# Properties

 Label 1344.4.a.bo.1.2 Level $1344$ Weight $4$ Character 1344.1 Self dual yes Analytic conductor $79.299$ Analytic rank $0$ 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] = [1344,4,Mod(1,1344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1344, 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("1344.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1344 = 2^{6} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1344.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: $$79.2985670477$$ Analytic rank: $$0$$ 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 21) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-3.27492$$ of defining polynomial Character $$\chi$$ $$=$$ 1344.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.00000 q^{3} +4.54983 q^{5} -7.00000 q^{7} +9.00000 q^{9} +O(q^{10})$$ $$q+3.00000 q^{3} +4.54983 q^{5} -7.00000 q^{7} +9.00000 q^{9} -40.7492 q^{11} -53.2990 q^{13} +13.6495 q^{15} +4.54983 q^{17} +122.598 q^{19} -21.0000 q^{21} -131.347 q^{23} -104.299 q^{25} +27.0000 q^{27} +216.598 q^{29} +251.794 q^{31} -122.248 q^{33} -31.8488 q^{35} -11.8970 q^{37} -159.897 q^{39} -111.752 q^{41} +369.196 q^{43} +40.9485 q^{45} +262.694 q^{47} +49.0000 q^{49} +13.6495 q^{51} +567.100 q^{53} -185.402 q^{55} +367.794 q^{57} +839.890 q^{59} +485.794 q^{61} -63.0000 q^{63} -242.502 q^{65} -333.691 q^{67} -394.042 q^{69} -590.248 q^{71} +490.701 q^{73} -312.897 q^{75} +285.244 q^{77} -121.691 q^{79} +81.0000 q^{81} +609.608 q^{83} +20.7010 q^{85} +649.794 q^{87} +719.038 q^{89} +373.093 q^{91} +755.382 q^{93} +557.801 q^{95} -637.877 q^{97} -366.743 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{3} - 6 q^{5} - 14 q^{7} + 18 q^{9}+O(q^{10})$$ 2 * q + 6 * q^3 - 6 * q^5 - 14 * q^7 + 18 * q^9 $$2 q + 6 q^{3} - 6 q^{5} - 14 q^{7} + 18 q^{9} - 6 q^{11} - 16 q^{13} - 18 q^{15} - 6 q^{17} + 64 q^{19} - 42 q^{21} - 6 q^{23} - 118 q^{25} + 54 q^{27} + 252 q^{29} - 40 q^{31} - 18 q^{33} + 42 q^{35} + 248 q^{37} - 48 q^{39} - 450 q^{41} + 376 q^{43} - 54 q^{45} + 12 q^{47} + 98 q^{49} - 18 q^{51} + 1104 q^{53} - 552 q^{55} + 192 q^{57} + 804 q^{59} + 428 q^{61} - 126 q^{63} - 636 q^{65} + 148 q^{67} - 18 q^{69} - 954 q^{71} + 1072 q^{73} - 354 q^{75} + 42 q^{77} + 572 q^{79} + 162 q^{81} + 1944 q^{83} + 132 q^{85} + 756 q^{87} + 366 q^{89} + 112 q^{91} - 120 q^{93} + 1176 q^{95} + 808 q^{97} - 54 q^{99}+O(q^{100})$$ 2 * q + 6 * q^3 - 6 * q^5 - 14 * q^7 + 18 * q^9 - 6 * q^11 - 16 * q^13 - 18 * q^15 - 6 * q^17 + 64 * q^19 - 42 * q^21 - 6 * q^23 - 118 * q^25 + 54 * q^27 + 252 * q^29 - 40 * q^31 - 18 * q^33 + 42 * q^35 + 248 * q^37 - 48 * q^39 - 450 * q^41 + 376 * q^43 - 54 * q^45 + 12 * q^47 + 98 * q^49 - 18 * q^51 + 1104 * q^53 - 552 * q^55 + 192 * q^57 + 804 * q^59 + 428 * q^61 - 126 * q^63 - 636 * q^65 + 148 * q^67 - 18 * q^69 - 954 * q^71 + 1072 * q^73 - 354 * q^75 + 42 * q^77 + 572 * q^79 + 162 * q^81 + 1944 * q^83 + 132 * q^85 + 756 * q^87 + 366 * q^89 + 112 * q^91 - 120 * q^93 + 1176 * q^95 + 808 * q^97 - 54 * 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
$$5$$ 4.54983 0.406950 0.203475 0.979080i $$-0.434777\pi$$
0.203475 + 0.979080i $$0.434777\pi$$
$$6$$ 0 0
$$7$$ −7.00000 −0.377964
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ −40.7492 −1.11694 −0.558470 0.829525i $$-0.688611\pi$$
−0.558470 + 0.829525i $$0.688611\pi$$
$$12$$ 0 0
$$13$$ −53.2990 −1.13711 −0.568557 0.822644i $$-0.692498\pi$$
−0.568557 + 0.822644i $$0.692498\pi$$
$$14$$ 0 0
$$15$$ 13.6495 0.234952
$$16$$ 0 0
$$17$$ 4.54983 0.0649116 0.0324558 0.999473i $$-0.489667\pi$$
0.0324558 + 0.999473i $$0.489667\pi$$
$$18$$ 0 0
$$19$$ 122.598 1.48031 0.740156 0.672436i $$-0.234752\pi$$
0.740156 + 0.672436i $$0.234752\pi$$
$$20$$ 0 0
$$21$$ −21.0000 −0.218218
$$22$$ 0 0
$$23$$ −131.347 −1.19077 −0.595387 0.803439i $$-0.703001\pi$$
−0.595387 + 0.803439i $$0.703001\pi$$
$$24$$ 0 0
$$25$$ −104.299 −0.834392
$$26$$ 0 0
$$27$$ 27.0000 0.192450
$$28$$ 0 0
$$29$$ 216.598 1.38694 0.693470 0.720486i $$-0.256081\pi$$
0.693470 + 0.720486i $$0.256081\pi$$
$$30$$ 0 0
$$31$$ 251.794 1.45882 0.729412 0.684075i $$-0.239794\pi$$
0.729412 + 0.684075i $$0.239794\pi$$
$$32$$ 0 0
$$33$$ −122.248 −0.644865
$$34$$ 0 0
$$35$$ −31.8488 −0.153812
$$36$$ 0 0
$$37$$ −11.8970 −0.0528610 −0.0264305 0.999651i $$-0.508414\pi$$
−0.0264305 + 0.999651i $$0.508414\pi$$
$$38$$ 0 0
$$39$$ −159.897 −0.656513
$$40$$ 0 0
$$41$$ −111.752 −0.425678 −0.212839 0.977087i $$-0.568271\pi$$
