# Properties

 Label 7360.2.a.by.1.3 Level $7360$ Weight $2$ Character 7360.1 Self dual yes Analytic conductor $58.770$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$7360 = 2^{6} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7360.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$58.7698958877$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.2597.1 Defining polynomial: $$x^{3} - x^{2} - 9x + 8$$ x^3 - x^2 - 9*x + 8 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 920) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$-2.95759$$ of defining polynomial Character $$\chi$$ $$=$$ 7360.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.95759 q^{3} -1.00000 q^{5} +3.95759 q^{7} +5.74732 q^{9} +O(q^{10})$$ $$q+2.95759 q^{3} -1.00000 q^{5} +3.95759 q^{7} +5.74732 q^{9} +0.957587 q^{11} +2.74732 q^{13} -2.95759 q^{15} +5.74732 q^{17} +6.74732 q^{19} +11.7049 q^{21} +1.00000 q^{23} +1.00000 q^{25} +8.12544 q^{27} -5.21027 q^{29} -5.95759 q^{31} +2.83215 q^{33} -3.95759 q^{35} -9.12544 q^{37} +8.12544 q^{39} +0.252679 q^{41} -8.00000 q^{43} -5.74732 q^{45} +5.49464 q^{47} +8.66249 q^{49} +16.9982 q^{51} +7.12544 q^{53} -0.957587 q^{55} +19.9558 q^{57} +4.78973 q^{59} -12.4522 q^{61} +22.7455 q^{63} -2.74732 q^{65} -9.12544 q^{67} +2.95759 q^{69} +1.66249 q^{71} +12.3357 q^{73} +2.95759 q^{75} +3.78973 q^{77} +11.8303 q^{79} +6.78973 q^{81} -0.704908 q^{83} -5.74732 q^{85} -15.4098 q^{87} -15.8303 q^{89} +10.8728 q^{91} -17.6201 q^{93} -6.74732 q^{95} -10.8728 q^{97} +5.50356 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - q^{3} - 3 q^{5} + 2 q^{7} + 10 q^{9}+O(q^{10})$$ 3 * q - q^3 - 3 * q^5 + 2 * q^7 + 10 * q^9 $$3 q - q^{3} - 3 q^{5} + 2 q^{7} + 10 q^{9} - 7 q^{11} + q^{13} + q^{15} + 10 q^{17} + 13 q^{19} + 18 q^{21} + 3 q^{23} + 3 q^{25} + 2 q^{27} - 13 q^{29} - 8 q^{31} + 21 q^{33} - 2 q^{35} - 5 q^{37} + 2 q^{39} + 8 q^{41} - 24 q^{43} - 10 q^{45} + 2 q^{47} - q^{49} - q^{51} - q^{53} + 7 q^{55} - 2 q^{57} + 17 q^{59} - 13 q^{61} + 9 q^{63} - q^{65} - 5 q^{67} - q^{69} - 22 q^{71} + 12 q^{73} - q^{75} + 14 q^{77} - 4 q^{79} + 23 q^{81} + 15 q^{83} - 10 q^{85} - 12 q^{87} - 8 q^{89} + 3 q^{91} - 16 q^{93} - 13 q^{95} - 3 q^{97} - 21 q^{99}+O(q^{100})$$ 3 * q - q^3 - 3 * q^5 + 2 * q^7 + 10 * q^9 - 7 * q^11 + q^13 + q^15 + 10 * q^17 + 13 * q^19 + 18 * q^21 + 3 * q^23 + 3 * q^25 + 2 * q^27 - 13 * q^29 - 8 * q^31 + 21 * q^33 - 2 * q^35 - 5 * q^37 + 2 * q^39 + 8 * q^41 - 24 * q^43 - 10 * q^45 + 2 * q^47 - q^49 - q^51 - q^53 + 7 * q^55 - 2 * q^57 + 17 * q^59 - 13 * q^61 + 9 * q^63 - q^65 - 5 * q^67 - q^69 - 22 * q^71 + 12 * q^73 - q^75 + 14 * q^77 - 4 * q^79 + 23 * q^81 + 15 * q^83 - 10 * q^85 - 12 * q^87 - 8 * q^89 + 3 * q^91 - 16 * q^93 - 13 * q^95 - 3 * q^97 - 21 * 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
$$3$$ 2.95759 1.70756 0.853782 0.520631i $$-0.174303\pi$$
0.853782 + 0.520631i $$0.174303\pi$$
$$4$$ 0 0
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ 3.95759 1.49583 0.747914 0.663796i $$-0.231056\pi$$
0.747914 + 0.663796i $$0.231056\pi$$
$$8$$ 0 0
$$9$$ 5.74732 1.91577
$$10$$ 0 0
$$11$$ 0.957587 0.288723 0.144362 0.989525i $$-0.453887\pi$$
0.144362 + 0.989525i $$0.453887\pi$$
$$12$$ 0 0
$$13$$ 2.74732 0.761970 0.380985 0.924581i $$-0.375585\pi$$
0.380985 + 0.924581i $$0.375585\pi$$
$$14$$ 0 0
$$15$$ −2.95759 −0.763646
$$16$$ 0 0
$$17$$ 5.74732 1.39393 0.696965 0.717105i $$-0.254533\pi$$
0.696965 + 0.717105i $$0.254533\pi$$
$$18$$ 0 0
$$19$$ 6.74732 1.54794 0.773971 0.633221i $$-0.218268\pi$$
0.773971 + 0.633221i $$0.218268\pi$$
$$20$$ 0 0
$$21$$ 11.7049 2.55422
$$22$$ 0 0
$$23$$ 1.00000 0.208514
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 8.12544 1.56374
$$28$$ 0 0
$$29$$ −5.21027 −0.967522 −0.483761 0.875200i $$-0.660730\pi$$
−0.483761 + 0.875200i $$0.660730\pi$$
$$30$$ 0 0
$$31$$ −5.95759 −1.07001 −0.535007 0.844848i $$-0.679691\pi$$
−0.535007 + 0.844848i $$0.679691\pi$$
$$32$$ 0 0
$$33$$ 2.83215 0.493013
$$34$$ 0 0
$$35$$ −3.95759 −0.668954
$$36$$ 0 0
$$37$$ −9.12544 −1.50021 −0.750107 0.661317i $$-0.769998\pi$$
−0.750107 + 0.661317i $$0.769998\pi$$
$$38$$ 0 0
$$39$$ 8.12544 1.30111
$$40$$ 0 0
$$41$$ 0.252679 0.0394619 0.0197309 0.999805i $$-0.493719\pi$$
0.0197309 + 0.999805i $$0.493719\pi$$
$$42$$ 0 0
$$43$$ −8.00000 −1.21999 −0.609994 0.792406i $$-0.708828\pi$$
