# Properties

 Label 260.2.a.b.1.3 Level $260$ Weight $2$ Character 260.1 Self dual yes Analytic conductor $2.076$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [260,2,Mod(1,260)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(260, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("260.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$260 = 2^{2} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 260.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: $$2.07611045255$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.564.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 5x + 3$$ x^3 - x^2 - 5*x + 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$2.51414$$ of defining polynomial Character $$\chi$$ $$=$$ 260.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.32088 q^{3} +1.00000 q^{5} -5.02827 q^{7} +8.02827 q^{9} +O(q^{10})$$ $$q+3.32088 q^{3} +1.00000 q^{5} -5.02827 q^{7} +8.02827 q^{9} +1.70739 q^{11} +1.00000 q^{13} +3.32088 q^{15} -4.64177 q^{17} -4.34916 q^{19} -16.6983 q^{21} -0.679116 q^{23} +1.00000 q^{25} +16.6983 q^{27} -1.02827 q^{29} -2.29261 q^{31} +5.67004 q^{33} -5.02827 q^{35} -1.61350 q^{37} +3.32088 q^{39} +4.64177 q^{41} -3.32088 q^{43} +8.02827 q^{45} +1.02827 q^{47} +18.2835 q^{49} -15.4148 q^{51} -9.41478 q^{53} +1.70739 q^{55} -14.4431 q^{57} -8.93438 q^{59} +9.02827 q^{61} -40.3684 q^{63} +1.00000 q^{65} +5.61350 q^{67} -2.25526 q^{69} +1.70739 q^{71} +12.4431 q^{73} +3.32088 q^{75} -8.58522 q^{77} -2.64177 q^{79} +31.3684 q^{81} -8.25526 q^{83} -4.64177 q^{85} -3.41478 q^{87} +1.22699 q^{89} -5.02827 q^{91} -7.61350 q^{93} -4.34916 q^{95} -0.0565477 q^{97} +13.7074 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 2 q^{3} + 3 q^{5} - 2 q^{7} + 11 q^{9}+O(q^{10})$$ 3 * q + 2 * q^3 + 3 * q^5 - 2 * q^7 + 11 * q^9 $$3 q + 2 q^{3} + 3 q^{5} - 2 q^{7} + 11 q^{9} + 3 q^{13} + 2 q^{15} + 2 q^{17} + 8 q^{19} - 8 q^{21} - 10 q^{23} + 3 q^{25} + 8 q^{27} + 10 q^{29} - 12 q^{31} - 12 q^{33} - 2 q^{35} - 2 q^{37} + 2 q^{39} - 2 q^{41} - 2 q^{43} + 11 q^{45} - 10 q^{47} + 23 q^{49} - 36 q^{51} - 18 q^{53} - 20 q^{57} - 16 q^{59} + 14 q^{61} - 50 q^{63} + 3 q^{65} + 14 q^{67} + 12 q^{69} + 14 q^{73} + 2 q^{75} - 36 q^{77} + 8 q^{79} + 23 q^{81} - 6 q^{83} + 2 q^{85} - 2 q^{89} - 2 q^{91} - 20 q^{93} + 8 q^{95} + 26 q^{97} + 36 q^{99}+O(q^{100})$$ 3 * q + 2 * q^3 + 3 * q^5 - 2 * q^7 + 11 * q^9 + 3 * q^13 + 2 * q^15 + 2 * q^17 + 8 * q^19 - 8 * q^21 - 10 * q^23 + 3 * q^25 + 8 * q^27 + 10 * q^29 - 12 * q^31 - 12 * q^33 - 2 * q^35 - 2 * q^37 + 2 * q^39 - 2 * q^41 - 2 * q^43 + 11 * q^45 - 10 * q^47 + 23 * q^49 - 36 * q^51 - 18 * q^53 - 20 * q^57 - 16 * q^59 + 14 * q^61 - 50 * q^63 + 3 * q^65 + 14 * q^67 + 12 * q^69 + 14 * q^73 + 2 * q^75 - 36 * q^77 + 8 * q^79 + 23 * q^81 - 6 * q^83 + 2 * q^85 - 2 * q^89 - 2 * q^91 - 20 * q^93 + 8 * q^95 + 26 * q^97 + 36 * 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
$$3$$ 3.32088 1.91731 0.958657 0.284565i $$-0.0918491\pi$$
0.958657 + 0.284565i $$0.0918491\pi$$
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ −5.02827 −1.90051 −0.950254 0.311475i $$-0.899177\pi$$
−0.950254 + 0.311475i $$0.899177\pi$$
$$8$$ 0 0
$$9$$ 8.02827 2.67609
$$10$$ 0 0
$$11$$ 1.70739 0.514797 0.257399 0.966305i $$-0.417135\pi$$
0.257399 + 0.966305i $$0.417135\pi$$
$$12$$ 0 0
$$13$$ 1.00000 0.277350
$$14$$ 0 0
$$15$$ 3.32088 0.857449
$$16$$ 0 0
$$17$$ −4.64177 −1.12579 −0.562897 0.826527i $$-0.690313\pi$$
−0.562897 + 0.826527i $$0.690313\pi$$
$$18$$ 0 0
$$19$$ −4.34916 −0.997765 −0.498883 0.866670i $$-0.666256\pi$$
−0.498883 + 0.866670i $$0.666256\pi$$
$$20$$ 0 0
$$21$$ −16.6983 −3.64387
$$22$$ 0 0
$$23$$ −0.679116 −0.141605 −0.0708027 0.997490i $$-0.522556\pi$$
−0.0708027 + 0.997490i $$0.522556\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 16.6983 3.21359
$$28$$ 0 0
$$29$$ −1.02827 −0.190946 −0.0954728 0.995432i $$-0.530436\pi$$
−0.0954728 + 0.995432i $$0.530436\pi$$
$$30$$ 0 0
$$31$$ −2.29261 −0.411765 −0.205883 0.978577i $$-0.566006\pi$$
−0.205883 + 0.978577i $$0.566006\pi$$
$$32$$ 0 0
$$33$$ 5.67004 0.987028
$$34$$ 0 0
$$35$$ −5.02827 −0.849933
$$36$$ 0 0
$$37$$ −1.61350 −0.265257 −0.132628 0.991166i $$-0.542342\pi$$
−0.132628 + 0.991166i $$0.542342\pi$$
$$38$$ 0 0
$$39$$ 3.32088 0.531767
$$40$$ 0 0
$$41$$ 4.64177 0.724923 0.362461 0.931999i $$-0.381937\pi$$
0.362461 + 0.931999i $$0.381937\pi$$
