0.773124 0.634255i $$-0.218693\pi$$
0.773124 + 0.634255i $$0.218693\pi$$
$$402$$ 0 0
$$403$$ 331.233 0.0409426
$$404$$ 0 0
$$405$$ 155.017 0.0190195
$$406$$ 0 0
$$407$$ −6163.99 −0.750707
$$408$$ 0 0
$$409$$ 11615.8 1.40432 0.702159 0.712020i $$-0.252220\pi$$
0.702159 + 0.712020i $$0.252220\pi$$
$$410$$ 0 0
$$411$$ 6932.73 0.832035
$$412$$ 0 0
$$413$$ −2926.36 −0.348660
$$414$$ 0 0
$$415$$ 4821.25 0.570279
$$416$$ 0 0
$$417$$ −3654.47 −0.429161
$$418$$ 0 0
$$419$$ −16739.9 −1.95179 −0.975893 0.218248i $$-0.929966\pi$$
−0.975893 + 0.218248i $$0.929966\pi$$
$$420$$ 0 0
$$421$$ 6261.41 0.724852 0.362426 0.932013i $$-0.381949\pi$$
0.362426 + 0.932013i $$0.381949\pi$$
$$422$$ 0 0
$$423$$ 4986.17 0.573134
$$424$$ 0 0
$$425$$ 1220.23 0.139271
$$426$$ 0 0
$$427$$ 4969.64 0.563226
$$428$$ 0 0
$$429$$ 341.696 0.0384551
$$430$$ 0 0
$$431$$ 8893.01 0.993878 0.496939 0.867786i $$-0.334457\pi$$
0.496939 + 0.867786i $$0.334457\pi$$
$$432$$ 0 0
$$433$$ 6326.61 0.702166 0.351083 0.936344i $$-0.385814\pi$$
0.351083 + 0.936344i $$0.385814\pi$$
$$434$$ 0 0
$$435$$ −709.498 −0.0782019
$$436$$ 0 0
$$437$$ −2167.37 −0.237253
$$438$$ 0 0
$$439$$ −15823.4 −1.72030 −0.860149 0.510043i $$-0.829630\pi$$
−0.860149 + 0.510043i $$0.829630\pi$$
$$440$$ 0 0
$$441$$ 250.666 0.0270668
$$442$$ 0 0
$$443$$ 14056.3 1.50753 0.753763 0.657147i $$-0.228237\pi$$
0.753763 + 0.657147i $$0.228237\pi$$
$$444$$ 0 0
$$445$$ −2873.18 −0.306072
$$446$$ 0 0
$$447$$ −1548.03 −0.163802
$$448$$ 0 0
$$449$$ 15412.9 1.62000 0.809999 0.586431i $$-0.199468\pi$$
0.809999 + 0.586431i $$0.199468\pi$$
$$450$$ 0 0
$$451$$ −15087.6 −1.57527
$$452$$ 0 0
$$453$$ −5355.81 −0.555492
$$454$$ 0 0
$$455$$ 349.306 0.0359906
$$456$$ 0 0
$$457$$ 13336.9 1.36515 0.682573 0.730817i $$-0.260861\pi$$
0.682573 + 0.730817i $$0.260861\pi$$
$$458$$ 0 0
$$459$$ −2019.65 −0.205380
$$460$$ 0 0
$$461$$ −10226.9 −1.03322 −0.516612 0.856220i $$-0.672807\pi$$
−0.516612 + 0.856220i $$0.672807\pi$$
$$462$$ 0 0
$$463$$ 17797.4 1.78642 0.893212 0.449636i $$-0.148446\pi$$
0.893212 + 0.449636i $$0.148446\pi$$
$$464$$ 0 0
$$465$$ −2213.17 −0.220717
$$466$$ 0 0
$$467$$ 3100.03 0.307179 0.153589 0.988135i $$-0.450917\pi$$
0.153589 + 0.988135i $$0.450917\pi$$
$$468$$ 0 0
$$469$$ −1477.39 −0.145458
$$470$$ 0 0
$$471$$ 2573.77 0.251790
$$472$$ 0 0
