# Properties

 Label 9280.2.a.br.1.2 Level $9280$ Weight $2$ Character 9280.1 Self dual yes Analytic conductor $74.101$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9280,2,Mod(1,9280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9280, 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("9280.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9280 = 2^{6} \cdot 5 \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9280.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: $$74.1011730757$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.148.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 3x + 1$$ x^3 - x^2 - 3*x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 145) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-1.48119$$ of defining polynomial Character $$\chi$$ $$=$$ 9280.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+0.806063 q^{3} -1.00000 q^{5} -1.19394 q^{7} -2.35026 q^{9} +O(q^{10})$$ $$q+0.806063 q^{3} -1.00000 q^{5} -1.19394 q^{7} -2.35026 q^{9} +4.15633 q^{11} -2.96239 q^{13} -0.806063 q^{15} +5.50659 q^{17} -3.19394 q^{19} -0.962389 q^{21} -1.84367 q^{23} +1.00000 q^{25} -4.31265 q^{27} +1.00000 q^{29} +4.80606 q^{31} +3.35026 q^{33} +1.19394 q^{35} +9.50659 q^{37} -2.38787 q^{39} -11.2750 q^{41} -0.0303172 q^{43} +2.35026 q^{45} -4.80606 q^{47} -5.57452 q^{49} +4.43866 q^{51} +1.35026 q^{53} -4.15633 q^{55} -2.57452 q^{57} +13.2750 q^{59} -8.88717 q^{61} +2.80606 q^{63} +2.96239 q^{65} +5.84367 q^{67} -1.48612 q^{69} +1.27504 q^{71} -15.2447 q^{73} +0.806063 q^{75} -4.96239 q^{77} +4.93207 q^{79} +3.57452 q^{81} +4.41819 q^{83} -5.50659 q^{85} +0.806063 q^{87} -3.61213 q^{89} +3.53690 q^{91} +3.87399 q^{93} +3.19394 q^{95} -1.38058 q^{97} -9.76845 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 2 q^{3} - 3 q^{5} - 4 q^{7} + 3 q^{9}+O(q^{10})$$ 3 * q + 2 * q^3 - 3 * q^5 - 4 * q^7 + 3 * q^9 $$3 q + 2 q^{3} - 3 q^{5} - 4 q^{7} + 3 q^{9} + 2 q^{11} + 2 q^{13} - 2 q^{15} - 4 q^{17} - 10 q^{19} + 8 q^{21} - 16 q^{23} + 3 q^{25} + 8 q^{27} + 3 q^{29} + 14 q^{31} + 4 q^{35} + 8 q^{37} - 8 q^{39} - 2 q^{41} + 2 q^{43} - 3 q^{45} - 14 q^{47} - 5 q^{49} - 16 q^{51} - 6 q^{53} - 2 q^{55} + 4 q^{57} + 8 q^{59} + 6 q^{61} + 8 q^{63} - 2 q^{65} + 28 q^{67} - 12 q^{69} - 28 q^{71} - 16 q^{73} + 2 q^{75} - 4 q^{77} + 6 q^{79} - q^{81} + 12 q^{83} + 4 q^{85} + 2 q^{87} - 10 q^{89} - 12 q^{91} + 20 q^{93} + 10 q^{95} + 8 q^{97} - 18 q^{99}+O(q^{100})$$ 3 * q + 2 * q^3 - 3 * q^5 - 4 * q^7 + 3 * q^9 + 2 * q^11 + 2 * q^13 - 2 * q^15 - 4 * q^17 - 10 * q^19 + 8 * q^21 - 16 * q^23 + 3 * q^25 + 8 * q^27 + 3 * q^29 + 14 * q^31 + 4 * q^35 + 8 * q^37 - 8 * q^39 - 2 * q^41 + 2 * q^43 - 3 * q^45 - 14 * q^47 - 5 * q^49 - 16 * q^51 - 6 * q^53 - 2 * q^55 + 4 * q^57 + 8 * q^59 + 6 * q^61 + 8 * q^63 - 2 * q^65 + 28 * q^67 - 12 * q^69 - 28 * q^71 - 16 * q^73 + 2 * q^75 - 4 * q^77 + 6 * q^79 - q^81 + 12 * q^83 + 4 * q^85 + 2 * q^87 - 10 * q^89 - 12 * q^91 + 20 * q^93 + 10 * q^95 + 8 * q^97 - 18 * 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
$$3$$ 0.806063 0.465381 0.232690 0.972551i $$-0.425247\pi$$
0.232690 + 0.972551i $$0.425247\pi$$
$$4$$ 0 0
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ −1.19394 −0.451266 −0.225633 0.974212i $$-0.572445\pi$$
−0.225633 + 0.974212i $$0.572445\pi$$
$$8$$ 0 0
$$9$$ −2.35026 −0.783421
$$10$$ 0 0
$$11$$ 4.15633 1.25318 0.626590 0.779349i $$-0.284450\pi$$
0.626590 + 0.779349i $$0.284450\pi$$
$$12$$ 0 0
$$13$$ −2.96239 −0.821619 −0.410809 0.911721i $$-0.634754\pi$$
−0.410809 + 0.911721i $$0.634754\pi$$
$$14$$ 0 0
$$15$$ −0.806063 −0.208125
$$16$$ 0 0
$$17$$ 5.50659 1.33554 0.667772 0.744366i $$-0.267248\pi$$
0.667772 + 0.744366i $$0.267248\pi$$
$$18$$ 0 0
$$19$$ −3.19394 −0.732739 −0.366370 0.930469i $$-0.619399\pi$$
−0.366370 + 0.930469i $$0.619399\pi$$
$$20$$ 0 0
$$21$$ −0.962389 −0.210010
$$22$$ 0 0
$$23$$ −1.84367 −0.384433 −0.192216 0.981353i $$-0.561568\pi$$
−0.192216 + 0.981353i $$0.561568\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −4.31265 −0.829970
$$28$$ 0 0
$$29$$ 1.00000 0.185695
$$30$$ 0 0
$$31$$ 4.80606 0.863194 0.431597 0.902066i $$-0.357950\pi$$
0.431597 + 0.902066i $$0.357950\pi$$
$$32$$ 0 0
$$33$$ 3.35026 0.583206
$$34$$ 0 0
$$35$$ 1.19394 0.201812
$$36$$ 0 0
$$37$$ 9.50659 1.56287 0.781437 0.623985i $$-0.214487\pi$$
0.781437 + 0.623985i $$0.214487\pi$$
$$38$$ 0 0
$$39$$ −2.38787 −0.382366