−0.609994 + 0.792406i $$0.708828\pi$$
$$44$$ 0 0
$$45$$ −5.74732 −0.856760
$$46$$ 0 0
$$47$$ 5.49464 0.801476 0.400738 0.916193i $$-0.368754\pi$$
0.400738 + 0.916193i $$0.368754\pi$$
$$48$$ 0 0
$$49$$ 8.66249 1.23750
$$50$$ 0 0
$$51$$ 16.9982 2.38022
$$52$$ 0 0
$$53$$ 7.12544 0.978754 0.489377 0.872072i $$-0.337224\pi$$
0.489377 + 0.872072i $$0.337224\pi$$
$$54$$ 0 0
$$55$$ −0.957587 −0.129121
$$56$$ 0 0
$$57$$ 19.9558 2.64321
$$58$$ 0 0
$$59$$ 4.78973 0.623570 0.311785 0.950153i $$-0.399073\pi$$
0.311785 + 0.950153i $$0.399073\pi$$
$$60$$ 0 0
$$61$$ −12.4522 −1.59434 −0.797172 0.603752i $$-0.793672\pi$$
−0.797172 + 0.603752i $$0.793672\pi$$
$$62$$ 0 0
$$63$$ 22.7455 2.86567
$$64$$ 0 0
$$65$$ −2.74732 −0.340763
$$66$$ 0 0
$$67$$ −9.12544 −1.11485 −0.557425 0.830227i $$-0.688211\pi$$
−0.557425 + 0.830227i $$0.688211\pi$$
$$68$$ 0 0
$$69$$ 2.95759 0.356052
$$70$$ 0 0
$$71$$ 1.66249 0.197302 0.0986509 0.995122i $$-0.468547\pi$$
0.0986509 + 0.995122i $$0.468547\pi$$
$$72$$ 0 0
$$73$$ 12.3357 1.44379 0.721893 0.692005i $$-0.243272\pi$$
0.721893 + 0.692005i $$0.243272\pi$$
$$74$$ 0 0
$$75$$ 2.95759 0.341513
$$76$$ 0 0
$$77$$ 3.78973 0.431880
$$78$$ 0 0
$$79$$ 11.8303 1.33102 0.665509 0.746390i $$-0.268214\pi$$
0.665509 + 0.746390i $$0.268214\pi$$
$$80$$ 0 0
$$81$$ 6.78973 0.754415
$$82$$ 0 0
$$83$$ −0.704908 −0.0773737 −0.0386868 0.999251i $$-0.512317\pi$$
−0.0386868 + 0.999251i $$0.512317\pi$$
$$84$$ 0 0
$$85$$ −5.74732 −0.623384
$$86$$ 0 0
$$87$$ −15.4098 −1.65211
$$88$$ 0 0
$$89$$ −15.8303 −1.67801 −0.839007 0.544121i $$-0.816863\pi$$
−0.839007 + 0.544121i $$0.816863\pi$$
$$90$$ 0 0
$$91$$ 10.8728 1.13978
$$92$$ 0 0
$$93$$ −17.6201 −1.82712
$$94$$ 0 0
$$95$$ −6.74732 −0.692261
$$96$$ 0 0
$$97$$ −10.8728 −1.10396 −0.551981 0.833857i $$-0.686128\pi$$
−0.551981 + 0.833857i $$0.686128\pi$$
$$98$$ 0 0
$$99$$ 5.50356 0.553129
$$100$$ 0 0
$$101$$ −9.21027 −0.916456 −0.458228 0.888835i $$-0.651516\pi$$
−0.458228 + 0.888835i $$0.651516\pi$$
$$102$$ 0 0
$$103$$ 12.4522 1.22695 0.613477 0.789712i $$-0.289770\pi$$
0.613477 + 0.789712i $$0.289770\pi$$
$$104$$ 0 0
$$105$$ −11.7049 −1.14228
$$106$$ 0 0
$$107$$ −3.63080 −0.351003 −0.175501 0.984479i $$-0.556155\pi$$
−0.175501 + 0.984479i $$0.556155\pi$$
$$108$$ 0 0
$$109$$ −16.6625 −1.59598 −0.797989 0.602672i $$-0.794103\pi$$
−0.797989 + 0.602672i $$0.794103\pi$$
$$110$$ 0 0
$$111$$ −26.9893 −2.56171
$$112$$ 0 0
$$113$$ −18.1147 −1.70409 −0.852045 0.523469i $$-0.824638\pi$$
−0.852045 + 0.523469i $$0.824638\pi$$
$$114$$ 0 0
$$115$$ −1.00000 −0.0932505
$$116$$ 0 0
$$117$$ 15.7897 1.45976
$$118$$ 0 0
$$119$$ 22.7455 2.08508
$$120$$ 0 0
$$121$$ −10.0830 −0.916639
$$122$$ 0 0
$$123$$ 0.747321 0.0673836
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ −5.91517 −0.524887 −0.262443 0.964947i $$-0.584528\pi$$
−0.262443 + 0.964947i $$0.584528\pi$$
$$128$$ 0 0
$$129$$ −23.6607 −2.08321
$$130$$ 0 0
$$131$$ −3.40982 −0.297917 −0.148958 0.988843i $$-0.547592\pi$$
−0.148958 + 0.988843i $$0.547592\pi$$
$$132$$ 0 0
$$133$$ 26.7031 2.31545
$$134$$ 0 0
$$135$$ −8.12544 −0.699327
$$136$$ 0 0
$$137$$ −1.67321 −0.142952 −0.0714761 0.997442i $$-0.522771\pi$$
−0.0714761 + 0.997442i $$0.522771\pi$$
$$138$$ 0 0
$$139$$ −8.62008 −0.731146 −0.365573 0.930783i $$-0.619127\pi$$
−0.365573 + 0.930783i $$0.619127\pi$$
$$140$$ 0 0
$$141$$ 16.2509 1.36857
$$142$$ 0 0
$$143$$ 2.63080 0.219998
$$144$$ 0 0
$$145$$ 5.21027 0.432689
$$146$$ 0 0
$$147$$ 25.6201 2.11311
$$148$$ 0 0
$$149$$ 23.0830 1.89104 0.945518 0.325571i $$-0.105556\pi$$
0.945518 + 0.325571i $$0.105556\pi$$
$$150$$ 0 0
$$151$$ 10.7879 0.877910 0.438955 0.898509i $$-0.355349\pi$$
0.438955 + 0.898509i $$0.355349\pi$$
$$152$$ 0 0
$$153$$ 33.0317 2.67045
$$154$$ 0 0
$$155$$ 5.95759 0.478525
$$156$$ 0 0
$$157$$ −8.19955 −0.654395 −0.327198 0.944956i $$-0.606104\pi$$
−0.327198 + 0.944956i $$0.606104\pi$$
$$158$$ 0 0
$$159$$ 21.0741 1.67129
$$160$$ 0 0
$$161$$ 3.95759 0.311902
$$162$$ 0 0
$$163$$ −16.6625 −1.30511 −0.652554 0.757743i $$-0.726302\pi$$
−0.652554 + 0.757743i $$0.726302\pi$$
$$164$$ 0 0
$$165$$ −2.83215 −0.220482
$$166$$ 0 0
$$167$$ −8.00000 −0.619059 −0.309529 0.950890i $$-0.600171\pi$$
−0.309529 + 0.950890i $$0.600171\pi$$
$$168$$ 0 0
$$169$$ −5.45223 −0.419402
$$170$$ 0 0
$$171$$ 38.7790 2.96551
$$172$$ 0 0