$$473$$ −4081.89 −0.396798
$$474$$ 0 0
$$475$$ −1626.88 −0.157151
$$476$$ 0 0
$$477$$ −1885.04 −0.180943
$$478$$ 0 0
$$479$$ −6280.67 −0.599105 −0.299552 0.954080i $$-0.596837\pi$$
−0.299552 + 0.954080i $$0.596837\pi$$
$$480$$ 0 0
$$481$$ 558.826 0.0529735
$$482$$ 0 0
$$483$$ 6761.62 0.636987
$$484$$ 0 0
$$485$$ 9417.52 0.881706
$$486$$ 0 0
$$487$$ −7940.07 −0.738807 −0.369403 0.929269i $$-0.620438\pi$$
−0.369403 + 0.929269i $$0.620438\pi$$
$$488$$ 0 0
$$489$$ 4011.61 0.370984
$$490$$ 0 0
$$491$$ −17276.3 −1.58792 −0.793959 0.607971i $$-0.791984\pi$$
−0.793959 + 0.607971i $$0.791984\pi$$
$$492$$ 0 0
$$493$$ 492.020 0.0449482
$$494$$ 0 0
$$495$$ 3464.49 0.314580
$$496$$ 0 0
$$497$$ −13999.9 −1.26355
$$498$$ 0 0
$$499$$ −14939.6 −1.34026 −0.670128 0.742246i $$-0.733761\pi$$
−0.670128 + 0.742246i $$0.733761\pi$$
$$500$$ 0 0
$$501$$ 3508.28 0.312851
$$502$$ 0 0
$$503$$ 5559.39 0.492805 0.246402 0.969168i $$-0.420752\pi$$
0.246402 + 0.969168i $$0.420752\pi$$
$$504$$ 0 0
$$505$$ 10862.3 0.957157
$$506$$ 0 0
$$507$$ 7164.02 0.627545
$$508$$ 0 0
$$509$$ 11029.4 0.960452 0.480226 0.877145i $$-0.340555\pi$$
0.480226 + 0.877145i $$0.340555\pi$$
$$510$$ 0 0
$$511$$ 2693.12 0.233144
$$512$$ 0 0
$$513$$ 2692.71 0.231747
$$514$$ 0 0
$$515$$ −11845.6 −1.01355
$$516$$ 0 0
$$517$$ −10393.5 −0.884147
$$518$$ 0 0
$$519$$ −2704.49 −0.228736
$$520$$ 0 0
$$521$$ −7802.02 −0.656070 −0.328035 0.944665i $$-0.606386\pi$$
−0.328035 + 0.944665i $$0.606386\pi$$
$$522$$ 0 0
$$523$$ 23625.2 1.97525 0.987627 0.156820i $$-0.0501241\pi$$
0.987627 + 0.156820i $$0.0501241\pi$$
$$524$$ 0 0
$$525$$ 5075.44 0.421924
$$526$$ 0 0
$$527$$ 1534.78 0.126862
$$528$$ 0 0
$$529$$ 845.482 0.0694898
$$530$$ 0 0
$$531$$ 2631.33 0.215047
$$532$$ 0 0
$$533$$ 1367.84 0.111159
$$534$$ 0 0
$$535$$ −2581.93 −0.208647
$$536$$ 0 0
$$537$$ −1525.46 −0.122586
$$538$$ 0 0
$$539$$ −522.503 −0.0417548
$$540$$ 0 0
$$541$$ −24753.3 −1.96715 −0.983573 0.180509i $$-0.942226\pi$$
−0.983573 + 0.180509i $$0.942226\pi$$
$$542$$ 0 0
$$543$$ 3978.19 0.314402
$$544$$ 0 0
$$545$$ −7447.43 −0.585344
$$546$$ 0 0
$$547$$ −5655.38 −0.442060 −0.221030 0.975267i $$-0.570942\pi$$
−0.221030 + 0.975267i $$0.570942\pi$$
$$548$$ 0 0
$$549$$ −4468.61 −0.347388
$$550$$ 0 0
$$551$$ −655.989 −0.0507188
$$552$$ 0 0
$$553$$ 10087.7 0.775717
$$554$$ 0 0