$$42$$ 0 0
$$43$$ −3.32088 −0.506430 −0.253215 0.967410i $$-0.581488\pi$$
−0.253215 + 0.967410i $$0.581488\pi$$
$$44$$ 0 0
$$45$$ 8.02827 1.19678
$$46$$ 0 0
$$47$$ 1.02827 0.149989 0.0749946 0.997184i $$-0.476106\pi$$
0.0749946 + 0.997184i $$0.476106\pi$$
$$48$$ 0 0
$$49$$ 18.2835 2.61193
$$50$$ 0 0
$$51$$ −15.4148 −2.15850
$$52$$ 0 0
$$53$$ −9.41478 −1.29322 −0.646610 0.762821i $$-0.723814\pi$$
−0.646610 + 0.762821i $$0.723814\pi$$
$$54$$ 0 0
$$55$$ 1.70739 0.230224
$$56$$ 0 0
$$57$$ −14.4431 −1.91303
$$58$$ 0 0
$$59$$ −8.93438 −1.16316 −0.581579 0.813490i $$-0.697565\pi$$
−0.581579 + 0.813490i $$0.697565\pi$$
$$60$$ 0 0
$$61$$ 9.02827 1.15595 0.577976 0.816054i $$-0.303843\pi$$
0.577976 + 0.816054i $$0.303843\pi$$
$$62$$ 0 0
$$63$$ −40.3684 −5.08594
$$64$$ 0 0
$$65$$ 1.00000 0.124035
$$66$$ 0 0
$$67$$ 5.61350 0.685798 0.342899 0.939372i $$-0.388591\pi$$
0.342899 + 0.939372i $$0.388591\pi$$
$$68$$ 0 0
$$69$$ −2.25526 −0.271502
$$70$$ 0 0
$$71$$ 1.70739 0.202630 0.101315 0.994854i $$-0.467695\pi$$
0.101315 + 0.994854i $$0.467695\pi$$
$$72$$ 0 0
$$73$$ 12.4431 1.45635 0.728175 0.685392i $$-0.240369\pi$$
0.728175 + 0.685392i $$0.240369\pi$$
$$74$$ 0 0
$$75$$ 3.32088 0.383463
$$76$$ 0 0
$$77$$ −8.58522 −0.978377
$$78$$ 0 0
$$79$$ −2.64177 −0.297222 −0.148611 0.988896i $$-0.547480\pi$$
−0.148611 + 0.988896i $$0.547480\pi$$
$$80$$ 0 0
$$81$$ 31.3684 3.48537
$$82$$ 0 0
$$83$$ −8.25526 −0.906133 −0.453066 0.891477i $$-0.649670\pi$$
−0.453066 + 0.891477i $$0.649670\pi$$
$$84$$ 0 0
$$85$$ −4.64177 −0.503471
$$86$$ 0 0
$$87$$ −3.41478 −0.366103
$$88$$ 0 0
$$89$$ 1.22699 0.130061 0.0650304 0.997883i $$-0.479286\pi$$
0.0650304 + 0.997883i $$0.479286\pi$$
$$90$$ 0 0
$$91$$ −5.02827 −0.527106
$$92$$ 0 0
$$93$$ −7.61350 −0.789483
$$94$$ 0 0
$$95$$ −4.34916 −0.446214
$$96$$ 0 0
$$97$$ −0.0565477 −0.00574155 −0.00287078 0.999996i $$-0.500914\pi$$
−0.00287078 + 0.999996i $$0.500914\pi$$
$$98$$ 0 0
$$99$$ 13.7074 1.37764
$$100$$ 0 0
$$101$$ 6.00000 0.597022 0.298511 0.954406i $$-0.403510\pi$$
0.298511 + 0.954406i $$0.403510\pi$$
$$102$$ 0 0
$$103$$ 9.37743 0.923986 0.461993 0.886884i $$-0.347135\pi$$
0.461993 + 0.886884i $$0.347135\pi$$
$$104$$ 0 0
$$105$$ −16.6983 −1.62959
$$106$$ 0 0
$$107$$ −13.3774 −1.29325 −0.646623 0.762810i $$-0.723819\pi$$
−0.646623 + 0.762810i $$0.723819\pi$$
$$108$$ 0 0
$$109$$ 11.2835 1.08077 0.540383 0.841419i $$-0.318279\pi$$
0.540383 + 0.841419i $$0.318279\pi$$
$$110$$ 0 0
$$111$$ −5.35823 −0.508581
$$112$$ 0 0
$$113$$ −18.6983 −1.75899 −0.879495 0.475908i $$-0.842119\pi$$
−0.879495 + 0.475908i $$0.842119\pi$$
$$114$$ 0 0
$$115$$ −0.679116 −0.0633278
$$116$$ 0 0
$$117$$ 8.02827 0.742214
$$118$$ 0 0
$$119$$ 23.3401 2.13958
$$120$$ 0 0
$$121$$ −8.08482 −0.734984
$$122$$ 0 0
$$123$$ 15.4148 1.38990
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ 10.7357 0.952636 0.476318 0.879273i $$-0.341971\pi$$
0.476318 + 0.879273i $$0.341971\pi$$
$$128$$ 0 0
$$129$$ −11.0283 −0.970985
$$130$$ 0 0
$$131$$ 7.22699 0.631425 0.315713 0.948855i $$-0.397756\pi$$
0.315713 + 0.948855i $$0.397756\pi$$
$$132$$ 0 0
$$133$$ 21.8688 1.89626
$$134$$ 0 0
$$135$$ 16.6983 1.43716
$$136$$ 0 0
$$137$$ 17.3401 1.48146 0.740732 0.671801i $$-0.234479\pi$$
0.740732 + 0.671801i $$0.234479\pi$$
$$138$$ 0 0
$$139$$ 9.35823 0.793755 0.396877 0.917872i $$-0.370094\pi$$
0.396877 + 0.917872i $$0.370094\pi$$
$$140$$ 0 0
$$141$$ 3.41478 0.287576
$$142$$ 0 0
$$143$$ 1.70739 0.142779
$$144$$ 0 0
$$145$$ −1.02827 −0.0853935
$$146$$ 0 0
$$147$$ 60.7175 5.00790
$$148$$ 0 0
$$149$$ 17.3401 1.42056 0.710278 0.703922i $$-0.248569\pi$$
0.710278 + 0.703922i $$0.248569\pi$$
$$150$$ 0 0
$$151$$ 3.57615 0.291023 0.145511 0.989357i $$-0.453517\pi$$
0.145511 + 0.989357i $$0.453517\pi$$
$$152$$ 0 0
$$153$$ −37.2654 −3.01273
$$154$$ 0 0
$$155$$ −2.29261 −0.184147
$$156$$ 0 0
$$157$$ −7.28354 −0.581290 −0.290645 0.956831i $$-0.593870\pi$$
−0.290645 + 0.956831i $$0.593870\pi$$
$$158$$ 0 0
$$159$$ −31.2654 −2.47951
$$160$$ 0 0
$$161$$ 3.41478 0.269122
$$162$$ 0 0
$$163$$ 0.840485 0.0658319 0.0329159 0.999458i $$-0.489521\pi$$
0.0329159 + 0.999458i $$0.489521\pi$$
$$164$$ 0 0
$$165$$ 5.67004 0.441412
$$166$$ 0 0
$$167$$ −13.0283 −1.00816 −0.504079 0.863658i $$-0.668168\pi$$
−0.504079 + 0.863658i $$0.668168\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 0 0
$$171$$ −34.9162 −2.67011
$$172$$ 0 0
