# Properties

 Label 7360.2.a.bi.1.1 Level $7360$ Weight $2$ Character 7360.1 Self dual yes Analytic conductor $58.770$ 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] = [7360,2,Mod(1,7360)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7360, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("7360.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$7360 = 2^{6} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7360.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: $$58.7698958877$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 460) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$2.56155$$ of defining polynomial Character $$\chi$$ $$=$$ 7360.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.56155 q^{3} -1.00000 q^{5} -1.56155 q^{7} +3.56155 q^{9} +O(q^{10})$$ $$q-2.56155 q^{3} -1.00000 q^{5} -1.56155 q^{7} +3.56155 q^{9} -2.00000 q^{11} -0.561553 q^{13} +2.56155 q^{15} -1.56155 q^{17} -6.00000 q^{19} +4.00000 q^{21} +1.00000 q^{23} +1.00000 q^{25} -1.43845 q^{27} +2.12311 q^{29} -9.24621 q^{31} +5.12311 q^{33} +1.56155 q^{35} +0.438447 q^{37} +1.43845 q^{39} -4.12311 q^{41} -3.56155 q^{45} -7.68466 q^{47} -4.56155 q^{49} +4.00000 q^{51} +0.438447 q^{53} +2.00000 q^{55} +15.3693 q^{57} -8.68466 q^{59} -1.12311 q^{61} -5.56155 q^{63} +0.561553 q^{65} +4.43845 q^{67} -2.56155 q^{69} +1.87689 q^{71} -8.56155 q^{73} -2.56155 q^{75} +3.12311 q^{77} +13.1231 q^{79} -7.00000 q^{81} -14.9309 q^{83} +1.56155 q^{85} -5.43845 q^{87} -2.24621 q^{89} +0.876894 q^{91} +23.6847 q^{93} +6.00000 q^{95} -4.87689 q^{97} -7.12311 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{3} - 2 q^{5} + q^{7} + 3 q^{9}+O(q^{10})$$ 2 * q - q^3 - 2 * q^5 + q^7 + 3 * q^9 $$2 q - q^{3} - 2 q^{5} + q^{7} + 3 q^{9} - 4 q^{11} + 3 q^{13} + q^{15} + q^{17} - 12 q^{19} + 8 q^{21} + 2 q^{23} + 2 q^{25} - 7 q^{27} - 4 q^{29} - 2 q^{31} + 2 q^{33} - q^{35} + 5 q^{37} + 7 q^{39} - 3 q^{45} - 3 q^{47} - 5 q^{49} + 8 q^{51} + 5 q^{53} + 4 q^{55} + 6 q^{57} - 5 q^{59} + 6 q^{61} - 7 q^{63} - 3 q^{65} + 13 q^{67} - q^{69} + 12 q^{71} - 13 q^{73} - q^{75} - 2 q^{77} + 18 q^{79} - 14 q^{81} - q^{83} - q^{85} - 15 q^{87} + 12 q^{89} + 10 q^{91} + 35 q^{93} + 12 q^{95} - 18 q^{97} - 6 q^{99}+O(q^{100})$$ 2 * q - q^3 - 2 * q^5 + q^7 + 3 * q^9 - 4 * q^11 + 3 * q^13 + q^15 + q^17 - 12 * q^19 + 8 * q^21 + 2 * q^23 + 2 * q^25 - 7 * q^27 - 4 * q^29 - 2 * q^31 + 2 * q^33 - q^35 + 5 * q^37 + 7 * q^39 - 3 * q^45 - 3 * q^47 - 5 * q^49 + 8 * q^51 + 5 * q^53 + 4 * q^55 + 6 * q^57 - 5 * q^59 + 6 * q^61 - 7 * q^63 - 3 * q^65 + 13 * q^67 - q^69 + 12 * q^71 - 13 * q^73 - q^75 - 2 * q^77 + 18 * q^79 - 14 * q^81 - q^83 - q^85 - 15 * q^87 + 12 * q^89 + 10 * q^91 + 35 * q^93 + 12 * q^95 - 18 * q^97 - 6 * 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
$$3$$ −2.56155 −1.47891 −0.739457 0.673204i $$-0.764917\pi$$
−0.739457 + 0.673204i $$0.764917\pi$$
$$4$$ 0 0
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ −1.56155 −0.590211 −0.295106 0.955465i $$-0.595355\pi$$
−0.295106 + 0.955465i $$0.595355\pi$$
$$8$$ 0 0
$$9$$ 3.56155 1.18718
$$10$$ 0 0
$$11$$ −2.00000 −0.603023 −0.301511 0.953463i $$-0.597491\pi$$
−0.301511 + 0.953463i $$0.597491\pi$$
$$12$$ 0 0
$$13$$ −0.561553 −0.155747 −0.0778734 0.996963i $$-0.524813\pi$$
−0.0778734 + 0.996963i $$0.524813\pi$$
$$14$$ 0 0
$$15$$ 2.56155 0.661390
$$16$$ 0 0
$$17$$ −1.56155 −0.378732 −0.189366 0.981907i $$-0.560643\pi$$
−0.189366 + 0.981907i $$0.560643\pi$$
$$18$$ 0 0
$$19$$ −6.00000 −1.37649 −0.688247 0.725476i $$-0.741620\pi$$
−0.688247 + 0.725476i $$0.741620\pi$$
$$20$$ 0 0
$$21$$ 4.00000 0.872872
$$22$$ 0 0
$$23$$ 1.00000 0.208514
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −1.43845 −0.276829
$$28$$ 0 0
$$29$$ 2.12311 0.394251 0.197125 0.980378i $$-0.436839\pi$$
0.197125 + 0.980378i $$0.436839\pi$$
$$30$$ 0 0
$$31$$ −9.24621 −1.66067 −0.830334 0.557266i $$-0.811851\pi$$
−0.830334 + 0.557266i $$0.811851\pi$$
$$32$$ 0 0
$$33$$ 5.12311 0.891818
$$34$$ 0 0
$$35$$ 1.56155 0.263951
$$36$$ 0 0
$$37$$ 0.438447 0.0720803 0.0360401 0.999350i $$-0.488526\pi$$
0.0360401 + 0.999350i $$0.488526\pi$$
$$38$$ 0 0
$$39$$ 1.43845 0.230336
$$40$$ 0 0
$$41$$ −4.12311 −0.643921 −0.321960 0.946753i $$-0.604342\pi$$
−0.321960 + 0.946753i $$0.604342\pi$$
$$42$$ 0 0
$$43$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$44$$ 0 0