$$555$$ −3733.86 −0.285574
$$556$$ 0 0
$$557$$ −14716.3 −1.11948 −0.559741 0.828668i $$-0.689099\pi$$
−0.559741 + 0.828668i $$0.689099\pi$$
$$558$$ 0 0
$$559$$ 370.063 0.0280000
$$560$$ 0 0
$$561$$ 1583.26 0.119154
$$562$$ 0 0
$$563$$ 4400.91 0.329443 0.164721 0.986340i $$-0.447328\pi$$
0.164721 + 0.986340i $$0.447328\pi$$
$$564$$ 0 0
$$565$$ −3196.94 −0.238046
$$566$$ 0 0
$$567$$ −447.140 −0.0331183
$$568$$ 0 0
$$569$$ 10176.3 0.749758 0.374879 0.927074i $$-0.377684\pi$$
0.374879 + 0.927074i $$0.377684\pi$$
$$570$$ 0 0
$$571$$ −6775.74 −0.496595 −0.248298 0.968684i $$-0.579871\pi$$
−0.248298 + 0.968684i $$0.579871\pi$$
$$572$$ 0 0
$$573$$ 507.427 0.0369949
$$574$$ 0 0
$$575$$ 9767.49 0.708404
$$576$$ 0 0
$$577$$ −3714.70 −0.268016 −0.134008 0.990980i $$-0.542785\pi$$
−0.134008 + 0.990980i $$0.542785\pi$$
$$578$$ 0 0
$$579$$ 9093.26 0.652683
$$580$$ 0 0
$$581$$ −13906.6 −0.993021
$$582$$ 0 0
$$583$$ 3929.29 0.279133
$$584$$ 0 0
$$585$$ −314.090 −0.0221983
$$586$$ 0 0
$$587$$ 9362.24 0.658298 0.329149 0.944278i $$-0.393238\pi$$
0.329149 + 0.944278i $$0.393238\pi$$
$$588$$ 0 0
$$589$$ −2046.26 −0.143149
$$590$$ 0 0
$$591$$ −15932.8 −1.10894
$$592$$ 0 0
$$593$$ −14437.5 −0.999793 −0.499897 0.866085i $$-0.666629\pi$$
−0.499897 + 0.866085i $$0.666629\pi$$
$$594$$ 0 0
$$595$$ 1618.52 0.111518
$$596$$ 0 0
$$597$$ −9872.02 −0.676775
$$598$$ 0 0
$$599$$ 19992.7 1.36374 0.681871 0.731473i $$-0.261167\pi$$
0.681871 + 0.731473i $$0.261167\pi$$
$$600$$ 0 0
$$601$$ 17371.4 1.17902 0.589511 0.807760i $$-0.299320\pi$$
0.589511 + 0.807760i $$0.299320\pi$$
$$602$$ 0 0
$$603$$ 1328.45 0.0897157
$$604$$ 0 0
$$605$$ 1130.32 0.0759574
$$606$$ 0 0
$$607$$ 14626.3 0.978025 0.489013 0.872277i $$-0.337357\pi$$
0.489013 + 0.872277i $$0.337357\pi$$
$$608$$ 0 0
$$609$$ 2046.51 0.136172
$$610$$ 0 0
$$611$$ 942.269 0.0623897
$$612$$ 0 0
$$613$$ −21052.2 −1.38710 −0.693549 0.720409i $$-0.743954\pi$$
−0.693549 + 0.720409i $$0.743954\pi$$
$$614$$ 0 0
$$615$$ −9139.39 −0.599245
$$616$$ 0 0
$$617$$ 1605.72 0.104771 0.0523855 0.998627i $$-0.483318\pi$$
0.0523855 + 0.998627i $$0.483318\pi$$
$$618$$ 0 0
$$619$$ −21457.6 −1.39330 −0.696652 0.717410i $$-0.745328\pi$$
−0.696652 + 0.717410i $$0.745328\pi$$
$$620$$ 0 0
$$621$$ −16166.5 −1.04467
$$622$$ 0 0
$$623$$ 8287.54 0.532959
$$624$$ 0 0
$$625$$ 2409.89 0.154233