$$173$$ 25.9253 1.97106 0.985532 0.169488i $$-0.0542113\pi$$
0.985532 + 0.169488i $$0.0542113\pi$$
$$174$$ 0 0
$$175$$ −5.02827 −0.380102
$$176$$ 0 0
$$177$$ −29.6700 −2.23014
$$178$$ 0 0
$$179$$ −4.77301 −0.356751 −0.178376 0.983962i $$-0.557084\pi$$
−0.178376 + 0.983962i $$0.557084\pi$$
$$180$$ 0 0
$$181$$ −14.3118 −1.06379 −0.531894 0.846811i $$-0.678520\pi$$
−0.531894 + 0.846811i $$0.678520\pi$$
$$182$$ 0 0
$$183$$ 29.9819 2.21632
$$184$$ 0 0
$$185$$ −1.61350 −0.118627
$$186$$ 0 0
$$187$$ −7.92531 −0.579556
$$188$$ 0 0
$$189$$ −83.9637 −6.10746
$$190$$ 0 0
$$191$$ 12.0000 0.868290 0.434145 0.900843i $$-0.357051\pi$$
0.434145 + 0.900843i $$0.357051\pi$$
$$192$$ 0 0
$$193$$ 4.05655 0.291997 0.145998 0.989285i $$-0.453361\pi$$
0.145998 + 0.989285i $$0.453361\pi$$
$$194$$ 0 0
$$195$$ 3.32088 0.237813
$$196$$ 0 0
$$197$$ −15.2835 −1.08891 −0.544453 0.838791i $$-0.683263\pi$$
−0.544453 + 0.838791i $$0.683263\pi$$
$$198$$ 0 0
$$199$$ −8.77301 −0.621902 −0.310951 0.950426i $$-0.600648\pi$$
−0.310951 + 0.950426i $$0.600648\pi$$
$$200$$ 0 0
$$201$$ 18.6418 1.31489
$$202$$ 0 0
$$203$$ 5.17044 0.362894
$$204$$ 0 0
$$205$$ 4.64177 0.324195
$$206$$ 0 0
$$207$$ −5.45213 −0.378949
$$208$$ 0 0
$$209$$ −7.42571 −0.513647
$$210$$ 0 0
$$211$$ 0.773010 0.0532162 0.0266081 0.999646i $$-0.491529\pi$$
0.0266081 + 0.999646i $$0.491529\pi$$
$$212$$ 0 0
$$213$$ 5.67004 0.388505
$$214$$ 0 0
$$215$$ −3.32088 −0.226482
$$216$$ 0 0
$$217$$ 11.5279 0.782563
$$218$$ 0 0
$$219$$ 41.3219 2.79228
$$220$$ 0 0
$$221$$ −4.64177 −0.312239
$$222$$ 0 0
$$223$$ −14.3118 −0.958390 −0.479195 0.877709i $$-0.659071\pi$$
−0.479195 + 0.877709i $$0.659071\pi$$
$$224$$ 0 0
$$225$$ 8.02827 0.535218
$$226$$ 0 0
$$227$$ −13.7266 −0.911066 −0.455533 0.890219i $$-0.650551\pi$$
−0.455533 + 0.890219i $$0.650551\pi$$
$$228$$ 0 0
$$229$$ 21.9253 1.44887 0.724433 0.689346i $$-0.242102\pi$$
0.724433 + 0.689346i $$0.242102\pi$$
$$230$$ 0 0
$$231$$ −28.5105 −1.87586
$$232$$ 0 0
$$233$$ −6.00000 −0.393073 −0.196537 0.980497i $$-0.562969\pi$$
−0.196537 + 0.980497i $$0.562969\pi$$
$$234$$ 0 0
$$235$$ 1.02827 0.0670772
$$236$$ 0 0
$$237$$ −8.77301 −0.569868
$$238$$ 0 0
$$239$$ −15.7639 −1.01968 −0.509842 0.860268i $$-0.670296\pi$$
−0.509842 + 0.860268i $$0.670296\pi$$
$$240$$ 0 0
$$241$$ 9.92531 0.639345 0.319673 0.947528i $$-0.396427\pi$$
0.319673 + 0.947528i $$0.396427\pi$$
$$242$$ 0 0
$$243$$ 54.0757 3.46896
$$244$$ 0 0
$$245$$ 18.2835 1.16809
$$246$$ 0 0
$$247$$ −4.34916 −0.276730
$$248$$ 0 0
$$249$$ −27.4148 −1.73734
$$250$$ 0 0
$$251$$ 24.6983 1.55894 0.779472 0.626437i $$-0.215487\pi$$
0.779472 + 0.626437i $$0.215487\pi$$
$$252$$ 0 0
$$253$$ −1.15951 −0.0728981
$$254$$ 0 0
$$255$$ −15.4148 −0.965311
$$256$$ 0 0
$$257$$ −20.0565 −1.25109 −0.625547 0.780187i $$-0.715124\pi$$
−0.625547 + 0.780187i $$0.715124\pi$$
$$258$$ 0 0
$$259$$ 8.11310 0.504123
$$260$$ 0 0
$$261$$ −8.25526 −0.510988
$$262$$ 0 0
$$263$$ 26.0757 1.60790 0.803950 0.594697i $$-0.202728\pi$$
0.803950 + 0.594697i $$0.202728\pi$$
$$264$$ 0 0
$$265$$ −9.41478 −0.578345
$$266$$ 0 0
$$267$$ 4.07469 0.249367
$$268$$ 0 0
$$269$$ −12.8296 −0.782232 −0.391116 0.920341i $$-0.627911\pi$$
−0.391116 + 0.920341i $$0.627911\pi$$
$$270$$ 0 0
$$271$$ −17.7074 −1.07565 −0.537824 0.843057i $$-0.680753\pi$$
−0.537824 + 0.843057i $$0.680753\pi$$
$$272$$ 0 0
$$273$$ −16.6983 −1.01063
$$274$$ 0 0
$$275$$ 1.70739 0.102959
$$276$$ 0 0
$$277$$ −15.4713 −0.929582 −0.464791 0.885420i $$-0.653871\pi$$
−0.464791 + 0.885420i $$0.653871\pi$$
$$278$$ 0 0
$$279$$ −18.4057 −1.10192
$$280$$ 0 0
$$281$$ 7.35823 0.438955 0.219478 0.975618i $$-0.429565\pi$$
0.219478 + 0.975618i $$0.429565\pi$$
$$282$$ 0 0
$$283$$ −0.604422 −0.0359292 −0.0179646 0.999839i $$-0.505719\pi$$
−0.0179646 + 0.999839i $$0.505719\pi$$
$$284$$ 0 0
$$285$$ −14.4431 −0.855533
$$286$$ 0 0
$$287$$ −23.3401 −1.37772
$$288$$ 0 0
$$289$$ 4.54602 0.267413
$$290$$ 0 0
$$291$$ −0.187788 −0.0110084
$$292$$ 0 0
$$293$$ 23.0101 1.34427 0.672133 0.740430i $$-0.265378\pi$$
0.672133 + 0.740430i $$0.265378\pi$$
$$294$$ 0 0
$$295$$ −8.93438 −0.520180
$$296$$ 0 0
$$297$$ 28.5105 1.65435
$$298$$ 0 0
$$299$$ −0.679116 −0.0392743
$$300$$ 0 0
$$301$$ 16.6983 0.962475
$$302$$ 0 0
$$303$$ 19.9253 1.14468
$$304$$ 0 0
$$305$$ 9.02827 0.516957
$$306$$ 0 0
$$307$$ 20.3684 1.16248 0.581242 0.813731i $$-0.302567\pi$$
0.581242 + 0.813731i $$0.302567\pi$$
$$308$$ 0 0