$$173$$ −8.87276 −0.674584 −0.337292 0.941400i $$-0.609511\pi$$
−0.337292 + 0.941400i $$0.609511\pi$$
$$174$$ 0 0
$$175$$ 3.95759 0.299165
$$176$$ 0 0
$$177$$ 14.1661 1.06479
$$178$$ 0 0
$$179$$ −1.49464 −0.111715 −0.0558574 0.998439i $$-0.517789\pi$$
−0.0558574 + 0.998439i $$0.517789\pi$$
$$180$$ 0 0
$$181$$ 12.5777 0.934891 0.467445 0.884022i $$-0.345174\pi$$
0.467445 + 0.884022i $$0.345174\pi$$
$$182$$ 0 0
$$183$$ −36.8285 −2.72244
$$184$$ 0 0
$$185$$ 9.12544 0.670916
$$186$$ 0 0
$$187$$ 5.50356 0.402460
$$188$$ 0 0
$$189$$ 32.1571 2.33909
$$190$$ 0 0
$$191$$ 16.9045 1.22316 0.611582 0.791181i $$-0.290534\pi$$
0.611582 + 0.791181i $$0.290534\pi$$
$$192$$ 0 0
$$193$$ 9.91517 0.713710 0.356855 0.934160i $$-0.383849\pi$$
0.356855 + 0.934160i $$0.383849\pi$$
$$194$$ 0 0
$$195$$ −8.12544 −0.581875
$$196$$ 0 0
$$197$$ −4.53705 −0.323252 −0.161626 0.986852i $$-0.551674\pi$$
−0.161626 + 0.986852i $$0.551674\pi$$
$$198$$ 0 0
$$199$$ −14.0000 −0.992434 −0.496217 0.868199i $$-0.665278\pi$$
−0.496217 + 0.868199i $$0.665278\pi$$
$$200$$ 0 0
$$201$$ −26.9893 −1.90368
$$202$$ 0 0
$$203$$ −20.6201 −1.44725
$$204$$ 0 0
$$205$$ −0.252679 −0.0176479
$$206$$ 0 0
$$207$$ 5.74732 0.399466
$$208$$ 0 0
$$209$$ 6.46115 0.446927
$$210$$ 0 0
$$211$$ −8.70491 −0.599271 −0.299635 0.954054i $$-0.596865\pi$$
−0.299635 + 0.954054i $$0.596865\pi$$
$$212$$ 0 0
$$213$$ 4.91697 0.336905
$$214$$ 0 0
$$215$$ 8.00000 0.545595
$$216$$ 0 0
$$217$$ −23.5777 −1.60056
$$218$$ 0 0
$$219$$ 36.4839 2.46536
$$220$$ 0 0
$$221$$ 15.7897 1.06213
$$222$$ 0 0
$$223$$ −5.40982 −0.362268 −0.181134 0.983458i $$-0.557977\pi$$
−0.181134 + 0.983458i $$0.557977\pi$$
$$224$$ 0 0
$$225$$ 5.74732 0.383155
$$226$$ 0 0
$$227$$ 17.3250 1.14990 0.574950 0.818189i $$-0.305022\pi$$
0.574950 + 0.818189i $$0.305022\pi$$
$$228$$ 0 0
$$229$$ 11.4098 0.753982 0.376991 0.926217i $$-0.376959\pi$$
0.376991 + 0.926217i $$0.376959\pi$$
$$230$$ 0 0
$$231$$ 11.2085 0.737463
$$232$$ 0 0
$$233$$ 2.08483 0.136581 0.0682907 0.997665i $$-0.478245\pi$$
0.0682907 + 0.997665i $$0.478245\pi$$
$$234$$ 0 0
$$235$$ −5.49464 −0.358431
$$236$$ 0 0
$$237$$ 34.9893 2.27280
$$238$$ 0 0
$$239$$ −18.5353 −1.19895 −0.599473 0.800395i $$-0.704623\pi$$
−0.599473 + 0.800395i $$0.704623\pi$$
$$240$$ 0 0
$$241$$ 10.2509 0.660317 0.330159 0.943925i $$-0.392898\pi$$
0.330159 + 0.943925i $$0.392898\pi$$
$$242$$ 0 0
$$243$$ −4.29509 −0.275530
$$244$$ 0 0
$$245$$ −8.66249 −0.553426
$$246$$ 0 0
$$247$$ 18.5371 1.17948
$$248$$ 0 0
$$249$$ −2.08483 −0.132120
$$250$$ 0 0
$$251$$ 17.2527 1.08898 0.544490 0.838768i $$-0.316723\pi$$
0.544490 + 0.838768i $$0.316723\pi$$
$$252$$ 0 0
$$253$$ 0.957587 0.0602030
$$254$$ 0 0
$$255$$ −16.9982 −1.06447
$$256$$ 0 0
$$257$$ 28.9045 1.80301 0.901505 0.432768i $$-0.142463\pi$$
0.901505 + 0.432768i $$0.142463\pi$$
$$258$$ 0 0
$$259$$ −36.1147 −2.24406
$$260$$ 0 0
$$261$$ −29.9451 −1.85355
$$262$$ 0 0
$$263$$ 5.24196 0.323233 0.161617 0.986854i $$-0.448329\pi$$
0.161617 + 0.986854i $$0.448329\pi$$
$$264$$ 0 0
$$265$$ −7.12544 −0.437712
$$266$$ 0 0
$$267$$ −46.8196 −2.86531
$$268$$ 0 0
$$269$$ 5.37992 0.328019 0.164010 0.986459i $$-0.447557\pi$$
0.164010 + 0.986459i $$0.447557\pi$$
$$270$$ 0 0
$$271$$ −11.5371 −0.700826 −0.350413 0.936595i $$-0.613959\pi$$
−0.350413 + 0.936595i $$0.613959\pi$$
$$272$$ 0 0
$$273$$ 32.1571 1.94624
$$274$$ 0 0
$$275$$ 0.957587 0.0577447
$$276$$ 0 0
$$277$$ −5.83035 −0.350312 −0.175156 0.984541i $$-0.556043\pi$$
−0.175156 + 0.984541i $$0.556043\pi$$
$$278$$ 0 0
$$279$$ −34.2402 −2.04990
$$280$$ 0 0
$$281$$ 11.0741 0.660626 0.330313 0.943871i $$-0.392846\pi$$
0.330313 + 0.943871i $$0.392846\pi$$
$$282$$ 0 0
$$283$$ 3.86384 0.229682 0.114841 0.993384i $$-0.463364\pi$$
0.114841 + 0.993384i $$0.463364\pi$$
$$284$$ 0 0
$$285$$ −19.9558 −1.18208
$$286$$ 0 0
$$287$$ 1.00000 0.0590281
$$288$$ 0 0
$$289$$ 16.0317 0.943041
$$290$$ 0 0
$$291$$ −32.1571 −1.88508
$$292$$ 0 0
$$293$$ 1.54597 0.0903167 0.0451583 0.998980i $$-0.485621\pi$$
0.0451583 + 0.998980i $$0.485621\pi$$
$$294$$ 0 0
$$295$$ −4.78973 −0.278869
$$296$$ 0 0
$$297$$ 7.78082 0.451489
$$298$$ 0 0
$$299$$ 2.74732 0.158882
$$300$$ 0 0
$$301$$ −31.6607 −1.82489
$$302$$ 0 0
$$303$$ −27.2402 −1.56491
$$304$$ 0 0
$$305$$ 12.4522 0.713013
$$306$$ 0 0
$$307$$ 6.45223 0.368248 0.184124 0.982903i $$-0.441055\pi$$