$$40$$ 0 0
$$41$$ −11.2750 −1.76087 −0.880433 0.474171i $$-0.842748\pi$$
−0.880433 + 0.474171i $$0.842748\pi$$
$$42$$ 0 0
$$43$$ −0.0303172 −0.00462332 −0.00231166 0.999997i $$-0.500736\pi$$
−0.00231166 + 0.999997i $$0.500736\pi$$
$$44$$ 0 0
$$45$$ 2.35026 0.350356
$$46$$ 0 0
$$47$$ −4.80606 −0.701036 −0.350518 0.936556i $$-0.613995\pi$$
−0.350518 + 0.936556i $$0.613995\pi$$
$$48$$ 0 0
$$49$$ −5.57452 −0.796359
$$50$$ 0 0
$$51$$ 4.43866 0.621536
$$52$$ 0 0
$$53$$ 1.35026 0.185473 0.0927364 0.995691i $$-0.470439\pi$$
0.0927364 + 0.995691i $$0.470439\pi$$
$$54$$ 0 0
$$55$$ −4.15633 −0.560439
$$56$$ 0 0
$$57$$ −2.57452 −0.341003
$$58$$ 0 0
$$59$$ 13.2750 1.72826 0.864131 0.503266i $$-0.167868\pi$$
0.864131 + 0.503266i $$0.167868\pi$$
$$60$$ 0 0
$$61$$ −8.88717 −1.13788 −0.568942 0.822377i $$-0.692647\pi$$
−0.568942 + 0.822377i $$0.692647\pi$$
$$62$$ 0 0
$$63$$ 2.80606 0.353531
$$64$$ 0 0
$$65$$ 2.96239 0.367439
$$66$$ 0 0
$$67$$ 5.84367 0.713919 0.356959 0.934120i $$-0.383813\pi$$
0.356959 + 0.934120i $$0.383813\pi$$
$$68$$ 0 0
$$69$$ −1.48612 −0.178908
$$70$$ 0 0
$$71$$ 1.27504 0.151319 0.0756596 0.997134i $$-0.475894\pi$$
0.0756596 + 0.997134i $$0.475894\pi$$
$$72$$ 0 0
$$73$$ −15.2447 −1.78426 −0.892130 0.451779i $$-0.850790\pi$$
−0.892130 + 0.451779i $$0.850790\pi$$
$$74$$ 0 0
$$75$$ 0.806063 0.0930762
$$76$$ 0 0
$$77$$ −4.96239 −0.565517
$$78$$ 0 0
$$79$$ 4.93207 0.554901 0.277451 0.960740i $$-0.410510\pi$$
0.277451 + 0.960740i $$0.410510\pi$$
$$80$$ 0 0
$$81$$ 3.57452 0.397168
$$82$$ 0 0
$$83$$ 4.41819 0.484959 0.242480 0.970156i $$-0.422039\pi$$
0.242480 + 0.970156i $$0.422039\pi$$
$$84$$ 0 0
$$85$$ −5.50659 −0.597273
$$86$$ 0 0
$$87$$ 0.806063 0.0864191
$$88$$ 0 0
$$89$$ −3.61213 −0.382885 −0.191442 0.981504i $$-0.561316\pi$$
−0.191442 + 0.981504i $$0.561316\pi$$
$$90$$ 0 0
$$91$$ 3.53690 0.370768
$$92$$ 0 0
$$93$$ 3.87399 0.401714
$$94$$ 0 0
$$95$$ 3.19394 0.327691
$$96$$ 0 0
$$97$$ −1.38058 −0.140177 −0.0700883 0.997541i $$-0.522328\pi$$
−0.0700883 + 0.997541i $$0.522328\pi$$
$$98$$ 0 0
$$99$$ −9.76845 −0.981766
$$100$$ 0 0
$$101$$ 13.0132 1.29486 0.647430 0.762125i $$-0.275844\pi$$
0.647430 + 0.762125i $$0.275844\pi$$
$$102$$ 0 0
$$103$$ −5.31994 −0.524190 −0.262095 0.965042i $$-0.584413\pi$$
−0.262095 + 0.965042i $$0.584413\pi$$
$$104$$ 0 0
$$105$$ 0.962389 0.0939195
$$106$$ 0 0
$$107$$ −13.8192 −1.33596 −0.667978 0.744181i $$-0.732840\pi$$
−0.667978 + 0.744181i $$0.732840\pi$$
$$108$$ 0 0
$$109$$ 1.87399 0.179496 0.0897479 0.995965i $$-0.471394\pi$$
0.0897479 + 0.995965i $$0.471394\pi$$
$$110$$ 0 0
$$111$$ 7.66291 0.727331
$$112$$ 0 0
$$113$$ −11.7685 −1.10708 −0.553541 0.832822i $$-0.686724\pi$$
−0.553541 + 0.832822i $$0.686724\pi$$
$$114$$ 0 0
$$115$$ 1.84367 0.171924
$$116$$ 0 0
$$117$$ 6.96239 0.643673
$$118$$ 0 0
$$119$$ −6.57452 −0.602685
$$120$$ 0 0
$$121$$ 6.27504 0.570458
$$122$$ 0 0
$$123$$ −9.08840 −0.819473
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ −14.2677 −1.26606 −0.633029 0.774128i $$-0.718189\pi$$
−0.633029 + 0.774128i $$0.718189\pi$$
$$128$$ 0 0
$$129$$ −0.0244376 −0.00215161
$$130$$ 0 0
$$131$$ 5.89446 0.515001 0.257501 0.966278i $$-0.417101\pi$$
0.257501 + 0.966278i $$0.417101\pi$$
$$132$$ 0 0
$$133$$ 3.81336 0.330660
$$134$$ 0 0
$$135$$ 4.31265 0.371174
$$136$$ 0 0
$$137$$ 18.2823 1.56197 0.780983 0.624553i $$-0.214719\pi$$
0.780983 + 0.624553i $$0.214719\pi$$
$$138$$ 0 0
$$139$$ −11.5369 −0.978547 −0.489274 0.872130i $$-0.662738\pi$$
−0.489274 + 0.872130i $$0.662738\pi$$
$$140$$ 0 0
$$141$$ −3.87399 −0.326249
$$142$$ 0 0
$$143$$ −12.3127 −1.02964
$$144$$ 0 0
$$145$$ −1.00000 −0.0830455
$$146$$ 0 0
$$147$$ −4.49341 −0.370610
$$148$$ 0 0
$$149$$ −2.77575 −0.227398 −0.113699 0.993515i $$-0.536270\pi$$
−0.113699 + 0.993515i $$0.536270\pi$$
$$150$$ 0 0
$$151$$ −1.79877 −0.146382 −0.0731909 0.997318i $$-0.523318\pi$$
−0.0731909 + 0.997318i $$0.523318\pi$$
$$152$$ 0 0
$$153$$ −12.9419 −1.04629
$$154$$ 0 0
$$155$$ −4.80606 −0.386032
$$156$$ 0 0
$$157$$ −3.76845 −0.300755 −0.150378 0.988629i $$-0.548049\pi$$
−0.150378 + 0.988629i $$0.548049\pi$$
$$158$$ 0 0
$$159$$ 1.08840 0.0863155
$$160$$ 0 0
$$161$$ 2.20123 0.173481
$$162$$ 0 0
$$163$$ 1.64244 0.128646 0.0643231 0.997929i $$-0.479511\pi$$
0.0643231 + 0.997929i $$0.479511\pi$$
$$164$$ 0 0
$$165$$ −3.35026 −0.260818
$$166$$ 0 0
$$167$$ −8.08110 −0.625334 −0.312667 0.949863i $$-0.601222\pi$$