$$309$$ 31.1414 1.77157
$$310$$ 0 0
$$311$$ −19.9253 −1.12986 −0.564930 0.825139i $$-0.691097\pi$$
−0.564930 + 0.825139i $$0.691097\pi$$
$$312$$ 0 0
$$313$$ 31.4713 1.77886 0.889432 0.457067i $$-0.151100\pi$$
0.889432 + 0.457067i $$0.151100\pi$$
$$314$$ 0 0
$$315$$ −40.3684 −2.27450
$$316$$ 0 0
$$317$$ 1.72659 0.0969750 0.0484875 0.998824i $$-0.484560\pi$$
0.0484875 + 0.998824i $$0.484560\pi$$
$$318$$ 0 0
$$319$$ −1.75566 −0.0982983
$$320$$ 0 0
$$321$$ −44.4249 −2.47956
$$322$$ 0 0
$$323$$ 20.1878 1.12328
$$324$$ 0 0
$$325$$ 1.00000 0.0554700
$$326$$ 0 0
$$327$$ 37.4713 2.07217
$$328$$ 0 0
$$329$$ −5.17044 −0.285056
$$330$$ 0 0
$$331$$ 10.4057 0.571949 0.285975 0.958237i $$-0.407683\pi$$
0.285975 + 0.958237i $$0.407683\pi$$
$$332$$ 0 0
$$333$$ −12.9536 −0.709852
$$334$$ 0 0
$$335$$ 5.61350 0.306698
$$336$$ 0 0
$$337$$ −10.6983 −0.582774 −0.291387 0.956605i $$-0.594117\pi$$
−0.291387 + 0.956605i $$0.594117\pi$$
$$338$$ 0 0
$$339$$ −62.0950 −3.37253
$$340$$ 0 0
$$341$$ −3.91438 −0.211976
$$342$$ 0 0
$$343$$ −56.7367 −3.06349
$$344$$ 0 0
$$345$$ −2.25526 −0.121419
$$346$$ 0 0
$$347$$ −32.6044 −1.75030 −0.875149 0.483854i $$-0.839236\pi$$
−0.875149 + 0.483854i $$0.839236\pi$$
$$348$$ 0 0
$$349$$ −13.4148 −0.718077 −0.359038 0.933323i $$-0.616895\pi$$
−0.359038 + 0.933323i $$0.616895\pi$$
$$350$$ 0 0
$$351$$ 16.6983 0.891290
$$352$$ 0 0
$$353$$ −35.0101 −1.86340 −0.931701 0.363227i $$-0.881675\pi$$
−0.931701 + 0.363227i $$0.881675\pi$$
$$354$$ 0 0
$$355$$ 1.70739 0.0906188
$$356$$ 0 0
$$357$$ 77.5097 4.10225
$$358$$ 0 0
$$359$$ 20.9344 1.10487 0.552437 0.833555i $$-0.313698\pi$$
0.552437 + 0.833555i $$0.313698\pi$$
$$360$$ 0 0
$$361$$ −0.0848216 −0.00446429
$$362$$ 0 0
$$363$$ −26.8488 −1.40919
$$364$$ 0 0
$$365$$ 12.4431 0.651299
$$366$$ 0 0
$$367$$ 11.4340 0.596849 0.298424 0.954433i $$-0.403539\pi$$
0.298424 + 0.954433i $$0.403539\pi$$
$$368$$ 0 0
$$369$$ 37.2654 1.93996
$$370$$ 0 0
$$371$$ 47.3401 2.45777
$$372$$ 0 0
$$373$$ −14.1131 −0.730748 −0.365374 0.930861i $$-0.619059\pi$$
−0.365374 + 0.930861i $$0.619059\pi$$
$$374$$ 0 0
$$375$$ 3.32088 0.171490
$$376$$ 0 0
$$377$$ −1.02827 −0.0529588
$$378$$ 0 0
$$379$$ −1.89518 −0.0973487 −0.0486744 0.998815i $$-0.515500\pi$$
−0.0486744 + 0.998815i $$0.515500\pi$$
$$380$$ 0 0
$$381$$ 35.6519 1.82650
$$382$$ 0 0
$$383$$ −22.3118 −1.14008 −0.570040 0.821617i $$-0.693072\pi$$
−0.570040 + 0.821617i $$0.693072\pi$$
$$384$$ 0 0
$$385$$ −8.58522 −0.437543
$$386$$ 0 0
$$387$$ −26.6610 −1.35525
$$388$$ 0 0
$$389$$ −28.6802 −1.45414 −0.727071 0.686562i $$-0.759119\pi$$
−0.727071 + 0.686562i $$0.759119\pi$$
$$390$$ 0 0
$$391$$ 3.15230 0.159419
$$392$$ 0 0
$$393$$ 24.0000 1.21064
$$394$$ 0 0
$$395$$ −2.64177 −0.132922
$$396$$ 0 0
$$397$$ 15.5569 0.780781 0.390390 0.920649i $$-0.372340\pi$$
0.390390 + 0.920649i $$0.372340\pi$$
$$398$$ 0 0
$$399$$ 72.6236 3.63573
$$400$$ 0 0
$$401$$ −24.1696 −1.20697 −0.603487 0.797373i $$-0.706223\pi$$
−0.603487 + 0.797373i $$0.706223\pi$$
$$402$$ 0 0
$$403$$ −2.29261 −0.114203
$$404$$ 0 0
$$405$$ 31.3684 1.55871
$$406$$ 0 0
$$407$$ −2.75486 −0.136554
$$408$$ 0 0
$$409$$ −29.9253 −1.47971 −0.739856 0.672766i $$-0.765106\pi$$
−0.739856 + 0.672766i $$0.765106\pi$$
$$410$$ 0 0
$$411$$ 57.5844 2.84043
$$412$$ 0 0
$$413$$ 44.9245 2.21059
$$414$$ 0 0
$$415$$ −8.25526 −0.405235
$$416$$ 0 0
$$417$$ 31.0776 1.52188
$$418$$ 0 0
$$419$$ −8.58522 −0.419416 −0.209708 0.977764i $$-0.567251\pi$$
−0.209708 + 0.977764i $$0.567251\pi$$
$$420$$ 0 0
$$421$$ −21.7375 −1.05942 −0.529711 0.848178i $$-0.677700\pi$$
−0.529711 + 0.848178i $$0.677700\pi$$
$$422$$ 0 0
$$423$$ 8.25526 0.401385
$$424$$ 0 0
$$425$$ −4.64177 −0.225159
$$426$$ 0 0
$$427$$ −45.3966 −2.19690
$$428$$ 0 0
$$429$$ 5.67004 0.273752
$$430$$ 0 0
$$431$$ 13.7074 0.660262 0.330131 0.943935i $$-0.392907\pi$$
0.330131 + 0.943935i $$0.392907\pi$$
$$432$$ 0 0
$$433$$ −40.5671 −1.94953 −0.974765 0.223235i $$-0.928338\pi$$
−0.974765 + 0.223235i $$0.928338\pi$$
$$434$$ 0 0
$$435$$ −3.41478 −0.163726
$$436$$ 0 0
$$437$$ 2.95358 0.141289
$$438$$ 0 0
$$439$$ 26.5671 1.26798 0.633989 0.773342i $$-0.281417\pi$$
0.633989 + 0.773342i $$0.281417\pi$$
$$440$$ 0 0
$$441$$ 146.785 6.98977
$$442$$ 0 0
$$443$$ −16.7922 −0.797822 −0.398911 0.916990i $$-0.630612\pi$$
−0.398911 + 0.916990i $$0.630612\pi$$
$$444$$ 0 0
$$445$$ 1.22699 0.0581649
$$446$$ 0 0
$$447$$ 57.5844 2.72365