0.184124 + 0.982903i $$0.441055\pi$$
$$308$$ 0 0
$$309$$ 36.8285 2.09510
$$310$$ 0 0
$$311$$ 22.8196 1.29398 0.646991 0.762497i $$-0.276027\pi$$
0.646991 + 0.762497i $$0.276027\pi$$
$$312$$ 0 0
$$313$$ −16.8620 −0.953099 −0.476550 0.879148i $$-0.658113\pi$$
−0.476550 + 0.879148i $$0.658113\pi$$
$$314$$ 0 0
$$315$$ −22.7455 −1.28156
$$316$$ 0 0
$$317$$ 1.33751 0.0751218 0.0375609 0.999294i $$-0.488041\pi$$
0.0375609 + 0.999294i $$0.488041\pi$$
$$318$$ 0 0
$$319$$ −4.98928 −0.279346
$$320$$ 0 0
$$321$$ −10.7384 −0.599359
$$322$$ 0 0
$$323$$ 38.7790 2.15772
$$324$$ 0 0
$$325$$ 2.74732 0.152394
$$326$$ 0 0
$$327$$ −49.2808 −2.72523
$$328$$ 0 0
$$329$$ 21.7455 1.19887
$$330$$ 0 0
$$331$$ 32.2808 1.77431 0.887156 0.461470i $$-0.152678\pi$$
0.887156 + 0.461470i $$0.152678\pi$$
$$332$$ 0 0
$$333$$ −52.4468 −2.87407
$$334$$ 0 0
$$335$$ 9.12544 0.498576
$$336$$ 0 0
$$337$$ −4.83215 −0.263224 −0.131612 0.991301i $$-0.542015\pi$$
−0.131612 + 0.991301i $$0.542015\pi$$
$$338$$ 0 0
$$339$$ −53.5759 −2.90984
$$340$$ 0 0
$$341$$ −5.70491 −0.308938
$$342$$ 0 0
$$343$$ 6.57947 0.355258
$$344$$ 0 0
$$345$$ −2.95759 −0.159231
$$346$$ 0 0
$$347$$ 34.6183 1.85841 0.929203 0.369569i $$-0.120495\pi$$
0.929203 + 0.369569i $$0.120495\pi$$
$$348$$ 0 0
$$349$$ −26.9558 −1.44291 −0.721455 0.692461i $$-0.756526\pi$$
−0.721455 + 0.692461i $$0.756526\pi$$
$$350$$ 0 0
$$351$$ 22.3232 1.19152
$$352$$ 0 0
$$353$$ −8.00000 −0.425797 −0.212899 0.977074i $$-0.568290\pi$$
−0.212899 + 0.977074i $$0.568290\pi$$
$$354$$ 0 0
$$355$$ −1.66249 −0.0882361
$$356$$ 0 0
$$357$$ 67.2719 3.56040
$$358$$ 0 0
$$359$$ 0.420532 0.0221949 0.0110974 0.999938i $$-0.496468\pi$$
0.0110974 + 0.999938i $$0.496468\pi$$
$$360$$ 0 0
$$361$$ 26.5263 1.39612
$$362$$ 0 0
$$363$$ −29.8214 −1.56522
$$364$$ 0 0
$$365$$ −12.3357 −0.645680
$$366$$ 0 0
$$367$$ 11.7156 0.611551 0.305775 0.952104i $$-0.401084\pi$$
0.305775 + 0.952104i $$0.401084\pi$$
$$368$$ 0 0
$$369$$ 1.45223 0.0756000
$$370$$ 0 0
$$371$$ 28.1995 1.46405
$$372$$ 0 0
$$373$$ 28.1661 1.45838 0.729192 0.684310i $$-0.239896\pi$$
0.729192 + 0.684310i $$0.239896\pi$$
$$374$$ 0 0
$$375$$ −2.95759 −0.152729
$$376$$ 0 0
$$377$$ −14.3143 −0.737223
$$378$$ 0 0
$$379$$ −27.3781 −1.40632 −0.703160 0.711032i $$-0.748228\pi$$
−0.703160 + 0.711032i $$0.748228\pi$$
$$380$$ 0 0
$$381$$ −17.4946 −0.896278
$$382$$ 0 0
$$383$$ −31.7754 −1.62365 −0.811824 0.583902i $$-0.801525\pi$$
−0.811824 + 0.583902i $$0.801525\pi$$
$$384$$ 0 0
$$385$$ −3.78973 −0.193143
$$386$$ 0 0
$$387$$ −45.9786 −2.33722
$$388$$ 0 0
$$389$$ 18.7031 0.948285 0.474143 0.880448i $$-0.342758\pi$$
0.474143 + 0.880448i $$0.342758\pi$$
$$390$$ 0 0
$$391$$ 5.74732 0.290655
$$392$$ 0 0
$$393$$ −10.0848 −0.508712
$$394$$ 0 0
$$395$$ −11.8303 −0.595249
$$396$$ 0 0
$$397$$ 39.6924 1.99210 0.996052 0.0887714i $$-0.0282941\pi$$
0.996052 + 0.0887714i $$0.0282941\pi$$
$$398$$ 0 0
$$399$$ 78.9768 3.95378
$$400$$ 0 0
$$401$$ −11.3250 −0.565543 −0.282771 0.959187i $$-0.591254\pi$$
−0.282771 + 0.959187i $$0.591254\pi$$
$$402$$ 0 0
$$403$$ −16.3674 −0.815318
$$404$$ 0 0
$$405$$ −6.78973 −0.337385
$$406$$ 0 0
$$407$$ −8.73840 −0.433147
$$408$$ 0 0
$$409$$ 9.45223 0.467383 0.233691 0.972311i $$-0.424919\pi$$
0.233691 + 0.972311i $$0.424919\pi$$
$$410$$ 0 0
$$411$$ −4.94867 −0.244100
$$412$$ 0 0
$$413$$ 18.9558 0.932753
$$414$$ 0 0
$$415$$ 0.704908 0.0346026
$$416$$ 0 0
$$417$$ −25.4946 −1.24848
$$418$$ 0 0
$$419$$ 33.2402 1.62389 0.811944 0.583735i $$-0.198409\pi$$
0.811944 + 0.583735i $$0.198409\pi$$
$$420$$ 0 0
$$421$$ 38.8285 1.89239 0.946194 0.323600i $$-0.104893\pi$$
0.946194 + 0.323600i $$0.104893\pi$$
$$422$$ 0 0
$$423$$ 31.5795 1.53545
$$424$$ 0 0
$$425$$ 5.74732 0.278786
$$426$$ 0 0
$$427$$ −49.2808 −2.38486
$$428$$ 0 0
$$429$$ 7.78082 0.375661
$$430$$ 0 0
$$431$$ −12.4839 −0.601329 −0.300665 0.953730i $$-0.597209\pi$$
−0.300665 + 0.953730i $$0.597209\pi$$
$$432$$ 0 0
$$433$$ −5.78793 −0.278150 −0.139075 0.990282i $$-0.544413\pi$$
−0.139075 + 0.990282i $$0.544413\pi$$
$$434$$ 0 0
$$435$$ 15.4098 0.738844
$$436$$ 0 0
$$437$$ 6.74732 0.322768
$$438$$ 0 0
$$439$$ −11.9241 −0.569106 −0.284553 0.958660i $$-0.591845\pi$$
−0.284553 + 0.958660i $$0.591845\pi$$
$$440$$ 0 0
$$441$$ 49.7861 2.37077
$$442$$ 0 0
$$443$$ −39.3973 −1.87182 −0.935911 0.352236i $$-0.885421\pi$$
−0.935911 + 0.352236i $$0.885421\pi$$