$$626$$ 0 0
$$627$$ −2110.89 −0.134451
$$628$$ 0 0
$$629$$ 2589.34 0.164140
$$630$$ 0 0
$$631$$ −9732.44 −0.614013 −0.307007 0.951707i $$-0.599327\pi$$
−0.307007 + 0.951707i $$0.599327\pi$$
$$632$$ 0 0
$$633$$ 11247.6 0.706242
$$634$$ 0 0
$$635$$ −7339.84 −0.458697
$$636$$ 0 0
$$637$$ 47.3700 0.00294642
$$638$$ 0 0
$$639$$ 12588.5 0.779333
$$640$$ 0 0
$$641$$ −25594.0 −1.57707 −0.788536 0.614989i $$-0.789161\pi$$
−0.788536 + 0.614989i $$0.789161\pi$$
$$642$$ 0 0
$$643$$ 11938.5 0.732207 0.366104 0.930574i $$-0.380692\pi$$
0.366104 + 0.930574i $$0.380692\pi$$
$$644$$ 0 0
$$645$$ −2472.62 −0.150945
$$646$$ 0 0
$$647$$ −15685.4 −0.953105 −0.476552 0.879146i $$-0.658114\pi$$
−0.476552 + 0.879146i $$0.658114\pi$$
$$648$$ 0 0
$$649$$ −5484.91 −0.331743
$$650$$ 0 0
$$651$$ 6383.77 0.384331
$$652$$ 0 0
$$653$$ 11414.3 0.684037 0.342019 0.939693i $$-0.388889\pi$$
0.342019 + 0.939693i $$0.388889\pi$$
$$654$$ 0 0
$$655$$ −6603.16 −0.393903
$$656$$ 0 0
$$657$$ −2421.61 −0.143799
$$658$$ 0 0
$$659$$ −24500.2 −1.44824 −0.724122 0.689672i $$-0.757755\pi$$
−0.724122 + 0.689672i $$0.757755\pi$$
$$660$$ 0 0
$$661$$ 24821.0 1.46055 0.730276 0.683152i $$-0.239391\pi$$
0.730276 + 0.683152i $$0.239391\pi$$
$$662$$ 0 0
$$663$$ −143.538 −0.00840808
$$664$$ 0 0
$$665$$ −2157.90 −0.125835
$$666$$ 0 0
$$667$$ 3938.43 0.228631
$$668$$ 0 0
$$669$$ −7361.14 −0.425408
$$670$$ 0 0
$$671$$ 9314.65 0.535899
$$672$$ 0 0
$$673$$ −14862.9 −0.851297 −0.425649 0.904888i $$-0.639954\pi$$
−0.425649 + 0.904888i $$0.639954\pi$$
$$674$$ 0 0
$$675$$ −12135.0 −0.691964
$$676$$ 0 0
$$677$$ −1218.79 −0.0691902 −0.0345951 0.999401i $$-0.511014\pi$$
−0.0345951 + 0.999401i $$0.511014\pi$$
$$678$$ 0 0
$$679$$ −27164.3 −1.53530
$$680$$ 0 0
$$681$$ 6169.33 0.347150
$$682$$ 0 0
$$683$$ 10180.0 0.570319 0.285159 0.958480i $$-0.407953\pi$$
0.285159 + 0.958480i $$0.407953\pi$$
$$684$$ 0 0
$$685$$ 13283.5 0.740929
$$686$$ 0 0
$$687$$ −11771.7 −0.653736
$$688$$ 0 0
$$689$$ −356.228 −0.0196970
$$690$$ 0 0
$$691$$ −13712.9 −0.754941 −0.377470 0.926022i $$-0.623206\pi$$
−0.377470 + 0.926022i $$0.623206\pi$$
$$692$$ 0 0
$$693$$ −9993.14 −0.547775
$$694$$ 0 0
$$695$$ −7002.16 −0.382168
$$696$$ 0 0
$$697$$ 6337.95 0.344429
$$698$$ 0 0
$$699$$ 18110.3 0.979964
$$700$$ 0 0
$$701$$ −29853.2 −1.60848 −0.804238 0.594307i $$-0.797426\pi$$