$$448$$ 0 0
$$449$$ −16.2443 −0.766618 −0.383309 0.923620i $$-0.625215\pi$$
−0.383309 + 0.923620i $$0.625215\pi$$
$$450$$ 0 0
$$451$$ 7.92531 0.373188
$$452$$ 0 0
$$453$$ 11.8760 0.557982
$$454$$ 0 0
$$455$$ −5.02827 −0.235729
$$456$$ 0 0
$$457$$ 6.77301 0.316828 0.158414 0.987373i $$-0.449362\pi$$
0.158414 + 0.987373i $$0.449362\pi$$
$$458$$ 0 0
$$459$$ −77.5097 −3.61784
$$460$$ 0 0
$$461$$ −22.8114 −1.06243 −0.531217 0.847236i $$-0.678265\pi$$
−0.531217 + 0.847236i $$0.678265\pi$$
$$462$$ 0 0
$$463$$ −16.3300 −0.758917 −0.379459 0.925209i $$-0.623890\pi$$
−0.379459 + 0.925209i $$0.623890\pi$$
$$464$$ 0 0
$$465$$ −7.61350 −0.353067
$$466$$ 0 0
$$467$$ 10.6610 0.493331 0.246665 0.969101i $$-0.420665\pi$$
0.246665 + 0.969101i $$0.420665\pi$$
$$468$$ 0 0
$$469$$ −28.2262 −1.30336
$$470$$ 0 0
$$471$$ −24.1878 −1.11451
$$472$$ 0 0
$$473$$ −5.67004 −0.260709
$$474$$ 0 0
$$475$$ −4.34916 −0.199553
$$476$$ 0 0
$$477$$ −75.5844 −3.46077
$$478$$ 0 0
$$479$$ 6.48040 0.296097 0.148048 0.988980i $$-0.452701\pi$$
0.148048 + 0.988980i $$0.452701\pi$$
$$480$$ 0 0
$$481$$ −1.61350 −0.0735690
$$482$$ 0 0
$$483$$ 11.3401 0.515992
$$484$$ 0 0
$$485$$ −0.0565477 −0.00256770
$$486$$ 0 0
$$487$$ 23.7831 1.07772 0.538858 0.842396i $$-0.318856\pi$$
0.538858 + 0.842396i $$0.318856\pi$$
$$488$$ 0 0
$$489$$ 2.79116 0.126220
$$490$$ 0 0
$$491$$ −6.13124 −0.276699 −0.138350 0.990383i $$-0.544180\pi$$
−0.138350 + 0.990383i $$0.544180\pi$$
$$492$$ 0 0
$$493$$ 4.77301 0.214966
$$494$$ 0 0
$$495$$ 13.7074 0.616101
$$496$$ 0 0
$$497$$ −8.58522 −0.385100
$$498$$ 0 0
$$499$$ 21.0475 0.942214 0.471107 0.882076i $$-0.343854\pi$$
0.471107 + 0.882076i $$0.343854\pi$$
$$500$$ 0 0
$$501$$ −43.2654 −1.93296
$$502$$ 0 0
$$503$$ 33.5652 1.49660 0.748300 0.663361i $$-0.230871\pi$$
0.748300 + 0.663361i $$0.230871\pi$$
$$504$$ 0 0
$$505$$ 6.00000 0.266996
$$506$$ 0 0
$$507$$ 3.32088 0.147486
$$508$$ 0 0
$$509$$ −14.5852 −0.646479 −0.323239 0.946317i $$-0.604772\pi$$
−0.323239 + 0.946317i $$0.604772\pi$$
$$510$$ 0 0
$$511$$ −62.5671 −2.76780
$$512$$ 0 0
$$513$$ −72.6236 −3.20641
$$514$$ 0 0
$$515$$ 9.37743 0.413219
$$516$$ 0 0
$$517$$ 1.75566 0.0772140
$$518$$ 0 0
$$519$$ 86.0950 3.77915
$$520$$ 0 0
$$521$$ −3.48225 −0.152560 −0.0762802 0.997086i $$-0.524304\pi$$
−0.0762802 + 0.997086i $$0.524304\pi$$
$$522$$ 0 0
$$523$$ −11.5087 −0.503239 −0.251620 0.967826i $$-0.580963\pi$$
−0.251620 + 0.967826i $$0.580963\pi$$
$$524$$ 0 0
$$525$$ −16.6983 −0.728774
$$526$$ 0 0
$$527$$ 10.6418 0.463563
$$528$$ 0 0
$$529$$ −22.5388 −0.979948
$$530$$ 0 0
$$531$$ −71.7276 −3.11271
$$532$$ 0 0
$$533$$ 4.64177 0.201057
$$534$$ 0 0
$$535$$ −13.3774 −0.578357
$$536$$ 0 0
$$537$$ −15.8506 −0.684004
$$538$$ 0 0
$$539$$ 31.2171 1.34462
$$540$$ 0 0
$$541$$ −13.8122 −0.593833 −0.296917 0.954903i $$-0.595958\pi$$
−0.296917 + 0.954903i $$0.595958\pi$$
$$542$$ 0 0
$$543$$ −47.5279 −2.03962
$$544$$ 0 0
$$545$$ 11.2835 0.483334
$$546$$ 0 0
$$547$$ −5.37743 −0.229922 −0.114961 0.993370i $$-0.536674\pi$$
−0.114961 + 0.993370i $$0.536674\pi$$
$$548$$ 0 0
$$549$$ 72.4815 3.09343
$$550$$ 0 0
$$551$$ 4.47213 0.190519
$$552$$ 0 0
$$553$$ 13.2835 0.564873
$$554$$ 0 0
$$555$$ −5.35823 −0.227444
$$556$$ 0 0
$$557$$ −0.0674757 −0.00285904 −0.00142952 0.999999i $$-0.500455\pi$$
−0.00142952 + 0.999999i $$0.500455\pi$$
$$558$$ 0 0
$$559$$ −3.32088 −0.140458
$$560$$ 0 0
$$561$$ −26.3190 −1.11119
$$562$$ 0 0
$$563$$ −7.20779 −0.303772 −0.151886 0.988398i $$-0.548535\pi$$
−0.151886 + 0.988398i $$0.548535\pi$$
$$564$$ 0 0
$$565$$ −18.6983 −0.786644
$$566$$ 0 0
$$567$$ −157.729 −6.62398
$$568$$ 0 0
$$569$$ −7.85783 −0.329417 −0.164709 0.986342i $$-0.552668\pi$$
−0.164709 + 0.986342i $$0.552668\pi$$
$$570$$ 0 0
$$571$$ −23.9253 −1.00124 −0.500621 0.865666i $$-0.666895\pi$$
−0.500621 + 0.865666i $$0.666895\pi$$
$$572$$ 0 0
$$573$$ 39.8506 1.66478
$$574$$ 0 0
$$575$$ −0.679116 −0.0283211
$$576$$ 0 0
$$577$$ 31.0101 1.29097 0.645484 0.763774i $$-0.276656\pi$$
0.645484 + 0.763774i $$0.276656\pi$$
$$578$$ 0 0
$$579$$ 13.4713 0.559849
$$580$$ 0 0
$$581$$ 41.5097 1.72211
$$582$$ 0 0
$$583$$ −16.0747 −0.665746
$$584$$ 0 0
$$585$$ 8.02827 0.331928
$$586$$ 0 0
$$587$$ 47.4076 1.95672 0.978360 0.206911i $$-0.0663411\pi$$
0.978360 + 0.206911i $$0.0663411\pi$$
$$588$$ 0 0
$$589$$ 9.97093 0.410845
$$590$$ 0 0
$$591$$ −50.7549 −2.08778
$$592$$ 0 0
$$593$$ −14.8861 −0.611299 −0.305650 0.952144i $$-0.598874\pi$$