−0.312667 + 0.949863i $$0.601222\pi$$
$$168$$ 0 0
$$169$$ −4.22425 −0.324943
$$170$$ 0 0
$$171$$ 7.50659 0.574043
$$172$$ 0 0
$$173$$ 7.73813 0.588320 0.294160 0.955756i $$-0.404960\pi$$
0.294160 + 0.955756i $$0.404960\pi$$
$$174$$ 0 0
$$175$$ −1.19394 −0.0902531
$$176$$ 0 0
$$177$$ 10.7005 0.804301
$$178$$ 0 0
$$179$$ −21.4010 −1.59959 −0.799795 0.600274i $$-0.795058\pi$$
−0.799795 + 0.600274i $$0.795058\pi$$
$$180$$ 0 0
$$181$$ −15.2750 −1.13538 −0.567692 0.823241i $$-0.692164\pi$$
−0.567692 + 0.823241i $$0.692164\pi$$
$$182$$ 0 0
$$183$$ −7.16362 −0.529550
$$184$$ 0 0
$$185$$ −9.50659 −0.698938
$$186$$ 0 0
$$187$$ 22.8872 1.67368
$$188$$ 0 0
$$189$$ 5.14903 0.374537
$$190$$ 0 0
$$191$$ −3.31994 −0.240223 −0.120111 0.992760i $$-0.538325\pi$$
−0.120111 + 0.992760i $$0.538325\pi$$
$$192$$ 0 0
$$193$$ −4.88129 −0.351363 −0.175681 0.984447i $$-0.556213\pi$$
−0.175681 + 0.984447i $$0.556213\pi$$
$$194$$ 0 0
$$195$$ 2.38787 0.170999
$$196$$ 0 0
$$197$$ 24.2374 1.72685 0.863423 0.504481i $$-0.168316\pi$$
0.863423 + 0.504481i $$0.168316\pi$$
$$198$$ 0 0
$$199$$ −16.7513 −1.18747 −0.593734 0.804661i $$-0.702347\pi$$
−0.593734 + 0.804661i $$0.702347\pi$$
$$200$$ 0 0
$$201$$ 4.71037 0.332244
$$202$$ 0 0
$$203$$ −1.19394 −0.0837979
$$204$$ 0 0
$$205$$ 11.2750 0.787483
$$206$$ 0 0
$$207$$ 4.33312 0.301173
$$208$$ 0 0
$$209$$ −13.2750 −0.918254
$$210$$ 0 0
$$211$$ −25.3054 −1.74209 −0.871046 0.491201i $$-0.836558\pi$$
−0.871046 + 0.491201i $$0.836558\pi$$
$$212$$ 0 0
$$213$$ 1.02776 0.0704211
$$214$$ 0 0
$$215$$ 0.0303172 0.00206761
$$216$$ 0 0
$$217$$ −5.73813 −0.389530
$$218$$ 0 0
$$219$$ −12.2882 −0.830360
$$220$$ 0 0
$$221$$ −16.3127 −1.09731
$$222$$ 0 0
$$223$$ −17.6932 −1.18483 −0.592413 0.805634i $$-0.701825\pi$$
−0.592413 + 0.805634i $$0.701825\pi$$
$$224$$ 0 0
$$225$$ −2.35026 −0.156684
$$226$$ 0 0
$$227$$ 26.8423 1.78158 0.890792 0.454412i $$-0.150151\pi$$
0.890792 + 0.454412i $$0.150151\pi$$
$$228$$ 0 0
$$229$$ 17.2243 1.13821 0.569105 0.822265i $$-0.307290\pi$$
0.569105 + 0.822265i $$0.307290\pi$$
$$230$$ 0 0
$$231$$ −4.00000 −0.263181
$$232$$ 0 0
$$233$$ −9.07381 −0.594445 −0.297222 0.954808i $$-0.596060\pi$$
−0.297222 + 0.954808i $$0.596060\pi$$
$$234$$ 0 0
$$235$$ 4.80606 0.313513
$$236$$ 0 0
$$237$$ 3.97556 0.258241
$$238$$ 0 0
$$239$$ −20.4993 −1.32599 −0.662995 0.748624i $$-0.730715\pi$$
−0.662995 + 0.748624i $$0.730715\pi$$
$$240$$ 0 0
$$241$$ 5.47627 0.352758 0.176379 0.984322i $$-0.443562\pi$$
0.176379 + 0.984322i $$0.443562\pi$$
$$242$$ 0 0
$$243$$ 15.8192 1.01480
$$244$$ 0 0
$$245$$ 5.57452 0.356143
$$246$$ 0 0
$$247$$ 9.46168 0.602032
$$248$$ 0 0
$$249$$ 3.56134 0.225691
$$250$$ 0 0
$$251$$ −29.6180 −1.86947 −0.934736 0.355343i $$-0.884364\pi$$
−0.934736 + 0.355343i $$0.884364\pi$$
$$252$$ 0 0
$$253$$ −7.66291 −0.481763
$$254$$ 0 0
$$255$$ −4.43866 −0.277960
$$256$$ 0 0
$$257$$ 17.6629 1.10178 0.550891 0.834577i $$-0.314288\pi$$
0.550891 + 0.834577i $$0.314288\pi$$
$$258$$ 0 0
$$259$$ −11.3503 −0.705271
$$260$$ 0 0
$$261$$ −2.35026 −0.145478
$$262$$ 0 0
$$263$$ −27.3561 −1.68685 −0.843426 0.537245i $$-0.819465\pi$$
−0.843426 + 0.537245i $$0.819465\pi$$
$$264$$ 0 0
$$265$$ −1.35026 −0.0829459
$$266$$ 0 0
$$267$$ −2.91160 −0.178187
$$268$$ 0 0
$$269$$ −10.4993 −0.640153 −0.320077 0.947392i $$-0.603709\pi$$
−0.320077 + 0.947392i $$0.603709\pi$$
$$270$$ 0 0
$$271$$ 9.61801 0.584252 0.292126 0.956380i $$-0.405637\pi$$
0.292126 + 0.956380i $$0.405637\pi$$
$$272$$ 0 0
$$273$$ 2.85097 0.172548
$$274$$ 0 0
$$275$$ 4.15633 0.250636
$$276$$ 0 0
$$277$$ −13.3503 −0.802139 −0.401070 0.916048i $$-0.631362\pi$$
−0.401070 + 0.916048i $$0.631362\pi$$
$$278$$ 0 0
$$279$$ −11.2955 −0.676244
$$280$$ 0 0
$$281$$ 20.4241 1.21840 0.609199 0.793017i $$-0.291491\pi$$
0.609199 + 0.793017i $$0.291491\pi$$
$$282$$ 0 0
$$283$$ −8.02047 −0.476767 −0.238384 0.971171i $$-0.576618\pi$$
−0.238384 + 0.971171i $$0.576618\pi$$
$$284$$ 0 0
$$285$$ 2.57452 0.152501
$$286$$ 0 0
$$287$$ 13.4617 0.794618
$$288$$ 0 0
$$289$$ 13.3225 0.783676
$$290$$ 0 0
$$291$$ −1.11283 −0.0652355
$$292$$ 0 0
$$293$$ 23.3054 1.36151 0.680757 0.732510i $$-0.261651\pi$$
0.680757 + 0.732510i $$0.261651\pi$$
$$294$$ 0 0
$$295$$ −13.2750 −0.772903
$$296$$ 0 0
$$297$$ −17.9248 −1.04010
$$298$$ 0 0
$$299$$ 5.46168 0.315857
$$300$$ 0 0
$$301$$ 0.0361968 0.00208635
$$302$$ 0 0
$$303$$ 10.4894 0.602603
$$304$$ 0 0
$$305$$ 8.88717 0.508878
$$306$$ 0 0