−0.804238 + 0.594307i $$0.797426\pi$$
$$702$$ 0 0
$$703$$ −3452.26 −0.185212
$$704$$ 0 0
$$705$$ −6295.88 −0.336335
$$706$$ 0 0
$$707$$ −31331.6 −1.66669
$$708$$ 0 0
$$709$$ −11488.4 −0.608541 −0.304271 0.952586i $$-0.598413\pi$$
−0.304271 + 0.952586i $$0.598413\pi$$
$$710$$ 0 0
$$711$$ −9070.67 −0.478448
$$712$$ 0 0
$$713$$ 12285.3 0.645286
$$714$$ 0 0
$$715$$ 654.708 0.0342443
$$716$$ 0 0
$$717$$ 8474.91 0.441424
$$718$$ 0 0
$$719$$ 17536.5 0.909596 0.454798 0.890595i $$-0.349711\pi$$
0.454798 + 0.890595i $$0.349711\pi$$
$$720$$ 0 0
$$721$$ 34168.1 1.76489
$$722$$ 0 0
$$723$$ 11712.6 0.602484
$$724$$ 0 0
$$725$$ 2956.28 0.151439
$$726$$ 0 0
$$727$$ −27607.0 −1.40837 −0.704185 0.710016i $$-0.748688\pi$$
−0.704185 + 0.710016i $$0.748688\pi$$
$$728$$ 0 0
$$729$$ 12933.5 0.657089
$$730$$ 0 0
$$731$$ 1714.70 0.0867587
$$732$$ 0 0
$$733$$ −3519.63 −0.177354 −0.0886770 0.996060i $$-0.528264\pi$$
−0.0886770 + 0.996060i $$0.528264\pi$$
$$734$$ 0 0
$$735$$ −316.508 −0.0158838
$$736$$ 0 0
$$737$$ −2769.09 −0.138400
$$738$$ 0 0
$$739$$ −4557.41 −0.226857 −0.113428 0.993546i $$-0.536183\pi$$
−0.113428 + 0.993546i $$0.536183\pi$$
$$740$$ 0 0
$$741$$ 191.373 0.00948754
$$742$$ 0 0
$$743$$ −5330.53 −0.263201 −0.131600 0.991303i $$-0.542012\pi$$
−0.131600 + 0.991303i $$0.542012\pi$$
$$744$$ 0 0
$$745$$ −2966.11 −0.145866
$$746$$ 0 0
$$747$$ 12504.6 0.612477
$$748$$ 0 0
$$749$$ 7447.43 0.363315
$$750$$ 0 0
$$751$$ −22386.8 −1.08776 −0.543879 0.839163i $$-0.683045\pi$$
−0.543879 + 0.839163i $$0.683045\pi$$
$$752$$ 0 0
$$753$$ 19541.0 0.945703
$$754$$ 0 0
$$755$$ −10262.0 −0.494666
$$756$$ 0 0
$$757$$ 25315.6 1.21547 0.607736 0.794139i $$-0.292078\pi$$
0.607736 + 0.794139i $$0.292078\pi$$
$$758$$ 0 0
$$759$$ 12673.4 0.606080
$$760$$ 0 0
$$761$$ −18715.0 −0.891485 −0.445742 0.895161i $$-0.647060\pi$$
−0.445742 + 0.895161i $$0.647060\pi$$
$$762$$ 0 0
$$763$$ 21481.7 1.01925
$$764$$ 0 0
$$765$$ −1455.35 −0.0687820
$$766$$ 0 0
$$767$$ 497.260 0.0234094
$$768$$ 0 0
$$769$$ 20985.3 0.984070 0.492035 0.870575i $$-0.336253\pi$$
0.492035 + 0.870575i $$0.336253\pi$$
$$770$$ 0 0
$$771$$ −12579.0 −0.587576
$$772$$ 0 0
$$773$$ 6302.27 0.293243 0.146622 0.989193i $$-0.453160\pi$$
0.146622 + 0.989193i $$0.453160\pi$$
$$774$$ 0 0
$$775$$ 9221.66 0.427422
$$776$$ 0 0
$$777$$ 10770.1 0.497266
