# Properties

 Label 380.2.a.d.1.2 Level $380$ Weight $2$ Character 380.1 Self dual yes Analytic conductor $3.034$ 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] = [380,2,Mod(1,380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(380, 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("380.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$380 = 2^{2} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 380.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: $$3.03431527681$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{12})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.73205$$ of defining polynomial Character $$\chi$$ $$=$$ 380.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.73205 q^{3} +1.00000 q^{5} +2.00000 q^{7} +4.46410 q^{9} +O(q^{10})$$ $$q+2.73205 q^{3} +1.00000 q^{5} +2.00000 q^{7} +4.46410 q^{9} -3.46410 q^{11} -2.73205 q^{13} +2.73205 q^{15} -3.46410 q^{17} +1.00000 q^{19} +5.46410 q^{21} -3.46410 q^{23} +1.00000 q^{25} +4.00000 q^{27} +3.46410 q^{29} -1.46410 q^{31} -9.46410 q^{33} +2.00000 q^{35} +6.73205 q^{37} -7.46410 q^{39} -6.00000 q^{41} -4.92820 q^{43} +4.46410 q^{45} +12.9282 q^{47} -3.00000 q^{49} -9.46410 q^{51} -10.7321 q^{53} -3.46410 q^{55} +2.73205 q^{57} +6.92820 q^{59} +12.3923 q^{61} +8.92820 q^{63} -2.73205 q^{65} +6.73205 q^{67} -9.46410 q^{69} -2.53590 q^{71} -0.535898 q^{73} +2.73205 q^{75} -6.92820 q^{77} +2.92820 q^{79} -2.46410 q^{81} +3.46410 q^{83} -3.46410 q^{85} +9.46410 q^{87} -15.4641 q^{89} -5.46410 q^{91} -4.00000 q^{93} +1.00000 q^{95} -16.5885 q^{97} -15.4641 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} + 2 q^{5} + 4 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 + 2 * q^5 + 4 * q^7 + 2 * q^9 $$2 q + 2 q^{3} + 2 q^{5} + 4 q^{7} + 2 q^{9} - 2 q^{13} + 2 q^{15} + 2 q^{19} + 4 q^{21} + 2 q^{25} + 8 q^{27} + 4 q^{31} - 12 q^{33} + 4 q^{35} + 10 q^{37} - 8 q^{39} - 12 q^{41} + 4 q^{43} + 2 q^{45} + 12 q^{47} - 6 q^{49} - 12 q^{51} - 18 q^{53} + 2 q^{57} + 4 q^{61} + 4 q^{63} - 2 q^{65} + 10 q^{67} - 12 q^{69} - 12 q^{71} - 8 q^{73} + 2 q^{75} - 8 q^{79} + 2 q^{81} + 12 q^{87} - 24 q^{89} - 4 q^{91} - 8 q^{93} + 2 q^{95} - 2 q^{97} - 24 q^{99}+O(q^{100})$$ 2 * q + 2 * q^3 + 2 * q^5 + 4 * q^7 + 2 * q^9 - 2 * q^13 + 2 * q^15 + 2 * q^19 + 4 * q^21 + 2 * q^25 + 8 * q^27 + 4 * q^31 - 12 * q^33 + 4 * q^35 + 10 * q^37 - 8 * q^39 - 12 * q^41 + 4 * q^43 + 2 * q^45 + 12 * q^47 - 6 * q^49 - 12 * q^51 - 18 * q^53 + 2 * q^57 + 4 * q^61 + 4 * q^63 - 2 * q^65 + 10 * q^67 - 12 * q^69 - 12 * q^71 - 8 * q^73 + 2 * q^75 - 8 * q^79 + 2 * q^81 + 12 * q^87 - 24 * q^89 - 4 * q^91 - 8 * q^93 + 2 * q^95 - 2 * q^97 - 24 * 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
$$3$$ 2.73205 1.57735 0.788675 0.614810i $$-0.210767\pi$$
0.788675 + 0.614810i $$0.210767\pi$$
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ 2.00000 0.755929 0.377964 0.925820i $$-0.376624\pi$$
0.377964 + 0.925820i $$0.376624\pi$$
$$8$$ 0 0
$$9$$ 4.46410 1.48803
$$10$$ 0 0
$$11$$ −3.46410 −1.04447 −0.522233 0.852803i $$-0.674901\pi$$
−0.522233 + 0.852803i $$0.674901\pi$$
$$12$$ 0 0
$$13$$ −2.73205 −0.757735 −0.378867 0.925451i $$-0.623686\pi$$
−0.378867 + 0.925451i $$0.623686\pi$$
$$14$$ 0 0
$$15$$ 2.73205 0.705412
$$16$$ 0 0
$$17$$ −3.46410 −0.840168 −0.420084 0.907485i $$-0.637999\pi$$
−0.420084 + 0.907485i $$0.637999\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ 5.46410 1.19236
$$22$$ 0 0
$$23$$ −3.46410 −0.722315 −0.361158 0.932505i $$-0.617618\pi$$
−0.361158 + 0.932505i $$0.617618\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 4.00000 0.769800
$$28$$ 0 0
$$29$$ 3.46410 0.643268 0.321634 0.946864i $$-0.395768\pi$$
0.321634 + 0.946864i $$0.395768\pi$$
$$30$$ 0 0
$$31$$ −1.46410 −0.262960 −0.131480 0.991319i $$-0.541973\pi$$
−0.131480 + 0.991319i $$0.541973\pi$$
$$32$$ 0 0
$$33$$ −9.46410 −1.64749
$$34$$ 0 0
$$35$$ 2.00000 0.338062
$$36$$ 0 0
$$37$$ 6.73205 1.10674 0.553371 0.832935i $$-0.313341\pi$$
0.553371 + 0.832935i $$0.313341\pi$$
$$38$$ 0 0
$$39$$ −7.46410 −1.19521
$$40$$ 0 0
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ 0 0
$$43$$ −4.92820 −0.751544 −0.375772 0.926712i $$-0.622622\pi$$
−0.375772 + 0.926712i $$0.622622\pi$$
$$44$$ 0 0
$$45$$ 4.46410 0.665469
$$46$$ 0 0
$$47$$ 12.9282 1.88577 0.942886 0.333115i $$-0.108100\pi$$