$$45$$ −3.56155 −0.530925
$$46$$ 0 0
$$47$$ −7.68466 −1.12092 −0.560461 0.828181i $$-0.689376\pi$$
−0.560461 + 0.828181i $$0.689376\pi$$
$$48$$ 0 0
$$49$$ −4.56155 −0.651650
$$50$$ 0 0
$$51$$ 4.00000 0.560112
$$52$$ 0 0
$$53$$ 0.438447 0.0602254 0.0301127 0.999547i $$-0.490413\pi$$
0.0301127 + 0.999547i $$0.490413\pi$$
$$54$$ 0 0
$$55$$ 2.00000 0.269680
$$56$$ 0 0
$$57$$ 15.3693 2.03572
$$58$$ 0 0
$$59$$ −8.68466 −1.13065 −0.565323 0.824870i $$-0.691249\pi$$
−0.565323 + 0.824870i $$0.691249\pi$$
$$60$$ 0 0
$$61$$ −1.12311 −0.143799 −0.0718995 0.997412i $$-0.522906\pi$$
−0.0718995 + 0.997412i $$0.522906\pi$$
$$62$$ 0 0
$$63$$ −5.56155 −0.700690
$$64$$ 0 0
$$65$$ 0.561553 0.0696521
$$66$$ 0 0
$$67$$ 4.43845 0.542243 0.271121 0.962545i $$-0.412606\pi$$
0.271121 + 0.962545i $$0.412606\pi$$
$$68$$ 0 0
$$69$$ −2.56155 −0.308375
$$70$$ 0 0
$$71$$ 1.87689 0.222746 0.111373 0.993779i $$-0.464475\pi$$
0.111373 + 0.993779i $$0.464475\pi$$
$$72$$ 0 0
$$73$$ −8.56155 −1.00205 −0.501027 0.865432i $$-0.667044\pi$$
−0.501027 + 0.865432i $$0.667044\pi$$
$$74$$ 0 0
$$75$$ −2.56155 −0.295783
$$76$$ 0 0
$$77$$ 3.12311 0.355911
$$78$$ 0 0
$$79$$ 13.1231 1.47646 0.738232 0.674546i $$-0.235661\pi$$
0.738232 + 0.674546i $$0.235661\pi$$
$$80$$ 0 0
$$81$$ −7.00000 −0.777778
$$82$$ 0 0
$$83$$ −14.9309 −1.63888 −0.819438 0.573168i $$-0.805714\pi$$
−0.819438 + 0.573168i $$0.805714\pi$$
$$84$$ 0 0
$$85$$ 1.56155 0.169374
$$86$$ 0 0
$$87$$ −5.43845 −0.583063
$$88$$ 0 0
$$89$$ −2.24621 −0.238098 −0.119049 0.992888i $$-0.537985\pi$$
−0.119049 + 0.992888i $$0.537985\pi$$
$$90$$ 0 0
$$91$$ 0.876894 0.0919235
$$92$$ 0 0
$$93$$ 23.6847 2.45598
$$94$$ 0 0
$$95$$ 6.00000 0.615587
$$96$$ 0 0
$$97$$ −4.87689 −0.495174 −0.247587 0.968866i $$-0.579638\pi$$
−0.247587 + 0.968866i $$0.579638\pi$$
$$98$$ 0 0
$$99$$ −7.12311 −0.715899
$$100$$ 0 0
$$101$$ 5.31534 0.528896 0.264448 0.964400i $$-0.414810\pi$$
0.264448 + 0.964400i $$0.414810\pi$$
$$102$$ 0 0
$$103$$ −16.0000 −1.57653 −0.788263 0.615338i $$-0.789020\pi$$
−0.788263 + 0.615338i $$0.789020\pi$$
$$104$$ 0 0
$$105$$ −4.00000 −0.390360
$$106$$ 0 0
$$107$$ −7.56155 −0.731003 −0.365501 0.930811i $$-0.619102\pi$$
−0.365501 + 0.930811i $$0.619102\pi$$
$$108$$ 0 0
$$109$$ 9.36932 0.897418 0.448709 0.893678i $$-0.351884\pi$$
0.448709 + 0.893678i $$0.351884\pi$$
$$110$$ 0 0
$$111$$ −1.12311 −0.106600
$$112$$ 0 0
$$113$$ −7.80776 −0.734493 −0.367246 0.930124i $$-0.619699\pi$$
−0.367246 + 0.930124i $$0.619699\pi$$
$$114$$ 0 0
$$115$$ −1.00000 −0.0932505
$$116$$ 0 0
$$117$$ −2.00000 −0.184900
$$118$$ 0 0
$$119$$ 2.43845 0.223532
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ 0 0
$$123$$ 10.5616 0.952303
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ −12.8078 −1.13651 −0.568253 0.822854i $$-0.692380\pi$$
−0.568253 + 0.822854i $$0.692380\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −20.8078 −1.81798 −0.908991 0.416815i $$-0.863146\pi$$
−0.908991 + 0.416815i $$0.863146\pi$$
$$132$$ 0 0
$$133$$ 9.36932 0.812423
$$134$$ 0 0
$$135$$ 1.43845 0.123802
$$136$$ 0 0
$$137$$ 15.1231 1.29205 0.646027 0.763315i $$-0.276429\pi$$
0.646027 + 0.763315i $$0.276429\pi$$
$$138$$ 0 0
$$139$$ −5.24621 −0.444978 −0.222489 0.974935i $$-0.571418\pi$$
−0.222489 + 0.974935i $$0.571418\pi$$
$$140$$ 0 0
$$141$$ 19.6847 1.65775
$$142$$ 0 0
$$143$$ 1.12311 0.0939188
$$144$$ 0 0
$$145$$ −2.12311 −0.176314
$$146$$ 0 0
$$147$$ 11.6847 0.963734
$$148$$ 0 0
$$149$$ −15.3693 −1.25910 −0.629552 0.776959i $$-0.716761\pi$$
−0.629552 + 0.776959i $$0.716761\pi$$
$$150$$ 0 0
$$151$$ 23.0540 1.87611 0.938053 0.346492i $$-0.112627\pi$$
0.938053 + 0.346492i $$0.112627\pi$$
$$152$$ 0 0
$$153$$ −5.56155 −0.449625
$$154$$ 0 0
$$155$$ 9.24621 0.742674
$$156$$ 0 0
$$157$$ 16.6847 1.33158 0.665790 0.746139i $$-0.268095\pi$$
0.665790 + 0.746139i $$0.268095\pi$$
$$158$$ 0 0
$$159$$ −1.12311 −0.0890681
$$160$$ 0 0
$$161$$ −1.56155 −0.123068
$$162$$ 0 0
$$163$$ 11.9309 0.934498 0.467249 0.884126i $$-0.345245\pi$$
0.467249 + 0.884126i $$0.345245\pi$$
$$164$$ 0 0
$$165$$ −5.12311 −0.398833
$$166$$ 0 0
$$167$$ −8.00000 −0.619059 −0.309529 0.950890i $$-0.600171\pi$$
−0.309529 + 0.950890i $$0.600171\pi$$
$$168$$ 0 0
$$169$$ −12.6847 −0.975743
$$170$$ 0 0
$$171$$ −21.3693 −1.63415
$$172$$ 0 0
$$173$$ 19.3693 1.47262 0.736311 0.676643i $$-0.236566\pi$$