$$444$$ 0 0
$$445$$ 15.8303 0.750430
$$446$$ 0 0
$$447$$ 68.2701 3.22906
$$448$$ 0 0
$$449$$ 21.3674 1.00839 0.504195 0.863590i $$-0.331789\pi$$
0.504195 + 0.863590i $$0.331789\pi$$
$$450$$ 0 0
$$451$$ 0.241962 0.0113936
$$452$$ 0 0
$$453$$ 31.9063 1.49909
$$454$$ 0 0
$$455$$ −10.8728 −0.509723
$$456$$ 0 0
$$457$$ −20.3656 −0.952663 −0.476331 0.879266i $$-0.658034\pi$$
−0.476331 + 0.879266i $$0.658034\pi$$
$$458$$ 0 0
$$459$$ 46.6995 2.17975
$$460$$ 0 0
$$461$$ 27.4946 1.28055 0.640277 0.768144i $$-0.278820\pi$$
0.640277 + 0.768144i $$0.278820\pi$$
$$462$$ 0 0
$$463$$ −16.8196 −0.781675 −0.390837 0.920460i $$-0.627814\pi$$
−0.390837 + 0.920460i $$0.627814\pi$$
$$464$$ 0 0
$$465$$ 17.6201 0.817112
$$466$$ 0 0
$$467$$ 24.4504 1.13143 0.565715 0.824601i $$-0.308600\pi$$
0.565715 + 0.824601i $$0.308600\pi$$
$$468$$ 0 0
$$469$$ −36.1147 −1.66762
$$470$$ 0 0
$$471$$ −24.2509 −1.11742
$$472$$ 0 0
$$473$$ −7.66070 −0.352239
$$474$$ 0 0
$$475$$ 6.74732 0.309588
$$476$$ 0 0
$$477$$ 40.9522 1.87507
$$478$$ 0 0
$$479$$ 10.5688 0.482899 0.241449 0.970413i $$-0.422377\pi$$
0.241449 + 0.970413i $$0.422377\pi$$
$$480$$ 0 0
$$481$$ −25.0705 −1.14312
$$482$$ 0 0
$$483$$ 11.7049 0.532592
$$484$$ 0 0
$$485$$ 10.8728 0.493707
$$486$$ 0 0
$$487$$ −13.0741 −0.592444 −0.296222 0.955119i $$-0.595727\pi$$
−0.296222 + 0.955119i $$0.595727\pi$$
$$488$$ 0 0
$$489$$ −49.2808 −2.22855
$$490$$ 0 0
$$491$$ −2.22098 −0.100232 −0.0501158 0.998743i $$-0.515959\pi$$
−0.0501158 + 0.998743i $$0.515959\pi$$
$$492$$ 0 0
$$493$$ −29.9451 −1.34866
$$494$$ 0 0
$$495$$ −5.50356 −0.247367
$$496$$ 0 0
$$497$$ 6.57947 0.295129
$$498$$ 0 0
$$499$$ −2.87456 −0.128683 −0.0643415 0.997928i $$-0.520495\pi$$
−0.0643415 + 0.997928i $$0.520495\pi$$
$$500$$ 0 0
$$501$$ −23.6607 −1.05708
$$502$$ 0 0
$$503$$ 9.36740 0.417672 0.208836 0.977951i $$-0.433033\pi$$
0.208836 + 0.977951i $$0.433033\pi$$
$$504$$ 0 0
$$505$$ 9.21027 0.409851
$$506$$ 0 0
$$507$$ −16.1254 −0.716156
$$508$$ 0 0
$$509$$ 40.5866 1.79897 0.899484 0.436953i $$-0.143942\pi$$
0.899484 + 0.436953i $$0.143942\pi$$
$$510$$ 0 0
$$511$$ 48.8196 2.15965
$$512$$ 0 0
$$513$$ 54.8250 2.42058
$$514$$ 0 0
$$515$$ −12.4522 −0.548711
$$516$$ 0 0
$$517$$ 5.26160 0.231405
$$518$$ 0 0
$$519$$ −26.2420 −1.15189
$$520$$ 0 0
$$521$$ 34.6714 1.51898 0.759491 0.650518i $$-0.225448\pi$$
0.759491 + 0.650518i $$0.225448\pi$$
$$522$$ 0 0
$$523$$ −35.4098 −1.54836 −0.774182 0.632964i $$-0.781838\pi$$
−0.774182 + 0.632964i $$0.781838\pi$$
$$524$$ 0 0
$$525$$ 11.7049 0.510844
$$526$$ 0 0
$$527$$ −34.2402 −1.49152
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ 27.5281 1.19462
$$532$$ 0 0
$$533$$ 0.694191 0.0300687
$$534$$ 0 0
$$535$$ 3.63080 0.156973
$$536$$ 0 0
$$537$$ −4.42053 −0.190760
$$538$$ 0 0
$$539$$ 8.29509 0.357295
$$540$$ 0 0
$$541$$ −37.0705 −1.59379 −0.796893 0.604121i $$-0.793525\pi$$
−0.796893 + 0.604121i $$0.793525\pi$$
$$542$$ 0 0
$$543$$ 37.1995 1.59639
$$544$$ 0 0
$$545$$ 16.6625 0.713743
$$546$$ 0 0
$$547$$ −21.4187 −0.915799 −0.457899 0.889004i $$-0.651398\pi$$
−0.457899 + 0.889004i $$0.651398\pi$$
$$548$$ 0 0
$$549$$ −71.5670 −3.05440
$$550$$ 0 0
$$551$$ −35.1553 −1.49767
$$552$$ 0 0
$$553$$ 46.8196 1.99097
$$554$$ 0 0
$$555$$ 26.9893 1.14563
$$556$$ 0 0
$$557$$ −20.9558 −0.887925 −0.443963 0.896045i $$-0.646428\pi$$
−0.443963 + 0.896045i $$0.646428\pi$$
$$558$$ 0 0
$$559$$ −21.9786 −0.929594
$$560$$ 0 0
$$561$$ 16.2773 0.687226
$$562$$ 0 0
$$563$$ −6.70491 −0.282578 −0.141289 0.989968i $$-0.545125\pi$$
−0.141289 + 0.989968i $$0.545125\pi$$
$$564$$ 0 0
$$565$$ 18.1147 0.762092
$$566$$ 0 0
$$567$$ 26.8710 1.12847
$$568$$ 0 0
$$569$$ 6.00000 0.251533 0.125767 0.992060i $$-0.459861\pi$$
0.125767 + 0.992060i $$0.459861\pi$$
$$570$$ 0 0
$$571$$ −13.2933 −0.556307 −0.278154 0.960537i $$-0.589722\pi$$
−0.278154 + 0.960537i $$0.589722\pi$$
$$572$$ 0 0
$$573$$ 49.9964 2.08863
$$574$$ 0 0
$$575$$ 1.00000 0.0417029
$$576$$ 0 0
$$577$$ 23.2402 0.967501 0.483750 0.875206i $$-0.339274\pi$$
0.483750 + 0.875206i $$0.339274\pi$$
$$578$$ 0 0
$$579$$ 29.3250 1.21870
$$580$$ 0 0
$$581$$ −2.78973 −0.115738
$$582$$ 0 0
$$583$$ 6.82323 0.282589
$$584$$ 0 0
$$585$$ −15.7897 −0.652825
$$586$$ 0 0
$$587$$ −17.6112 −0.726891 −0.363445 0.931616i $$-0.618400\pi$$
−0.363445 + 0.931616i $$0.618400\pi$$
$$588$$ 0 0
$$589$$ −40.1978 −1.65632