−0.212839 + 0.977087i $$0.568271\pi$$
$$42$$ 0 0
$$43$$ 369.196 1.30935 0.654673 0.755912i $$-0.272806\pi$$
0.654673 + 0.755912i $$0.272806\pi$$
$$44$$ 0 0
$$45$$ 40.9485 0.135650
$$46$$ 0 0
$$47$$ 262.694 0.815275 0.407637 0.913144i $$-0.366353\pi$$
0.407637 + 0.913144i $$0.366353\pi$$
$$48$$ 0 0
$$49$$ 49.0000 0.142857
$$50$$ 0 0
$$51$$ 13.6495 0.0374767
$$52$$ 0 0
$$53$$ 567.100 1.46976 0.734879 0.678199i $$-0.237239\pi$$
0.734879 + 0.678199i $$0.237239\pi$$
$$54$$ 0 0
$$55$$ −185.402 −0.454538
$$56$$ 0 0
$$57$$ 367.794 0.854658
$$58$$ 0 0
$$59$$ 839.890 1.85330 0.926648 0.375931i $$-0.122677\pi$$
0.926648 + 0.375931i $$0.122677\pi$$
$$60$$ 0 0
$$61$$ 485.794 1.01966 0.509832 0.860274i $$-0.329707\pi$$
0.509832 + 0.860274i $$0.329707\pi$$
$$62$$ 0 0
$$63$$ −63.0000 −0.125988
$$64$$ 0 0
$$65$$ −242.502 −0.462748
$$66$$ 0 0
$$67$$ −333.691 −0.608460 −0.304230 0.952599i $$-0.598399\pi$$
−0.304230 + 0.952599i $$0.598399\pi$$
$$68$$ 0 0
$$69$$ −394.042 −0.687493
$$70$$ 0 0
$$71$$ −590.248 −0.986613 −0.493306 0.869856i $$-0.664212\pi$$
−0.493306 + 0.869856i $$0.664212\pi$$
$$72$$ 0 0
$$73$$ 490.701 0.786743 0.393371 0.919380i $$-0.371309\pi$$
0.393371 + 0.919380i $$0.371309\pi$$
$$74$$ 0 0
$$75$$ −312.897 −0.481736
$$76$$ 0 0
$$77$$ 285.244 0.422164
$$78$$ 0 0
$$79$$ −121.691 −0.173308 −0.0866539 0.996238i $$-0.527617\pi$$
−0.0866539 + 0.996238i $$0.527617\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ 609.608 0.806183 0.403091 0.915160i $$-0.367936\pi$$
0.403091 + 0.915160i $$0.367936\pi$$
$$84$$ 0 0
$$85$$ 20.7010 0.0264157
$$86$$ 0 0
$$87$$ 649.794 0.800750
$$88$$ 0 0
$$89$$ 719.038 0.856381 0.428190 0.903689i $$-0.359151\pi$$
0.428190 + 0.903689i $$0.359151\pi$$
$$90$$ 0 0
$$91$$ 373.093 0.429789
$$92$$ 0 0
$$93$$ 755.382 0.842252
$$94$$ 0 0
$$95$$ 557.801 0.602412
$$96$$ 0 0
$$97$$ −637.877 −0.667697 −0.333849 0.942627i $$-0.608347\pi$$
−0.333849 + 0.942627i $$0.608347\pi$$
$$98$$ 0 0
$$99$$ −366.743 −0.372313
$$100$$ 0 0
$$101$$ −671.148 −0.661205 −0.330603 0.943770i $$-0.607252\pi$$
−0.330603 + 0.943770i $$0.607252\pi$$
$$102$$ 0 0
$$103$$ 912.412 0.872841 0.436420 0.899743i $$-0.356246\pi$$
0.436420 + 0.899743i $$0.356246\pi$$
$$104$$ 0 0
$$105$$ −95.5465 −0.0888037
$$106$$ 0 0
$$107$$ −116.736 −0.105470 −0.0527350 0.998609i $$-0.516794\pi$$
−0.0527350 + 0.998609i $$0.516794\pi$$
$$108$$ 0 0
$$109$$ −837.176 −0.735660 −0.367830 0.929893i $$-0.619899\pi$$
−0.367830 + 0.929893i $$0.619899\pi$$
$$110$$ 0 0
$$111$$ −35.6911 −0.0305193
$$112$$ 0 0
$$113$$ −1086.58 −0.904572 −0.452286 0.891873i $$-0.649391\pi$$
−0.452286 + 0.891873i $$0.649391\pi$$
$$114$$ 0 0
$$115$$ −597.608 −0.484585
$$116$$ 0 0
$$117$$ −479.691 −0.379038
$$118$$ 0 0
$$119$$ −31.8488 −0.0245343
$$120$$ 0 0
$$121$$ 329.495 0.247554
$$122$$ 0 0
$$123$$ −335.257 −0.245765
$$124$$ 0 0
$$125$$ −1043.27 −0.746505
$$126$$ 0 0
$$127$$ 537.113 0.375284 0.187642 0.982237i $$-0.439916\pi$$
0.187642 + 0.982237i $$0.439916\pi$$
$$128$$ 0 0
$$129$$ 1107.59 0.755951
$$130$$ 0 0
$$131$$ 1497.39 0.998683 0.499341 0.866405i $$-0.333575\pi$$
0.499341 + 0.866405i $$0.333575\pi$$
$$132$$ 0 0
$$133$$ −858.186 −0.559505
$$134$$ 0 0
$$135$$ 122.846 0.0783175
$$136$$ 0 0
$$137$$ −1380.09 −0.860650 −0.430325 0.902674i $$-0.641601\pi$$
−0.430325 + 0.902674i $$0.641601\pi$$
$$138$$ 0 0
$$139$$ −141.980 −0.0866374 −0.0433187 0.999061i $$-0.513793\pi$$
−0.0433187 + 0.999061i $$0.513793\pi$$
$$140$$ 0 0
$$141$$ 788.083 0.470699
$$142$$ 0 0
$$143$$ 2171.89 1.27009
$$144$$ 0 0
$$145$$ 985.485 0.564414
$$146$$ 0 0
$$147$$ 147.000 0.0824786
$$148$$ 0 0
$$149$$ 1943.87 1.06878 0.534390 0.845238i $$-0.320542\pi$$
0.534390 + 0.845238i $$0.320542\pi$$
$$150$$ 0 0
$$151$$ 2654.76 1.43074 0.715370 0.698746i $$-0.246258\pi$$
0.715370 + 0.698746i $$0.246258\pi$$
$$152$$ 0 0
$$153$$ 40.9485 0.0216372
$$154$$ 0 0
$$155$$ 1145.62 0.593668
$$156$$ 0 0
$$157$$ −1665.22 −0.846489 −0.423244 0.906016i $$-0.639109\pi$$
−0.423244 + 0.906016i $$0.639109\pi$$
$$158$$ 0 0
$$159$$ 1701.30 0.848565
$$160$$ 0 0
$$161$$ 919.430 0.450070
$$162$$ 0 0
$$163$$ −33.0732 −0.0158926 −0.00794629 0.999968i $$-0.502529\pi$$
−0.00794629 + 0.999968i $$0.502529\pi$$
$$164$$ 0 0
$$165$$ −556.206 −0.262428
$$166$$ 0 0
$$167$$ −1654.48 −0.766630 −0.383315 0.923618i $$-0.625218\pi$$
−0.383315 + 0.923618i $$0.625218\pi$$
$$168$$ 0 0
$$169$$ 643.784 0.293029
$$170$$ 0 0
$$171$$ 1103.38 0.493437
$$172$$ 0 0
$$173$$ −64.1909 −0.0282101 −0.0141050 0.999901i $$-0.504490\pi$$