$$307$$ −6.73084 −0.384149 −0.192075 0.981380i $$-0.561522\pi$$
−0.192075 + 0.981380i $$0.561522\pi$$
$$308$$ 0 0
$$309$$ −4.28821 −0.243948
$$310$$ 0 0
$$311$$ 22.0567 1.25072 0.625359 0.780337i $$-0.284952\pi$$
0.625359 + 0.780337i $$0.284952\pi$$
$$312$$ 0 0
$$313$$ 5.03761 0.284743 0.142371 0.989813i $$-0.454527\pi$$
0.142371 + 0.989813i $$0.454527\pi$$
$$314$$ 0 0
$$315$$ −2.80606 −0.158104
$$316$$ 0 0
$$317$$ −34.2941 −1.92615 −0.963074 0.269237i $$-0.913229\pi$$
−0.963074 + 0.269237i $$0.913229\pi$$
$$318$$ 0 0
$$319$$ 4.15633 0.232710
$$320$$ 0 0
$$321$$ −11.1392 −0.621729
$$322$$ 0 0
$$323$$ −17.5877 −0.978605
$$324$$ 0 0
$$325$$ −2.96239 −0.164324
$$326$$ 0 0
$$327$$ 1.51056 0.0835340
$$328$$ 0 0
$$329$$ 5.73813 0.316354
$$330$$ 0 0
$$331$$ −34.8324 −1.91456 −0.957281 0.289159i $$-0.906625\pi$$
−0.957281 + 0.289159i $$0.906625\pi$$
$$332$$ 0 0
$$333$$ −22.3430 −1.22439
$$334$$ 0 0
$$335$$ −5.84367 −0.319274
$$336$$ 0 0
$$337$$ −17.6326 −0.960509 −0.480254 0.877129i $$-0.659456\pi$$
−0.480254 + 0.877129i $$0.659456\pi$$
$$338$$ 0 0
$$339$$ −9.48612 −0.515215
$$340$$ 0 0
$$341$$ 19.9756 1.08174
$$342$$ 0 0
$$343$$ 15.0132 0.810635
$$344$$ 0 0
$$345$$ 1.48612 0.0800099
$$346$$ 0 0
$$347$$ −3.11871 −0.167421 −0.0837107 0.996490i $$-0.526677\pi$$
−0.0837107 + 0.996490i $$0.526677\pi$$
$$348$$ 0 0
$$349$$ 13.0738 0.699825 0.349912 0.936782i $$-0.386211\pi$$
0.349912 + 0.936782i $$0.386211\pi$$
$$350$$ 0 0
$$351$$ 12.7757 0.681919
$$352$$ 0 0
$$353$$ 5.19982 0.276758 0.138379 0.990379i $$-0.455811\pi$$
0.138379 + 0.990379i $$0.455811\pi$$
$$354$$ 0 0
$$355$$ −1.27504 −0.0676720
$$356$$ 0 0
$$357$$ −5.29948 −0.280478
$$358$$ 0 0
$$359$$ −30.4182 −1.60541 −0.802705 0.596376i $$-0.796607\pi$$
−0.802705 + 0.596376i $$0.796607\pi$$
$$360$$ 0 0
$$361$$ −8.79877 −0.463093
$$362$$ 0 0
$$363$$ 5.05808 0.265480
$$364$$ 0 0
$$365$$ 15.2447 0.797945
$$366$$ 0 0
$$367$$ −20.6556 −1.07821 −0.539107 0.842237i $$-0.681238\pi$$
−0.539107 + 0.842237i $$0.681238\pi$$
$$368$$ 0 0
$$369$$ 26.4993 1.37950
$$370$$ 0 0
$$371$$ −1.61213 −0.0836975
$$372$$ 0 0
$$373$$ −11.0884 −0.574135 −0.287068 0.957910i $$-0.592680\pi$$
−0.287068 + 0.957910i $$0.592680\pi$$
$$374$$ 0 0
$$375$$ −0.806063 −0.0416249
$$376$$ 0 0
$$377$$ −2.96239 −0.152571
$$378$$ 0 0
$$379$$ 10.0811 0.517831 0.258916 0.965900i $$-0.416635\pi$$
0.258916 + 0.965900i $$0.416635\pi$$
$$380$$ 0 0
$$381$$ −11.5007 −0.589199
$$382$$ 0 0
$$383$$ −16.3576 −0.835832 −0.417916 0.908486i $$-0.637239\pi$$
−0.417916 + 0.908486i $$0.637239\pi$$
$$384$$ 0 0
$$385$$ 4.96239 0.252907
$$386$$ 0 0
$$387$$ 0.0712533 0.00362201
$$388$$ 0 0
$$389$$ −31.9003 −1.61741 −0.808706 0.588213i $$-0.799831\pi$$
−0.808706 + 0.588213i $$0.799831\pi$$
$$390$$ 0 0
$$391$$ −10.1524 −0.513427
$$392$$ 0 0
$$393$$ 4.75131 0.239672
$$394$$ 0 0
$$395$$ −4.93207 −0.248159
$$396$$ 0 0
$$397$$ −2.98683 −0.149905 −0.0749523 0.997187i $$-0.523880\pi$$
−0.0749523 + 0.997187i $$0.523880\pi$$
$$398$$ 0 0
$$399$$ 3.07381 0.153883
$$400$$ 0 0
$$401$$ −21.9756 −1.09741 −0.548704 0.836017i $$-0.684878\pi$$
−0.548704 + 0.836017i $$0.684878\pi$$
$$402$$ 0 0
$$403$$ −14.2374 −0.709217
$$404$$ 0 0
$$405$$ −3.57452 −0.177619
$$406$$ 0 0
$$407$$ 39.5125 1.95856
$$408$$ 0 0
$$409$$ −22.4387 −1.10952 −0.554760 0.832010i $$-0.687190\pi$$
−0.554760 + 0.832010i $$0.687190\pi$$
$$410$$ 0 0
$$411$$ 14.7367 0.726909
$$412$$ 0 0
$$413$$ −15.8496 −0.779906
$$414$$ 0 0
$$415$$ −4.41819 −0.216880
$$416$$ 0 0
$$417$$ −9.29948 −0.455397
$$418$$ 0 0
$$419$$ 10.3634 0.506287 0.253143 0.967429i $$-0.418536\pi$$
0.253143 + 0.967429i $$0.418536\pi$$
$$420$$ 0 0
$$421$$ −34.0362 −1.65882 −0.829411 0.558638i $$-0.811324\pi$$
−0.829411 + 0.558638i $$0.811324\pi$$
$$422$$ 0 0
$$423$$ 11.2955 0.549206
$$424$$ 0 0
$$425$$ 5.50659 0.267109
$$426$$ 0 0
$$427$$ 10.6107 0.513488
$$428$$ 0 0
$$429$$ −9.92478 −0.479173
$$430$$ 0 0
$$431$$ −25.7743 −1.24151 −0.620753 0.784006i $$-0.713173\pi$$
−0.620753 + 0.784006i $$0.713173\pi$$
$$432$$ 0 0
$$433$$ 2.18076 0.104801 0.0524004 0.998626i $$-0.483313\pi$$
0.0524004 + 0.998626i $$0.483313\pi$$
$$434$$ 0 0
$$435$$ −0.806063 −0.0386478
$$436$$ 0 0
$$437$$ 5.88858 0.281689
$$438$$ 0 0
$$439$$ 35.5125 1.69492 0.847459 0.530861i $$-0.178131\pi$$
0.847459 + 0.530861i $$0.178131\pi$$
$$440$$ 0 0
$$441$$ 13.1016 0.623884
$$442$$ 0 0
$$443$$ −4.34297 −0.206341 −0.103170 0.994664i $$-0.532899\pi$$