0.942886 + 0.333115i $$0.108100\pi$$
$$48$$ 0 0
$$49$$ −3.00000 −0.428571
$$50$$ 0 0
$$51$$ −9.46410 −1.32524
$$52$$ 0 0
$$53$$ −10.7321 −1.47416 −0.737080 0.675805i $$-0.763796\pi$$
−0.737080 + 0.675805i $$0.763796\pi$$
$$54$$ 0 0
$$55$$ −3.46410 −0.467099
$$56$$ 0 0
$$57$$ 2.73205 0.361869
$$58$$ 0 0
$$59$$ 6.92820 0.901975 0.450988 0.892530i $$-0.351072\pi$$
0.450988 + 0.892530i $$0.351072\pi$$
$$60$$ 0 0
$$61$$ 12.3923 1.58667 0.793336 0.608784i $$-0.208342\pi$$
0.793336 + 0.608784i $$0.208342\pi$$
$$62$$ 0 0
$$63$$ 8.92820 1.12485
$$64$$ 0 0
$$65$$ −2.73205 −0.338869
$$66$$ 0 0
$$67$$ 6.73205 0.822451 0.411225 0.911534i $$-0.365101\pi$$
0.411225 + 0.911534i $$0.365101\pi$$
$$68$$ 0 0
$$69$$ −9.46410 −1.13934
$$70$$ 0 0
$$71$$ −2.53590 −0.300956 −0.150478 0.988613i $$-0.548081\pi$$
−0.150478 + 0.988613i $$0.548081\pi$$
$$72$$ 0 0
$$73$$ −0.535898 −0.0627222 −0.0313611 0.999508i $$-0.509984\pi$$
−0.0313611 + 0.999508i $$0.509984\pi$$
$$74$$ 0 0
$$75$$ 2.73205 0.315470
$$76$$ 0 0
$$77$$ −6.92820 −0.789542
$$78$$ 0 0
$$79$$ 2.92820 0.329449 0.164724 0.986340i $$-0.447327\pi$$
0.164724 + 0.986340i $$0.447327\pi$$
$$80$$ 0 0
$$81$$ −2.46410 −0.273789
$$82$$ 0 0
$$83$$ 3.46410 0.380235 0.190117 0.981761i $$-0.439113\pi$$
0.190117 + 0.981761i $$0.439113\pi$$
$$84$$ 0 0
$$85$$ −3.46410 −0.375735
$$86$$ 0 0
$$87$$ 9.46410 1.01466
$$88$$ 0 0
$$89$$ −15.4641 −1.63919 −0.819596 0.572942i $$-0.805802\pi$$
−0.819596 + 0.572942i $$0.805802\pi$$
$$90$$ 0 0
$$91$$ −5.46410 −0.572793
$$92$$ 0 0
$$93$$ −4.00000 −0.414781
$$94$$ 0 0
$$95$$ 1.00000 0.102598
$$96$$ 0 0
$$97$$ −16.5885 −1.68430 −0.842151 0.539241i $$-0.818711\pi$$
−0.842151 + 0.539241i $$0.818711\pi$$
$$98$$ 0 0
$$99$$ −15.4641 −1.55420
$$100$$ 0 0
$$101$$ −16.3923 −1.63110 −0.815548 0.578690i $$-0.803564\pi$$
−0.815548 + 0.578690i $$0.803564\pi$$
$$102$$ 0 0
$$103$$ 13.6603 1.34598 0.672992 0.739649i $$-0.265009\pi$$
0.672992 + 0.739649i $$0.265009\pi$$
$$104$$ 0 0
$$105$$ 5.46410 0.533242
$$106$$ 0 0
$$107$$ −5.66025 −0.547197 −0.273599 0.961844i $$-0.588214\pi$$
−0.273599 + 0.961844i $$0.588214\pi$$
$$108$$ 0 0
$$109$$ −14.3923 −1.37853 −0.689266 0.724508i $$-0.742067\pi$$
−0.689266 + 0.724508i $$0.742067\pi$$
$$110$$ 0 0
$$111$$ 18.3923 1.74572
$$112$$ 0 0
$$113$$ 12.5885 1.18422 0.592111 0.805856i $$-0.298295\pi$$
0.592111 + 0.805856i $$0.298295\pi$$
$$114$$ 0 0
$$115$$ −3.46410 −0.323029
$$116$$ 0 0
$$117$$ −12.1962 −1.12753
$$118$$ 0 0
$$119$$ −6.92820 −0.635107
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 0 0
$$123$$ −16.3923 −1.47804
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ 20.5885 1.82693 0.913465 0.406917i $$-0.133396\pi$$
0.913465 + 0.406917i $$0.133396\pi$$
$$128$$ 0 0
$$129$$ −13.4641 −1.18545
$$130$$ 0 0
$$131$$ −18.9282 −1.65376 −0.826882 0.562375i $$-0.809888\pi$$
−0.826882 + 0.562375i $$0.809888\pi$$
$$132$$ 0 0
$$133$$ 2.00000 0.173422
$$134$$ 0 0
$$135$$ 4.00000 0.344265
$$136$$ 0 0
$$137$$ 17.3205 1.47979 0.739895 0.672722i $$-0.234875\pi$$
0.739895 + 0.672722i $$0.234875\pi$$
$$138$$ 0 0
$$139$$ 11.4641 0.972372 0.486186 0.873855i $$-0.338388\pi$$
0.486186 + 0.873855i $$0.338388\pi$$
$$140$$ 0 0
$$141$$ 35.3205 2.97452
$$142$$ 0 0
$$143$$ 9.46410 0.791428
$$144$$ 0 0
$$145$$ 3.46410 0.287678
$$146$$ 0 0
$$147$$ −8.19615 −0.676007
$$148$$ 0 0
$$149$$ −4.39230 −0.359832 −0.179916 0.983682i $$-0.557583\pi$$
−0.179916 + 0.983682i $$0.557583\pi$$
$$150$$ 0 0
$$151$$ −8.39230 −0.682956 −0.341478 0.939890i $$-0.610927\pi$$
−0.341478 + 0.939890i $$0.610927\pi$$
$$152$$ 0 0
$$153$$ −15.4641 −1.25020
$$154$$ 0 0
$$155$$ −1.46410 −0.117599
$$156$$ 0 0
$$157$$ 23.4641 1.87264 0.936320 0.351149i $$-0.114209\pi$$
0.936320 + 0.351149i $$0.114209\pi$$
$$158$$ 0 0
$$159$$ −29.3205 −2.32527
$$160$$ 0 0
$$161$$ −6.92820 −0.546019
$$162$$ 0 0
$$163$$ 7.07180 0.553906 0.276953 0.960883i $$-0.410675\pi$$
0.276953 + 0.960883i $$0.410675\pi$$
$$164$$ 0 0
$$165$$ −9.46410 −0.736779
$$166$$ 0 0
$$167$$ −10.7321 −0.830471 −0.415236 0.909714i $$-0.636301\pi$$
−0.415236 + 0.909714i $$0.636301\pi$$
$$168$$ 0 0
$$169$$ −5.53590 −0.425838
$$170$$ 0 0
$$171$$ 4.46410 0.341378
$$172$$ 0 0
$$173$$ 3.80385 0.289201 0.144601 0.989490i $$-0.453810\pi$$
0.144601 + 0.989490i $$0.453810\pi$$
$$174$$ 0 0
$$175$$ 2.00000 0.151186
$$176$$ 0 0