0.736311 + 0.676643i $$0.236566\pi$$
$$174$$ 0 0
$$175$$ −1.56155 −0.118042
$$176$$ 0 0
$$177$$ 22.2462 1.67213
$$178$$ 0 0
$$179$$ −21.9309 −1.63919 −0.819595 0.572943i $$-0.805802\pi$$
−0.819595 + 0.572943i $$0.805802\pi$$
$$180$$ 0 0
$$181$$ −19.3693 −1.43971 −0.719855 0.694124i $$-0.755792\pi$$
−0.719855 + 0.694124i $$0.755792\pi$$
$$182$$ 0 0
$$183$$ 2.87689 0.212666
$$184$$ 0 0
$$185$$ −0.438447 −0.0322353
$$186$$ 0 0
$$187$$ 3.12311 0.228384
$$188$$ 0 0
$$189$$ 2.24621 0.163388
$$190$$ 0 0
$$191$$ −7.36932 −0.533225 −0.266613 0.963804i $$-0.585904\pi$$
−0.266613 + 0.963804i $$0.585904\pi$$
$$192$$ 0 0
$$193$$ 6.56155 0.472311 0.236155 0.971715i $$-0.424113\pi$$
0.236155 + 0.971715i $$0.424113\pi$$
$$194$$ 0 0
$$195$$ −1.43845 −0.103009
$$196$$ 0 0
$$197$$ 6.31534 0.449949 0.224975 0.974365i $$-0.427770\pi$$
0.224975 + 0.974365i $$0.427770\pi$$
$$198$$ 0 0
$$199$$ −10.0000 −0.708881 −0.354441 0.935079i $$-0.615329\pi$$
−0.354441 + 0.935079i $$0.615329\pi$$
$$200$$ 0 0
$$201$$ −11.3693 −0.801930
$$202$$ 0 0
$$203$$ −3.31534 −0.232691
$$204$$ 0 0
$$205$$ 4.12311 0.287970
$$206$$ 0 0
$$207$$ 3.56155 0.247545
$$208$$ 0 0
$$209$$ 12.0000 0.830057
$$210$$ 0 0
$$211$$ −12.6847 −0.873248 −0.436624 0.899644i $$-0.643826\pi$$
−0.436624 + 0.899644i $$0.643826\pi$$
$$212$$ 0 0
$$213$$ −4.80776 −0.329423
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 14.4384 0.980146
$$218$$ 0 0
$$219$$ 21.9309 1.48195
$$220$$ 0 0
$$221$$ 0.876894 0.0589863
$$222$$ 0 0
$$223$$ 23.6155 1.58141 0.790706 0.612196i $$-0.209713\pi$$
0.790706 + 0.612196i $$0.209713\pi$$
$$224$$ 0 0
$$225$$ 3.56155 0.237437
$$226$$ 0 0
$$227$$ −9.75379 −0.647382 −0.323691 0.946163i $$-0.604924\pi$$
−0.323691 + 0.946163i $$0.604924\pi$$
$$228$$ 0 0
$$229$$ −22.7386 −1.50261 −0.751306 0.659954i $$-0.770576\pi$$
−0.751306 + 0.659954i $$0.770576\pi$$
$$230$$ 0 0
$$231$$ −8.00000 −0.526361
$$232$$ 0 0
$$233$$ 11.6847 0.765487 0.382744 0.923855i $$-0.374979\pi$$
0.382744 + 0.923855i $$0.374979\pi$$
$$234$$ 0 0
$$235$$ 7.68466 0.501292
$$236$$ 0 0
$$237$$ −33.6155 −2.18356
$$238$$ 0 0
$$239$$ 19.2462 1.24493 0.622467 0.782646i $$-0.286131\pi$$
0.622467 + 0.782646i $$0.286131\pi$$
$$240$$ 0 0
$$241$$ 6.00000 0.386494 0.193247 0.981150i $$-0.438098\pi$$
0.193247 + 0.981150i $$0.438098\pi$$
$$242$$ 0 0
$$243$$ 22.2462 1.42710
$$244$$ 0 0
$$245$$ 4.56155 0.291427
$$246$$ 0 0
$$247$$ 3.36932 0.214384
$$248$$ 0 0
$$249$$ 38.2462 2.42376
$$250$$ 0 0
$$251$$ −17.3693 −1.09634 −0.548171 0.836366i $$-0.684676\pi$$
−0.548171 + 0.836366i $$0.684676\pi$$
$$252$$ 0 0
$$253$$ −2.00000 −0.125739
$$254$$ 0 0
$$255$$ −4.00000 −0.250490
$$256$$ 0 0
$$257$$ 11.1922 0.698152 0.349076 0.937094i $$-0.386495\pi$$
0.349076 + 0.937094i $$0.386495\pi$$
$$258$$ 0 0
$$259$$ −0.684658 −0.0425426
$$260$$ 0 0
$$261$$ 7.56155 0.468048
$$262$$ 0 0
$$263$$ 28.9309 1.78395 0.891977 0.452081i $$-0.149318\pi$$
0.891977 + 0.452081i $$0.149318\pi$$
$$264$$ 0 0
$$265$$ −0.438447 −0.0269336
$$266$$ 0 0
$$267$$ 5.75379 0.352126
$$268$$ 0 0
$$269$$ −6.75379 −0.411786 −0.205893 0.978575i $$-0.566010\pi$$
−0.205893 + 0.978575i $$0.566010\pi$$
$$270$$ 0 0
$$271$$ −6.93087 −0.421020 −0.210510 0.977592i $$-0.567513\pi$$
−0.210510 + 0.977592i $$0.567513\pi$$
$$272$$ 0 0
$$273$$ −2.24621 −0.135947
$$274$$ 0 0
$$275$$ −2.00000 −0.120605
$$276$$ 0 0
$$277$$ −5.68466 −0.341558 −0.170779 0.985309i $$-0.554628\pi$$
−0.170779 + 0.985309i $$0.554628\pi$$
$$278$$ 0 0
$$279$$ −32.9309 −1.97152
$$280$$ 0 0
$$281$$ −15.1231 −0.902169 −0.451084 0.892481i $$-0.648963\pi$$
−0.451084 + 0.892481i $$0.648963\pi$$
$$282$$ 0 0
$$283$$ 5.80776 0.345236 0.172618 0.984989i $$-0.444777\pi$$
0.172618 + 0.984989i $$0.444777\pi$$
$$284$$ 0 0
$$285$$ −15.3693 −0.910400
$$286$$ 0 0
$$287$$ 6.43845 0.380050
$$288$$ 0 0
$$289$$ −14.5616 −0.856562
$$290$$ 0 0
$$291$$ 12.4924 0.732319
$$292$$ 0 0
$$293$$ −3.80776 −0.222452 −0.111226 0.993795i $$-0.535478\pi$$
−0.111226 + 0.993795i $$0.535478\pi$$
$$294$$ 0 0
$$295$$ 8.68466 0.505640
$$296$$ 0 0
$$297$$ 2.87689 0.166934
$$298$$ 0 0
$$299$$ −0.561553 −0.0324754
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ −13.6155 −0.782192
$$304$$ 0 0
$$305$$ 1.12311 0.0643088
$$306$$ 0 0
$$307$$ 12.8769 0.734923 0.367462 0.930039i $$-0.380227\pi$$
0.367462 + 0.930039i $$0.380227\pi$$