$$778$$ 0 0
$$779$$ −8450.11 −0.388648
$$780$$ 0 0
$$781$$ −26240.3 −1.20224
$$782$$ 0 0
$$783$$ −4893.05 −0.223325
$$784$$ 0 0
$$785$$ 4931.48 0.224219
$$786$$ 0 0
$$787$$ −4124.73 −0.186825 −0.0934123 0.995628i $$-0.529777\pi$$
−0.0934123 + 0.995628i $$0.529777\pi$$
$$788$$ 0 0
$$789$$ 12956.9 0.584636
$$790$$ 0 0
$$791$$ 9221.39 0.414507
$$792$$ 0 0
$$793$$ −844.464 −0.0378156
$$794$$ 0 0
$$795$$ 2380.18 0.106184
$$796$$ 0 0
$$797$$ 8090.48 0.359573 0.179786 0.983706i $$-0.442459\pi$$
0.179786 + 0.983706i $$0.442459\pi$$
$$798$$ 0 0
$$799$$ 4366.04 0.193316
$$800$$ 0 0
$$801$$ −7452.02 −0.328719
$$802$$ 0 0
$$803$$ 5047.75 0.221832
$$804$$ 0 0
$$805$$ 12955.6 0.567237
$$806$$ 0 0
$$807$$ −7069.44 −0.308372
$$808$$ 0 0
$$809$$ −16735.9 −0.727323 −0.363662 0.931531i $$-0.618474\pi$$
−0.363662 + 0.931531i $$0.618474\pi$$
$$810$$ 0 0
$$811$$ −8476.59 −0.367020 −0.183510 0.983018i $$-0.558746\pi$$
−0.183510 + 0.983018i $$0.558746\pi$$
$$812$$ 0 0
$$813$$ −7603.62 −0.328008
$$814$$ 0 0
$$815$$ 7686.46 0.330362
$$816$$ 0 0
$$817$$ −2286.14 −0.0978970
$$818$$ 0 0
$$819$$ 905.975 0.0386537
$$820$$ 0 0
$$821$$ 902.399 0.0383605 0.0191802 0.999816i $$-0.493894\pi$$
0.0191802 + 0.999816i $$0.493894\pi$$
$$822$$ 0 0
$$823$$ 23266.2 0.985431 0.492716 0.870190i $$-0.336004\pi$$
0.492716 + 0.870190i $$0.336004\pi$$
$$824$$ 0 0
$$825$$ 9512.96 0.401453
$$826$$ 0 0
$$827$$ 17103.5 0.719160 0.359580 0.933114i $$-0.382920\pi$$
0.359580 + 0.933114i $$0.382920\pi$$
$$828$$ 0 0
$$829$$ −6125.30 −0.256623 −0.128312 0.991734i $$-0.540956\pi$$
−0.128312 + 0.991734i $$0.540956\pi$$
$$830$$ 0 0
$$831$$ 20252.9 0.845445
$$832$$ 0 0
$$833$$ 219.491 0.00912955
$$834$$ 0 0
$$835$$ 6722.05 0.278594
$$836$$ 0 0
$$837$$ −15263.1 −0.630311
$$838$$ 0 0
$$839$$ −19918.5 −0.819620 −0.409810 0.912171i $$-0.634405\pi$$
−0.409810 + 0.912171i $$0.634405\pi$$
$$840$$ 0 0
$$841$$ −23197.0 −0.951124
$$842$$ 0 0
$$843$$ 26137.8 1.06789
$$844$$ 0 0
$$845$$ 13726.6 0.558829
$$846$$ 0 0
$$847$$ −3260.36 −0.132264
$$848$$ 0 0
$$849$$ −15889.0 −0.642294
$$850$$ 0 0
$$851$$ 20726.7 0.834901
$$852$$ 0 0
$$853$$ 18785.3 0.754042 0.377021 0.926205i $$-0.376948\pi$$
0.377021 + 0.926205i $$0.376948\pi$$
$$854$$ 0 0
$$855$$ 1940.35 0.0776124
$$856$$ 0 0
$$857$$ −33483.4 −1.33462 −0.667311 0.744779i $$-0.732555\pi$$