−0.0141050 + 0.999901i $$0.504490\pi$$
$$174$$ 0 0
$$175$$ 730.093 0.315371
$$176$$ 0 0
$$177$$ 2519.67 1.07000
$$178$$ 0 0
$$179$$ 3914.68 1.63462 0.817309 0.576200i $$-0.195465\pi$$
0.817309 + 0.576200i $$0.195465\pi$$
$$180$$ 0 0
$$181$$ 2058.04 0.845156 0.422578 0.906327i $$-0.361125\pi$$
0.422578 + 0.906327i $$0.361125\pi$$
$$182$$ 0 0
$$183$$ 1457.38 0.588704
$$184$$ 0 0
$$185$$ −54.1295 −0.0215118
$$186$$ 0 0
$$187$$ −185.402 −0.0725023
$$188$$ 0 0
$$189$$ −189.000 −0.0727393
$$190$$ 0 0
$$191$$ −428.048 −0.162160 −0.0810798 0.996708i $$-0.525837\pi$$
−0.0810798 + 0.996708i $$0.525837\pi$$
$$192$$ 0 0
$$193$$ 1604.93 0.598576 0.299288 0.954163i $$-0.403251\pi$$
0.299288 + 0.954163i $$0.403251\pi$$
$$194$$ 0 0
$$195$$ −727.505 −0.267168
$$196$$ 0 0
$$197$$ −3738.83 −1.35218 −0.676092 0.736817i $$-0.736328\pi$$
−0.676092 + 0.736817i $$0.736328\pi$$
$$198$$ 0 0
$$199$$ 349.030 0.124332 0.0621660 0.998066i $$-0.480199\pi$$
0.0621660 + 0.998066i $$0.480199\pi$$
$$200$$ 0 0
$$201$$ −1001.07 −0.351295
$$202$$ 0 0
$$203$$ −1516.19 −0.524214
$$204$$ 0 0
$$205$$ −508.455 −0.173230
$$206$$ 0 0
$$207$$ −1182.12 −0.396924
$$208$$ 0 0
$$209$$ −4995.77 −1.65342
$$210$$ 0 0
$$211$$ 2588.58 0.844574 0.422287 0.906462i $$-0.361227\pi$$
0.422287 + 0.906462i $$0.361227\pi$$
$$212$$ 0 0
$$213$$ −1770.74 −0.569621
$$214$$ 0 0
$$215$$ 1679.78 0.532838
$$216$$ 0 0
$$217$$ −1762.56 −0.551384
$$218$$ 0 0
$$219$$ 1472.10 0.454226
$$220$$ 0 0
$$221$$ −242.502 −0.0738119
$$222$$ 0 0
$$223$$ 3236.21 0.971804 0.485902 0.874013i $$-0.338491\pi$$
0.485902 + 0.874013i $$0.338491\pi$$
$$224$$ 0 0
$$225$$ −938.691 −0.278131
$$226$$ 0 0
$$227$$ −5631.62 −1.64662 −0.823312 0.567589i $$-0.807876\pi$$
−0.823312 + 0.567589i $$0.807876\pi$$
$$228$$ 0 0
$$229$$ −3770.25 −1.08797 −0.543985 0.839095i $$-0.683085\pi$$
−0.543985 + 0.839095i $$0.683085\pi$$
$$230$$ 0 0
$$231$$ 855.733 0.243736
$$232$$ 0 0
$$233$$ −6560.90 −1.84472 −0.922358 0.386336i $$-0.873741\pi$$
−0.922358 + 0.386336i $$0.873741\pi$$
$$234$$ 0 0
$$235$$ 1195.22 0.331776
$$236$$ 0 0
$$237$$ −365.073 −0.100059
$$238$$ 0 0
$$239$$ 771.444 0.208789 0.104394 0.994536i $$-0.466710\pi$$
0.104394 + 0.994536i $$0.466710\pi$$
$$240$$ 0 0
$$241$$ 1252.10 0.334668 0.167334 0.985900i $$-0.446484\pi$$
0.167334 + 0.985900i $$0.446484\pi$$
$$242$$ 0 0
$$243$$ 243.000 0.0641500
$$244$$ 0 0
$$245$$ 222.942 0.0581357
$$246$$ 0 0
$$247$$ −6534.35 −1.68328
$$248$$ 0 0
$$249$$ 1828.82 0.465450
$$250$$ 0 0
$$251$$ 5166.27 1.29917 0.649586 0.760288i $$-0.274942\pi$$
0.649586 + 0.760288i $$0.274942\pi$$
$$252$$ 0 0
$$253$$ 5352.29 1.33002
$$254$$ 0 0
$$255$$ 62.1030 0.0152511
$$256$$ 0 0
$$257$$ 2767.45 0.671707 0.335854 0.941914i $$-0.390975\pi$$
0.335854 + 0.941914i $$0.390975\pi$$
$$258$$ 0 0
$$259$$ 83.2791 0.0199796
$$260$$ 0 0
$$261$$ 1949.38 0.462313
$$262$$ 0 0
$$263$$ 4101.78 0.961699 0.480849 0.876803i $$-0.340328\pi$$
0.480849 + 0.876803i $$0.340328\pi$$
$$264$$ 0 0
$$265$$ 2580.21 0.598117
$$266$$ 0 0
$$267$$ 2157.11 0.494432
$$268$$ 0 0
$$269$$ 6950.84 1.57546 0.787732 0.616018i $$-0.211255\pi$$
0.787732 + 0.616018i $$0.211255\pi$$
$$270$$ 0 0
$$271$$ 7140.29 1.60052 0.800262 0.599651i $$-0.204694\pi$$
0.800262 + 0.599651i $$0.204694\pi$$
$$272$$ 0 0
$$273$$ 1119.28 0.248139
$$274$$ 0 0
$$275$$ 4250.10 0.931966
$$276$$ 0 0
$$277$$ −1320.51 −0.286433 −0.143217 0.989691i $$-0.545745\pi$$
−0.143217 + 0.989691i $$0.545745\pi$$
$$278$$ 0 0
$$279$$ 2266.15 0.486275
$$280$$ 0 0
$$281$$ −204.309 −0.0433738 −0.0216869 0.999765i $$-0.506904\pi$$
−0.0216869 + 0.999765i $$0.506904\pi$$
$$282$$ 0 0
$$283$$ 975.794 0.204964 0.102482 0.994735i $$-0.467322\pi$$
0.102482 + 0.994735i $$0.467322\pi$$
$$284$$ 0 0
$$285$$ 1673.40 0.347803
$$286$$ 0 0
$$287$$ 782.267 0.160891
$$288$$ 0 0
$$289$$ −4892.30 −0.995786
$$290$$ 0 0
$$291$$ −1913.63 −0.385495
$$292$$ 0 0
$$293$$ −607.919 −0.121212 −0.0606058 0.998162i $$-0.519303\pi$$
−0.0606058 + 0.998162i $$0.519303\pi$$
$$294$$ 0 0
$$295$$ 3821.36 0.754198
$$296$$ 0 0
$$297$$ −1100.23 −0.214955
$$298$$ 0 0
$$299$$ 7000.67 1.35405
$$300$$ 0 0
$$301$$ −2584.37 −0.494886
$$302$$ 0 0
$$303$$ −2013.44 −0.381747
$$304$$ 0 0
$$305$$ 2210.28 0.414952
$$306$$ 0 0
$$307$$ 8037.08 1.49414 0.747069 0.664747i $$-0.231461\pi$$
0.747069 + 0.664747i $$0.231461\pi$$
$$308$$ 0 0
$$309$$ 2737.24 0.503935
$$310$$ 0 0
$$311$$ −5311.60 −0.968468 −0.484234 0.874939i $$-0.660902\pi$$
−0.484234 + 0.874939i $$0.660902\pi$$