$$177$$ 18.9282 1.42273
$$178$$ 0 0
$$179$$ 20.7846 1.55351 0.776757 0.629800i $$-0.216863\pi$$
0.776757 + 0.629800i $$0.216863\pi$$
$$180$$ 0 0
$$181$$ 14.0000 1.04061 0.520306 0.853980i $$-0.325818\pi$$
0.520306 + 0.853980i $$0.325818\pi$$
$$182$$ 0 0
$$183$$ 33.8564 2.50274
$$184$$ 0 0
$$185$$ 6.73205 0.494950
$$186$$ 0 0
$$187$$ 12.0000 0.877527
$$188$$ 0 0
$$189$$ 8.00000 0.581914
$$190$$ 0 0
$$191$$ 6.92820 0.501307 0.250654 0.968077i $$-0.419354\pi$$
0.250654 + 0.968077i $$0.419354\pi$$
$$192$$ 0 0
$$193$$ −7.80385 −0.561733 −0.280867 0.959747i $$-0.590622\pi$$
−0.280867 + 0.959747i $$0.590622\pi$$
$$194$$ 0 0
$$195$$ −7.46410 −0.534515
$$196$$ 0 0
$$197$$ 0.928203 0.0661317 0.0330659 0.999453i $$-0.489473\pi$$
0.0330659 + 0.999453i $$0.489473\pi$$
$$198$$ 0 0
$$199$$ 26.9282 1.90889 0.954445 0.298387i $$-0.0964487\pi$$
0.954445 + 0.298387i $$0.0964487\pi$$
$$200$$ 0 0
$$201$$ 18.3923 1.29729
$$202$$ 0 0
$$203$$ 6.92820 0.486265
$$204$$ 0 0
$$205$$ −6.00000 −0.419058
$$206$$ 0 0
$$207$$ −15.4641 −1.07483
$$208$$ 0 0
$$209$$ −3.46410 −0.239617
$$210$$ 0 0
$$211$$ 19.3205 1.33008 0.665039 0.746808i $$-0.268415\pi$$
0.665039 + 0.746808i $$0.268415\pi$$
$$212$$ 0 0
$$213$$ −6.92820 −0.474713
$$214$$ 0 0
$$215$$ −4.92820 −0.336101
$$216$$ 0 0
$$217$$ −2.92820 −0.198779
$$218$$ 0 0
$$219$$ −1.46410 −0.0989348
$$220$$ 0 0
$$221$$ 9.46410 0.636624
$$222$$ 0 0
$$223$$ −9.66025 −0.646898 −0.323449 0.946246i $$-0.604842\pi$$
−0.323449 + 0.946246i $$0.604842\pi$$
$$224$$ 0 0
$$225$$ 4.46410 0.297607
$$226$$ 0 0
$$227$$ 8.19615 0.543998 0.271999 0.962298i $$-0.412315\pi$$
0.271999 + 0.962298i $$0.412315\pi$$
$$228$$ 0 0
$$229$$ 17.4641 1.15406 0.577030 0.816723i $$-0.304212\pi$$
0.577030 + 0.816723i $$0.304212\pi$$
$$230$$ 0 0
$$231$$ −18.9282 −1.24538
$$232$$ 0 0
$$233$$ −0.928203 −0.0608086 −0.0304043 0.999538i $$-0.509679\pi$$
−0.0304043 + 0.999538i $$0.509679\pi$$
$$234$$ 0 0
$$235$$ 12.9282 0.843343
$$236$$ 0 0
$$237$$ 8.00000 0.519656
$$238$$ 0 0
$$239$$ −13.8564 −0.896296 −0.448148 0.893959i $$-0.647916\pi$$
−0.448148 + 0.893959i $$0.647916\pi$$
$$240$$ 0 0
$$241$$ −11.8564 −0.763738 −0.381869 0.924216i $$-0.624719\pi$$
−0.381869 + 0.924216i $$0.624719\pi$$
$$242$$ 0 0
$$243$$ −18.7321 −1.20166
$$244$$ 0 0
$$245$$ −3.00000 −0.191663
$$246$$ 0 0
$$247$$ −2.73205 −0.173836
$$248$$ 0 0
$$249$$ 9.46410 0.599763
$$250$$ 0 0
$$251$$ −24.0000 −1.51487 −0.757433 0.652913i $$-0.773547\pi$$
−0.757433 + 0.652913i $$0.773547\pi$$
$$252$$ 0 0
$$253$$ 12.0000 0.754434
$$254$$ 0 0
$$255$$ −9.46410 −0.592665
$$256$$ 0 0
$$257$$ −15.1244 −0.943431 −0.471716 0.881751i $$-0.656365\pi$$
−0.471716 + 0.881751i $$0.656365\pi$$
$$258$$ 0 0
$$259$$ 13.4641 0.836619
$$260$$ 0 0
$$261$$ 15.4641 0.957204
$$262$$ 0 0
$$263$$ 8.53590 0.526346 0.263173 0.964749i $$-0.415231\pi$$
0.263173 + 0.964749i $$0.415231\pi$$
$$264$$ 0 0
$$265$$ −10.7321 −0.659265
$$266$$ 0 0
$$267$$ −42.2487 −2.58558
$$268$$ 0 0
$$269$$ 10.3923 0.633630 0.316815 0.948487i $$-0.397387\pi$$
0.316815 + 0.948487i $$0.397387\pi$$
$$270$$ 0 0
$$271$$ −2.39230 −0.145322 −0.0726611 0.997357i $$-0.523149\pi$$
−0.0726611 + 0.997357i $$0.523149\pi$$
$$272$$ 0 0
$$273$$ −14.9282 −0.903496
$$274$$ 0 0
$$275$$ −3.46410 −0.208893
$$276$$ 0 0
$$277$$ 18.3923 1.10509 0.552543 0.833484i $$-0.313657\pi$$
0.552543 + 0.833484i $$0.313657\pi$$
$$278$$ 0 0
$$279$$ −6.53590 −0.391294
$$280$$ 0 0
$$281$$ 0.928203 0.0553720 0.0276860 0.999617i $$-0.491186\pi$$
0.0276860 + 0.999617i $$0.491186\pi$$
$$282$$ 0 0
$$283$$ 18.3923 1.09331 0.546655 0.837358i $$-0.315901\pi$$
0.546655 + 0.837358i $$0.315901\pi$$
$$284$$ 0 0
$$285$$ 2.73205 0.161833
$$286$$ 0 0
$$287$$ −12.0000 −0.708338
$$288$$ 0 0
$$289$$ −5.00000 −0.294118
$$290$$ 0 0
$$291$$ −45.3205 −2.65674
$$292$$ 0 0
$$293$$ −5.66025 −0.330676 −0.165338 0.986237i $$-0.552871\pi$$
−0.165338 + 0.986237i $$0.552871\pi$$
$$294$$ 0 0
$$295$$ 6.92820 0.403376
$$296$$ 0 0
$$297$$ −13.8564 −0.804030
$$298$$ 0 0
$$299$$ 9.46410 0.547323
$$300$$ 0 0
$$301$$ −9.85641 −0.568114
$$302$$ 0 0
$$303$$ −44.7846 −2.57281
$$304$$ 0 0
$$305$$ 12.3923 0.709581
$$306$$ 0 0
$$307$$ −7.80385 −0.445389 −0.222695 0.974888i $$-0.571485\pi$$
−0.222695 + 0.974888i $$0.571485\pi$$
$$308$$ 0 0
$$309$$ 37.3205 2.12309
$$310$$ 0 0
$$311$$ −1.60770 −0.0911640 −0.0455820 0.998961i $$-0.514514\pi$$