$$308$$ 0 0
$$309$$ 40.9848 2.33155
$$310$$ 0 0
$$311$$ −3.68466 −0.208938 −0.104469 0.994528i $$-0.533314\pi$$
−0.104469 + 0.994528i $$0.533314\pi$$
$$312$$ 0 0
$$313$$ 19.8078 1.11960 0.559801 0.828627i $$-0.310878\pi$$
0.559801 + 0.828627i $$0.310878\pi$$
$$314$$ 0 0
$$315$$ 5.56155 0.313358
$$316$$ 0 0
$$317$$ −6.87689 −0.386245 −0.193122 0.981175i $$-0.561861\pi$$
−0.193122 + 0.981175i $$0.561861\pi$$
$$318$$ 0 0
$$319$$ −4.24621 −0.237742
$$320$$ 0 0
$$321$$ 19.3693 1.08109
$$322$$ 0 0
$$323$$ 9.36932 0.521323
$$324$$ 0 0
$$325$$ −0.561553 −0.0311493
$$326$$ 0 0
$$327$$ −24.0000 −1.32720
$$328$$ 0 0
$$329$$ 12.0000 0.661581
$$330$$ 0 0
$$331$$ 12.6155 0.693412 0.346706 0.937974i $$-0.387300\pi$$
0.346706 + 0.937974i $$0.387300\pi$$
$$332$$ 0 0
$$333$$ 1.56155 0.0855726
$$334$$ 0 0
$$335$$ −4.43845 −0.242498
$$336$$ 0 0
$$337$$ −27.6155 −1.50431 −0.752157 0.658984i $$-0.770986\pi$$
−0.752157 + 0.658984i $$0.770986\pi$$
$$338$$ 0 0
$$339$$ 20.0000 1.08625
$$340$$ 0 0
$$341$$ 18.4924 1.00142
$$342$$ 0 0
$$343$$ 18.0540 0.974823
$$344$$ 0 0
$$345$$ 2.56155 0.137909
$$346$$ 0 0
$$347$$ 16.4924 0.885360 0.442680 0.896680i $$-0.354028\pi$$
0.442680 + 0.896680i $$0.354028\pi$$
$$348$$ 0 0
$$349$$ 1.63068 0.0872885 0.0436442 0.999047i $$-0.486103\pi$$
0.0436442 + 0.999047i $$0.486103\pi$$
$$350$$ 0 0
$$351$$ 0.807764 0.0431153
$$352$$ 0 0
$$353$$ 35.0540 1.86573 0.932867 0.360220i $$-0.117298\pi$$
0.932867 + 0.360220i $$0.117298\pi$$
$$354$$ 0 0
$$355$$ −1.87689 −0.0996152
$$356$$ 0 0
$$357$$ −6.24621 −0.330585
$$358$$ 0 0
$$359$$ −25.6155 −1.35194 −0.675968 0.736931i $$-0.736274\pi$$
−0.675968 + 0.736931i $$0.736274\pi$$
$$360$$ 0 0
$$361$$ 17.0000 0.894737
$$362$$ 0 0
$$363$$ 17.9309 0.941127
$$364$$ 0 0
$$365$$ 8.56155 0.448132
$$366$$ 0 0
$$367$$ −18.4384 −0.962479 −0.481240 0.876589i $$-0.659813\pi$$
−0.481240 + 0.876589i $$0.659813\pi$$
$$368$$ 0 0
$$369$$ −14.6847 −0.764453
$$370$$ 0 0
$$371$$ −0.684658 −0.0355457
$$372$$ 0 0
$$373$$ 3.75379 0.194364 0.0971819 0.995267i $$-0.469017\pi$$
0.0971819 + 0.995267i $$0.469017\pi$$
$$374$$ 0 0
$$375$$ 2.56155 0.132278
$$376$$ 0 0
$$377$$ −1.19224 −0.0614033
$$378$$ 0 0
$$379$$ 3.50758 0.180172 0.0900861 0.995934i $$-0.471286\pi$$
0.0900861 + 0.995934i $$0.471286\pi$$
$$380$$ 0 0
$$381$$ 32.8078 1.68079
$$382$$ 0 0
$$383$$ 23.8078 1.21652 0.608260 0.793738i $$-0.291868\pi$$
0.608260 + 0.793738i $$0.291868\pi$$
$$384$$ 0 0
$$385$$ −3.12311 −0.159168
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 19.1231 0.969580 0.484790 0.874631i $$-0.338896\pi$$
0.484790 + 0.874631i $$0.338896\pi$$
$$390$$ 0 0
$$391$$ −1.56155 −0.0789711
$$392$$ 0 0
$$393$$ 53.3002 2.68864
$$394$$ 0 0
$$395$$ −13.1231 −0.660295
$$396$$ 0 0
$$397$$ 35.5464 1.78402 0.892011 0.452013i $$-0.149294\pi$$
0.892011 + 0.452013i $$0.149294\pi$$
$$398$$ 0 0
$$399$$ −24.0000 −1.20150
$$400$$ 0 0
$$401$$ 2.00000 0.0998752 0.0499376 0.998752i $$-0.484098\pi$$
0.0499376 + 0.998752i $$0.484098\pi$$
$$402$$ 0 0
$$403$$ 5.19224 0.258644
$$404$$ 0 0
$$405$$ 7.00000 0.347833
$$406$$ 0 0
$$407$$ −0.876894 −0.0434660
$$408$$ 0 0
$$409$$ 29.9848 1.48266 0.741328 0.671143i $$-0.234197\pi$$
0.741328 + 0.671143i $$0.234197\pi$$
$$410$$ 0 0
$$411$$ −38.7386 −1.91084
$$412$$ 0 0
$$413$$ 13.5616 0.667320
$$414$$ 0 0
$$415$$ 14.9309 0.732928
$$416$$ 0 0
$$417$$ 13.4384 0.658084
$$418$$ 0 0
$$419$$ 27.1231 1.32505 0.662525 0.749040i $$-0.269485\pi$$
0.662525 + 0.749040i $$0.269485\pi$$
$$420$$ 0 0
$$421$$ −16.8769 −0.822530 −0.411265 0.911516i $$-0.634913\pi$$
−0.411265 + 0.911516i $$0.634913\pi$$
$$422$$ 0 0
$$423$$ −27.3693 −1.33074
$$424$$ 0 0
$$425$$ −1.56155 −0.0757464
$$426$$ 0 0
$$427$$ 1.75379 0.0848718
$$428$$ 0 0
$$429$$ −2.87689 −0.138898
$$430$$ 0 0
$$431$$ 6.24621 0.300869 0.150435 0.988620i $$-0.451933\pi$$
0.150435 + 0.988620i $$0.451933\pi$$
$$432$$ 0 0
$$433$$ 11.0691 0.531948 0.265974 0.963980i $$-0.414306\pi$$
0.265974 + 0.963980i $$0.414306\pi$$
$$434$$ 0 0
$$435$$ 5.43845 0.260754
$$436$$ 0 0
$$437$$ −6.00000 −0.287019
$$438$$ 0 0
$$439$$ −0.807764 −0.0385525 −0.0192762 0.999814i $$-0.506136\pi$$
−0.0192762 + 0.999814i $$0.506136\pi$$
$$440$$ 0 0
$$441$$ −16.2462 −0.773629
$$442$$ 0 0
$$443$$ −8.80776 −0.418469 −0.209235 0.977865i $$-0.567097\pi$$
−0.209235 + 0.977865i $$0.567097\pi$$
$$444$$ 0 0
$$445$$ 2.24621 0.106481