$$312$$ 0 0
$$313$$ −1531.61 −0.276587 −0.138293 0.990391i $$-0.544162\pi$$
−0.138293 + 0.990391i $$0.544162\pi$$
$$314$$ 0 0
$$315$$ −286.640 −0.0512708
$$316$$ 0 0
$$317$$ −4219.19 −0.747549 −0.373775 0.927520i $$-0.621937\pi$$
−0.373775 + 0.927520i $$0.621937\pi$$
$$318$$ 0 0
$$319$$ −8826.19 −1.54913
$$320$$ 0 0
$$321$$ −350.208 −0.0608931
$$322$$ 0 0
$$323$$ 557.801 0.0960893
$$324$$ 0 0
$$325$$ 5559.03 0.948799
$$326$$ 0 0
$$327$$ −2511.53 −0.424733
$$328$$ 0 0
$$329$$ −1838.86 −0.308145
$$330$$ 0 0
$$331$$ 8298.19 1.37797 0.688987 0.724773i $$-0.258056\pi$$
0.688987 + 0.724773i $$0.258056\pi$$
$$332$$ 0 0
$$333$$ −107.073 −0.0176203
$$334$$ 0 0
$$335$$ −1518.24 −0.247613
$$336$$ 0 0
$$337$$ −4348.44 −0.702892 −0.351446 0.936208i $$-0.614310\pi$$
−0.351446 + 0.936208i $$0.614310\pi$$
$$338$$ 0 0
$$339$$ −3259.73 −0.522255
$$340$$ 0 0
$$341$$ −10260.4 −1.62942
$$342$$ 0 0
$$343$$ −343.000 −0.0539949
$$344$$ 0 0
$$345$$ −1792.82 −0.279775
$$346$$ 0 0
$$347$$ −8345.54 −1.29110 −0.645550 0.763718i $$-0.723372\pi$$
−0.645550 + 0.763718i $$0.723372\pi$$
$$348$$ 0 0
$$349$$ 9982.54 1.53110 0.765549 0.643378i $$-0.222468\pi$$
0.765549 + 0.643378i $$0.222468\pi$$
$$350$$ 0 0
$$351$$ −1439.07 −0.218838
$$352$$ 0 0
$$353$$ 8801.59 1.32709 0.663543 0.748138i $$-0.269052\pi$$
0.663543 + 0.748138i $$0.269052\pi$$
$$354$$ 0 0
$$355$$ −2685.53 −0.401502
$$356$$ 0 0
$$357$$ −95.5465 −0.0141649
$$358$$ 0 0
$$359$$ 524.039 0.0770409 0.0385205 0.999258i $$-0.487736\pi$$
0.0385205 + 0.999258i $$0.487736\pi$$
$$360$$ 0 0
$$361$$ 8171.27 1.19132
$$362$$ 0 0
$$363$$ 988.485 0.142926
$$364$$ 0 0
$$365$$ 2232.61 0.320165
$$366$$ 0 0
$$367$$ 6362.72 0.904991 0.452495 0.891767i $$-0.350534\pi$$
0.452495 + 0.891767i $$0.350534\pi$$
$$368$$ 0 0
$$369$$ −1005.77 −0.141893
$$370$$ 0 0
$$371$$ −3969.70 −0.555516
$$372$$ 0 0
$$373$$ 11265.8 1.56387 0.781935 0.623361i $$-0.214233\pi$$
0.781935 + 0.623361i $$0.214233\pi$$
$$374$$ 0 0
$$375$$ −3129.82 −0.430995
$$376$$ 0 0
$$377$$ −11544.5 −1.57711
$$378$$ 0 0
$$379$$ −1151.71 −0.156094 −0.0780470 0.996950i $$-0.524868\pi$$
−0.0780470 + 0.996950i $$0.524868\pi$$
$$380$$ 0 0
$$381$$ 1611.34 0.216670
$$382$$ 0 0
$$383$$ 151.554 0.0202195 0.0101097 0.999949i $$-0.496782\pi$$
0.0101097 + 0.999949i $$0.496782\pi$$
$$384$$ 0 0
$$385$$ 1297.81 0.171799
$$386$$ 0 0
$$387$$ 3322.76 0.436449
$$388$$ 0 0
$$389$$ −4794.18 −0.624870 −0.312435 0.949939i $$-0.601145\pi$$
−0.312435 + 0.949939i $$0.601145\pi$$
$$390$$ 0 0
$$391$$ −597.608 −0.0772950
$$392$$ 0 0
$$393$$ 4492.17 0.576590
$$394$$ 0 0
$$395$$ −553.674 −0.0705275
$$396$$ 0 0
$$397$$ 4623.94 0.584556 0.292278 0.956333i $$-0.405587\pi$$
0.292278 + 0.956333i $$0.405587\pi$$
$$398$$ 0 0
$$399$$ −2574.56 −0.323030
$$400$$ 0 0
$$401$$ −3610.63 −0.449642 −0.224821 0.974400i $$-0.572180\pi$$
−0.224821 + 0.974400i $$0.572180\pi$$
$$402$$ 0 0
$$403$$ −13420.4 −1.65885
$$404$$ 0 0
$$405$$ 368.537 0.0452166
$$406$$ 0 0
$$407$$ 484.794 0.0590426
$$408$$ 0 0
$$409$$ 8959.57 1.08318 0.541592 0.840641i $$-0.317822\pi$$
0.541592 + 0.840641i $$0.317822\pi$$
$$410$$ 0 0
$$411$$ −4140.27 −0.496896
$$412$$ 0 0
$$413$$ −5879.23 −0.700480
$$414$$ 0 0
$$415$$ 2773.62 0.328076
$$416$$ 0 0
$$417$$ −425.940 −0.0500201
$$418$$ 0 0
$$419$$ −7078.28 −0.825290 −0.412645 0.910892i $$-0.635395\pi$$
−0.412645 + 0.910892i $$0.635395\pi$$
$$420$$ 0 0
$$421$$ −11551.5 −1.33725 −0.668626 0.743599i $$-0.733117\pi$$
−0.668626 + 0.743599i $$0.733117\pi$$
$$422$$ 0 0
$$423$$ 2364.25 0.271758
$$424$$ 0 0
$$425$$ −474.543 −0.0541617
$$426$$ 0 0
$$427$$ −3400.56 −0.385397
$$428$$ 0 0
$$429$$ 6515.67 0.733286
$$430$$ 0 0
$$431$$ 4064.38 0.454232 0.227116 0.973868i $$-0.427070\pi$$
0.227116 + 0.973868i $$0.427070\pi$$
$$432$$ 0 0
$$433$$ 17456.3 1.93740 0.968701 0.248229i $$-0.0798487\pi$$
0.968701 + 0.248229i $$0.0798487\pi$$
$$434$$ 0 0
$$435$$ 2956.46 0.325865
$$436$$ 0 0
$$437$$ −16102.9 −1.76271
$$438$$ 0 0
$$439$$ −4595.39 −0.499604 −0.249802 0.968297i $$-0.580365\pi$$
−0.249802 + 0.968297i $$0.580365\pi$$
$$440$$ 0 0
$$441$$ 441.000 0.0476190
$$442$$ 0 0
$$443$$ −306.214 −0.0328412 −0.0164206 0.999865i $$-0.505227\pi$$
−0.0164206 + 0.999865i $$0.505227\pi$$
$$444$$ 0 0
$$445$$ 3271.50 0.348504
$$446$$ 0 0
$$447$$ 5831.61 0.617060
$$448$$ 0 0
$$449$$ 9229.22 0.970053 0.485026 0.874500i $$-0.338810\pi$$
0.485026 + 0.874500i $$0.338810\pi$$
$$450$$ 0 0
$$451$$ 4553.82 0.475457
$$452$$ 0 0
$$453$$ 7964.29 0.826038
$$454$$ 0 0