−0.667311 + 0.744779i $$0.732555\pi$$
$$858$$ 0 0
$$859$$ −7322.74 −0.290860 −0.145430 0.989369i $$-0.546457\pi$$
−0.145430 + 0.989369i $$0.546457\pi$$
$$860$$ 0 0
$$861$$ 26362.1 1.04346
$$862$$ 0 0
$$863$$ −905.563 −0.0357193 −0.0178596 0.999841i $$-0.505685\pi$$
−0.0178596 + 0.999841i $$0.505685\pi$$
$$864$$ 0 0
$$865$$ −5181.96 −0.203690
$$866$$ 0 0
$$867$$ 15424.6 0.604205
$$868$$ 0 0
$$869$$ 18907.5 0.738080
$$870$$ 0 0
$$871$$ 251.045 0.00976619
$$872$$ 0 0
$$873$$ 24425.7 0.946947
$$874$$ 0 0
$$875$$ 23921.6 0.924224
$$876$$ 0 0
$$877$$ −27155.7 −1.04559 −0.522795 0.852459i $$-0.675111\pi$$
−0.522795 + 0.852459i $$0.675111\pi$$
$$878$$ 0 0
$$879$$ 4158.32 0.159564
$$880$$ 0 0
$$881$$ 26094.1 0.997881 0.498941 0.866636i $$-0.333723\pi$$
0.498941 + 0.866636i $$0.333723\pi$$
$$882$$ 0 0
$$883$$ −29674.6 −1.13095 −0.565476 0.824765i $$-0.691307\pi$$
−0.565476 + 0.824765i $$0.691307\pi$$
$$884$$ 0 0
$$885$$ −3322.50 −0.126197
$$886$$ 0 0
$$887$$ 1734.72 0.0656665 0.0328333 0.999461i $$-0.489547\pi$$
0.0328333 + 0.999461i $$0.489547\pi$$
$$888$$ 0 0
$$889$$ 21171.4 0.798724
$$890$$ 0 0
$$891$$ −838.079 −0.0315115
$$892$$ 0 0
$$893$$ −5821.05 −0.218135
$$894$$ 0 0
$$895$$ −2922.87 −0.109163
$$896$$ 0 0
$$897$$ −1148.97 −0.0427680
$$898$$ 0 0
$$899$$ 3718.34 0.137946
$$900$$ 0 0
$$901$$ −1650.60 −0.0610315
$$902$$ 0 0
$$903$$ 7132.14 0.262838
$$904$$ 0 0
$$905$$ 7622.42 0.279975
$$906$$ 0 0
$$907$$ −6755.01 −0.247295 −0.123647 0.992326i $$-0.539459\pi$$
−0.123647 + 0.992326i $$0.539459\pi$$
$$908$$ 0 0
$$909$$ 28172.9 1.02798
$$910$$ 0 0
$$911$$ 35150.7 1.27837 0.639184 0.769054i $$-0.279272\pi$$
0.639184 + 0.769054i $$0.279272\pi$$
$$912$$ 0 0
$$913$$ −26065.4 −0.944840
$$914$$ 0 0
$$915$$ 5642.38 0.203859
$$916$$ 0 0
$$917$$ 19046.5 0.685899
$$918$$ 0 0
$$919$$ 28684.2 1.02960 0.514802 0.857309i $$-0.327866\pi$$
0.514802 + 0.857309i $$0.327866\pi$$
$$920$$ 0 0
$$921$$ −31764.6 −1.13646
$$922$$ 0 0
$$923$$ 2378.94 0.0848360
$$924$$ 0 0
$$925$$ 15557.9 0.553018
$$926$$ 0 0
$$927$$ −30723.4 −1.08855
$$928$$ 0 0
$$929$$ −3762.58 −0.132881 −0.0664405 0.997790i $$-0.521164\pi$$
−0.0664405 + 0.997790i $$0.521164\pi$$
$$930$$ 0 0
$$931$$ −292.638 −0.0103016
$$932$$ 0 0
$$933$$ −1058.73 −0.0371503
$$934$$ 0 0
$$935$$ 3033.62 0.106107
$$936$$ 0 0