$$446$$ 0 0
$$447$$ 39.3693 1.86210
$$448$$ 0 0
$$449$$ −9.31534 −0.439618 −0.219809 0.975543i $$-0.570543\pi$$
−0.219809 + 0.975543i $$0.570543\pi$$
$$450$$ 0 0
$$451$$ 8.24621 0.388299
$$452$$ 0 0
$$453$$ −59.0540 −2.77460
$$454$$ 0 0
$$455$$ −0.876894 −0.0411094
$$456$$ 0 0
$$457$$ 13.5616 0.634383 0.317191 0.948362i $$-0.397260\pi$$
0.317191 + 0.948362i $$0.397260\pi$$
$$458$$ 0 0
$$459$$ 2.24621 0.104844
$$460$$ 0 0
$$461$$ 12.0691 0.562115 0.281058 0.959691i $$-0.409315\pi$$
0.281058 + 0.959691i $$0.409315\pi$$
$$462$$ 0 0
$$463$$ −6.63068 −0.308154 −0.154077 0.988059i $$-0.549240\pi$$
−0.154077 + 0.988059i $$0.549240\pi$$
$$464$$ 0 0
$$465$$ −23.6847 −1.09835
$$466$$ 0 0
$$467$$ 34.6847 1.60501 0.802507 0.596642i $$-0.203499\pi$$
0.802507 + 0.596642i $$0.203499\pi$$
$$468$$ 0 0
$$469$$ −6.93087 −0.320038
$$470$$ 0 0
$$471$$ −42.7386 −1.96929
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ −6.00000 −0.275299
$$476$$ 0 0
$$477$$ 1.56155 0.0714986
$$478$$ 0 0
$$479$$ 39.2311 1.79251 0.896256 0.443536i $$-0.146276\pi$$
0.896256 + 0.443536i $$0.146276\pi$$
$$480$$ 0 0
$$481$$ −0.246211 −0.0112263
$$482$$ 0 0
$$483$$ 4.00000 0.182006
$$484$$ 0 0
$$485$$ 4.87689 0.221448
$$486$$ 0 0
$$487$$ −12.8078 −0.580375 −0.290188 0.956970i $$-0.593718\pi$$
−0.290188 + 0.956970i $$0.593718\pi$$
$$488$$ 0 0
$$489$$ −30.5616 −1.38204
$$490$$ 0 0
$$491$$ 21.4924 0.969939 0.484970 0.874531i $$-0.338831\pi$$
0.484970 + 0.874531i $$0.338831\pi$$
$$492$$ 0 0
$$493$$ −3.31534 −0.149315
$$494$$ 0 0
$$495$$ 7.12311 0.320160
$$496$$ 0 0
$$497$$ −2.93087 −0.131467
$$498$$ 0 0
$$499$$ 18.6155 0.833345 0.416673 0.909057i $$-0.363196\pi$$
0.416673 + 0.909057i $$0.363196\pi$$
$$500$$ 0 0
$$501$$ 20.4924 0.915534
$$502$$ 0 0
$$503$$ 24.9309 1.11161 0.555806 0.831312i $$-0.312410\pi$$
0.555806 + 0.831312i $$0.312410\pi$$
$$504$$ 0 0
$$505$$ −5.31534 −0.236530
$$506$$ 0 0
$$507$$ 32.4924 1.44304
$$508$$ 0 0
$$509$$ −18.1771 −0.805685 −0.402842 0.915269i $$-0.631978\pi$$
−0.402842 + 0.915269i $$0.631978\pi$$
$$510$$ 0 0
$$511$$ 13.3693 0.591424
$$512$$ 0 0
$$513$$ 8.63068 0.381054
$$514$$ 0 0
$$515$$ 16.0000 0.705044
$$516$$ 0 0
$$517$$ 15.3693 0.675942
$$518$$ 0 0
$$519$$ −49.6155 −2.17788
$$520$$ 0 0
$$521$$ 27.6155 1.20986 0.604929 0.796279i $$-0.293201\pi$$
0.604929 + 0.796279i $$0.293201\pi$$
$$522$$ 0 0
$$523$$ −34.7386 −1.51901 −0.759507 0.650499i $$-0.774560\pi$$
−0.759507 + 0.650499i $$0.774560\pi$$
$$524$$ 0 0
$$525$$ 4.00000 0.174574
$$526$$ 0 0
$$527$$ 14.4384 0.628949
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ −30.9309 −1.34229
$$532$$ 0 0
$$533$$ 2.31534 0.100289
$$534$$ 0 0
$$535$$ 7.56155 0.326914
$$536$$ 0 0
$$537$$ 56.1771 2.42422
$$538$$ 0 0
$$539$$ 9.12311 0.392960
$$540$$ 0 0
$$541$$ −9.68466 −0.416376 −0.208188 0.978089i $$-0.566757\pi$$
−0.208188 + 0.978089i $$0.566757\pi$$
$$542$$ 0 0
$$543$$ 49.6155 2.12921
$$544$$ 0 0
$$545$$ −9.36932 −0.401337
$$546$$ 0 0
$$547$$ −24.1771 −1.03374 −0.516869 0.856065i $$-0.672902\pi$$
−0.516869 + 0.856065i $$0.672902\pi$$
$$548$$ 0 0
$$549$$ −4.00000 −0.170716
$$550$$ 0 0
$$551$$ −12.7386 −0.542684
$$552$$ 0 0
$$553$$ −20.4924 −0.871426
$$554$$ 0 0
$$555$$ 1.12311 0.0476732
$$556$$ 0 0
$$557$$ 3.31534 0.140476 0.0702378 0.997530i $$-0.477624\pi$$
0.0702378 + 0.997530i $$0.477624\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ −8.00000 −0.337760
$$562$$ 0 0
$$563$$ 45.6695 1.92474 0.962370 0.271742i $$-0.0875998\pi$$
0.962370 + 0.271742i $$0.0875998\pi$$
$$564$$ 0 0
$$565$$ 7.80776 0.328475
$$566$$ 0 0
$$567$$ 10.9309 0.459053
$$568$$ 0 0
$$569$$ −32.7386 −1.37247 −0.686237 0.727378i $$-0.740739\pi$$
−0.686237 + 0.727378i $$0.740739\pi$$
$$570$$ 0 0
$$571$$ −36.0000 −1.50655 −0.753277 0.657704i $$-0.771528\pi$$
−0.753277 + 0.657704i $$0.771528\pi$$
$$572$$ 0 0
$$573$$ 18.8769 0.788594
$$574$$ 0 0
$$575$$ 1.00000 0.0417029
$$576$$ 0 0
$$577$$ −25.4384 −1.05902 −0.529508 0.848305i $$-0.677624\pi$$
−0.529508 + 0.848305i $$0.677624\pi$$
$$578$$ 0 0
$$579$$ −16.8078 −0.698507
$$580$$ 0 0
$$581$$ 23.3153 0.967283
$$582$$ 0 0
$$583$$ −0.876894 −0.0363173
$$584$$ 0 0
$$585$$ 2.00000 0.0826898
$$586$$ 0 0
$$587$$ −35.9309 −1.48303 −0.741513 0.670939i $$-0.765891\pi$$
−0.741513 + 0.670939i $$0.765891\pi$$
$$588$$ 0 0
$$589$$ 55.4773 2.28590
$$590$$ 0 0
$$591$$ −16.1771 −0.665436