$$455$$ 1697.51 0.174902
$$456$$ 0 0
$$457$$ −10992.2 −1.12515 −0.562577 0.826745i $$-0.690190\pi$$
−0.562577 + 0.826745i $$0.690190\pi$$
$$458$$ 0 0
$$459$$ 122.846 0.0124922
$$460$$ 0 0
$$461$$ −7387.88 −0.746394 −0.373197 0.927752i $$-0.621739\pi$$
−0.373197 + 0.927752i $$0.621739\pi$$
$$462$$ 0 0
$$463$$ −10163.8 −1.02020 −0.510101 0.860114i $$-0.670392\pi$$
−0.510101 + 0.860114i $$0.670392\pi$$
$$464$$ 0 0
$$465$$ 3436.86 0.342754
$$466$$ 0 0
$$467$$ −15814.6 −1.56705 −0.783524 0.621362i $$-0.786580\pi$$
−0.783524 + 0.621362i $$0.786580\pi$$
$$468$$ 0 0
$$469$$ 2335.84 0.229976
$$470$$ 0 0
$$471$$ −4995.65 −0.488720
$$472$$ 0 0
$$473$$ −15044.4 −1.46246
$$474$$ 0 0
$$475$$ −12786.9 −1.23516
$$476$$ 0 0
$$477$$ 5103.90 0.489919
$$478$$ 0 0
$$479$$ 1444.85 0.137823 0.0689113 0.997623i $$-0.478047\pi$$
0.0689113 + 0.997623i $$0.478047\pi$$
$$480$$ 0 0
$$481$$ 634.099 0.0601090
$$482$$ 0 0
$$483$$ 2758.29 0.259848
$$484$$ 0 0
$$485$$ −2902.24 −0.271719
$$486$$ 0 0
$$487$$ 489.402 0.0455378 0.0227689 0.999741i $$-0.492752\pi$$
0.0227689 + 0.999741i $$0.492752\pi$$
$$488$$ 0 0
$$489$$ −99.2195 −0.00917559
$$490$$ 0 0
$$491$$ −3941.30 −0.362257 −0.181129 0.983459i $$-0.557975\pi$$
−0.181129 + 0.983459i $$0.557975\pi$$
$$492$$ 0 0
$$493$$ 985.485 0.0900284
$$494$$ 0 0
$$495$$ −1668.62 −0.151513
$$496$$ 0 0
$$497$$ 4131.73 0.372905
$$498$$ 0 0
$$499$$ 11.0894 0.000994850 0 0.000497425 1.00000i $$-0.499842\pi$$
0.000497425 1.00000i $$0.499842\pi$$
$$500$$ 0 0
$$501$$ −4963.43 −0.442614
$$502$$ 0 0
$$503$$ −7088.41 −0.628343 −0.314172 0.949366i $$-0.601727\pi$$
−0.314172 + 0.949366i $$0.601727\pi$$
$$504$$ 0 0
$$505$$ −3053.61 −0.269077
$$506$$ 0 0
$$507$$ 1931.35 0.169180
$$508$$ 0 0
$$509$$ −17588.4 −1.53162 −0.765810 0.643067i $$-0.777662\pi$$
−0.765810 + 0.643067i $$0.777662\pi$$
$$510$$ 0 0
$$511$$ −3434.91 −0.297361
$$512$$ 0 0
$$513$$ 3310.15 0.284886
$$514$$ 0 0
$$515$$ 4151.32 0.355202
$$516$$ 0 0
$$517$$ −10704.6 −0.910613
$$518$$ 0 0
$$519$$ −192.573 −0.0162871
$$520$$ 0 0
$$521$$ −11646.6 −0.979360 −0.489680 0.871902i $$-0.662886\pi$$
−0.489680 + 0.871902i $$0.662886\pi$$
$$522$$ 0 0
$$523$$ 8965.82 0.749614 0.374807 0.927103i $$-0.377709\pi$$
0.374807 + 0.927103i $$0.377709\pi$$
$$524$$ 0 0
$$525$$ 2190.28 0.182079
$$526$$ 0 0
$$527$$ 1145.62 0.0946946
$$528$$ 0 0
$$529$$ 5085.08 0.417941
$$530$$ 0 0
$$531$$ 7559.01 0.617765
$$532$$ 0 0
$$533$$ 5956.30 0.484045
$$534$$ 0 0
$$535$$ −531.129 −0.0429210
$$536$$ 0 0
$$537$$ 11744.0 0.943747
$$538$$ 0 0
$$539$$ −1996.71 −0.159563
$$540$$ 0 0
$$541$$ 195.272 0.0155183 0.00775914 0.999970i $$-0.497530\pi$$
0.00775914 + 0.999970i $$0.497530\pi$$
$$542$$ 0 0
$$543$$ 6174.13 0.487951
$$544$$ 0 0
$$545$$ −3809.01 −0.299376
$$546$$ 0 0
$$547$$ −1399.26 −0.109375 −0.0546874 0.998504i $$-0.517416\pi$$
−0.0546874 + 0.998504i $$0.517416\pi$$
$$548$$ 0 0
$$549$$ 4372.15 0.339888
$$550$$ 0 0
$$551$$ 26554.5 2.05310
$$552$$ 0 0
$$553$$ 851.837 0.0655042
$$554$$ 0 0
$$555$$ −162.388 −0.0124198
$$556$$ 0 0
$$557$$ −43.0467 −0.00327459 −0.00163730 0.999999i $$-0.500521\pi$$
−0.00163730 + 0.999999i $$0.500521\pi$$
$$558$$ 0 0
$$559$$ −19677.8 −1.48888
$$560$$ 0 0
$$561$$ −556.206 −0.0418592
$$562$$ 0 0
$$563$$ −19232.9 −1.43973 −0.719865 0.694114i $$-0.755797\pi$$
−0.719865 + 0.694114i $$0.755797\pi$$
$$564$$ 0 0
$$565$$ −4943.75 −0.368115
$$566$$ 0 0
$$567$$ −567.000 −0.0419961
$$568$$ 0 0
$$569$$ 5163.98 0.380466 0.190233 0.981739i $$-0.439076\pi$$
0.190233 + 0.981739i $$0.439076\pi$$
$$570$$ 0 0
$$571$$ −10231.9 −0.749899 −0.374950 0.927045i $$-0.622340\pi$$
−0.374950 + 0.927045i $$0.622340\pi$$
$$572$$ 0 0
$$573$$ −1284.14 −0.0936229
$$574$$ 0 0
$$575$$ 13699.4 0.993572
$$576$$ 0 0
$$577$$ 16563.7 1.19507 0.597537 0.801842i $$-0.296146\pi$$
0.597537 + 0.801842i $$0.296146\pi$$
$$578$$ 0 0
$$579$$ 4814.78 0.345588
$$580$$ 0 0
$$581$$ −4267.26 −0.304708
$$582$$ 0 0
$$583$$ −23108.8 −1.64163
$$584$$ 0 0
$$585$$ −2182.51 −0.154249
$$586$$ 0 0
$$587$$ 16020.6 1.12648 0.563239 0.826294i $$-0.309555\pi$$
0.563239 + 0.826294i $$0.309555\pi$$
$$588$$ 0 0
$$589$$ 30869.4 2.15951
$$590$$ 0 0
$$591$$ −11216.5 −0.780684
$$592$$ 0 0
$$593$$ −6771.14 −0.468900 −0.234450 0.972128i $$-0.575329\pi$$
−0.234450 + 0.972128i $$0.575329\pi$$
$$594$$ 0 0
$$595$$ −144.907 −0.00998421
$$596$$ 0 0
$$597$$ 1047.09 0.0717831
$$598$$ 0 0
$$599$$ −11070.2 −0.755120 −0.377560 0.925985i $$-0.623237\pi$$
−0.377560 + 0.925985i $$0.623237\pi$$
$$600$$ 0 0