−0.0455820 + 0.998961i $$0.514514\pi$$
$$312$$ 0 0
$$313$$ 10.7846 0.609582 0.304791 0.952419i $$-0.401413\pi$$
0.304791 + 0.952419i $$0.401413\pi$$
$$314$$ 0 0
$$315$$ 8.92820 0.503047
$$316$$ 0 0
$$317$$ 3.12436 0.175481 0.0877406 0.996143i $$-0.472035\pi$$
0.0877406 + 0.996143i $$0.472035\pi$$
$$318$$ 0 0
$$319$$ −12.0000 −0.671871
$$320$$ 0 0
$$321$$ −15.4641 −0.863122
$$322$$ 0 0
$$323$$ −3.46410 −0.192748
$$324$$ 0 0
$$325$$ −2.73205 −0.151547
$$326$$ 0 0
$$327$$ −39.3205 −2.17443
$$328$$ 0 0
$$329$$ 25.8564 1.42551
$$330$$ 0 0
$$331$$ −22.2487 −1.22290 −0.611450 0.791283i $$-0.709413\pi$$
−0.611450 + 0.791283i $$0.709413\pi$$
$$332$$ 0 0
$$333$$ 30.0526 1.64687
$$334$$ 0 0
$$335$$ 6.73205 0.367811
$$336$$ 0 0
$$337$$ −17.2679 −0.940645 −0.470323 0.882495i $$-0.655862\pi$$
−0.470323 + 0.882495i $$0.655862\pi$$
$$338$$ 0 0
$$339$$ 34.3923 1.86793
$$340$$ 0 0
$$341$$ 5.07180 0.274653
$$342$$ 0 0
$$343$$ −20.0000 −1.07990
$$344$$ 0 0
$$345$$ −9.46410 −0.509530
$$346$$ 0 0
$$347$$ 31.1769 1.67366 0.836832 0.547459i $$-0.184405\pi$$
0.836832 + 0.547459i $$0.184405\pi$$
$$348$$ 0 0
$$349$$ 20.9282 1.12026 0.560131 0.828404i $$-0.310751\pi$$
0.560131 + 0.828404i $$0.310751\pi$$
$$350$$ 0 0
$$351$$ −10.9282 −0.583304
$$352$$ 0 0
$$353$$ 1.60770 0.0855690 0.0427845 0.999084i $$-0.486377\pi$$
0.0427845 + 0.999084i $$0.486377\pi$$
$$354$$ 0 0
$$355$$ −2.53590 −0.134592
$$356$$ 0 0
$$357$$ −18.9282 −1.00179
$$358$$ 0 0
$$359$$ −8.53590 −0.450507 −0.225254 0.974300i $$-0.572321\pi$$
−0.225254 + 0.974300i $$0.572321\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 2.73205 0.143395
$$364$$ 0 0
$$365$$ −0.535898 −0.0280502
$$366$$ 0 0
$$367$$ 3.85641 0.201303 0.100651 0.994922i $$-0.467907\pi$$
0.100651 + 0.994922i $$0.467907\pi$$
$$368$$ 0 0
$$369$$ −26.7846 −1.39435
$$370$$ 0 0
$$371$$ −21.4641 −1.11436
$$372$$ 0 0
$$373$$ −16.5885 −0.858918 −0.429459 0.903086i $$-0.641296\pi$$
−0.429459 + 0.903086i $$0.641296\pi$$
$$374$$ 0 0
$$375$$ 2.73205 0.141082
$$376$$ 0 0
$$377$$ −9.46410 −0.487426
$$378$$ 0 0
$$379$$ 14.9282 0.766810 0.383405 0.923580i $$-0.374751\pi$$
0.383405 + 0.923580i $$0.374751\pi$$
$$380$$ 0 0
$$381$$ 56.2487 2.88171
$$382$$ 0 0
$$383$$ 6.33975 0.323946 0.161973 0.986795i $$-0.448214\pi$$
0.161973 + 0.986795i $$0.448214\pi$$
$$384$$ 0 0
$$385$$ −6.92820 −0.353094
$$386$$ 0 0
$$387$$ −22.0000 −1.11832
$$388$$ 0 0
$$389$$ 18.0000 0.912636 0.456318 0.889817i $$-0.349168\pi$$
0.456318 + 0.889817i $$0.349168\pi$$
$$390$$ 0 0
$$391$$ 12.0000 0.606866
$$392$$ 0 0
$$393$$ −51.7128 −2.60857
$$394$$ 0 0
$$395$$ 2.92820 0.147334
$$396$$ 0 0
$$397$$ −19.4641 −0.976875 −0.488438 0.872599i $$-0.662433\pi$$
−0.488438 + 0.872599i $$0.662433\pi$$
$$398$$ 0 0
$$399$$ 5.46410 0.273547
$$400$$ 0 0
$$401$$ 38.7846 1.93681 0.968405 0.249381i $$-0.0802271\pi$$
0.968405 + 0.249381i $$0.0802271\pi$$
$$402$$ 0 0
$$403$$ 4.00000 0.199254
$$404$$ 0 0
$$405$$ −2.46410 −0.122442
$$406$$ 0 0
$$407$$ −23.3205 −1.15595
$$408$$ 0 0
$$409$$ −14.3923 −0.711654 −0.355827 0.934552i $$-0.615801\pi$$
−0.355827 + 0.934552i $$0.615801\pi$$
$$410$$ 0 0
$$411$$ 47.3205 2.33415
$$412$$ 0 0
$$413$$ 13.8564 0.681829
$$414$$ 0 0
$$415$$ 3.46410 0.170046
$$416$$ 0 0
$$417$$ 31.3205 1.53377
$$418$$ 0 0
$$419$$ −12.0000 −0.586238 −0.293119 0.956076i $$-0.594693\pi$$
−0.293119 + 0.956076i $$0.594693\pi$$
$$420$$ 0 0
$$421$$ −34.0000 −1.65706 −0.828529 0.559946i $$-0.810822\pi$$
−0.828529 + 0.559946i $$0.810822\pi$$
$$422$$ 0 0
$$423$$ 57.7128 2.80609
$$424$$ 0 0
$$425$$ −3.46410 −0.168034
$$426$$ 0 0
$$427$$ 24.7846 1.19941
$$428$$ 0 0
$$429$$ 25.8564 1.24836
$$430$$ 0 0
$$431$$ −33.4641 −1.61191 −0.805955 0.591977i $$-0.798347\pi$$
−0.805955 + 0.591977i $$0.798347\pi$$
$$432$$ 0 0
$$433$$ −22.3397 −1.07358 −0.536790 0.843716i $$-0.680363\pi$$
−0.536790 + 0.843716i $$0.680363\pi$$
$$434$$ 0 0
$$435$$ 9.46410 0.453769
$$436$$ 0 0
$$437$$ −3.46410 −0.165710
$$438$$ 0 0
$$439$$ −31.7128 −1.51357 −0.756785 0.653664i $$-0.773231\pi$$
−0.756785 + 0.653664i $$0.773231\pi$$
$$440$$ 0 0
$$441$$ −13.3923 −0.637729
$$442$$ 0 0
$$443$$ 13.6077 0.646521 0.323261 0.946310i $$-0.395221\pi$$
0.323261 + 0.946310i $$0.395221\pi$$
$$444$$ 0 0
$$445$$ −15.4641 −0.733069
$$446$$ 0 0
$$447$$ −12.0000 −0.567581
$$448$$ 0 0