−0.103170 + 0.994664i $$0.532899\pi$$
$$444$$ 0 0
$$445$$ 3.61213 0.171231
$$446$$ 0 0
$$447$$ −2.23743 −0.105827
$$448$$ 0 0
$$449$$ 31.3357 1.47882 0.739411 0.673254i $$-0.235104\pi$$
0.739411 + 0.673254i $$0.235104\pi$$
$$450$$ 0 0
$$451$$ −46.8627 −2.20668
$$452$$ 0 0
$$453$$ −1.44992 −0.0681233
$$454$$ 0 0
$$455$$ −3.53690 −0.165813
$$456$$ 0 0
$$457$$ 34.3488 1.60677 0.803386 0.595459i $$-0.203030\pi$$
0.803386 + 0.595459i $$0.203030\pi$$
$$458$$ 0 0
$$459$$ −23.7480 −1.10846
$$460$$ 0 0
$$461$$ −11.8641 −0.552568 −0.276284 0.961076i $$-0.589103\pi$$
−0.276284 + 0.961076i $$0.589103\pi$$
$$462$$ 0 0
$$463$$ −40.4953 −1.88198 −0.940989 0.338438i $$-0.890101\pi$$
−0.940989 + 0.338438i $$0.890101\pi$$
$$464$$ 0 0
$$465$$ −3.87399 −0.179652
$$466$$ 0 0
$$467$$ −30.2071 −1.39782 −0.698909 0.715210i $$-0.746331\pi$$
−0.698909 + 0.715210i $$0.746331\pi$$
$$468$$ 0 0
$$469$$ −6.97698 −0.322167
$$470$$ 0 0
$$471$$ −3.03761 −0.139966
$$472$$ 0 0
$$473$$ −0.126008 −0.00579385
$$474$$ 0 0
$$475$$ −3.19394 −0.146548
$$476$$ 0 0
$$477$$ −3.17347 −0.145303
$$478$$ 0 0
$$479$$ −0.0547547 −0.00250181 −0.00125090 0.999999i $$-0.500398\pi$$
−0.00125090 + 0.999999i $$0.500398\pi$$
$$480$$ 0 0
$$481$$ −28.1622 −1.28409
$$482$$ 0 0
$$483$$ 1.77433 0.0807349
$$484$$ 0 0
$$485$$ 1.38058 0.0626889
$$486$$ 0 0
$$487$$ 0.881286 0.0399349 0.0199674 0.999801i $$-0.493644\pi$$
0.0199674 + 0.999801i $$0.493644\pi$$
$$488$$ 0 0
$$489$$ 1.32391 0.0598695
$$490$$ 0 0
$$491$$ 41.0698 1.85346 0.926728 0.375733i $$-0.122609\pi$$
0.926728 + 0.375733i $$0.122609\pi$$
$$492$$ 0 0
$$493$$ 5.50659 0.248004
$$494$$ 0 0
$$495$$ 9.76845 0.439059
$$496$$ 0 0
$$497$$ −1.52232 −0.0682852
$$498$$ 0 0
$$499$$ 12.3733 0.553904 0.276952 0.960884i $$-0.410676\pi$$
0.276952 + 0.960884i $$0.410676\pi$$
$$500$$ 0 0
$$501$$ −6.51388 −0.291019
$$502$$ 0 0
$$503$$ 2.26774 0.101114 0.0505569 0.998721i $$-0.483900\pi$$
0.0505569 + 0.998721i $$0.483900\pi$$
$$504$$ 0 0
$$505$$ −13.0132 −0.579079
$$506$$ 0 0
$$507$$ −3.40502 −0.151222
$$508$$ 0 0
$$509$$ 10.9018 0.483212 0.241606 0.970374i $$-0.422326\pi$$
0.241606 + 0.970374i $$0.422326\pi$$
$$510$$ 0 0
$$511$$ 18.2012 0.805175
$$512$$ 0 0
$$513$$ 13.7743 0.608152
$$514$$ 0 0
$$515$$ 5.31994 0.234425
$$516$$ 0 0
$$517$$ −19.9756 −0.878524
$$518$$ 0 0
$$519$$ 6.23743 0.273793
$$520$$ 0 0
$$521$$ −4.72496 −0.207004 −0.103502 0.994629i $$-0.533005\pi$$
−0.103502 + 0.994629i $$0.533005\pi$$
$$522$$ 0 0
$$523$$ −1.06793 −0.0466973 −0.0233486 0.999727i $$-0.507433\pi$$
−0.0233486 + 0.999727i $$0.507433\pi$$
$$524$$ 0 0
$$525$$ −0.962389 −0.0420021
$$526$$ 0 0
$$527$$ 26.4650 1.15283
$$528$$ 0 0
$$529$$ −19.6009 −0.852211
$$530$$ 0 0
$$531$$ −31.1998 −1.35396
$$532$$ 0 0
$$533$$ 33.4010 1.44676
$$534$$ 0 0
$$535$$ 13.8192 0.597458
$$536$$ 0 0
$$537$$ −17.2506 −0.744418
$$538$$ 0 0
$$539$$ −23.1695 −0.997981
$$540$$ 0 0
$$541$$ 7.46168 0.320803 0.160401 0.987052i $$-0.448721\pi$$
0.160401 + 0.987052i $$0.448721\pi$$
$$542$$ 0 0
$$543$$ −12.3127 −0.528386
$$544$$ 0 0
$$545$$ −1.87399 −0.0802730
$$546$$ 0 0
$$547$$ −38.9683 −1.66616 −0.833081 0.553150i $$-0.813425\pi$$
−0.833081 + 0.553150i $$0.813425\pi$$
$$548$$ 0 0
$$549$$ 20.8872 0.891443
$$550$$ 0 0
$$551$$ −3.19394 −0.136066
$$552$$ 0 0
$$553$$ −5.88858 −0.250408
$$554$$ 0 0
$$555$$ −7.66291 −0.325273
$$556$$ 0 0
$$557$$ 22.9986 0.974481 0.487241 0.873268i $$-0.338003\pi$$
0.487241 + 0.873268i $$0.338003\pi$$
$$558$$ 0 0
$$559$$ 0.0898112 0.00379861
$$560$$ 0 0
$$561$$ 18.4485 0.778897
$$562$$ 0 0
$$563$$ 11.6688 0.491781 0.245890 0.969298i $$-0.420920\pi$$
0.245890 + 0.969298i $$0.420920\pi$$
$$564$$ 0 0
$$565$$ 11.7685 0.495102
$$566$$ 0 0
$$567$$ −4.26774 −0.179228
$$568$$ 0 0
$$569$$ 11.3357 0.475216 0.237608 0.971361i $$-0.423637\pi$$
0.237608 + 0.971361i $$0.423637\pi$$
$$570$$ 0 0
$$571$$ 27.1754 1.13725 0.568627 0.822595i $$-0.307475\pi$$
0.568627 + 0.822595i $$0.307475\pi$$
$$572$$ 0 0
$$573$$ −2.67609 −0.111795
$$574$$ 0 0
$$575$$ −1.84367 −0.0768866
$$576$$ 0 0
$$577$$ 22.5950 0.940641 0.470321 0.882496i $$-0.344138\pi$$
0.470321 + 0.882496i $$0.344138\pi$$
$$578$$ 0 0
$$579$$ −3.93463 −0.163517
$$580$$ 0 0
$$581$$ −5.27504 −0.218845
$$582$$ 0 0
$$583$$ 5.61213 0.232431
$$584$$ 0 0
$$585$$ −6.96239 −0.287859
$$586$$ 0 0
$$587$$ 9.31994 0.384675 0.192338 0.981329i $$-0.438393\pi$$
0.192338 + 0.981329i $$0.438393\pi$$
$$588$$ 0 0