−0.305650 + 0.952144i $$0.598874\pi$$
$$594$$ 0 0
$$595$$ 23.3401 0.956850
$$596$$ 0 0
$$597$$ −29.1342 −1.19238
$$598$$ 0 0
$$599$$ 42.8680 1.75154 0.875769 0.482731i $$-0.160355\pi$$
0.875769 + 0.482731i $$0.160355\pi$$
$$600$$ 0 0
$$601$$ −10.0000 −0.407909 −0.203954 0.978980i $$-0.565379\pi$$
−0.203954 + 0.978980i $$0.565379\pi$$
$$602$$ 0 0
$$603$$ 45.0667 1.83526
$$604$$ 0 0
$$605$$ −8.08482 −0.328695
$$606$$ 0 0
$$607$$ −18.7357 −0.760457 −0.380229 0.924893i $$-0.624155\pi$$
−0.380229 + 0.924893i $$0.624155\pi$$
$$608$$ 0 0
$$609$$ 17.1704 0.695781
$$610$$ 0 0
$$611$$ 1.02827 0.0415995
$$612$$ 0 0
$$613$$ 36.6802 1.48150 0.740749 0.671782i $$-0.234471\pi$$
0.740749 + 0.671782i $$0.234471\pi$$
$$614$$ 0 0
$$615$$ 15.4148 0.621584
$$616$$ 0 0
$$617$$ −8.05655 −0.324344 −0.162172 0.986762i $$-0.551850\pi$$
−0.162172 + 0.986762i $$0.551850\pi$$
$$618$$ 0 0
$$619$$ −33.1606 −1.33284 −0.666418 0.745578i $$-0.732173\pi$$
−0.666418 + 0.745578i $$0.732173\pi$$
$$620$$ 0 0
$$621$$ −11.3401 −0.455062
$$622$$ 0 0
$$623$$ −6.16964 −0.247182
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ −24.6599 −0.984822
$$628$$ 0 0
$$629$$ 7.48947 0.298625
$$630$$ 0 0
$$631$$ −33.8205 −1.34637 −0.673186 0.739473i $$-0.735075\pi$$
−0.673186 + 0.739473i $$0.735075\pi$$
$$632$$ 0 0
$$633$$ 2.56708 0.102032
$$634$$ 0 0
$$635$$ 10.7357 0.426032
$$636$$ 0 0
$$637$$ 18.2835 0.724420
$$638$$ 0 0
$$639$$ 13.7074 0.542256
$$640$$ 0 0
$$641$$ 12.8296 0.506737 0.253369 0.967370i $$-0.418461\pi$$
0.253369 + 0.967370i $$0.418461\pi$$
$$642$$ 0 0
$$643$$ 43.7084 1.72369 0.861846 0.507169i $$-0.169308\pi$$
0.861846 + 0.507169i $$0.169308\pi$$
$$644$$ 0 0
$$645$$ −11.0283 −0.434238
$$646$$ 0 0
$$647$$ 25.4158 0.999200 0.499600 0.866256i $$-0.333480\pi$$
0.499600 + 0.866256i $$0.333480\pi$$
$$648$$ 0 0
$$649$$ −15.2545 −0.598790
$$650$$ 0 0
$$651$$ 38.2827 1.50042
$$652$$ 0 0
$$653$$ −3.94345 −0.154319 −0.0771596 0.997019i $$-0.524585\pi$$
−0.0771596 + 0.997019i $$0.524585\pi$$
$$654$$ 0 0
$$655$$ 7.22699 0.282382
$$656$$ 0 0
$$657$$ 99.8962 3.89732
$$658$$ 0 0
$$659$$ −38.0950 −1.48397 −0.741984 0.670417i $$-0.766115\pi$$
−0.741984 + 0.670417i $$0.766115\pi$$
$$660$$ 0 0
$$661$$ 18.1131 0.704518 0.352259 0.935903i $$-0.385414\pi$$
0.352259 + 0.935903i $$0.385414\pi$$
$$662$$ 0 0
$$663$$ −15.4148 −0.598660
$$664$$ 0 0
$$665$$ 21.8688 0.848034
$$666$$ 0 0
$$667$$ 0.698317 0.0270389
$$668$$ 0 0
$$669$$ −47.5279 −1.83753
$$670$$ 0 0
$$671$$ 15.4148 0.595081
$$672$$ 0 0
$$673$$ 43.2088 1.66558 0.832789 0.553590i $$-0.186743\pi$$
0.832789 + 0.553590i $$0.186743\pi$$
$$674$$ 0 0
$$675$$ 16.6983 0.642719
$$676$$ 0 0
$$677$$ −30.6983 −1.17983 −0.589916 0.807465i $$-0.700839\pi$$
−0.589916 + 0.807465i $$0.700839\pi$$
$$678$$ 0 0
$$679$$ 0.284337 0.0109119
$$680$$ 0 0
$$681$$ −45.5844 −1.74680
$$682$$ 0 0
$$683$$ −32.9536 −1.26093 −0.630467 0.776216i $$-0.717137\pi$$
−0.630467 + 0.776216i $$0.717137\pi$$
$$684$$ 0 0
$$685$$ 17.3401 0.662531
$$686$$ 0 0
$$687$$ 72.8114 2.77793
$$688$$ 0 0
$$689$$ −9.41478 −0.358675
$$690$$ 0 0
$$691$$ 37.1222 1.41219 0.706097 0.708115i $$-0.250454\pi$$
0.706097 + 0.708115i $$0.250454\pi$$
$$692$$ 0 0
$$693$$ −68.9245 −2.61823
$$694$$ 0 0
$$695$$ 9.35823 0.354978
$$696$$ 0 0
$$697$$ −21.5460 −0.816114
$$698$$ 0 0
$$699$$ −19.9253 −0.753644
$$700$$ 0 0
$$701$$ −7.39663 −0.279367 −0.139683 0.990196i $$-0.544609\pi$$
−0.139683 + 0.990196i $$0.544609\pi$$
$$702$$ 0 0
$$703$$ 7.01735 0.264664
$$704$$ 0 0
$$705$$ 3.41478 0.128608
$$706$$ 0 0
$$707$$ −30.1696 −1.13465
$$708$$ 0 0
$$709$$ 28.7549 1.07991 0.539956 0.841693i $$-0.318441\pi$$
0.539956 + 0.841693i $$0.318441\pi$$
$$710$$ 0 0
$$711$$ −21.2088 −0.795394
$$712$$ 0 0
$$713$$ 1.55695 0.0583081
$$714$$ 0 0
$$715$$ 1.70739 0.0638527
$$716$$ 0 0
$$717$$ −52.3502 −1.95505
$$718$$ 0 0
$$719$$ −39.4532 −1.47136 −0.735678 0.677332i $$-0.763136\pi$$
−0.735678 + 0.677332i $$0.763136\pi$$
$$720$$ 0 0
$$721$$ −47.1523 −1.75604
$$722$$ 0 0
$$723$$ 32.9608 1.22583
$$724$$ 0 0
$$725$$ −1.02827 −0.0381891
$$726$$ 0 0
$$727$$ −0.866904 −0.0321517 −0.0160758 0.999871i $$-0.505117\pi$$
−0.0160758 + 0.999871i $$0.505117\pi$$
$$728$$ 0 0
$$729$$ 85.4742 3.16571
$$730$$ 0 0
$$731$$ 15.4148 0.570136
$$732$$ 0 0
$$733$$ −10.0000 −0.369358 −0.184679 0.982799i $$-0.559125\pi$$
−0.184679 + 0.982799i $$0.559125\pi$$
$$734$$ 0 0