$$449$$ −3.46410 −0.163481 −0.0817405 0.996654i $$-0.526048\pi$$
−0.0817405 + 0.996654i $$0.526048\pi$$
$$450$$ 0 0
$$451$$ 20.7846 0.978709
$$452$$ 0 0
$$453$$ −22.9282 −1.07726
$$454$$ 0 0
$$455$$ −5.46410 −0.256161
$$456$$ 0 0
$$457$$ 34.7846 1.62716 0.813578 0.581456i $$-0.197517\pi$$
0.813578 + 0.581456i $$0.197517\pi$$
$$458$$ 0 0
$$459$$ −13.8564 −0.646762
$$460$$ 0 0
$$461$$ −21.7128 −1.01127 −0.505633 0.862749i $$-0.668741\pi$$
−0.505633 + 0.862749i $$0.668741\pi$$
$$462$$ 0 0
$$463$$ 4.53590 0.210801 0.105401 0.994430i $$-0.466388\pi$$
0.105401 + 0.994430i $$0.466388\pi$$
$$464$$ 0 0
$$465$$ −4.00000 −0.185496
$$466$$ 0 0
$$467$$ −29.3205 −1.35679 −0.678396 0.734697i $$-0.737324\pi$$
−0.678396 + 0.734697i $$0.737324\pi$$
$$468$$ 0 0
$$469$$ 13.4641 0.621714
$$470$$ 0 0
$$471$$ 64.1051 2.95381
$$472$$ 0 0
$$473$$ 17.0718 0.784962
$$474$$ 0 0
$$475$$ 1.00000 0.0458831
$$476$$ 0 0
$$477$$ −47.9090 −2.19360
$$478$$ 0 0
$$479$$ −22.3923 −1.02313 −0.511565 0.859244i $$-0.670934\pi$$
−0.511565 + 0.859244i $$0.670934\pi$$
$$480$$ 0 0
$$481$$ −18.3923 −0.838617
$$482$$ 0 0
$$483$$ −18.9282 −0.861263
$$484$$ 0 0
$$485$$ −16.5885 −0.753243
$$486$$ 0 0
$$487$$ −12.1962 −0.552660 −0.276330 0.961063i $$-0.589118\pi$$
−0.276330 + 0.961063i $$0.589118\pi$$
$$488$$ 0 0
$$489$$ 19.3205 0.873704
$$490$$ 0 0
$$491$$ 18.9282 0.854218 0.427109 0.904200i $$-0.359532\pi$$
0.427109 + 0.904200i $$0.359532\pi$$
$$492$$ 0 0
$$493$$ −12.0000 −0.540453
$$494$$ 0 0
$$495$$ −15.4641 −0.695060
$$496$$ 0 0
$$497$$ −5.07180 −0.227501
$$498$$ 0 0
$$499$$ 23.4641 1.05040 0.525199 0.850980i $$-0.323991\pi$$
0.525199 + 0.850980i $$0.323991\pi$$
$$500$$ 0 0
$$501$$ −29.3205 −1.30994
$$502$$ 0 0
$$503$$ 15.4641 0.689510 0.344755 0.938693i $$-0.387962\pi$$
0.344755 + 0.938693i $$0.387962\pi$$
$$504$$ 0 0
$$505$$ −16.3923 −0.729448
$$506$$ 0 0
$$507$$ −15.1244 −0.671696
$$508$$ 0 0
$$509$$ −5.32051 −0.235827 −0.117914 0.993024i $$-0.537621\pi$$
−0.117914 + 0.993024i $$0.537621\pi$$
$$510$$ 0 0
$$511$$ −1.07180 −0.0474135
$$512$$ 0 0
$$513$$ 4.00000 0.176604
$$514$$ 0 0
$$515$$ 13.6603 0.601943
$$516$$ 0 0
$$517$$ −44.7846 −1.96962
$$518$$ 0 0
$$519$$ 10.3923 0.456172
$$520$$ 0 0
$$521$$ −14.7846 −0.647726 −0.323863 0.946104i $$-0.604982\pi$$
−0.323863 + 0.946104i $$0.604982\pi$$
$$522$$ 0 0
$$523$$ −37.3731 −1.63421 −0.817105 0.576489i $$-0.804422\pi$$
−0.817105 + 0.576489i $$0.804422\pi$$
$$524$$ 0 0
$$525$$ 5.46410 0.238473
$$526$$ 0 0
$$527$$ 5.07180 0.220931
$$528$$ 0 0
$$529$$ −11.0000 −0.478261
$$530$$ 0 0
$$531$$ 30.9282 1.34217
$$532$$ 0 0
$$533$$ 16.3923 0.710030
$$534$$ 0 0
$$535$$ −5.66025 −0.244714
$$536$$ 0 0
$$537$$ 56.7846 2.45044
$$538$$ 0 0
$$539$$ 10.3923 0.447628
$$540$$ 0 0
$$541$$ −29.1769 −1.25441 −0.627207 0.778853i $$-0.715802\pi$$
−0.627207 + 0.778853i $$0.715802\pi$$
$$542$$ 0 0
$$543$$ 38.2487 1.64141
$$544$$ 0 0
$$545$$ −14.3923 −0.616499
$$546$$ 0 0
$$547$$ 6.73205 0.287842 0.143921 0.989589i $$-0.454029\pi$$
0.143921 + 0.989589i $$0.454029\pi$$
$$548$$ 0 0
$$549$$ 55.3205 2.36102
$$550$$ 0 0
$$551$$ 3.46410 0.147576
$$552$$ 0 0
$$553$$ 5.85641 0.249040
$$554$$ 0 0
$$555$$ 18.3923 0.780710
$$556$$ 0 0
$$557$$ 5.32051 0.225437 0.112719 0.993627i $$-0.464044\pi$$
0.112719 + 0.993627i $$0.464044\pi$$
$$558$$ 0 0
$$559$$ 13.4641 0.569471
$$560$$ 0 0
$$561$$ 32.7846 1.38417
$$562$$ 0 0
$$563$$ 44.1962 1.86265 0.931323 0.364195i $$-0.118656\pi$$
0.931323 + 0.364195i $$0.118656\pi$$
$$564$$ 0 0
$$565$$ 12.5885 0.529600
$$566$$ 0 0
$$567$$ −4.92820 −0.206965
$$568$$ 0 0
$$569$$ −29.3205 −1.22918 −0.614590 0.788847i $$-0.710678\pi$$
−0.614590 + 0.788847i $$0.710678\pi$$
$$570$$ 0 0
$$571$$ 18.3923 0.769694 0.384847 0.922980i $$-0.374254\pi$$
0.384847 + 0.922980i $$0.374254\pi$$
$$572$$ 0 0
$$573$$ 18.9282 0.790737
$$574$$ 0 0
$$575$$ −3.46410 −0.144463
$$576$$ 0 0
$$577$$ −6.78461 −0.282447 −0.141223 0.989978i $$-0.545104\pi$$
−0.141223 + 0.989978i $$0.545104\pi$$
$$578$$ 0 0
$$579$$ −21.3205 −0.886050
$$580$$ 0 0
$$581$$ 6.92820 0.287430
$$582$$ 0 0
$$583$$ 37.1769 1.53971
$$584$$ 0 0
$$585$$ −12.1962 −0.504249
$$586$$ 0 0
$$587$$ 43.1769 1.78210 0.891051 0.453903i $$-0.149969\pi$$
0.891051 + 0.453903i $$0.149969\pi$$
$$588$$ 0 0
$$589$$ −1.46410 −0.0603273
$$590$$ 0 0
$$591$$ 2.53590 0.104313
$$592$$ 0 0