$$589$$ −15.3503 −0.632497
$$590$$ 0 0
$$591$$ 19.5369 0.803641
$$592$$ 0 0
$$593$$ −15.1246 −0.621093 −0.310546 0.950558i $$-0.600512\pi$$
−0.310546 + 0.950558i $$0.600512\pi$$
$$594$$ 0 0
$$595$$ 6.57452 0.269529
$$596$$ 0 0
$$597$$ −13.5026 −0.552625
$$598$$ 0 0
$$599$$ −4.09569 −0.167345 −0.0836727 0.996493i $$-0.526665\pi$$
−0.0836727 + 0.996493i $$0.526665\pi$$
$$600$$ 0 0
$$601$$ 22.2276 0.906682 0.453341 0.891337i $$-0.350232\pi$$
0.453341 + 0.891337i $$0.350232\pi$$
$$602$$ 0 0
$$603$$ −13.7342 −0.559298
$$604$$ 0 0
$$605$$ −6.27504 −0.255117
$$606$$ 0 0
$$607$$ 48.2941 1.96020 0.980098 0.198512i $$-0.0636110\pi$$
0.980098 + 0.198512i $$0.0636110\pi$$
$$608$$ 0 0
$$609$$ −0.962389 −0.0389980
$$610$$ 0 0
$$611$$ 14.2374 0.575985
$$612$$ 0 0
$$613$$ 9.74798 0.393717 0.196859 0.980432i $$-0.436926\pi$$
0.196859 + 0.980432i $$0.436926\pi$$
$$614$$ 0 0
$$615$$ 9.08840 0.366480
$$616$$ 0 0
$$617$$ 18.2170 0.733387 0.366694 0.930342i $$-0.380490\pi$$
0.366694 + 0.930342i $$0.380490\pi$$
$$618$$ 0 0
$$619$$ 25.0943 1.00862 0.504312 0.863521i $$-0.331746\pi$$
0.504312 + 0.863521i $$0.331746\pi$$
$$620$$ 0 0
$$621$$ 7.95112 0.319068
$$622$$ 0 0
$$623$$ 4.31265 0.172783
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ −10.7005 −0.427338
$$628$$ 0 0
$$629$$ 52.3488 2.08729
$$630$$ 0 0
$$631$$ −21.4617 −0.854376 −0.427188 0.904163i $$-0.640496\pi$$
−0.427188 + 0.904163i $$0.640496\pi$$
$$632$$ 0 0
$$633$$ −20.3977 −0.810737
$$634$$ 0 0
$$635$$ 14.2677 0.566198
$$636$$ 0 0
$$637$$ 16.5139 0.654304
$$638$$ 0 0
$$639$$ −2.99668 −0.118547
$$640$$ 0 0
$$641$$ 3.17347 0.125344 0.0626722 0.998034i $$-0.480038\pi$$
0.0626722 + 0.998034i $$0.480038\pi$$
$$642$$ 0 0
$$643$$ −2.74069 −0.108082 −0.0540411 0.998539i $$-0.517210\pi$$
−0.0540411 + 0.998539i $$0.517210\pi$$
$$644$$ 0 0
$$645$$ 0.0244376 0.000962228 0
$$646$$ 0 0
$$647$$ −6.34297 −0.249368 −0.124684 0.992197i $$-0.539792\pi$$
−0.124684 + 0.992197i $$0.539792\pi$$
$$648$$ 0 0
$$649$$ 55.1754 2.16582
$$650$$ 0 0
$$651$$ −4.62530 −0.181280
$$652$$ 0 0
$$653$$ −4.08110 −0.159706 −0.0798529 0.996807i $$-0.525445\pi$$
−0.0798529 + 0.996807i $$0.525445\pi$$
$$654$$ 0 0
$$655$$ −5.89446 −0.230316
$$656$$ 0 0
$$657$$ 35.8291 1.39783
$$658$$ 0 0
$$659$$ 9.58181 0.373254 0.186627 0.982431i $$-0.440244\pi$$
0.186627 + 0.982431i $$0.440244\pi$$
$$660$$ 0 0
$$661$$ 27.5271 1.07068 0.535339 0.844637i $$-0.320184\pi$$
0.535339 + 0.844637i $$0.320184\pi$$
$$662$$ 0 0
$$663$$ −13.1490 −0.510666
$$664$$ 0 0
$$665$$ −3.81336 −0.147876
$$666$$ 0 0
$$667$$ −1.84367 −0.0713874
$$668$$ 0 0
$$669$$ −14.2619 −0.551396
$$670$$ 0 0
$$671$$ −36.9380 −1.42597
$$672$$ 0 0
$$673$$ 3.13727 0.120933 0.0604665 0.998170i $$-0.480741\pi$$
0.0604665 + 0.998170i $$0.480741\pi$$
$$674$$ 0 0
$$675$$ −4.31265 −0.165994
$$676$$ 0 0
$$677$$ 46.2579 1.77784 0.888918 0.458067i $$-0.151458\pi$$
0.888918 + 0.458067i $$0.151458\pi$$
$$678$$ 0 0
$$679$$ 1.64832 0.0632569
$$680$$ 0 0
$$681$$ 21.6366 0.829115
$$682$$ 0 0
$$683$$ −9.01905 −0.345104 −0.172552 0.985000i $$-0.555201\pi$$
−0.172552 + 0.985000i $$0.555201\pi$$
$$684$$ 0 0
$$685$$ −18.2823 −0.698532
$$686$$ 0 0
$$687$$ 13.8838 0.529702
$$688$$ 0 0
$$689$$ −4.00000 −0.152388
$$690$$ 0 0
$$691$$ −50.0625 −1.90447 −0.952234 0.305368i $$-0.901221\pi$$
−0.952234 + 0.305368i $$0.901221\pi$$
$$692$$ 0 0
$$693$$ 11.6629 0.443037
$$694$$ 0 0
$$695$$ 11.5369 0.437620
$$696$$ 0 0
$$697$$ −62.0870 −2.35171
$$698$$ 0 0
$$699$$ −7.31406 −0.276643
$$700$$ 0 0
$$701$$ −45.3014 −1.71101 −0.855505 0.517795i $$-0.826753\pi$$
−0.855505 + 0.517795i $$0.826753\pi$$
$$702$$ 0 0
$$703$$ −30.3634 −1.14518
$$704$$ 0 0
$$705$$ 3.87399 0.145903
$$706$$ 0 0
$$707$$ −15.5369 −0.584325
$$708$$ 0 0
$$709$$ 3.27504 0.122997 0.0614983 0.998107i $$-0.480412\pi$$
0.0614983 + 0.998107i $$0.480412\pi$$
$$710$$ 0 0
$$711$$ −11.5917 −0.434721
$$712$$ 0 0
$$713$$ −8.86082 −0.331840
$$714$$ 0 0
$$715$$ 12.3127 0.460467
$$716$$ 0 0
$$717$$ −16.5237 −0.617090
$$718$$ 0 0
$$719$$ −27.7235 −1.03391 −0.516957 0.856011i $$-0.672935\pi$$
−0.516957 + 0.856011i $$0.672935\pi$$
$$720$$ 0 0
$$721$$ 6.35168 0.236549
$$722$$ 0 0
$$723$$ 4.41422 0.164167
$$724$$ 0 0
$$725$$ 1.00000 0.0371391
$$726$$ 0 0
$$727$$ 26.8930 0.997408 0.498704 0.866772i $$-0.333810\pi$$
0.498704 + 0.866772i $$0.333810\pi$$
$$728$$ 0 0
$$729$$ 2.02776 0.0751023
$$730$$ 0 0
$$731$$ −0.166944 −0.00617465
$$732$$ 0 0