$$735$$ 60.7175 2.23960
$$736$$ 0 0
$$737$$ 9.58442 0.353047
$$738$$ 0 0
$$739$$ −31.5015 −1.15880 −0.579400 0.815043i $$-0.696713\pi$$
−0.579400 + 0.815043i $$0.696713\pi$$
$$740$$ 0 0
$$741$$ −14.4431 −0.530579
$$742$$ 0 0
$$743$$ −31.8578 −1.16875 −0.584375 0.811484i $$-0.698660\pi$$
−0.584375 + 0.811484i $$0.698660\pi$$
$$744$$ 0 0
$$745$$ 17.3401 0.635292
$$746$$ 0 0
$$747$$ −66.2755 −2.42489
$$748$$ 0 0
$$749$$ 67.2654 2.45782
$$750$$ 0 0
$$751$$ 46.0950 1.68203 0.841014 0.541013i $$-0.181959\pi$$
0.841014 + 0.541013i $$0.181959\pi$$
$$752$$ 0 0
$$753$$ 82.0203 2.98898
$$754$$ 0 0
$$755$$ 3.57615 0.130149
$$756$$ 0 0
$$757$$ 5.01735 0.182359 0.0911793 0.995834i $$-0.470936\pi$$
0.0911793 + 0.995834i $$0.470936\pi$$
$$758$$ 0 0
$$759$$ −3.85061 −0.139768
$$760$$ 0 0
$$761$$ 24.8296 0.900071 0.450035 0.893011i $$-0.351411\pi$$
0.450035 + 0.893011i $$0.351411\pi$$
$$762$$ 0 0
$$763$$ −56.7367 −2.05401
$$764$$ 0 0
$$765$$ −37.2654 −1.34733
$$766$$ 0 0
$$767$$ −8.93438 −0.322602
$$768$$ 0 0
$$769$$ 36.9427 1.33219 0.666093 0.745869i $$-0.267965\pi$$
0.666093 + 0.745869i $$0.267965\pi$$
$$770$$ 0 0
$$771$$ −66.6055 −2.39874
$$772$$ 0 0
$$773$$ −16.4431 −0.591415 −0.295708 0.955278i $$-0.595555\pi$$
−0.295708 + 0.955278i $$0.595555\pi$$
$$774$$ 0 0
$$775$$ −2.29261 −0.0823530
$$776$$ 0 0
$$777$$ 26.9427 0.966562
$$778$$ 0 0
$$779$$ −20.1878 −0.723303
$$780$$ 0 0
$$781$$ 2.91518 0.104313
$$782$$ 0 0
$$783$$ −17.1704 −0.613622
$$784$$ 0 0
$$785$$ −7.28354 −0.259961
$$786$$ 0 0
$$787$$ 40.2553 1.43495 0.717473 0.696587i $$-0.245299\pi$$
0.717473 + 0.696587i $$0.245299\pi$$
$$788$$ 0 0
$$789$$ 86.5946 3.08285
$$790$$ 0 0
$$791$$ 94.0203 3.34298
$$792$$ 0 0
$$793$$ 9.02827 0.320603
$$794$$ 0 0
$$795$$ −31.2654 −1.10887
$$796$$ 0 0
$$797$$ 27.2835 0.966432 0.483216 0.875501i $$-0.339468\pi$$
0.483216 + 0.875501i $$0.339468\pi$$
$$798$$ 0 0
$$799$$ −4.77301 −0.168857
$$800$$ 0 0
$$801$$ 9.85061 0.348054
$$802$$ 0 0
$$803$$ 21.2451 0.749725
$$804$$ 0 0
$$805$$ 3.41478 0.120355
$$806$$ 0 0
$$807$$ −42.6055 −1.49978
$$808$$ 0 0
$$809$$ −28.4815 −1.00135 −0.500677 0.865634i $$-0.666916\pi$$
−0.500677 + 0.865634i $$0.666916\pi$$
$$810$$ 0 0
$$811$$ 31.6892 1.11276 0.556380 0.830928i $$-0.312190\pi$$
0.556380 + 0.830928i $$0.312190\pi$$
$$812$$ 0 0
$$813$$ −58.8042 −2.06235
$$814$$ 0 0
$$815$$ 0.840485 0.0294409
$$816$$ 0 0
$$817$$ 14.4431 0.505298
$$818$$ 0 0
$$819$$ −40.3684 −1.41058
$$820$$ 0 0
$$821$$ −6.65991 −0.232433 −0.116216 0.993224i $$-0.537077\pi$$
−0.116216 + 0.993224i $$0.537077\pi$$
$$822$$ 0 0
$$823$$ −7.79301 −0.271647 −0.135824 0.990733i $$-0.543368\pi$$
−0.135824 + 0.990733i $$0.543368\pi$$
$$824$$ 0 0
$$825$$ 5.67004 0.197406
$$826$$ 0 0
$$827$$ 26.4249 0.918884 0.459442 0.888208i $$-0.348049\pi$$
0.459442 + 0.888208i $$0.348049\pi$$
$$828$$ 0 0
$$829$$ 11.7447 0.407912 0.203956 0.978980i $$-0.434620\pi$$
0.203956 + 0.978980i $$0.434620\pi$$
$$830$$ 0 0
$$831$$ −51.3785 −1.78230
$$832$$ 0 0
$$833$$ −84.8680 −2.94050
$$834$$ 0 0
$$835$$ −13.0283 −0.450862
$$836$$ 0 0
$$837$$ −38.2827 −1.32325
$$838$$ 0 0
$$839$$ −44.5753 −1.53891 −0.769456 0.638700i $$-0.779473\pi$$
−0.769456 + 0.638700i $$0.779473\pi$$
$$840$$ 0 0
$$841$$ −27.9427 −0.963540
$$842$$ 0 0
$$843$$ 24.4358 0.841615
$$844$$ 0 0
$$845$$ 1.00000 0.0344010
$$846$$ 0 0
$$847$$ 40.6527 1.39684
$$848$$ 0 0
$$849$$ −2.00722 −0.0688875
$$850$$ 0 0
$$851$$ 1.09575 0.0375618
$$852$$ 0 0
$$853$$ 29.6135 1.01395 0.506973 0.861962i $$-0.330764\pi$$
0.506973 + 0.861962i $$0.330764\pi$$
$$854$$ 0 0
$$855$$ −34.9162 −1.19411
$$856$$ 0 0
$$857$$ 22.5105 0.768945 0.384472 0.923136i $$-0.374383\pi$$
0.384472 + 0.923136i $$0.374383\pi$$
$$858$$ 0 0
$$859$$ −0.585221 −0.0199675 −0.00998375 0.999950i $$-0.503178\pi$$
−0.00998375 + 0.999950i $$0.503178\pi$$
$$860$$ 0 0
$$861$$ −77.5097 −2.64152
$$862$$ 0 0
$$863$$ 45.6519 1.55401 0.777004 0.629495i $$-0.216738\pi$$
0.777004 + 0.629495i $$0.216738\pi$$
$$864$$ 0 0
$$865$$ 25.9253 0.881487
$$866$$ 0 0
$$867$$ 15.0968 0.512714
$$868$$ 0 0
$$869$$ −4.51053 −0.153009
$$870$$ 0 0
$$871$$ 5.61350 0.190206
$$872$$ 0 0
$$873$$ −0.453981 −0.0153649
$$874$$ 0 0
$$875$$ −5.02827 −0.169987
$$876$$ 0 0
$$877$$ −10.0000 −0.337676 −0.168838 0.985644i $$-0.554001\pi$$
−0.168838 + 0.985644i $$0.554001\pi$$
$$878$$ 0 0
$$879$$ 76.4140 2.57738
$$880$$ 0 0
$$881$$ 42.5380 1.43314 0.716571 0.697514i $$-0.245711\pi$$