$$593$$ 31.8564 1.30819 0.654093 0.756414i $$-0.273051\pi$$
0.654093 + 0.756414i $$0.273051\pi$$
$$594$$ 0 0
$$595$$ −6.92820 −0.284029
$$596$$ 0 0
$$597$$ 73.5692 3.01099
$$598$$ 0 0
$$599$$ −20.7846 −0.849236 −0.424618 0.905373i $$-0.639592\pi$$
−0.424618 + 0.905373i $$0.639592\pi$$
$$600$$ 0 0
$$601$$ 24.6410 1.00513 0.502564 0.864540i $$-0.332390\pi$$
0.502564 + 0.864540i $$0.332390\pi$$
$$602$$ 0 0
$$603$$ 30.0526 1.22383
$$604$$ 0 0
$$605$$ 1.00000 0.0406558
$$606$$ 0 0
$$607$$ −39.9090 −1.61985 −0.809927 0.586530i $$-0.800494\pi$$
−0.809927 + 0.586530i $$0.800494\pi$$
$$608$$ 0 0
$$609$$ 18.9282 0.767010
$$610$$ 0 0
$$611$$ −35.3205 −1.42891
$$612$$ 0 0
$$613$$ 6.39230 0.258183 0.129091 0.991633i $$-0.458794\pi$$
0.129091 + 0.991633i $$0.458794\pi$$
$$614$$ 0 0
$$615$$ −16.3923 −0.661002
$$616$$ 0 0
$$617$$ 1.60770 0.0647234 0.0323617 0.999476i $$-0.489697\pi$$
0.0323617 + 0.999476i $$0.489697\pi$$
$$618$$ 0 0
$$619$$ −2.39230 −0.0961549 −0.0480774 0.998844i $$-0.515309\pi$$
−0.0480774 + 0.998844i $$0.515309\pi$$
$$620$$ 0 0
$$621$$ −13.8564 −0.556038
$$622$$ 0 0
$$623$$ −30.9282 −1.23911
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ −9.46410 −0.377960
$$628$$ 0 0
$$629$$ −23.3205 −0.929850
$$630$$ 0 0
$$631$$ 11.4641 0.456379 0.228189 0.973617i $$-0.426719\pi$$
0.228189 + 0.973617i $$0.426719\pi$$
$$632$$ 0 0
$$633$$ 52.7846 2.09800
$$634$$ 0 0
$$635$$ 20.5885 0.817028
$$636$$ 0 0
$$637$$ 8.19615 0.324743
$$638$$ 0 0
$$639$$ −11.3205 −0.447832
$$640$$ 0 0
$$641$$ −24.9282 −0.984605 −0.492302 0.870424i $$-0.663845\pi$$
−0.492302 + 0.870424i $$0.663845\pi$$
$$642$$ 0 0
$$643$$ −14.3923 −0.567577 −0.283789 0.958887i $$-0.591591\pi$$
−0.283789 + 0.958887i $$0.591591\pi$$
$$644$$ 0 0
$$645$$ −13.4641 −0.530148
$$646$$ 0 0
$$647$$ 15.4641 0.607957 0.303978 0.952679i $$-0.401685\pi$$
0.303978 + 0.952679i $$0.401685\pi$$
$$648$$ 0 0
$$649$$ −24.0000 −0.942082
$$650$$ 0 0
$$651$$ −8.00000 −0.313545
$$652$$ 0 0
$$653$$ 27.4641 1.07475 0.537377 0.843342i $$-0.319415\pi$$
0.537377 + 0.843342i $$0.319415\pi$$
$$654$$ 0 0
$$655$$ −18.9282 −0.739586
$$656$$ 0 0
$$657$$ −2.39230 −0.0933327
$$658$$ 0 0
$$659$$ 8.78461 0.342200 0.171100 0.985254i $$-0.445268\pi$$
0.171100 + 0.985254i $$0.445268\pi$$
$$660$$ 0 0
$$661$$ 5.21539 0.202855 0.101428 0.994843i $$-0.467659\pi$$
0.101428 + 0.994843i $$0.467659\pi$$
$$662$$ 0 0
$$663$$ 25.8564 1.00418
$$664$$ 0 0
$$665$$ 2.00000 0.0775567
$$666$$ 0 0
$$667$$ −12.0000 −0.464642
$$668$$ 0 0
$$669$$ −26.3923 −1.02039
$$670$$ 0 0
$$671$$ −42.9282 −1.65722
$$672$$ 0 0
$$673$$ −50.7321 −1.95558 −0.977788 0.209594i $$-0.932786\pi$$
−0.977788 + 0.209594i $$0.932786\pi$$
$$674$$ 0 0
$$675$$ 4.00000 0.153960
$$676$$ 0 0
$$677$$ 16.9808 0.652624 0.326312 0.945262i $$-0.394194\pi$$
0.326312 + 0.945262i $$0.394194\pi$$
$$678$$ 0 0
$$679$$ −33.1769 −1.27321
$$680$$ 0 0
$$681$$ 22.3923 0.858075
$$682$$ 0 0
$$683$$ −17.6603 −0.675751 −0.337875 0.941191i $$-0.609708\pi$$
−0.337875 + 0.941191i $$0.609708\pi$$
$$684$$ 0 0
$$685$$ 17.3205 0.661783
$$686$$ 0 0
$$687$$ 47.7128 1.82036
$$688$$ 0 0
$$689$$ 29.3205 1.11702
$$690$$ 0 0
$$691$$ 51.1769 1.94686 0.973431 0.228981i $$-0.0735395\pi$$
0.973431 + 0.228981i $$0.0735395\pi$$
$$692$$ 0 0
$$693$$ −30.9282 −1.17487
$$694$$ 0 0
$$695$$ 11.4641 0.434858
$$696$$ 0 0
$$697$$ 20.7846 0.787273
$$698$$ 0 0
$$699$$ −2.53590 −0.0959165
$$700$$ 0 0
$$701$$ −26.5359 −1.00225 −0.501124 0.865376i $$-0.667080\pi$$
−0.501124 + 0.865376i $$0.667080\pi$$
$$702$$ 0 0
$$703$$ 6.73205 0.253904
$$704$$ 0 0
$$705$$ 35.3205 1.33025
$$706$$ 0 0
$$707$$ −32.7846 −1.23299
$$708$$ 0 0
$$709$$ −30.7846 −1.15614 −0.578070 0.815987i $$-0.696194\pi$$
−0.578070 + 0.815987i $$0.696194\pi$$
$$710$$ 0 0
$$711$$ 13.0718 0.490231
$$712$$ 0 0
$$713$$ 5.07180 0.189940
$$714$$ 0 0
$$715$$ 9.46410 0.353937
$$716$$ 0 0
$$717$$ −37.8564 −1.41377
$$718$$ 0 0
$$719$$ 17.3205 0.645946 0.322973 0.946408i $$-0.395318\pi$$
0.322973 + 0.946408i $$0.395318\pi$$
$$720$$ 0 0
$$721$$ 27.3205 1.01747
$$722$$ 0 0
$$723$$ −32.3923 −1.20468
$$724$$ 0 0
$$725$$ 3.46410 0.128654
$$726$$ 0 0
$$727$$ 0.143594 0.00532559 0.00266279 0.999996i $$-0.499152\pi$$
0.00266279 + 0.999996i $$0.499152\pi$$
$$728$$ 0 0
$$729$$ −43.7846 −1.62165
$$730$$ 0 0
$$731$$ 17.0718 0.631423
$$732$$ 0 0
$$733$$ 10.7846 0.398339 0.199169 0.979965i $$-0.436176\pi$$