$$592$$ 0 0
$$593$$ −25.6155 −1.05190 −0.525952 0.850514i $$-0.676291\pi$$
−0.525952 + 0.850514i $$0.676291\pi$$
$$594$$ 0 0
$$595$$ −2.43845 −0.0999666
$$596$$ 0 0
$$597$$ 25.6155 1.04837
$$598$$ 0 0
$$599$$ 32.9848 1.34772 0.673862 0.738857i $$-0.264634\pi$$
0.673862 + 0.738857i $$0.264634\pi$$
$$600$$ 0 0
$$601$$ 11.6307 0.474425 0.237213 0.971458i $$-0.423766\pi$$
0.237213 + 0.971458i $$0.423766\pi$$
$$602$$ 0 0
$$603$$ 15.8078 0.643742
$$604$$ 0 0
$$605$$ 7.00000 0.284590
$$606$$ 0 0
$$607$$ −24.4924 −0.994117 −0.497058 0.867717i $$-0.665587\pi$$
−0.497058 + 0.867717i $$0.665587\pi$$
$$608$$ 0 0
$$609$$ 8.49242 0.344130
$$610$$ 0 0
$$611$$ 4.31534 0.174580
$$612$$ 0 0
$$613$$ −35.6155 −1.43850 −0.719249 0.694753i $$-0.755514\pi$$
−0.719249 + 0.694753i $$0.755514\pi$$
$$614$$ 0 0
$$615$$ −10.5616 −0.425883
$$616$$ 0 0
$$617$$ −3.56155 −0.143383 −0.0716914 0.997427i $$-0.522840\pi$$
−0.0716914 + 0.997427i $$0.522840\pi$$
$$618$$ 0 0
$$619$$ 20.4924 0.823660 0.411830 0.911261i $$-0.364890\pi$$
0.411830 + 0.911261i $$0.364890\pi$$
$$620$$ 0 0
$$621$$ −1.43845 −0.0577229
$$622$$ 0 0
$$623$$ 3.50758 0.140528
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ −30.7386 −1.22758
$$628$$ 0 0
$$629$$ −0.684658 −0.0272991
$$630$$ 0 0
$$631$$ 13.7538 0.547530 0.273765 0.961797i $$-0.411731\pi$$
0.273765 + 0.961797i $$0.411731\pi$$
$$632$$ 0 0
$$633$$ 32.4924 1.29146
$$634$$ 0 0
$$635$$ 12.8078 0.508261
$$636$$ 0 0
$$637$$ 2.56155 0.101492
$$638$$ 0 0
$$639$$ 6.68466 0.264441
$$640$$ 0 0
$$641$$ 15.1231 0.597327 0.298663 0.954359i $$-0.403459\pi$$
0.298663 + 0.954359i $$0.403459\pi$$
$$642$$ 0 0
$$643$$ 41.4233 1.63358 0.816788 0.576939i $$-0.195753\pi$$
0.816788 + 0.576939i $$0.195753\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 17.6847 0.695256 0.347628 0.937633i $$-0.386987\pi$$
0.347628 + 0.937633i $$0.386987\pi$$
$$648$$ 0 0
$$649$$ 17.3693 0.681805
$$650$$ 0 0
$$651$$ −36.9848 −1.44955
$$652$$ 0 0
$$653$$ 38.6695 1.51325 0.756627 0.653846i $$-0.226846\pi$$
0.756627 + 0.653846i $$0.226846\pi$$
$$654$$ 0 0
$$655$$ 20.8078 0.813027
$$656$$ 0 0
$$657$$ −30.4924 −1.18962
$$658$$ 0 0
$$659$$ 9.36932 0.364977 0.182488 0.983208i $$-0.441585\pi$$
0.182488 + 0.983208i $$0.441585\pi$$
$$660$$ 0 0
$$661$$ −26.4924 −1.03044 −0.515218 0.857059i $$-0.672289\pi$$
−0.515218 + 0.857059i $$0.672289\pi$$
$$662$$ 0 0
$$663$$ −2.24621 −0.0872356
$$664$$ 0 0
$$665$$ −9.36932 −0.363327
$$666$$ 0 0
$$667$$ 2.12311 0.0822070
$$668$$ 0 0
$$669$$ −60.4924 −2.33877
$$670$$ 0 0
$$671$$ 2.24621 0.0867140
$$672$$ 0 0
$$673$$ −35.5464 −1.37021 −0.685106 0.728443i $$-0.740244\pi$$
−0.685106 + 0.728443i $$0.740244\pi$$
$$674$$ 0 0
$$675$$ −1.43845 −0.0553659
$$676$$ 0 0
$$677$$ −6.19224 −0.237987 −0.118993 0.992895i $$-0.537967\pi$$
−0.118993 + 0.992895i $$0.537967\pi$$
$$678$$ 0 0
$$679$$ 7.61553 0.292257
$$680$$ 0 0
$$681$$ 24.9848 0.957421
$$682$$ 0 0
$$683$$ −4.94602 −0.189254 −0.0946272 0.995513i $$-0.530166\pi$$
−0.0946272 + 0.995513i $$0.530166\pi$$
$$684$$ 0 0
$$685$$ −15.1231 −0.577824
$$686$$ 0 0
$$687$$ 58.2462 2.22223
$$688$$ 0 0
$$689$$ −0.246211 −0.00937990
$$690$$ 0 0
$$691$$ 16.4924 0.627401 0.313701 0.949522i $$-0.398431\pi$$
0.313701 + 0.949522i $$0.398431\pi$$
$$692$$ 0 0
$$693$$ 11.1231 0.422532
$$694$$ 0 0
$$695$$ 5.24621 0.199000
$$696$$ 0 0
$$697$$ 6.43845 0.243874
$$698$$ 0 0
$$699$$ −29.9309 −1.13209
$$700$$ 0 0
$$701$$ 18.2462 0.689150 0.344575 0.938759i $$-0.388023\pi$$
0.344575 + 0.938759i $$0.388023\pi$$
$$702$$ 0 0
$$703$$ −2.63068 −0.0992181
$$704$$ 0 0
$$705$$ −19.6847 −0.741367
$$706$$ 0 0
$$707$$ −8.30019 −0.312161
$$708$$ 0 0
$$709$$ −46.2462 −1.73681 −0.868406 0.495853i $$-0.834855\pi$$
−0.868406 + 0.495853i $$0.834855\pi$$
$$710$$ 0 0
$$711$$ 46.7386 1.75284
$$712$$ 0 0
$$713$$ −9.24621 −0.346273
$$714$$ 0 0
$$715$$ −1.12311 −0.0420018
$$716$$ 0 0
$$717$$ −49.3002 −1.84115
$$718$$ 0 0
$$719$$ −18.0540 −0.673300 −0.336650 0.941630i $$-0.609294\pi$$
−0.336650 + 0.941630i $$0.609294\pi$$
$$720$$ 0 0
$$721$$ 24.9848 0.930484
$$722$$ 0 0
$$723$$ −15.3693 −0.571591
$$724$$ 0 0
$$725$$ 2.12311 0.0788502
$$726$$ 0 0
$$727$$ −41.8078 −1.55056 −0.775282 0.631615i $$-0.782392\pi$$
−0.775282 + 0.631615i $$0.782392\pi$$
$$728$$ 0 0
$$729$$ −35.9848 −1.33277
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ −12.6847 −0.468519 −0.234259 0.972174i $$-0.575266\pi$$