$$733$$ −3.17935 −0.117432 −0.0587160 0.998275i $$-0.518701\pi$$
−0.0587160 + 0.998275i $$0.518701\pi$$
$$734$$ 0 0
$$735$$ 4.49341 0.165742
$$736$$ 0 0
$$737$$ 24.2882 0.894668
$$738$$ 0 0
$$739$$ −29.7440 −1.09415 −0.547076 0.837083i $$-0.684259\pi$$
−0.547076 + 0.837083i $$0.684259\pi$$
$$740$$ 0 0
$$741$$ 7.62672 0.280174
$$742$$ 0 0
$$743$$ 4.34297 0.159328 0.0796640 0.996822i $$-0.474615\pi$$
0.0796640 + 0.996822i $$0.474615\pi$$
$$744$$ 0 0
$$745$$ 2.77575 0.101695
$$746$$ 0 0
$$747$$ −10.3839 −0.379927
$$748$$ 0 0
$$749$$ 16.4993 0.602871
$$750$$ 0 0
$$751$$ −22.5804 −0.823970 −0.411985 0.911191i $$-0.635164\pi$$
−0.411985 + 0.911191i $$0.635164\pi$$
$$752$$ 0 0
$$753$$ −23.8740 −0.870017
$$754$$ 0 0
$$755$$ 1.79877 0.0654639
$$756$$ 0 0
$$757$$ −9.88461 −0.359262 −0.179631 0.983734i $$-0.557490\pi$$
−0.179631 + 0.983734i $$0.557490\pi$$
$$758$$ 0 0
$$759$$ −6.17679 −0.224203
$$760$$ 0 0
$$761$$ 13.6991 0.496592 0.248296 0.968684i $$-0.420129\pi$$
0.248296 + 0.968684i $$0.420129\pi$$
$$762$$ 0 0
$$763$$ −2.23743 −0.0810003
$$764$$ 0 0
$$765$$ 12.9419 0.467916
$$766$$ 0 0
$$767$$ −39.3258 −1.41997
$$768$$ 0 0
$$769$$ 25.0132 0.901998 0.450999 0.892524i $$-0.351068\pi$$
0.450999 + 0.892524i $$0.351068\pi$$
$$770$$ 0 0
$$771$$ 14.2374 0.512748
$$772$$ 0 0
$$773$$ 35.9062 1.29146 0.645728 0.763567i $$-0.276554\pi$$
0.645728 + 0.763567i $$0.276554\pi$$
$$774$$ 0 0
$$775$$ 4.80606 0.172639
$$776$$ 0 0
$$777$$ −9.14903 −0.328220
$$778$$ 0 0
$$779$$ 36.0118 1.29026
$$780$$ 0 0
$$781$$ 5.29948 0.189630
$$782$$ 0 0
$$783$$ −4.31265 −0.154122
$$784$$ 0 0
$$785$$ 3.76845 0.134502
$$786$$ 0 0
$$787$$ 50.3839 1.79599 0.897996 0.440003i $$-0.145023\pi$$
0.897996 + 0.440003i $$0.145023\pi$$
$$788$$ 0 0
$$789$$ −22.0508 −0.785029
$$790$$ 0 0
$$791$$ 14.0508 0.499588
$$792$$ 0 0
$$793$$ 26.3272 0.934908
$$794$$ 0 0
$$795$$ −1.08840 −0.0386014
$$796$$ 0 0
$$797$$ 5.69323 0.201665 0.100832 0.994903i $$-0.467849\pi$$
0.100832 + 0.994903i $$0.467849\pi$$
$$798$$ 0 0
$$799$$ −26.4650 −0.936265
$$800$$ 0 0
$$801$$ 8.48944 0.299960
$$802$$ 0 0
$$803$$ −63.3620 −2.23600
$$804$$ 0 0
$$805$$ −2.20123 −0.0775832
$$806$$ 0 0
$$807$$ −8.46310 −0.297915
$$808$$ 0 0
$$809$$ −7.76257 −0.272918 −0.136459 0.990646i $$-0.543572\pi$$
−0.136459 + 0.990646i $$0.543572\pi$$
$$810$$ 0 0
$$811$$ −26.4894 −0.930170 −0.465085 0.885266i $$-0.653976\pi$$
−0.465085 + 0.885266i $$0.653976\pi$$
$$812$$ 0 0
$$813$$ 7.75272 0.271900
$$814$$ 0 0
$$815$$ −1.64244 −0.0575323
$$816$$ 0 0
$$817$$ 0.0968311 0.00338769
$$818$$ 0 0
$$819$$ −8.31265 −0.290468
$$820$$ 0 0
$$821$$ −25.4763 −0.889128 −0.444564 0.895747i $$-0.646641\pi$$
−0.444564 + 0.895747i $$0.646641\pi$$
$$822$$ 0 0
$$823$$ 9.22028 0.321399 0.160699 0.987003i $$-0.448625\pi$$
0.160699 + 0.987003i $$0.448625\pi$$
$$824$$ 0 0
$$825$$ 3.35026 0.116641
$$826$$ 0 0
$$827$$ 24.5343 0.853143 0.426571 0.904454i $$-0.359721\pi$$
0.426571 + 0.904454i $$0.359721\pi$$
$$828$$ 0 0
$$829$$ −0.201231 −0.00698903 −0.00349452 0.999994i $$-0.501112\pi$$
−0.00349452 + 0.999994i $$0.501112\pi$$
$$830$$ 0 0
$$831$$ −10.7612 −0.373300
$$832$$ 0 0
$$833$$ −30.6966 −1.06357
$$834$$ 0 0
$$835$$ 8.08110 0.279658
$$836$$ 0 0
$$837$$ −20.7269 −0.716425
$$838$$ 0 0
$$839$$ −1.45580 −0.0502599 −0.0251299 0.999684i $$-0.508000\pi$$
−0.0251299 + 0.999684i $$0.508000\pi$$
$$840$$ 0 0
$$841$$ 1.00000 0.0344828
$$842$$ 0 0
$$843$$ 16.4631 0.567019
$$844$$ 0 0
$$845$$ 4.22425 0.145319
$$846$$ 0 0
$$847$$ −7.49200 −0.257428
$$848$$ 0 0
$$849$$ −6.46501 −0.221878
$$850$$ 0 0
$$851$$ −17.5271 −0.600820
$$852$$ 0 0
$$853$$ −43.1793 −1.47843 −0.739216 0.673468i $$-0.764804\pi$$
−0.739216 + 0.673468i $$0.764804\pi$$
$$854$$ 0 0
$$855$$ −7.50659 −0.256720
$$856$$ 0 0
$$857$$ −20.9887 −0.716962 −0.358481 0.933537i $$-0.616705\pi$$
−0.358481 + 0.933537i $$0.616705\pi$$
$$858$$ 0 0
$$859$$ −49.4069 −1.68574 −0.842871 0.538115i $$-0.819137\pi$$
−0.842871 + 0.538115i $$0.819137\pi$$
$$860$$ 0 0
$$861$$ 10.8510 0.369800
$$862$$ 0 0
$$863$$ −56.6820 −1.92948 −0.964738 0.263211i $$-0.915218\pi$$
−0.964738 + 0.263211i $$0.915218\pi$$
$$864$$ 0 0
$$865$$ −7.73813 −0.263104
$$866$$ 0 0
$$867$$ 10.7388 0.364708
$$868$$ 0 0
$$869$$ 20.4993 0.695391
$$870$$ 0 0
$$871$$ −17.3112 −0.586569
$$872$$ 0 0
$$873$$ 3.24472 0.109817
$$874$$ 0 0
$$875$$ 1.19394 0.0403624
$$876$$ 0 0
$$877$$ 13.1998 0.445726 0.222863 0.974850i $$-0.428460\pi$$