$$590$$ 0 0
$$591$$ −13.4187 −0.551973
$$592$$ 0 0
$$593$$ −41.5759 −1.70732 −0.853658 0.520834i $$-0.825621\pi$$
−0.853658 + 0.520834i $$0.825621\pi$$
$$594$$ 0 0
$$595$$ −22.7455 −0.932475
$$596$$ 0 0
$$597$$ −41.4062 −1.69464
$$598$$ 0 0
$$599$$ 17.7138 0.723767 0.361884 0.932223i $$-0.382134\pi$$
0.361884 + 0.932223i $$0.382134\pi$$
$$600$$ 0 0
$$601$$ −46.7772 −1.90808 −0.954041 0.299675i $$-0.903122\pi$$
−0.954041 + 0.299675i $$0.903122\pi$$
$$602$$ 0 0
$$603$$ −52.4468 −2.13580
$$604$$ 0 0
$$605$$ 10.0830 0.409933
$$606$$ 0 0
$$607$$ 42.9893 1.74488 0.872441 0.488720i $$-0.162536\pi$$
0.872441 + 0.488720i $$0.162536\pi$$
$$608$$ 0 0
$$609$$ −60.9857 −2.47126
$$610$$ 0 0
$$611$$ 15.0955 0.610700
$$612$$ 0 0
$$613$$ 32.6500 1.31872 0.659360 0.751827i $$-0.270827\pi$$
0.659360 + 0.751827i $$0.270827\pi$$
$$614$$ 0 0
$$615$$ −0.747321 −0.0301349
$$616$$ 0 0
$$617$$ −11.3232 −0.455854 −0.227927 0.973678i $$-0.573195\pi$$
−0.227927 + 0.973678i $$0.573195\pi$$
$$618$$ 0 0
$$619$$ 10.6625 0.428562 0.214281 0.976772i $$-0.431259\pi$$
0.214281 + 0.976772i $$0.431259\pi$$
$$620$$ 0 0
$$621$$ 8.12544 0.326063
$$622$$ 0 0
$$623$$ −62.6500 −2.51002
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 19.1094 0.763156
$$628$$ 0 0
$$629$$ −52.4468 −2.09119
$$630$$ 0 0
$$631$$ 5.24376 0.208751 0.104375 0.994538i $$-0.466716\pi$$
0.104375 + 0.994538i $$0.466716\pi$$
$$632$$ 0 0
$$633$$ −25.7455 −1.02329
$$634$$ 0 0
$$635$$ 5.91517 0.234737
$$636$$ 0 0
$$637$$ 23.7987 0.942937
$$638$$ 0 0
$$639$$ 9.55489 0.377986
$$640$$ 0 0
$$641$$ 30.4205 1.20154 0.600769 0.799422i $$-0.294861\pi$$
0.600769 + 0.799422i $$0.294861\pi$$
$$642$$ 0 0
$$643$$ 9.37992 0.369908 0.184954 0.982747i $$-0.440786\pi$$
0.184954 + 0.982747i $$0.440786\pi$$
$$644$$ 0 0
$$645$$ 23.6607 0.931639
$$646$$ 0 0
$$647$$ −18.6714 −0.734049 −0.367024 0.930211i $$-0.619623\pi$$
−0.367024 + 0.930211i $$0.619623\pi$$
$$648$$ 0 0
$$649$$ 4.58659 0.180039
$$650$$ 0 0
$$651$$ −69.7330 −2.73305
$$652$$ 0 0
$$653$$ −12.0723 −0.472426 −0.236213 0.971701i $$-0.575906\pi$$
−0.236213 + 0.971701i $$0.575906\pi$$
$$654$$ 0 0
$$655$$ 3.40982 0.133233
$$656$$ 0 0
$$657$$ 70.8973 2.76597
$$658$$ 0 0
$$659$$ −21.2402 −0.827399 −0.413700 0.910413i $$-0.635764\pi$$
−0.413700 + 0.910413i $$0.635764\pi$$
$$660$$ 0 0
$$661$$ 26.9362 1.04769 0.523847 0.851812i $$-0.324496\pi$$
0.523847 + 0.851812i $$0.324496\pi$$
$$662$$ 0 0
$$663$$ 46.6995 1.81366
$$664$$ 0 0
$$665$$ −26.7031 −1.03550
$$666$$ 0 0
$$667$$ −5.21027 −0.201742
$$668$$ 0 0
$$669$$ −16.0000 −0.618596
$$670$$ 0 0
$$671$$ −11.9241 −0.460324
$$672$$ 0 0
$$673$$ −11.4098 −0.439816 −0.219908 0.975521i $$-0.570576\pi$$
−0.219908 + 0.975521i $$0.570576\pi$$
$$674$$ 0 0
$$675$$ 8.12544 0.312748
$$676$$ 0 0
$$677$$ 11.8638 0.455965 0.227982 0.973665i $$-0.426787\pi$$
0.227982 + 0.973665i $$0.426787\pi$$
$$678$$ 0 0
$$679$$ −43.0299 −1.65134
$$680$$ 0 0
$$681$$ 51.2402 1.96353
$$682$$ 0 0
$$683$$ −10.0723 −0.385406 −0.192703 0.981257i $$-0.561725\pi$$
−0.192703 + 0.981257i $$0.561725\pi$$
$$684$$ 0 0
$$685$$ 1.67321 0.0639301
$$686$$ 0 0
$$687$$ 33.7455 1.28747
$$688$$ 0 0
$$689$$ 19.5759 0.745781
$$690$$ 0 0
$$691$$ 45.9964 1.74979 0.874893 0.484317i $$-0.160932\pi$$
0.874893 + 0.484317i $$0.160932\pi$$
$$692$$ 0 0
$$693$$ 21.7808 0.827385
$$694$$ 0 0
$$695$$ 8.62008 0.326978
$$696$$ 0 0
$$697$$ 1.45223 0.0550071
$$698$$ 0 0
$$699$$ 6.16605 0.233222
$$700$$ 0 0
$$701$$ 21.2312 0.801893 0.400947 0.916101i $$-0.368681\pi$$
0.400947 + 0.916101i $$0.368681\pi$$
$$702$$ 0 0
$$703$$ −61.5723 −2.32224
$$704$$ 0 0
$$705$$ −16.2509 −0.612044
$$706$$ 0 0
$$707$$ −36.4504 −1.37086
$$708$$ 0 0
$$709$$ −31.3781 −1.17843 −0.589215 0.807976i $$-0.700563\pi$$
−0.589215 + 0.807976i $$0.700563\pi$$
$$710$$ 0 0
$$711$$ 67.9928 2.54993
$$712$$ 0 0
$$713$$ −5.95759 −0.223113
$$714$$ 0 0
$$715$$ −2.63080 −0.0983863
$$716$$ 0 0
$$717$$ −54.8196 −2.04728
$$718$$ 0 0
$$719$$ −20.4629 −0.763139 −0.381570 0.924340i $$-0.624616\pi$$
−0.381570 + 0.924340i $$0.624616\pi$$
$$720$$ 0 0
$$721$$ 49.2808 1.83531
$$722$$ 0 0
$$723$$ 30.3179 1.12753
$$724$$ 0 0
$$725$$ −5.21027 −0.193504
$$726$$ 0 0
$$727$$ 14.9875 0.555855 0.277928 0.960602i $$-0.410352\pi$$
0.277928 + 0.960602i $$0.410352\pi$$
$$728$$ 0 0
$$729$$ −33.0723 −1.22490
$$730$$ 0 0
$$731$$ −45.9786 −1.70058