$$937$$ 25465.7 0.887863 0.443932 0.896061i $$-0.353583\pi$$
0.443932 + 0.896061i $$0.353583\pi$$
$$938$$ 0 0
$$939$$ 10647.7 0.370048
$$940$$ 0 0
$$941$$ 40865.6 1.41571 0.707854 0.706359i $$-0.249663\pi$$
0.707854 + 0.706359i $$0.249663\pi$$
$$942$$ 0 0
$$943$$ 50732.8 1.75195
$$944$$ 0 0
$$945$$ −16095.9 −0.554074
$$946$$ 0 0
$$947$$ 8047.62 0.276148 0.138074 0.990422i $$-0.455909\pi$$
0.138074 + 0.990422i $$0.455909\pi$$
$$948$$ 0 0
$$949$$ −457.628 −0.0156536
$$950$$ 0 0
$$951$$ 24602.0 0.838880
$$952$$ 0 0
$$953$$ −21562.2 −0.732915 −0.366458 0.930435i $$-0.619430\pi$$
−0.366458 + 0.930435i $$0.619430\pi$$
$$954$$ 0 0
$$955$$ 972.257 0.0329440
$$956$$ 0 0
$$957$$ 3835.80 0.129565
$$958$$ 0 0
$$959$$ −38315.5 −1.29017
$$960$$ 0 0
$$961$$ −18192.2 −0.610661
$$962$$ 0 0
$$963$$ −6696.60 −0.224086
$$964$$ 0 0
$$965$$ 17423.2 0.581215
$$966$$ 0 0
$$967$$ 14667.4 0.487767 0.243883 0.969805i $$-0.421579\pi$$
0.243883 + 0.969805i $$0.421579\pi$$
$$968$$ 0 0
$$969$$ 886.735 0.0293974
$$970$$ 0 0
$$971$$ 60154.7 1.98811 0.994056 0.108867i $$-0.0347221\pi$$
0.994056 + 0.108867i $$0.0347221\pi$$
$$972$$ 0 0
$$973$$ 20197.4 0.665465
$$974$$ 0 0
$$975$$ −862.442 −0.0283285
$$976$$ 0 0
$$977$$ −29474.6 −0.965176 −0.482588 0.875848i $$-0.660303\pi$$
−0.482588 + 0.875848i $$0.660303\pi$$
$$978$$ 0 0
$$979$$ 15533.4 0.507100
$$980$$ 0 0
$$981$$ −19316.0 −0.628657
$$982$$ 0 0
$$983$$ −6813.42 −0.221073 −0.110536 0.993872i $$-0.535257\pi$$
−0.110536 + 0.993872i $$0.535257\pi$$
$$984$$ 0 0
$$985$$ −30528.0 −0.987517
$$986$$ 0 0
$$987$$ 18160.1 0.585657
$$988$$ 0 0
$$989$$ 13725.5 0.441301
$$990$$ 0 0
$$991$$ 11660.9 0.373786 0.186893 0.982380i $$-0.440158\pi$$
0.186893 + 0.982380i $$0.440158\pi$$
$$992$$ 0 0
$$993$$ 9356.91 0.299026
$$994$$ 0 0
$$995$$ −18915.3 −0.602669
$$996$$ 0 0
$$997$$ −37537.3 −1.19240 −0.596198 0.802838i $$-0.703323\pi$$
−0.596198 + 0.802838i $$0.703323\pi$$
$$998$$ 0 0
$$999$$ −25750.5 −0.815526
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 304.4.a.e.1.1 2
4.3 odd 2 152.4.a.a.1.2 2
8.3 odd 2 1216.4.a.m.1.1 2
8.5 even 2 1216.4.a.k.1.2 2
12.11 even 2 1368.4.a.a.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
152.4.a.a.1.2 2 4.3 odd 2
304.4.a.e.1.1 2 1.1 even 1 trivial
1216.4.a.k.1.2 2 8.5 even 2
1216.4.a.m.1.1 2 8.3 odd 2
1368.4.a.a.1.2 2 12.11 even 2