$$601$$ −24187.7 −1.64166 −0.820830 0.571173i $$-0.806489\pi$$
−0.820830 + 0.571173i $$0.806489\pi$$
$$602$$ 0 0
$$603$$ −3003.22 −0.202820
$$604$$ 0 0
$$605$$ 1499.15 0.100742
$$606$$ 0 0
$$607$$ 10074.1 0.673631 0.336816 0.941571i $$-0.390650\pi$$
0.336816 + 0.941571i $$0.390650\pi$$
$$608$$ 0 0
$$609$$ −4548.56 −0.302655
$$610$$ 0 0
$$611$$ −14001.3 −0.927060
$$612$$ 0 0
$$613$$ 11114.6 0.732323 0.366161 0.930551i $$-0.380672\pi$$
0.366161 + 0.930551i $$0.380672\pi$$
$$614$$ 0 0
$$615$$ −1525.37 −0.100014
$$616$$ 0 0
$$617$$ 20496.4 1.33737 0.668683 0.743548i $$-0.266858\pi$$
0.668683 + 0.743548i $$0.266858\pi$$
$$618$$ 0 0
$$619$$ −16714.4 −1.08532 −0.542658 0.839954i $$-0.682582\pi$$
−0.542658 + 0.839954i $$0.682582\pi$$
$$620$$ 0 0
$$621$$ −3546.37 −0.229164
$$622$$ 0 0
$$623$$ −5033.27 −0.323682
$$624$$ 0 0
$$625$$ 8290.66 0.530602
$$626$$ 0 0
$$627$$ −14987.3 −0.954602
$$628$$ 0 0
$$629$$ −54.1295 −0.00343129
$$630$$ 0 0
$$631$$ −9168.53 −0.578437 −0.289218 0.957263i $$-0.593395\pi$$
−0.289218 + 0.957263i $$0.593395\pi$$
$$632$$ 0 0
$$633$$ 7765.73 0.487615
$$634$$ 0 0
$$635$$ 2443.77 0.152722
$$636$$ 0 0
$$637$$ −2611.65 −0.162445
$$638$$ 0 0
$$639$$ −5312.23 −0.328871
$$640$$ 0 0
$$641$$ −4273.37 −0.263319 −0.131660 0.991295i $$-0.542031\pi$$
−0.131660 + 0.991295i $$0.542031\pi$$
$$642$$ 0 0
$$643$$ 2955.75 0.181281 0.0906404 0.995884i $$-0.471109\pi$$
0.0906404 + 0.995884i $$0.471109\pi$$
$$644$$ 0 0
$$645$$ 5039.34 0.307634
$$646$$ 0 0
$$647$$ −22701.2 −1.37941 −0.689704 0.724091i $$-0.742259\pi$$
−0.689704 + 0.724091i $$0.742259\pi$$
$$648$$ 0 0
$$649$$ −34224.8 −2.07002
$$650$$ 0 0
$$651$$ −5287.67 −0.318341
$$652$$ 0 0
$$653$$ −1537.81 −0.0921582 −0.0460791 0.998938i $$-0.514673\pi$$
−0.0460791 + 0.998938i $$0.514673\pi$$
$$654$$ 0 0
$$655$$ 6812.87 0.406414
$$656$$ 0 0
$$657$$ 4416.31 0.262248
$$658$$ 0 0
$$659$$ 12338.1 0.729323 0.364661 0.931140i $$-0.381185\pi$$
0.364661 + 0.931140i $$0.381185\pi$$
$$660$$ 0 0
$$661$$ −1845.10 −0.108572 −0.0542859 0.998525i $$-0.517288\pi$$
−0.0542859 + 0.998525i $$0.517288\pi$$
$$662$$ 0 0
$$663$$ −727.505 −0.0426153
$$664$$ 0 0
$$665$$ −3904.60 −0.227690
$$666$$ 0 0
$$667$$ −28449.5 −1.65153
$$668$$ 0 0
$$669$$ 9708.62 0.561072
$$670$$ 0 0
$$671$$ −19795.7 −1.13890
$$672$$ 0 0
$$673$$ 23955.4 1.37208 0.686041 0.727563i $$-0.259347\pi$$
0.686041 + 0.727563i $$0.259347\pi$$
$$674$$ 0 0
$$675$$ −2816.07 −0.160579
$$676$$ 0 0
$$677$$ 3678.26 0.208814 0.104407 0.994535i $$-0.466706\pi$$
0.104407 + 0.994535i $$0.466706\pi$$
$$678$$ 0 0
$$679$$ 4465.14 0.252366
$$680$$ 0 0
$$681$$ −16894.9 −0.950679
$$682$$ 0 0
$$683$$ 4390.87 0.245991 0.122996 0.992407i $$-0.460750\pi$$
0.122996 + 0.992407i $$0.460750\pi$$
$$684$$ 0 0
$$685$$ −6279.18 −0.350241
$$686$$ 0 0
$$687$$ −11310.7 −0.628140
$$688$$ 0 0
$$689$$ −30225.8 −1.67128
$$690$$ 0 0
$$691$$ 10371.7 0.570994 0.285497 0.958380i $$-0.407841\pi$$
0.285497 + 0.958380i $$0.407841\pi$$
$$692$$ 0 0
$$693$$ 2567.20 0.140721
$$694$$ 0 0
$$695$$ −645.986 −0.0352570
$$696$$ 0 0
$$697$$ −508.455 −0.0276314
$$698$$ 0 0
$$699$$ −19682.7 −1.06505
$$700$$ 0 0
$$701$$ −109.675 −0.00590922 −0.00295461 0.999996i $$-0.500940\pi$$
−0.00295461 + 0.999996i $$0.500940\pi$$
$$702$$ 0 0
$$703$$ −1458.55 −0.0782508
$$704$$ 0 0
$$705$$ 3585.65 0.191551
$$706$$ 0 0
$$707$$ 4698.03 0.249912
$$708$$ 0 0
$$709$$ −26918.8 −1.42589 −0.712944 0.701221i $$-0.752639\pi$$
−0.712944 + 0.701221i $$0.752639\pi$$
$$710$$ 0 0
$$711$$ −1095.22 −0.0577693
$$712$$ 0 0
$$713$$ −33072.4 −1.73713
$$714$$ 0 0
$$715$$ 9881.74 0.516862
$$716$$ 0 0
$$717$$ 2314.33 0.120544
$$718$$ 0 0
$$719$$ −15170.8 −0.786889 −0.393445 0.919348i $$-0.628717\pi$$
−0.393445 + 0.919348i $$0.628717\pi$$
$$720$$ 0 0
$$721$$ −6386.88 −0.329903
$$722$$ 0 0
$$723$$ 3756.31 0.193221
$$724$$ 0 0
$$725$$ −22591.0 −1.15725
$$726$$ 0 0
$$727$$ 33286.9 1.69813 0.849066 0.528288i $$-0.177166\pi$$
0.849066 + 0.528288i $$0.177166\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ 1679.78 0.0849917
$$732$$ 0 0
$$733$$ −20544.0 −1.03521 −0.517607 0.855619i $$-0.673177\pi$$
−0.517607 + 0.855619i $$0.673177\pi$$
$$734$$ 0 0
$$735$$ 668.826 0.0335646
$$736$$ 0 0
$$737$$ 13597.6 0.679614
$$738$$ 0 0
$$739$$ 34357.2 1.71022 0.855109 0.518449i $$-0.173490\pi$$
0.855109 + 0.518449i $$0.173490\pi$$
$$740$$ 0 0
$$741$$ −19603.1 −0.971844
$$742$$ 0 0
$$743$$ −8166.99 −0.403254 −0.201627 0.979462i $$-0.564623\pi$$
−0.201627 + 0.979462i $$0.564623\pi$$