$$732$$ 0 0
$$733$$ 22.2844 0.823092 0.411546 0.911389i $$-0.364989\pi$$
0.411546 + 0.911389i $$0.364989\pi$$
$$734$$ 0 0
$$735$$ −25.6201 −0.945011
$$736$$ 0 0
$$737$$ −8.73840 −0.321883
$$738$$ 0 0
$$739$$ −30.9344 −1.13794 −0.568969 0.822359i $$-0.692658\pi$$
−0.568969 + 0.822359i $$0.692658\pi$$
$$740$$ 0 0
$$741$$ 54.8250 2.01404
$$742$$ 0 0
$$743$$ −44.8285 −1.64460 −0.822300 0.569054i $$-0.807309\pi$$
−0.822300 + 0.569054i $$0.807309\pi$$
$$744$$ 0 0
$$745$$ −23.0830 −0.845697
$$746$$ 0 0
$$747$$ −4.05133 −0.148230
$$748$$ 0 0
$$749$$ −14.3692 −0.525039
$$750$$ 0 0
$$751$$ 32.1661 1.17376 0.586878 0.809675i $$-0.300357\pi$$
0.586878 + 0.809675i $$0.300357\pi$$
$$752$$ 0 0
$$753$$ 51.0263 1.85950
$$754$$ 0 0
$$755$$ −10.7879 −0.392613
$$756$$ 0 0
$$757$$ −40.6835 −1.47867 −0.739333 0.673340i $$-0.764859\pi$$
−0.739333 + 0.673340i $$0.764859\pi$$
$$758$$ 0 0
$$759$$ 2.83215 0.102800
$$760$$ 0 0
$$761$$ 28.4415 1.03100 0.515502 0.856888i $$-0.327605\pi$$
0.515502 + 0.856888i $$0.327605\pi$$
$$762$$ 0 0
$$763$$ −65.9433 −2.38731
$$764$$ 0 0
$$765$$ −33.0317 −1.19426
$$766$$ 0 0
$$767$$ 13.1589 0.475142
$$768$$ 0 0
$$769$$ −40.4205 −1.45760 −0.728801 0.684726i $$-0.759922\pi$$
−0.728801 + 0.684726i $$0.759922\pi$$
$$770$$ 0 0
$$771$$ 85.4874 3.07876
$$772$$ 0 0
$$773$$ −24.5688 −0.883677 −0.441838 0.897095i $$-0.645673\pi$$
−0.441838 + 0.897095i $$0.645673\pi$$
$$774$$ 0 0
$$775$$ −5.95759 −0.214003
$$776$$ 0 0
$$777$$ −106.812 −3.83187
$$778$$ 0 0
$$779$$ 1.70491 0.0610847
$$780$$ 0 0
$$781$$ 1.59198 0.0569656
$$782$$ 0 0
$$783$$ −42.3357 −1.51295
$$784$$ 0 0
$$785$$ 8.19955 0.292654
$$786$$ 0 0
$$787$$ 11.7790 0.419877 0.209938 0.977715i $$-0.432674\pi$$
0.209938 + 0.977715i $$0.432674\pi$$
$$788$$ 0 0
$$789$$ 15.5036 0.551941
$$790$$ 0 0
$$791$$ −71.6906 −2.54902
$$792$$ 0 0
$$793$$ −34.2103 −1.21484
$$794$$ 0 0
$$795$$ −21.0741 −0.747422
$$796$$ 0 0
$$797$$ 13.8602 0.490955 0.245478 0.969402i $$-0.421055\pi$$
0.245478 + 0.969402i $$0.421055\pi$$
$$798$$ 0 0
$$799$$ 31.5795 1.11720
$$800$$ 0 0
$$801$$ −90.9821 −3.21469
$$802$$ 0 0
$$803$$ 11.8125 0.416854
$$804$$ 0 0
$$805$$ −3.95759 −0.139487
$$806$$ 0 0
$$807$$ 15.9116 0.560114
$$808$$ 0 0
$$809$$ −52.0830 −1.83114 −0.915571 0.402157i $$-0.868261\pi$$
−0.915571 + 0.402157i $$0.868261\pi$$
$$810$$ 0 0
$$811$$ 40.7013 1.42922 0.714608 0.699525i $$-0.246605\pi$$
0.714608 + 0.699525i $$0.246605\pi$$
$$812$$ 0 0
$$813$$ −34.1218 −1.19671
$$814$$ 0 0
$$815$$ 16.6625 0.583662
$$816$$ 0 0
$$817$$ −53.9786 −1.88847
$$818$$ 0 0
$$819$$ 62.4892 2.18355
$$820$$ 0 0
$$821$$ −10.0000 −0.349002 −0.174501 0.984657i $$-0.555831\pi$$
−0.174501 + 0.984657i $$0.555831\pi$$
$$822$$ 0 0
$$823$$ −42.9009 −1.49543 −0.747715 0.664020i $$-0.768849\pi$$
−0.747715 + 0.664020i $$0.768849\pi$$
$$824$$ 0 0
$$825$$ 2.83215 0.0986027
$$826$$ 0 0
$$827$$ −39.4397 −1.37145 −0.685727 0.727859i $$-0.740515\pi$$
−0.685727 + 0.727859i $$0.740515\pi$$
$$828$$ 0 0
$$829$$ −29.5460 −1.02617 −0.513087 0.858337i $$-0.671498\pi$$
−0.513087 + 0.858337i $$0.671498\pi$$
$$830$$ 0 0
$$831$$ −17.2438 −0.598179
$$832$$ 0 0
$$833$$ 49.7861 1.72499
$$834$$ 0 0
$$835$$ 8.00000 0.276851
$$836$$ 0 0
$$837$$ −48.4080 −1.67323
$$838$$ 0 0
$$839$$ −1.09554 −0.0378223 −0.0189112 0.999821i $$-0.506020\pi$$
−0.0189112 + 0.999821i $$0.506020\pi$$
$$840$$ 0 0
$$841$$ −1.85313 −0.0639009
$$842$$ 0 0
$$843$$ 32.7526 1.12806
$$844$$ 0 0
$$845$$ 5.45223 0.187562
$$846$$ 0 0
$$847$$ −39.9045 −1.37113
$$848$$ 0 0
$$849$$ 11.4277 0.392196
$$850$$ 0 0
$$851$$ −9.12544 −0.312816
$$852$$ 0 0
$$853$$ 10.0937 0.345603 0.172802 0.984957i $$-0.444718\pi$$
0.172802 + 0.984957i $$0.444718\pi$$
$$854$$ 0 0
$$855$$ −38.7790 −1.32621
$$856$$ 0 0
$$857$$ 41.8303 1.42890 0.714449 0.699688i $$-0.246678\pi$$
0.714449 + 0.699688i $$0.246678\pi$$
$$858$$ 0 0
$$859$$ 3.35848 0.114590 0.0572950 0.998357i $$-0.481752\pi$$
0.0572950 + 0.998357i $$0.481752\pi$$
$$860$$ 0 0
$$861$$ 2.95759 0.100794
$$862$$ 0 0
$$863$$ −3.83395 −0.130509 −0.0652545 0.997869i $$-0.520786\pi$$
−0.0652545 + 0.997869i $$0.520786\pi$$
$$864$$ 0 0
$$865$$ 8.87276 0.301683
$$866$$ 0 0
$$867$$ 47.4151 1.61030
$$868$$ 0 0
$$869$$ 11.3286 0.384296
$$870$$ 0 0
$$871$$ −25.0705 −0.849482
$$872$$ 0 0
$$873$$ −62.4892 −2.11494
$$874$$ 0 0
$$875$$ −3.95759 −0.133791