0.199169 + 0.979965i $$0.436176\pi$$
$$734$$ 0 0
$$735$$ −8.19615 −0.302320
$$736$$ 0 0
$$737$$ −23.3205 −0.859022
$$738$$ 0 0
$$739$$ −28.0000 −1.03000 −0.514998 0.857191i $$-0.672207\pi$$
−0.514998 + 0.857191i $$0.672207\pi$$
$$740$$ 0 0
$$741$$ −7.46410 −0.274201
$$742$$ 0 0
$$743$$ 20.8756 0.765853 0.382927 0.923779i $$-0.374916\pi$$
0.382927 + 0.923779i $$0.374916\pi$$
$$744$$ 0 0
$$745$$ −4.39230 −0.160922
$$746$$ 0 0
$$747$$ 15.4641 0.565802
$$748$$ 0 0
$$749$$ −11.3205 −0.413642
$$750$$ 0 0
$$751$$ −25.4641 −0.929198 −0.464599 0.885521i $$-0.653802\pi$$
−0.464599 + 0.885521i $$0.653802\pi$$
$$752$$ 0 0
$$753$$ −65.5692 −2.38948
$$754$$ 0 0
$$755$$ −8.39230 −0.305427
$$756$$ 0 0
$$757$$ −10.0000 −0.363456 −0.181728 0.983349i $$-0.558169\pi$$
−0.181728 + 0.983349i $$0.558169\pi$$
$$758$$ 0 0
$$759$$ 32.7846 1.19001
$$760$$ 0 0
$$761$$ 19.6077 0.710778 0.355389 0.934718i $$-0.384348\pi$$
0.355389 + 0.934718i $$0.384348\pi$$
$$762$$ 0 0
$$763$$ −28.7846 −1.04207
$$764$$ 0 0
$$765$$ −15.4641 −0.559106
$$766$$ 0 0
$$767$$ −18.9282 −0.683458
$$768$$ 0 0
$$769$$ −30.5359 −1.10115 −0.550576 0.834785i $$-0.685592\pi$$
−0.550576 + 0.834785i $$0.685592\pi$$
$$770$$ 0 0
$$771$$ −41.3205 −1.48812
$$772$$ 0 0
$$773$$ 22.0526 0.793175 0.396588 0.917997i $$-0.370194\pi$$
0.396588 + 0.917997i $$0.370194\pi$$
$$774$$ 0 0
$$775$$ −1.46410 −0.0525921
$$776$$ 0 0
$$777$$ 36.7846 1.31964
$$778$$ 0 0
$$779$$ −6.00000 −0.214972
$$780$$ 0 0
$$781$$ 8.78461 0.314338
$$782$$ 0 0
$$783$$ 13.8564 0.495188
$$784$$ 0 0
$$785$$ 23.4641 0.837470
$$786$$ 0 0
$$787$$ −12.1962 −0.434746 −0.217373 0.976089i $$-0.569749\pi$$
−0.217373 + 0.976089i $$0.569749\pi$$
$$788$$ 0 0
$$789$$ 23.3205 0.830232
$$790$$ 0 0
$$791$$ 25.1769 0.895188
$$792$$ 0 0
$$793$$ −33.8564 −1.20228
$$794$$ 0 0
$$795$$ −29.3205 −1.03989
$$796$$ 0 0
$$797$$ −5.66025 −0.200496 −0.100248 0.994962i $$-0.531964\pi$$
−0.100248 + 0.994962i $$0.531964\pi$$
$$798$$ 0 0
$$799$$ −44.7846 −1.58437
$$800$$ 0 0
$$801$$ −69.0333 −2.43917
$$802$$ 0 0
$$803$$ 1.85641 0.0655112
$$804$$ 0 0
$$805$$ −6.92820 −0.244187
$$806$$ 0 0
$$807$$ 28.3923 0.999456
$$808$$ 0 0
$$809$$ 9.71281 0.341484 0.170742 0.985316i $$-0.445383\pi$$
0.170742 + 0.985316i $$0.445383\pi$$
$$810$$ 0 0
$$811$$ 15.6077 0.548060 0.274030 0.961721i $$-0.411643\pi$$
0.274030 + 0.961721i $$0.411643\pi$$
$$812$$ 0 0
$$813$$ −6.53590 −0.229224
$$814$$ 0 0
$$815$$ 7.07180 0.247714
$$816$$ 0 0
$$817$$ −4.92820 −0.172416
$$818$$ 0 0
$$819$$ −24.3923 −0.852336
$$820$$ 0 0
$$821$$ −16.1436 −0.563415 −0.281708 0.959500i $$-0.590901\pi$$
−0.281708 + 0.959500i $$0.590901\pi$$
$$822$$ 0 0
$$823$$ −39.5692 −1.37930 −0.689648 0.724145i $$-0.742235\pi$$
−0.689648 + 0.724145i $$0.742235\pi$$
$$824$$ 0 0
$$825$$ −9.46410 −0.329498
$$826$$ 0 0
$$827$$ −29.6603 −1.03139 −0.515694 0.856773i $$-0.672466\pi$$
−0.515694 + 0.856773i $$0.672466\pi$$
$$828$$ 0 0
$$829$$ 8.24871 0.286490 0.143245 0.989687i $$-0.454246\pi$$
0.143245 + 0.989687i $$0.454246\pi$$
$$830$$ 0 0
$$831$$ 50.2487 1.74311
$$832$$ 0 0
$$833$$ 10.3923 0.360072
$$834$$ 0 0
$$835$$ −10.7321 −0.371398
$$836$$ 0 0
$$837$$ −5.85641 −0.202427
$$838$$ 0 0
$$839$$ 18.9282 0.653474 0.326737 0.945115i $$-0.394051\pi$$
0.326737 + 0.945115i $$0.394051\pi$$
$$840$$ 0 0
$$841$$ −17.0000 −0.586207
$$842$$ 0 0
$$843$$ 2.53590 0.0873410
$$844$$ 0 0
$$845$$ −5.53590 −0.190441
$$846$$ 0 0
$$847$$ 2.00000 0.0687208
$$848$$ 0 0
$$849$$ 50.2487 1.72453
$$850$$ 0 0
$$851$$ −23.3205 −0.799417
$$852$$ 0 0
$$853$$ −44.6410 −1.52848 −0.764240 0.644932i $$-0.776886\pi$$
−0.764240 + 0.644932i $$0.776886\pi$$
$$854$$ 0 0
$$855$$ 4.46410 0.152669
$$856$$ 0 0
$$857$$ 3.80385 0.129937 0.0649685 0.997887i $$-0.479305\pi$$
0.0649685 + 0.997887i $$0.479305\pi$$
$$858$$ 0 0
$$859$$ 47.7128 1.62794 0.813970 0.580907i $$-0.197302\pi$$
0.813970 + 0.580907i $$0.197302\pi$$
$$860$$ 0 0
$$861$$ −32.7846 −1.11730
$$862$$ 0 0
$$863$$ −37.2679 −1.26862 −0.634308 0.773081i $$-0.718715\pi$$
−0.634308 + 0.773081i $$0.718715\pi$$
$$864$$ 0 0
$$865$$ 3.80385 0.129335
$$866$$ 0 0
$$867$$ −13.6603 −0.463927
$$868$$ 0 0
$$869$$ −10.1436 −0.344098
$$870$$ 0 0
$$871$$ −18.3923 −0.623199
$$872$$ 0 0
$$873$$ −74.0526 −2.50630
$$874$$ 0 0
$$875$$ 2.00000 0.0676123
$$876$$ 0 0
$$877$$ 0.980762 0.0331180 0.0165590 0.999863i $$-0.494729\pi$$