0.222863 + 0.974850i $$0.428460\pi$$
$$878$$ 0 0
$$879$$ 18.7856 0.633622
$$880$$ 0 0
$$881$$ 6.37802 0.214881 0.107441 0.994212i $$-0.465734\pi$$
0.107441 + 0.994212i $$0.465734\pi$$
$$882$$ 0 0
$$883$$ 48.6213 1.63624 0.818119 0.575049i $$-0.195017\pi$$
0.818119 + 0.575049i $$0.195017\pi$$
$$884$$ 0 0
$$885$$ −10.7005 −0.359694
$$886$$ 0 0
$$887$$ 15.0317 0.504716 0.252358 0.967634i $$-0.418794\pi$$
0.252358 + 0.967634i $$0.418794\pi$$
$$888$$ 0 0
$$889$$ 17.0348 0.571328
$$890$$ 0 0
$$891$$ 14.8568 0.497723
$$892$$ 0 0
$$893$$ 15.3503 0.513677
$$894$$ 0 0
$$895$$ 21.4010 0.715358
$$896$$ 0 0
$$897$$ 4.40246 0.146994
$$898$$ 0 0
$$899$$ 4.80606 0.160291
$$900$$ 0 0
$$901$$ 7.43533 0.247707
$$902$$ 0 0
$$903$$ 0.0291769 0.000970946 0
$$904$$ 0 0
$$905$$ 15.2750 0.507759
$$906$$ 0 0
$$907$$ −0.342968 −0.0113880 −0.00569402 0.999984i $$-0.501812\pi$$
−0.00569402 + 0.999984i $$0.501812\pi$$
$$908$$ 0 0
$$909$$ −30.5844 −1.01442
$$910$$ 0 0
$$911$$ −20.9076 −0.692701 −0.346350 0.938105i $$-0.612579\pi$$
−0.346350 + 0.938105i $$0.612579\pi$$
$$912$$ 0 0
$$913$$ 18.3634 0.607741
$$914$$ 0 0
$$915$$ 7.16362 0.236822
$$916$$ 0 0
$$917$$ −7.03761 −0.232402
$$918$$ 0 0
$$919$$ 1.90034 0.0626864 0.0313432 0.999509i $$-0.490022\pi$$
0.0313432 + 0.999509i $$0.490022\pi$$
$$920$$ 0 0
$$921$$ −5.42548 −0.178776
$$922$$ 0 0
$$923$$ −3.77716 −0.124327
$$924$$ 0 0
$$925$$ 9.50659 0.312575
$$926$$ 0 0
$$927$$ 12.5033 0.410661
$$928$$ 0 0
$$929$$ 39.3522 1.29110 0.645551 0.763717i $$-0.276628\pi$$
0.645551 + 0.763717i $$0.276628\pi$$
$$930$$ 0 0
$$931$$ 17.8046 0.583524
$$932$$ 0 0
$$933$$ 17.7791 0.582061
$$934$$ 0 0
$$935$$ −22.8872 −0.748490
$$936$$ 0 0
$$937$$ −6.37802 −0.208361 −0.104180 0.994558i $$-0.533222\pi$$
−0.104180 + 0.994558i $$0.533222\pi$$
$$938$$ 0 0
$$939$$ 4.06063 0.132514
$$940$$ 0 0
$$941$$ 26.6253 0.867960 0.433980 0.900923i $$-0.357109\pi$$
0.433980 + 0.900923i $$0.357109\pi$$
$$942$$ 0 0
$$943$$ 20.7875 0.676934
$$944$$ 0 0
$$945$$ −5.14903 −0.167498
$$946$$ 0 0
$$947$$ −12.2823 −0.399122 −0.199561 0.979885i $$-0.563952\pi$$
−0.199561 + 0.979885i $$0.563952\pi$$
$$948$$ 0 0
$$949$$ 45.1608 1.46598
$$950$$ 0 0
$$951$$ −27.6432 −0.896393
$$952$$ 0 0
$$953$$ 0.821792 0.0266205 0.0133102 0.999911i $$-0.495763\pi$$
0.0133102 + 0.999911i $$0.495763\pi$$
$$954$$ 0 0
$$955$$ 3.31994 0.107431
$$956$$ 0 0
$$957$$ 3.35026 0.108299
$$958$$ 0 0
$$959$$ −21.8279 −0.704861
$$960$$ 0 0
$$961$$ −7.90175 −0.254895
$$962$$ 0 0
$$963$$ 32.4788 1.04662
$$964$$ 0 0
$$965$$ 4.88129 0.157134
$$966$$ 0 0
$$967$$ 37.4314 1.20371 0.601856 0.798605i $$-0.294428\pi$$
0.601856 + 0.798605i $$0.294428\pi$$
$$968$$ 0 0
$$969$$ −14.1768 −0.455424
$$970$$ 0 0
$$971$$ −8.71625 −0.279718 −0.139859 0.990171i $$-0.544665\pi$$
−0.139859 + 0.990171i $$0.544665\pi$$
$$972$$ 0 0
$$973$$ 13.7743 0.441585
$$974$$ 0 0
$$975$$ −2.38787 −0.0764731
$$976$$ 0 0
$$977$$ −33.7645 −1.08022 −0.540111 0.841594i $$-0.681618\pi$$
−0.540111 + 0.841594i $$0.681618\pi$$
$$978$$ 0 0
$$979$$ −15.0132 −0.479823
$$980$$ 0 0
$$981$$ −4.40437 −0.140621
$$982$$ 0 0
$$983$$ −43.6082 −1.39088 −0.695442 0.718582i $$-0.744791\pi$$
−0.695442 + 0.718582i $$0.744791\pi$$
$$984$$ 0 0
$$985$$ −24.2374 −0.772269
$$986$$ 0 0
$$987$$ 4.62530 0.147225
$$988$$ 0 0
$$989$$ 0.0558950 0.00177736
$$990$$ 0 0
$$991$$ 52.9741 1.68278 0.841390 0.540429i $$-0.181738\pi$$
0.841390 + 0.540429i $$0.181738\pi$$
$$992$$ 0 0
$$993$$ −28.0771 −0.891001
$$994$$ 0 0
$$995$$ 16.7513 0.531052
$$996$$ 0 0
$$997$$ −13.6326 −0.431749 −0.215874 0.976421i $$-0.569260\pi$$
−0.215874 + 0.976421i $$0.569260\pi$$
$$998$$ 0 0
$$999$$ −40.9986 −1.29714
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 9280.2.a.br.1.2 3
4.3 odd 2 9280.2.a.bj.1.2 3
8.3 odd 2 145.2.a.c.1.1 3
8.5 even 2 2320.2.a.n.1.2 3
24.11 even 2 1305.2.a.p.1.3 3
40.3 even 4 725.2.b.e.349.5 6
40.19 odd 2 725.2.a.e.1.3 3
40.27 even 4 725.2.b.e.349.2 6
56.27 even 2 7105.2.a.o.1.1 3
120.59 even 2 6525.2.a.be.1.1 3
232.115 odd 2 4205.2.a.f.1.3 3

By twisted newform
Twist Min Dim Char Parity Ord Type
145.2.a.c.1.1 3 8.3 odd 2
725.2.a.e.1.3 3 40.19 odd 2
725.2.b.e.349.2 6 40.27 even 4
725.2.b.e.349.5 6 40.3 even 4
1305.2.a.p.1.3 3 24.11 even 2
2320.2.a.n.1.2 3 8.5 even 2
4205.2.a.f.1.3 3 232.115 odd 2
6525.2.a.be.1.1 3 120.59 even 2
7105.2.a.o.1.1 3 56.27 even 2
9280.2.a.bj.1.2 3 4.3 odd 2
9280.2.a.br.1.2 3 1.1 even 1 trivial