$$876$$ 0 0
$$877$$ −15.5263 −0.524287 −0.262144 0.965029i $$-0.584429\pi$$
−0.262144 + 0.965029i $$0.584429\pi$$
$$878$$ 0 0
$$879$$ 4.57235 0.154221
$$880$$ 0 0
$$881$$ 26.9893 0.909292 0.454646 0.890672i $$-0.349766\pi$$
0.454646 + 0.890672i $$0.349766\pi$$
$$882$$ 0 0
$$883$$ −21.5884 −0.726507 −0.363254 0.931690i $$-0.618334\pi$$
−0.363254 + 0.931690i $$0.618334\pi$$
$$884$$ 0 0
$$885$$ −14.1661 −0.476187
$$886$$ 0 0
$$887$$ 49.4098 1.65902 0.829510 0.558492i $$-0.188620\pi$$
0.829510 + 0.558492i $$0.188620\pi$$
$$888$$ 0 0
$$889$$ −23.4098 −0.785140
$$890$$ 0 0
$$891$$ 6.50176 0.217817
$$892$$ 0 0
$$893$$ 37.0741 1.24064
$$894$$ 0 0
$$895$$ 1.49464 0.0499604
$$896$$ 0 0
$$897$$ 8.12544 0.271301
$$898$$ 0 0
$$899$$ 31.0406 1.03526
$$900$$ 0 0
$$901$$ 40.9522 1.36432
$$902$$ 0 0
$$903$$ −93.6393 −3.11612
$$904$$ 0 0
$$905$$ −12.5777 −0.418096
$$906$$ 0 0
$$907$$ 9.54957 0.317088 0.158544 0.987352i $$-0.449320\pi$$
0.158544 + 0.987352i $$0.449320\pi$$
$$908$$ 0 0
$$909$$ −52.9344 −1.75572
$$910$$ 0 0
$$911$$ −47.3036 −1.56724 −0.783618 0.621243i $$-0.786628\pi$$
−0.783618 + 0.621243i $$0.786628\pi$$
$$912$$ 0 0
$$913$$ −0.675011 −0.0223396
$$914$$ 0 0
$$915$$ 36.8285 1.21751
$$916$$ 0 0
$$917$$ −13.4946 −0.445632
$$918$$ 0 0
$$919$$ 8.00000 0.263896 0.131948 0.991257i $$-0.457877\pi$$
0.131948 + 0.991257i $$0.457877\pi$$
$$920$$ 0 0
$$921$$ 19.0830 0.628807
$$922$$ 0 0
$$923$$ 4.56741 0.150338
$$924$$ 0 0
$$925$$ −9.12544 −0.300043
$$926$$ 0 0
$$927$$ 71.5670 2.35057
$$928$$ 0 0
$$929$$ −19.6942 −0.646145 −0.323073 0.946374i $$-0.604716\pi$$
−0.323073 + 0.946374i $$0.604716\pi$$
$$930$$ 0 0
$$931$$ 58.4486 1.91558
$$932$$ 0 0
$$933$$ 67.4910 2.20956
$$934$$ 0 0
$$935$$ −5.50356 −0.179986
$$936$$ 0 0
$$937$$ −31.8214 −1.03956 −0.519780 0.854300i $$-0.673986\pi$$
−0.519780 + 0.854300i $$0.673986\pi$$
$$938$$ 0 0
$$939$$ −49.8710 −1.62748
$$940$$ 0 0
$$941$$ 2.36740 0.0771751 0.0385876 0.999255i $$-0.487714\pi$$
0.0385876 + 0.999255i $$0.487714\pi$$
$$942$$ 0 0
$$943$$ 0.252679 0.00822837
$$944$$ 0 0
$$945$$ −32.1571 −1.04607
$$946$$ 0 0
$$947$$ 9.90266 0.321793 0.160897 0.986971i $$-0.448561\pi$$
0.160897 + 0.986971i $$0.448561\pi$$
$$948$$ 0 0
$$949$$ 33.8901 1.10012
$$950$$ 0 0
$$951$$ 3.95579 0.128275
$$952$$ 0 0
$$953$$ −31.5442 −1.02182 −0.510908 0.859635i $$-0.670691\pi$$
−0.510908 + 0.859635i $$0.670691\pi$$
$$954$$ 0 0
$$955$$ −16.9045 −0.547015
$$956$$ 0 0
$$957$$ −14.7562 −0.477001
$$958$$ 0 0
$$959$$ −6.62188 −0.213832
$$960$$ 0 0
$$961$$ 4.49284 0.144930
$$962$$ 0 0
$$963$$ −20.8674 −0.672441
$$964$$ 0 0
$$965$$ −9.91517 −0.319181
$$966$$ 0 0
$$967$$ −15.5973 −0.501575 −0.250788 0.968042i $$-0.580690\pi$$
−0.250788 + 0.968042i $$0.580690\pi$$
$$968$$ 0 0
$$969$$ 114.692 3.68445
$$970$$ 0 0
$$971$$ 47.1022 1.51158 0.755791 0.654813i $$-0.227253\pi$$
0.755791 + 0.654813i $$0.227253\pi$$
$$972$$ 0 0
$$973$$ −34.1147 −1.09367
$$974$$ 0 0
$$975$$ 8.12544 0.260222
$$976$$ 0 0
$$977$$ 28.9469 0.926092 0.463046 0.886334i $$-0.346756\pi$$
0.463046 + 0.886334i $$0.346756\pi$$
$$978$$ 0 0
$$979$$ −15.1589 −0.484482
$$980$$ 0 0
$$981$$ −95.7647 −3.05753
$$982$$ 0 0
$$983$$ −36.2312 −1.15560 −0.577799 0.816179i $$-0.696088\pi$$
−0.577799 + 0.816179i $$0.696088\pi$$
$$984$$ 0 0
$$985$$ 4.53705 0.144563
$$986$$ 0 0
$$987$$ 64.3143 2.04715
$$988$$ 0 0
$$989$$ −8.00000 −0.254385
$$990$$ 0 0
$$991$$ −35.1750 −1.11737 −0.558685 0.829380i $$-0.688694\pi$$
−0.558685 + 0.829380i $$0.688694\pi$$
$$992$$ 0 0
$$993$$ 95.4732 3.02975
$$994$$ 0 0
$$995$$ 14.0000 0.443830
$$996$$ 0 0
$$997$$ −19.2402 −0.609342 −0.304671 0.952458i $$-0.598547\pi$$
−0.304671 + 0.952458i $$0.598547\pi$$
$$998$$ 0 0
$$999$$ −74.1482 −2.34595
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.by.1.3 3
4.3 odd 2 7360.2.a.cc.1.1 3
8.3 odd 2 1840.2.a.s.1.3 3
8.5 even 2 920.2.a.h.1.1 3
24.5 odd 2 8280.2.a.bj.1.3 3
40.13 odd 4 4600.2.e.p.4049.2 6
40.19 odd 2 9200.2.a.ce.1.1 3
40.29 even 2 4600.2.a.x.1.3 3
40.37 odd 4 4600.2.e.p.4049.5 6

By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.h.1.1 3 8.5 even 2
1840.2.a.s.1.3 3 8.3 odd 2
4600.2.a.x.1.3 3 40.29 even 2
4600.2.e.p.4049.2 6 40.13 odd 4
4600.2.e.p.4049.5 6 40.37 odd 4
7360.2.a.by.1.3 3 1.1 even 1 trivial
7360.2.a.cc.1.1 3 4.3 odd 2
8280.2.a.bj.1.3 3 24.5 odd 2
9200.2.a.ce.1.1 3 40.19 odd 2