−0.234259 + 0.972174i $$0.575266\pi$$
$$734$$ 0 0
$$735$$ −11.6847 −0.430995
$$736$$ 0 0
$$737$$ −8.87689 −0.326985
$$738$$ 0 0
$$739$$ 40.6155 1.49407 0.747033 0.664787i $$-0.231478\pi$$
0.747033 + 0.664787i $$0.231478\pi$$
$$740$$ 0 0
$$741$$ −8.63068 −0.317056
$$742$$ 0 0
$$743$$ 36.4924 1.33878 0.669389 0.742912i $$-0.266556\pi$$
0.669389 + 0.742912i $$0.266556\pi$$
$$744$$ 0 0
$$745$$ 15.3693 0.563088
$$746$$ 0 0
$$747$$ −53.1771 −1.94565
$$748$$ 0 0
$$749$$ 11.8078 0.431446
$$750$$ 0 0
$$751$$ −4.87689 −0.177960 −0.0889802 0.996033i $$-0.528361\pi$$
−0.0889802 + 0.996033i $$0.528361\pi$$
$$752$$ 0 0
$$753$$ 44.4924 1.62139
$$754$$ 0 0
$$755$$ −23.0540 −0.839020
$$756$$ 0 0
$$757$$ −8.43845 −0.306701 −0.153350 0.988172i $$-0.549006\pi$$
−0.153350 + 0.988172i $$0.549006\pi$$
$$758$$ 0 0
$$759$$ 5.12311 0.185957
$$760$$ 0 0
$$761$$ 21.9848 0.796950 0.398475 0.917179i $$-0.369540\pi$$
0.398475 + 0.917179i $$0.369540\pi$$
$$762$$ 0 0
$$763$$ −14.6307 −0.529666
$$764$$ 0 0
$$765$$ 5.56155 0.201078
$$766$$ 0 0
$$767$$ 4.87689 0.176094
$$768$$ 0 0
$$769$$ −35.3693 −1.27545 −0.637725 0.770264i $$-0.720124\pi$$
−0.637725 + 0.770264i $$0.720124\pi$$
$$770$$ 0 0
$$771$$ −28.6695 −1.03251
$$772$$ 0 0
$$773$$ 32.8769 1.18250 0.591250 0.806488i $$-0.298635\pi$$
0.591250 + 0.806488i $$0.298635\pi$$
$$774$$ 0 0
$$775$$ −9.24621 −0.332134
$$776$$ 0 0
$$777$$ 1.75379 0.0629168
$$778$$ 0 0
$$779$$ 24.7386 0.886354
$$780$$ 0 0
$$781$$ −3.75379 −0.134321
$$782$$ 0 0
$$783$$ −3.05398 −0.109140
$$784$$ 0 0
$$785$$ −16.6847 −0.595501
$$786$$ 0 0
$$787$$ 27.3153 0.973687 0.486843 0.873489i $$-0.338148\pi$$
0.486843 + 0.873489i $$0.338148\pi$$
$$788$$ 0 0
$$789$$ −74.1080 −2.63831
$$790$$ 0 0
$$791$$ 12.1922 0.433506
$$792$$ 0 0
$$793$$ 0.630683 0.0223962
$$794$$ 0 0
$$795$$ 1.12311 0.0398325
$$796$$ 0 0
$$797$$ −9.42329 −0.333790 −0.166895 0.985975i $$-0.553374\pi$$
−0.166895 + 0.985975i $$0.553374\pi$$
$$798$$ 0 0
$$799$$ 12.0000 0.424529
$$800$$ 0 0
$$801$$ −8.00000 −0.282666
$$802$$ 0 0
$$803$$ 17.1231 0.604261
$$804$$ 0 0
$$805$$ 1.56155 0.0550375
$$806$$ 0 0
$$807$$ 17.3002 0.608995
$$808$$ 0 0
$$809$$ −8.82292 −0.310197 −0.155099 0.987899i $$-0.549570\pi$$
−0.155099 + 0.987899i $$0.549570\pi$$
$$810$$ 0 0
$$811$$ −30.7538 −1.07991 −0.539956 0.841693i $$-0.681559\pi$$
−0.539956 + 0.841693i $$0.681559\pi$$
$$812$$ 0 0
$$813$$ 17.7538 0.622653
$$814$$ 0 0
$$815$$ −11.9309 −0.417920
$$816$$ 0 0
$$817$$ 0 0
$$818$$ 0 0
$$819$$ 3.12311 0.109130
$$820$$ 0 0
$$821$$ 1.50758 0.0526148 0.0263074 0.999654i $$-0.491625\pi$$
0.0263074 + 0.999654i $$0.491625\pi$$
$$822$$ 0 0
$$823$$ −0.946025 −0.0329763 −0.0164882 0.999864i $$-0.505249\pi$$
−0.0164882 + 0.999864i $$0.505249\pi$$
$$824$$ 0 0
$$825$$ 5.12311 0.178364
$$826$$ 0 0
$$827$$ −7.31534 −0.254379 −0.127190 0.991878i $$-0.540596\pi$$
−0.127190 + 0.991878i $$0.540596\pi$$
$$828$$ 0 0
$$829$$ −40.5464 −1.40823 −0.704117 0.710084i $$-0.748657\pi$$
−0.704117 + 0.710084i $$0.748657\pi$$
$$830$$ 0 0
$$831$$ 14.5616 0.505135
$$832$$ 0 0
$$833$$ 7.12311 0.246801
$$834$$ 0 0
$$835$$ 8.00000 0.276851
$$836$$ 0 0
$$837$$ 13.3002 0.459722
$$838$$ 0 0
$$839$$ −51.1231 −1.76497 −0.882483 0.470345i $$-0.844130\pi$$
−0.882483 + 0.470345i $$0.844130\pi$$
$$840$$ 0 0
$$841$$ −24.4924 −0.844566
$$842$$ 0 0
$$843$$ 38.7386 1.33423
$$844$$ 0 0
$$845$$ 12.6847 0.436366
$$846$$ 0 0
$$847$$ 10.9309 0.375589
$$848$$ 0 0
$$849$$ −14.8769 −0.510574
$$850$$ 0 0
$$851$$ 0.438447 0.0150298
$$852$$ 0 0
$$853$$ 3.75379 0.128527 0.0642636 0.997933i $$-0.479530\pi$$
0.0642636 + 0.997933i $$0.479530\pi$$
$$854$$ 0 0
$$855$$ 21.3693 0.730815
$$856$$ 0 0
$$857$$ −45.6847 −1.56056 −0.780279 0.625431i $$-0.784923\pi$$
−0.780279 + 0.625431i $$0.784923\pi$$
$$858$$ 0 0
$$859$$ 47.4924 1.62042 0.810210 0.586139i $$-0.199353\pi$$
0.810210 + 0.586139i $$0.199353\pi$$
$$860$$ 0 0
$$861$$ −16.4924 −0.562060
$$862$$ 0 0
$$863$$ −3.43845 −0.117046 −0.0585231 0.998286i $$-0.518639\pi$$
−0.0585231 + 0.998286i $$0.518639\pi$$
$$864$$ 0 0
$$865$$ −19.3693 −0.658577
$$866$$ 0 0
$$867$$ 37.3002 1.26678
$$868$$ 0 0
$$869$$ −26.2462 −0.890342
$$870$$ 0 0
$$871$$ −2.49242 −0.0844525
$$872$$ 0 0
$$873$$ −17.3693 −0.587862
$$874$$ 0 0
$$875$$ 1.56155 0.0527901
$$876$$ 0 0