$$744$$ 0 0
$$745$$ 8844.29 0.434939
$$746$$ 0 0
$$747$$ 5486.47 0.268728
$$748$$ 0 0
$$749$$ 817.151 0.0398639
$$750$$ 0 0
$$751$$ −17080.1 −0.829909 −0.414954 0.909842i $$-0.636202\pi$$
−0.414954 + 0.909842i $$0.636202\pi$$
$$752$$ 0 0
$$753$$ 15498.8 0.750077
$$754$$ 0 0
$$755$$ 12078.7 0.582239
$$756$$ 0 0
$$757$$ 16324.0 0.783758 0.391879 0.920017i $$-0.371825\pi$$
0.391879 + 0.920017i $$0.371825\pi$$
$$758$$ 0 0
$$759$$ 16056.9 0.767888
$$760$$ 0 0
$$761$$ −32366.2 −1.54175 −0.770875 0.636986i $$-0.780181\pi$$
−0.770875 + 0.636986i $$0.780181\pi$$
$$762$$ 0 0
$$763$$ 5860.23 0.278053
$$764$$ 0 0
$$765$$ 186.309 0.00880525
$$766$$ 0 0
$$767$$ −44765.3 −2.10741
$$768$$ 0 0
$$769$$ −7948.44 −0.372728 −0.186364 0.982481i $$-0.559670\pi$$
−0.186364 + 0.982481i $$0.559670\pi$$
$$770$$ 0 0
$$771$$ 8302.35 0.387810
$$772$$ 0 0
$$773$$ −17819.3 −0.829127 −0.414564 0.910020i $$-0.636066\pi$$
−0.414564 + 0.910020i $$0.636066\pi$$
$$774$$ 0 0
$$775$$ −26261.9 −1.21723
$$776$$ 0 0
$$777$$ 249.837 0.0115352
$$778$$ 0 0
$$779$$ −13700.6 −0.630136
$$780$$ 0 0
$$781$$ 24052.1 1.10199
$$782$$ 0 0
$$783$$ 5848.15 0.266917
$$784$$ 0 0
$$785$$ −7576.46 −0.344478
$$786$$ 0 0
$$787$$ 2912.38 0.131912 0.0659562 0.997823i $$-0.478990\pi$$
0.0659562 + 0.997823i $$0.478990\pi$$
$$788$$ 0 0
$$789$$ 12305.3 0.555237
$$790$$ 0 0
$$791$$ 7606.05 0.341896
$$792$$ 0 0
$$793$$ −25892.3 −1.15948
$$794$$ 0 0
$$795$$ 7740.63 0.345323
$$796$$ 0 0
$$797$$ 33789.1 1.50172 0.750861 0.660460i $$-0.229639\pi$$
0.750861 + 0.660460i $$0.229639\pi$$
$$798$$ 0 0
$$799$$ 1195.22 0.0529208
$$800$$ 0 0
$$801$$ 6471.34 0.285460
$$802$$ 0 0
$$803$$ −19995.7 −0.878744
$$804$$ 0 0
$$805$$ 4183.26 0.183156
$$806$$ 0 0
$$807$$ 20852.5 0.909595
$$808$$ 0 0
$$809$$ 1252.13 0.0544159 0.0272079 0.999630i $$-0.491338\pi$$
0.0272079 + 0.999630i $$0.491338\pi$$
$$810$$ 0 0
$$811$$ −31913.1 −1.38178 −0.690889 0.722961i $$-0.742781\pi$$
−0.690889 + 0.722961i $$0.742781\pi$$
$$812$$ 0 0
$$813$$ 21420.9 0.924063
$$814$$ 0 0
$$815$$ −150.477 −0.00646748
$$816$$ 0 0
$$817$$ 45262.7 1.93824
$$818$$ 0 0
$$819$$ 3357.84 0.143263
$$820$$ 0 0
$$821$$ −30742.4 −1.30684 −0.653421 0.756995i $$-0.726667\pi$$
−0.653421 + 0.756995i $$0.726667\pi$$
$$822$$ 0 0
$$823$$ 13822.6 0.585449 0.292724 0.956197i $$-0.405438\pi$$
0.292724 + 0.956197i $$0.405438\pi$$
$$824$$ 0 0
$$825$$ 12750.3 0.538071
$$826$$ 0 0
$$827$$ −42107.1 −1.77051 −0.885253 0.465110i $$-0.846015\pi$$
−0.885253 + 0.465110i $$0.846015\pi$$
$$828$$ 0 0
$$829$$ 38763.8 1.62403 0.812015 0.583636i $$-0.198371\pi$$
0.812015 + 0.583636i $$0.198371\pi$$
$$830$$ 0 0
$$831$$ −3961.54 −0.165372
$$832$$ 0 0
$$833$$ 222.942 0.00927308
$$834$$ 0 0
$$835$$ −7527.59 −0.311980
$$836$$ 0 0
$$837$$ 6798.44 0.280751
$$838$$ 0 0
$$839$$ −16896.3 −0.695262 −0.347631 0.937631i $$-0.613014\pi$$
−0.347631 + 0.937631i $$0.613014\pi$$
$$840$$ 0 0
$$841$$ 22525.7 0.923601
$$842$$ 0 0
$$843$$ −612.927 −0.0250419
$$844$$ 0 0
$$845$$ 2929.11 0.119248
$$846$$ 0 0
$$847$$ −2306.47 −0.0935668
$$848$$ 0 0
$$849$$ 2927.38 0.118336
$$850$$ 0 0
$$851$$ 1562.64 0.0629455
$$852$$ 0 0
$$853$$ −46429.3 −1.86367 −0.931833 0.362887i $$-0.881791\pi$$
−0.931833 + 0.362887i $$0.881791\pi$$
$$854$$ 0 0
$$855$$ 5020.21 0.200804
$$856$$ 0 0
$$857$$ 21206.4 0.845272 0.422636 0.906300i $$-0.361105\pi$$
0.422636 + 0.906300i $$0.361105\pi$$
$$858$$ 0 0
$$859$$ 13876.2 0.551163 0.275581 0.961278i $$-0.411130\pi$$
0.275581 + 0.961278i $$0.411130\pi$$
$$860$$ 0 0
$$861$$ 2346.80 0.0928906
$$862$$ 0 0
$$863$$ −14337.1 −0.565515 −0.282757 0.959191i $$-0.591249\pi$$
−0.282757 + 0.959191i $$0.591249\pi$$
$$864$$ 0 0
$$865$$ −292.058 −0.0114801
$$866$$ 0 0
$$867$$ −14676.9 −0.574918
$$868$$ 0 0
$$869$$ 4958.81 0.193574
$$870$$ 0 0
$$871$$ 17785.4 0.691889
$$872$$ 0 0
$$873$$ −5740.89 −0.222566
$$874$$ 0 0
$$875$$ 7302.91 0.282152
$$876$$ 0 0
$$877$$ 24369.3 0.938304 0.469152 0.883118i $$-0.344560\pi$$
0.469152 + 0.883118i $$0.344560\pi$$
$$878$$ 0 0
$$879$$ −1823.76 −0.0699815
$$880$$ 0 0
$$881$$ −26127.0 −0.999140 −0.499570 0.866273i $$-0.666509\pi$$
−0.499570 + 0.866273i $$0.666509\pi$$
$$882$$ 0 0
$$883$$ −15713.1 −0.598855 −0.299428 0.954119i $$-0.596796\pi$$
−0.299428 + 0.954119i $$0.596796\pi$$
$$884$$ 0 0
$$885$$ 11464.1 0.435436
$$886$$ 0 0
$$887$$ −13139.5 −0.497385 −0.248692 0.968583i $$-0.580001\pi$$
−0.248692 + 0.968583i $$0.580001\pi$$
$$888$$ 0 0
$$889$$ −3759.79 −0.141844
$$890$$ 0 0
$$891$$ −3300.68 −0.124104