0.716571 + 0.697514i $$0.245711\pi$$
$$882$$ 0 0
$$883$$ 6.22513 0.209492 0.104746 0.994499i $$-0.466597\pi$$
0.104746 + 0.994499i $$0.466597\pi$$
$$884$$ 0 0
$$885$$ −29.6700 −0.997348
$$886$$ 0 0
$$887$$ 22.0011 0.738723 0.369362 0.929286i $$-0.379576\pi$$
0.369362 + 0.929286i $$0.379576\pi$$
$$888$$ 0 0
$$889$$ −53.9819 −1.81049
$$890$$ 0 0
$$891$$ 53.5580 1.79426
$$892$$ 0 0
$$893$$ −4.47213 −0.149654
$$894$$ 0 0
$$895$$ −4.77301 −0.159544
$$896$$ 0 0
$$897$$ −2.25526 −0.0753011
$$898$$ 0 0
$$899$$ 2.35743 0.0786247
$$900$$ 0 0
$$901$$ 43.7012 1.45590
$$902$$ 0 0
$$903$$ 55.4532 1.84537
$$904$$ 0 0
$$905$$ −14.3118 −0.475741
$$906$$ 0 0
$$907$$ 28.2070 0.936598 0.468299 0.883570i $$-0.344867\pi$$
0.468299 + 0.883570i $$0.344867\pi$$
$$908$$ 0 0
$$909$$ 48.1696 1.59769
$$910$$ 0 0
$$911$$ 40.5105 1.34217 0.671087 0.741379i $$-0.265828\pi$$
0.671087 + 0.741379i $$0.265828\pi$$
$$912$$ 0 0
$$913$$ −14.0950 −0.466475
$$914$$ 0 0
$$915$$ 29.9819 0.991170
$$916$$ 0 0
$$917$$ −36.3393 −1.20003
$$918$$ 0 0
$$919$$ 0.0746930 0.00246389 0.00123195 0.999999i $$-0.499608\pi$$
0.00123195 + 0.999999i $$0.499608\pi$$
$$920$$ 0 0
$$921$$ 67.6410 2.22885
$$922$$ 0 0
$$923$$ 1.70739 0.0561994
$$924$$ 0 0
$$925$$ −1.61350 −0.0530514
$$926$$ 0 0
$$927$$ 75.2846 2.47267
$$928$$ 0 0
$$929$$ 53.6410 1.75990 0.879952 0.475063i $$-0.157575\pi$$
0.879952 + 0.475063i $$0.157575\pi$$
$$930$$ 0 0
$$931$$ −79.5180 −2.60610
$$932$$ 0 0
$$933$$ −66.1696 −2.16630
$$934$$ 0 0
$$935$$ −7.92531 −0.259185
$$936$$ 0 0
$$937$$ 26.0000 0.849383 0.424691 0.905338i $$-0.360383\pi$$
0.424691 + 0.905338i $$0.360383\pi$$
$$938$$ 0 0
$$939$$ 104.513 3.41064
$$940$$ 0 0
$$941$$ −49.9253 −1.62752 −0.813759 0.581202i $$-0.802583\pi$$
−0.813759 + 0.581202i $$0.802583\pi$$
$$942$$ 0 0
$$943$$ −3.15230 −0.102653
$$944$$ 0 0
$$945$$ −83.9637 −2.73134
$$946$$ 0 0
$$947$$ −5.80128 −0.188516 −0.0942582 0.995548i $$-0.530048\pi$$
−0.0942582 + 0.995548i $$0.530048\pi$$
$$948$$ 0 0
$$949$$ 12.4431 0.403919
$$950$$ 0 0
$$951$$ 5.73381 0.185931
$$952$$ 0 0
$$953$$ −13.2270 −0.428464 −0.214232 0.976783i $$-0.568725\pi$$
−0.214232 + 0.976783i $$0.568725\pi$$
$$954$$ 0 0
$$955$$ 12.0000 0.388311
$$956$$ 0 0
$$957$$ −5.83036 −0.188469
$$958$$ 0 0
$$959$$ −87.1907 −2.81553
$$960$$ 0 0
$$961$$ −25.7439 −0.830450
$$962$$ 0 0
$$963$$ −107.398 −3.46084
$$964$$ 0 0
$$965$$ 4.05655 0.130585
$$966$$ 0 0
$$967$$ −1.61350 −0.0518865 −0.0259433 0.999663i $$-0.508259\pi$$
−0.0259433 + 0.999663i $$0.508259\pi$$
$$968$$ 0 0
$$969$$ 67.0413 2.15368
$$970$$ 0 0
$$971$$ 16.7730 0.538271 0.269136 0.963102i $$-0.413262\pi$$
0.269136 + 0.963102i $$0.413262\pi$$
$$972$$ 0 0
$$973$$ −47.0557 −1.50854
$$974$$ 0 0
$$975$$ 3.32088 0.106353
$$976$$ 0 0
$$977$$ −47.6700 −1.52510 −0.762550 0.646929i $$-0.776053\pi$$
−0.762550 + 0.646929i $$0.776053\pi$$
$$978$$ 0 0
$$979$$ 2.09495 0.0669549
$$980$$ 0 0
$$981$$ 90.5873 2.89223
$$982$$ 0 0
$$983$$ 30.4996 0.972786 0.486393 0.873740i $$-0.338312\pi$$
0.486393 + 0.873740i $$0.338312\pi$$
$$984$$ 0 0
$$985$$ −15.2835 −0.486974
$$986$$ 0 0
$$987$$ −17.1704 −0.546541
$$988$$ 0 0
$$989$$ 2.25526 0.0717132
$$990$$ 0 0
$$991$$ −19.8506 −0.630576 −0.315288 0.948996i $$-0.602101\pi$$
−0.315288 + 0.948996i $$0.602101\pi$$
$$992$$ 0 0
$$993$$ 34.5561 1.09661
$$994$$ 0 0
$$995$$ −8.77301 −0.278123
$$996$$ 0 0
$$997$$ −33.6410 −1.06542 −0.532710 0.846298i $$-0.678826\pi$$
−0.532710 + 0.846298i $$0.678826\pi$$
$$998$$ 0 0
$$999$$ −26.9427 −0.852428
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 260.2.a.b.1.3 3
3.2 odd 2 2340.2.a.n.1.1 3
4.3 odd 2 1040.2.a.o.1.1 3
5.2 odd 4 1300.2.c.f.1249.1 6
5.3 odd 4 1300.2.c.f.1249.6 6
5.4 even 2 1300.2.a.i.1.1 3
8.3 odd 2 4160.2.a.br.1.3 3
8.5 even 2 4160.2.a.bo.1.1 3
12.11 even 2 9360.2.a.da.1.3 3
13.5 odd 4 3380.2.f.h.3041.5 6
13.8 odd 4 3380.2.f.h.3041.6 6
13.12 even 2 3380.2.a.o.1.3 3
20.19 odd 2 5200.2.a.ci.1.3 3

By twisted newform
Twist Min Dim Char Parity Ord Type
260.2.a.b.1.3 3 1.1 even 1 trivial
1040.2.a.o.1.1 3 4.3 odd 2
1300.2.a.i.1.1 3 5.4 even 2
1300.2.c.f.1249.1 6 5.2 odd 4
1300.2.c.f.1249.6 6 5.3 odd 4
2340.2.a.n.1.1 3 3.2 odd 2
3380.2.a.o.1.3 3 13.12 even 2
3380.2.f.h.3041.5 6 13.5 odd 4
3380.2.f.h.3041.6 6 13.8 odd 4
4160.2.a.bo.1.1 3 8.5 even 2
4160.2.a.br.1.3 3 8.3 odd 2
5200.2.a.ci.1.3 3 20.19 odd 2
9360.2.a.da.1.3 3 12.11 even 2