$$877$$ −46.9848 −1.58657 −0.793283 0.608853i $$-0.791630\pi$$
−0.793283 + 0.608853i $$0.791630\pi$$
$$878$$ 0 0
$$879$$ 9.75379 0.328987
$$880$$ 0 0
$$881$$ −57.4773 −1.93646 −0.968229 0.250065i $$-0.919548\pi$$
−0.968229 + 0.250065i $$0.919548\pi$$
$$882$$ 0 0
$$883$$ 38.7386 1.30366 0.651829 0.758366i $$-0.274002\pi$$
0.651829 + 0.758366i $$0.274002\pi$$
$$884$$ 0 0
$$885$$ −22.2462 −0.747798
$$886$$ 0 0
$$887$$ −43.7926 −1.47041 −0.735206 0.677844i $$-0.762915\pi$$
−0.735206 + 0.677844i $$0.762915\pi$$
$$888$$ 0 0
$$889$$ 20.0000 0.670778
$$890$$ 0 0
$$891$$ 14.0000 0.469018
$$892$$ 0 0
$$893$$ 46.1080 1.54294
$$894$$ 0 0
$$895$$ 21.9309 0.733068
$$896$$ 0 0
$$897$$ 1.43845 0.0480284
$$898$$ 0 0
$$899$$ −19.6307 −0.654720
$$900$$ 0 0
$$901$$ −0.684658 −0.0228093
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 19.3693 0.643858
$$906$$ 0 0
$$907$$ −9.31534 −0.309311 −0.154655 0.987968i $$-0.549427\pi$$
−0.154655 + 0.987968i $$0.549427\pi$$
$$908$$ 0 0
$$909$$ 18.9309 0.627897
$$910$$ 0 0
$$911$$ 5.12311 0.169736 0.0848680 0.996392i $$-0.472953\pi$$
0.0848680 + 0.996392i $$0.472953\pi$$
$$912$$ 0 0
$$913$$ 29.8617 0.988279
$$914$$ 0 0
$$915$$ −2.87689 −0.0951072
$$916$$ 0 0
$$917$$ 32.4924 1.07299
$$918$$ 0 0
$$919$$ −2.73863 −0.0903392 −0.0451696 0.998979i $$-0.514383\pi$$
−0.0451696 + 0.998979i $$0.514383\pi$$
$$920$$ 0 0
$$921$$ −32.9848 −1.08689
$$922$$ 0 0
$$923$$ −1.05398 −0.0346920
$$924$$ 0 0
$$925$$ 0.438447 0.0144161
$$926$$ 0 0
$$927$$ −56.9848 −1.87163
$$928$$ 0 0
$$929$$ 58.7235 1.92665 0.963327 0.268329i $$-0.0864713\pi$$
0.963327 + 0.268329i $$0.0864713\pi$$
$$930$$ 0 0
$$931$$ 27.3693 0.896993
$$932$$ 0 0
$$933$$ 9.43845 0.309001
$$934$$ 0 0
$$935$$ −3.12311 −0.102136
$$936$$ 0 0
$$937$$ 0.246211 0.00804337 0.00402169 0.999992i $$-0.498720\pi$$
0.00402169 + 0.999992i $$0.498720\pi$$
$$938$$ 0 0
$$939$$ −50.7386 −1.65579
$$940$$ 0 0
$$941$$ 54.2462 1.76838 0.884188 0.467131i $$-0.154712\pi$$
0.884188 + 0.467131i $$0.154712\pi$$
$$942$$ 0 0
$$943$$ −4.12311 −0.134267
$$944$$ 0 0
$$945$$ −2.24621 −0.0730693
$$946$$ 0 0
$$947$$ −44.3153 −1.44006 −0.720028 0.693945i $$-0.755871\pi$$
−0.720028 + 0.693945i $$0.755871\pi$$
$$948$$ 0 0
$$949$$ 4.80776 0.156067
$$950$$ 0 0
$$951$$ 17.6155 0.571223
$$952$$ 0 0
$$953$$ −5.50758 −0.178408 −0.0892040 0.996013i $$-0.528432\pi$$
−0.0892040 + 0.996013i $$0.528432\pi$$
$$954$$ 0 0
$$955$$ 7.36932 0.238465
$$956$$ 0 0
$$957$$ 10.8769 0.351600
$$958$$ 0 0
$$959$$ −23.6155 −0.762585
$$960$$ 0 0
$$961$$ 54.4924 1.75782
$$962$$ 0 0
$$963$$ −26.9309 −0.867835
$$964$$ 0 0
$$965$$ −6.56155 −0.211224
$$966$$ 0 0
$$967$$ −44.3153 −1.42509 −0.712543 0.701629i $$-0.752457\pi$$
−0.712543 + 0.701629i $$0.752457\pi$$
$$968$$ 0 0
$$969$$ −24.0000 −0.770991
$$970$$ 0 0
$$971$$ 47.3693 1.52015 0.760077 0.649833i $$-0.225161\pi$$
0.760077 + 0.649833i $$0.225161\pi$$
$$972$$ 0 0
$$973$$ 8.19224 0.262631
$$974$$ 0 0
$$975$$ 1.43845 0.0460672
$$976$$ 0 0
$$977$$ −26.7926 −0.857172 −0.428586 0.903501i $$-0.640988\pi$$
−0.428586 + 0.903501i $$0.640988\pi$$
$$978$$ 0 0
$$979$$ 4.49242 0.143578
$$980$$ 0 0
$$981$$ 33.3693 1.06540
$$982$$ 0 0
$$983$$ 0.0539753 0.00172155 0.000860773 1.00000i $$-0.499726\pi$$
0.000860773 1.00000i $$0.499726\pi$$
$$984$$ 0 0
$$985$$ −6.31534 −0.201224
$$986$$ 0 0
$$987$$ −30.7386 −0.978421
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ −0.300187 −0.00953574 −0.00476787 0.999989i $$-0.501518\pi$$
−0.00476787 + 0.999989i $$0.501518\pi$$
$$992$$ 0 0
$$993$$ −32.3153 −1.02550
$$994$$ 0 0
$$995$$ 10.0000 0.317021
$$996$$ 0 0
$$997$$ −14.3845 −0.455561 −0.227780 0.973713i $$-0.573147\pi$$
−0.227780 + 0.973713i $$0.573147\pi$$
$$998$$ 0 0
$$999$$ −0.630683 −0.0199539
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 7360.2.a.bi.1.1 2
4.3 odd 2 7360.2.a.bo.1.2 2
8.3 odd 2 1840.2.a.m.1.1 2
8.5 even 2 460.2.a.e.1.2 2
24.5 odd 2 4140.2.a.m.1.1 2
40.13 odd 4 2300.2.c.h.1749.4 4
40.19 odd 2 9200.2.a.bv.1.2 2
40.29 even 2 2300.2.a.i.1.1 2
40.37 odd 4 2300.2.c.h.1749.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
460.2.a.e.1.2 2 8.5 even 2
1840.2.a.m.1.1 2 8.3 odd 2
2300.2.a.i.1.1 2 40.29 even 2
2300.2.c.h.1749.1 4 40.37 odd 4
2300.2.c.h.1749.4 4 40.13 odd 4
4140.2.a.m.1.1 2 24.5 odd 2
7360.2.a.bi.1.1 2 1.1 even 1 trivial
7360.2.a.bo.1.2 2 4.3 odd 2
9200.2.a.bv.1.2 2 40.19 odd 2