0.0165590 + 0.999863i $$0.494729\pi$$
$$878$$ 0 0
$$879$$ −15.4641 −0.521591
$$880$$ 0 0
$$881$$ −0.679492 −0.0228927 −0.0114463 0.999934i $$-0.503644\pi$$
−0.0114463 + 0.999934i $$0.503644\pi$$
$$882$$ 0 0
$$883$$ −22.0000 −0.740359 −0.370179 0.928960i $$-0.620704\pi$$
−0.370179 + 0.928960i $$0.620704\pi$$
$$884$$ 0 0
$$885$$ 18.9282 0.636265
$$886$$ 0 0
$$887$$ 23.4115 0.786083 0.393041 0.919521i $$-0.371423\pi$$
0.393041 + 0.919521i $$0.371423\pi$$
$$888$$ 0 0
$$889$$ 41.1769 1.38103
$$890$$ 0 0
$$891$$ 8.53590 0.285963
$$892$$ 0 0
$$893$$ 12.9282 0.432626
$$894$$ 0 0
$$895$$ 20.7846 0.694753
$$896$$ 0 0
$$897$$ 25.8564 0.863320
$$898$$ 0 0
$$899$$ −5.07180 −0.169154
$$900$$ 0 0
$$901$$ 37.1769 1.23854
$$902$$ 0 0
$$903$$ −26.9282 −0.896114
$$904$$ 0 0
$$905$$ 14.0000 0.465376
$$906$$ 0 0
$$907$$ 7.41154 0.246096 0.123048 0.992401i $$-0.460733\pi$$
0.123048 + 0.992401i $$0.460733\pi$$
$$908$$ 0 0
$$909$$ −73.1769 −2.42713
$$910$$ 0 0
$$911$$ 14.5359 0.481596 0.240798 0.970575i $$-0.422591\pi$$
0.240798 + 0.970575i $$0.422591\pi$$
$$912$$ 0 0
$$913$$ −12.0000 −0.397142
$$914$$ 0 0
$$915$$ 33.8564 1.11926
$$916$$ 0 0
$$917$$ −37.8564 −1.25013
$$918$$ 0 0
$$919$$ 44.4974 1.46783 0.733917 0.679239i $$-0.237690\pi$$
0.733917 + 0.679239i $$0.237690\pi$$
$$920$$ 0 0
$$921$$ −21.3205 −0.702535
$$922$$ 0 0
$$923$$ 6.92820 0.228045
$$924$$ 0 0
$$925$$ 6.73205 0.221348
$$926$$ 0 0
$$927$$ 60.9808 2.00287
$$928$$ 0 0
$$929$$ −47.5692 −1.56070 −0.780348 0.625346i $$-0.784958\pi$$
−0.780348 + 0.625346i $$0.784958\pi$$
$$930$$ 0 0
$$931$$ −3.00000 −0.0983210
$$932$$ 0 0
$$933$$ −4.39230 −0.143798
$$934$$ 0 0
$$935$$ 12.0000 0.392442
$$936$$ 0 0
$$937$$ 51.1769 1.67188 0.835938 0.548823i $$-0.184924\pi$$
0.835938 + 0.548823i $$0.184924\pi$$
$$938$$ 0 0
$$939$$ 29.4641 0.961525
$$940$$ 0 0
$$941$$ 52.6410 1.71605 0.858024 0.513610i $$-0.171692\pi$$
0.858024 + 0.513610i $$0.171692\pi$$
$$942$$ 0 0
$$943$$ 20.7846 0.676840
$$944$$ 0 0
$$945$$ 8.00000 0.260240
$$946$$ 0 0
$$947$$ 6.67949 0.217054 0.108527 0.994093i $$-0.465387\pi$$
0.108527 + 0.994093i $$0.465387\pi$$
$$948$$ 0 0
$$949$$ 1.46410 0.0475267
$$950$$ 0 0
$$951$$ 8.53590 0.276795
$$952$$ 0 0
$$953$$ 60.5885 1.96265 0.981326 0.192350i $$-0.0616110\pi$$
0.981326 + 0.192350i $$0.0616110\pi$$
$$954$$ 0 0
$$955$$ 6.92820 0.224191
$$956$$ 0 0
$$957$$ −32.7846 −1.05978
$$958$$ 0 0
$$959$$ 34.6410 1.11862
$$960$$ 0 0
$$961$$ −28.8564 −0.930852
$$962$$ 0 0
$$963$$ −25.2679 −0.814248
$$964$$ 0 0
$$965$$ −7.80385 −0.251215
$$966$$ 0 0
$$967$$ −45.3205 −1.45741 −0.728705 0.684828i $$-0.759877\pi$$
−0.728705 + 0.684828i $$0.759877\pi$$
$$968$$ 0 0
$$969$$ −9.46410 −0.304031
$$970$$ 0 0
$$971$$ −14.5359 −0.466479 −0.233240 0.972419i $$-0.574933\pi$$
−0.233240 + 0.972419i $$0.574933\pi$$
$$972$$ 0 0
$$973$$ 22.9282 0.735044
$$974$$ 0 0
$$975$$ −7.46410 −0.239043
$$976$$ 0 0
$$977$$ 32.1962 1.03005 0.515023 0.857176i $$-0.327783\pi$$
0.515023 + 0.857176i $$0.327783\pi$$
$$978$$ 0 0
$$979$$ 53.5692 1.71208
$$980$$ 0 0
$$981$$ −64.2487 −2.05130
$$982$$ 0 0
$$983$$ −2.44486 −0.0779790 −0.0389895 0.999240i $$-0.512414\pi$$
−0.0389895 + 0.999240i $$0.512414\pi$$
$$984$$ 0 0
$$985$$ 0.928203 0.0295750
$$986$$ 0 0
$$987$$ 70.6410 2.24853
$$988$$ 0 0
$$989$$ 17.0718 0.542852
$$990$$ 0 0
$$991$$ −8.39230 −0.266590 −0.133295 0.991076i $$-0.542556\pi$$
−0.133295 + 0.991076i $$0.542556\pi$$
$$992$$ 0 0
$$993$$ −60.7846 −1.92894
$$994$$ 0 0
$$995$$ 26.9282 0.853681
$$996$$ 0 0
$$997$$ 30.3923 0.962534 0.481267 0.876574i $$-0.340177\pi$$
0.481267 + 0.876574i $$0.340177\pi$$
$$998$$ 0 0
$$999$$ 26.9282 0.851971
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 380.2.a.d.1.2 2
3.2 odd 2 3420.2.a.h.1.2 2
4.3 odd 2 1520.2.a.l.1.1 2
5.2 odd 4 1900.2.c.e.1749.1 4
5.3 odd 4 1900.2.c.e.1749.4 4
5.4 even 2 1900.2.a.d.1.1 2
8.3 odd 2 6080.2.a.bj.1.2 2
8.5 even 2 6080.2.a.z.1.1 2
19.18 odd 2 7220.2.a.h.1.1 2
20.19 odd 2 7600.2.a.bf.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.a.d.1.2 2 1.1 even 1 trivial
1520.2.a.l.1.1 2 4.3 odd 2
1900.2.a.d.1.1 2 5.4 even 2
1900.2.c.e.1749.1 4 5.2 odd 4
1900.2.c.e.1749.4 4 5.3 odd 4
3420.2.a.h.1.2 2 3.2 odd 2
6080.2.a.z.1.1 2 8.5 even 2
6080.2.a.bj.1.2 2 8.3 odd 2
7220.2.a.h.1.1 2 19.18 odd 2
7600.2.a.bf.1.2 2 20.19 odd 2