$$892$$ 0 0
$$893$$ 32205.8 1.20686
$$894$$ 0 0
$$895$$ 17811.1 0.665207
$$896$$ 0 0
$$897$$ 21002.0 0.781758
$$898$$ 0 0
$$899$$ 54538.1 2.02330
$$900$$ 0 0
$$901$$ 2580.21 0.0954043
$$902$$ 0 0
$$903$$ −7753.12 −0.285723
$$904$$ 0 0
$$905$$ 9363.76 0.343936
$$906$$ 0 0
$$907$$ −3799.71 −0.139104 −0.0695519 0.997578i $$-0.522157\pi$$
−0.0695519 + 0.997578i $$0.522157\pi$$
$$908$$ 0 0
$$909$$ −6040.33 −0.220402
$$910$$ 0 0
$$911$$ 51528.4 1.87400 0.936998 0.349334i $$-0.113592\pi$$
0.936998 + 0.349334i $$0.113592\pi$$
$$912$$ 0 0
$$913$$ −24841.0 −0.900458
$$914$$ 0 0
$$915$$ 6630.85 0.239573
$$916$$ 0 0
$$917$$ −10481.7 −0.377467
$$918$$ 0 0
$$919$$ 16984.7 0.609657 0.304828 0.952407i $$-0.401401\pi$$
0.304828 + 0.952407i $$0.401401\pi$$
$$920$$ 0 0
$$921$$ 24111.2 0.862641
$$922$$ 0 0
$$923$$ 31459.6 1.12189
$$924$$ 0 0
$$925$$ 1240.85 0.0441068
$$926$$ 0 0
$$927$$ 8211.71 0.290947
$$928$$ 0 0
$$929$$ −5451.85 −0.192540 −0.0962699 0.995355i $$-0.530691\pi$$
−0.0962699 + 0.995355i $$0.530691\pi$$
$$930$$ 0 0
$$931$$ 6007.30 0.211473
$$932$$ 0 0
$$933$$ −15934.8 −0.559145
$$934$$ 0 0
$$935$$ −843.548 −0.0295048
$$936$$ 0 0
$$937$$ 42429.4 1.47930 0.739652 0.672989i $$-0.234990\pi$$
0.739652 + 0.672989i $$0.234990\pi$$
$$938$$ 0 0
$$939$$ −4594.82 −0.159687
$$940$$ 0 0
$$941$$ 32977.9 1.14245 0.571226 0.820793i $$-0.306468\pi$$
0.571226 + 0.820793i $$0.306468\pi$$
$$942$$ 0 0
$$943$$ 14678.4 0.506886
$$944$$ 0 0
$$945$$ −859.919 −0.0296012
$$946$$ 0 0
$$947$$ −23753.4 −0.815082 −0.407541 0.913187i $$-0.633614\pi$$
−0.407541 + 0.913187i $$0.633614\pi$$
$$948$$ 0 0
$$949$$ −26153.9 −0.894616
$$950$$ 0 0
$$951$$ −12657.6 −0.431598
$$952$$ 0 0
$$953$$ −28074.3 −0.954267 −0.477134 0.878831i $$-0.658324\pi$$
−0.477134 + 0.878831i $$0.658324\pi$$
$$954$$ 0 0
$$955$$ −1947.55 −0.0659908
$$956$$ 0 0
$$957$$ −26478.6 −0.894389
$$958$$ 0 0
$$959$$ 9660.63 0.325295
$$960$$ 0 0
$$961$$ 33609.2 1.12817
$$962$$ 0 0
$$963$$ −1050.62 −0.0351567
$$964$$ 0 0
$$965$$ 7302.15 0.243590
$$966$$ 0 0
$$967$$ 11150.3 0.370806 0.185403 0.982663i $$-0.440641\pi$$
0.185403 + 0.982663i $$0.440641\pi$$
$$968$$ 0 0
$$969$$ 1673.40 0.0554772
$$970$$ 0 0
$$971$$ 6059.04 0.200251 0.100126 0.994975i $$-0.468076\pi$$
0.100126 + 0.994975i $$0.468076\pi$$
$$972$$ 0 0
$$973$$ 993.861 0.0327459
$$974$$ 0 0
$$975$$ 16677.1 0.547789
$$976$$ 0 0
$$977$$ 5700.49 0.186668 0.0933341 0.995635i $$-0.470248\pi$$
0.0933341 + 0.995635i $$0.470248\pi$$
$$978$$ 0 0
$$979$$ −29300.2 −0.956526
$$980$$ 0 0
$$981$$ −7534.59 −0.245220
$$982$$ 0 0
$$983$$ −197.480 −0.00640757 −0.00320378 0.999995i $$-0.501020\pi$$
−0.00320378 + 0.999995i $$0.501020\pi$$
$$984$$ 0 0
$$985$$ −17011.0 −0.550271
$$986$$ 0 0
$$987$$ −5516.58 −0.177908
$$988$$ 0 0
$$989$$ −48492.9 −1.55913
$$990$$ 0 0
$$991$$ −20620.8 −0.660990 −0.330495 0.943808i $$-0.607216\pi$$
−0.330495 + 0.943808i $$0.607216\pi$$
$$992$$ 0 0
$$993$$ 24894.6 0.795574
$$994$$ 0 0
$$995$$ 1588.03 0.0505969
$$996$$ 0 0
$$997$$ −19326.8 −0.613928 −0.306964 0.951721i $$-0.599313\pi$$
−0.306964 + 0.951721i $$0.599313\pi$$
$$998$$ 0 0
$$999$$ −321.220 −0.0101731
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 1344.4.a.bo.1.2 2
4.3 odd 2 1344.4.a.bg.1.2 2
8.3 odd 2 21.4.a.c.1.2 2
8.5 even 2 336.4.a.m.1.1 2
24.5 odd 2 1008.4.a.ba.1.2 2
24.11 even 2 63.4.a.e.1.1 2
40.3 even 4 525.4.d.g.274.2 4
40.19 odd 2 525.4.a.n.1.1 2
40.27 even 4 525.4.d.g.274.3 4
56.3 even 6 147.4.e.m.79.1 4
56.11 odd 6 147.4.e.l.79.1 4
56.13 odd 2 2352.4.a.bz.1.2 2
56.19 even 6 147.4.e.m.67.1 4
56.27 even 2 147.4.a.i.1.2 2
56.51 odd 6 147.4.e.l.67.1 4
120.59 even 2 1575.4.a.p.1.2 2
168.11 even 6 441.4.e.q.226.2 4
168.59 odd 6 441.4.e.p.226.2 4
168.83 odd 2 441.4.a.r.1.1 2
168.107 even 6 441.4.e.q.361.2 4
168.131 odd 6 441.4.e.p.361.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
21.4.a.c.1.2 2 8.3 odd 2
63.4.a.e.1.1 2 24.11 even 2
147.4.a.i.1.2 2 56.27 even 2
147.4.e.l.67.1 4 56.51 odd 6
147.4.e.l.79.1 4 56.11 odd 6
147.4.e.m.67.1 4 56.19 even 6
147.4.e.m.79.1 4 56.3 even 6
336.4.a.m.1.1 2 8.5 even 2
441.4.a.r.1.1 2 168.83 odd 2
441.4.e.p.226.2 4 168.59 odd 6
441.4.e.p.361.2 4 168.131 odd 6
441.4.e.q.226.2 4 168.11 even 6
441.4.e.q.361.2 4 168.107 even 6
525.4.a.n.1.1 2 40.19 odd 2
525.4.d.g.274.2 4 40.3 even 4
525.4.d.g.274.3 4 40.27 even 4
1008.4.a.ba.1.2 2 24.5 odd 2
1344.4.a.bg.1.2 2 4.3 odd 2
1344.4.a.bo.1.2 2 1.1 even 1 trivial
1575.4.a.p.1.2 2 120.59 even 2
2352.4.a.bz.1.2 2 56.13 odd 2