# Properties

 Label 9280.2.a.bd.1.1 Level $9280$ Weight $2$ Character 9280.1 Self dual yes Analytic conductor $74.101$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Learn more

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: $$2$$ Coefficient field: $$\Q(\zeta_{10})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 4640) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-0.618034$$ 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)

 $$f(q)$$ $$=$$ $$q-0.618034 q^{3} +1.00000 q^{5} -2.85410 q^{7} -2.61803 q^{9} +O(q^{10})$$ $$q-0.618034 q^{3} +1.00000 q^{5} -2.85410 q^{7} -2.61803 q^{9} +3.23607 q^{11} +5.09017 q^{13} -0.618034 q^{15} -3.61803 q^{17} -2.76393 q^{19} +1.76393 q^{21} +5.85410 q^{23} +1.00000 q^{25} +3.47214 q^{27} -1.00000 q^{29} -1.61803 q^{31} -2.00000 q^{33} -2.85410 q^{35} -9.23607 q^{37} -3.14590 q^{39} -3.70820 q^{41} +7.61803 q^{43} -2.61803 q^{45} -8.00000 q^{47} +1.14590 q^{49} +2.23607 q^{51} -9.32624 q^{53} +3.23607 q^{55} +1.70820 q^{57} +9.38197 q^{59} -2.14590 q^{61} +7.47214 q^{63} +5.09017 q^{65} +2.47214 q^{67} -3.61803 q^{69} +12.9443 q^{71} +1.90983 q^{73} -0.618034 q^{75} -9.23607 q^{77} -2.90983 q^{79} +5.70820 q^{81} -15.7082 q^{83} -3.61803 q^{85} +0.618034 q^{87} -1.70820 q^{89} -14.5279 q^{91} +1.00000 q^{93} -2.76393 q^{95} +8.79837 q^{97} -8.47214 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} + 2 q^{5} + q^{7} - 3 q^{9}+O(q^{10})$$ 2 * q + q^3 + 2 * q^5 + q^7 - 3 * q^9 $$2 q + q^{3} + 2 q^{5} + q^{7} - 3 q^{9} + 2 q^{11} - q^{13} + q^{15} - 5 q^{17} - 10 q^{19} + 8 q^{21} + 5 q^{23} + 2 q^{25} - 2 q^{27} - 2 q^{29} - q^{31} - 4 q^{33} + q^{35} - 14 q^{37} - 13 q^{39} + 6 q^{41} + 13 q^{43} - 3 q^{45} - 16 q^{47} + 9 q^{49} - 3 q^{53} + 2 q^{55} - 10 q^{57} + 21 q^{59} - 11 q^{61} + 6 q^{63} - q^{65} - 4 q^{67} - 5 q^{69} + 8 q^{71} + 15 q^{73} + q^{75} - 14 q^{77} - 17 q^{79} - 2 q^{81} - 18 q^{83} - 5 q^{85} - q^{87} + 10 q^{89} - 38 q^{91} + 2 q^{93} - 10 q^{95} - 7 q^{97} - 8 q^{99}+O(q^{100})$$ 2 * q + q^3 + 2 * q^5 + q^7 - 3 * q^9 + 2 * q^11 - q^13 + q^15 - 5 * q^17 - 10 * q^19 + 8 * q^21 + 5 * q^23 + 2 * q^25 - 2 * q^27 - 2 * q^29 - q^31 - 4 * q^33 + q^35 - 14 * q^37 - 13 * q^39 + 6 * q^41 + 13 * q^43 - 3 * q^45 - 16 * q^47 + 9 * q^49 - 3 * q^53 + 2 * q^55 - 10 * q^57 + 21 * q^59 - 11 * q^61 + 6 * q^63 - q^65 - 4 * q^67 - 5 * q^69 + 8 * q^71 + 15 * q^73 + q^75 - 14 * q^77 - 17 * q^79 - 2 * q^81 - 18 * q^83 - 5 * q^85 - q^87 + 10 * q^89 - 38 * q^91 + 2 * q^93 - 10 * q^95 - 7 * q^97 - 8 * 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
$$3$$ −0.618034 −0.356822 −0.178411 0.983956i $$-0.557096\pi$$
−0.178411 + 0.983956i $$0.557096\pi$$
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ −2.85410 −1.07875 −0.539375 0.842066i $$-0.681339\pi$$
−0.539375 + 0.842066i $$0.681339\pi$$
$$8$$ 0 0
$$9$$ −2.61803 −0.872678
$$10$$ 0 0
$$11$$ 3.23607 0.975711 0.487856 0.872924i $$-0.337779\pi$$
0.487856 + 0.872924i $$0.337779\pi$$
$$12$$ 0 0
$$13$$ 5.09017 1.41176 0.705880 0.708332i $$-0.250552\pi$$
0.705880 + 0.708332i $$0.250552\pi$$
$$14$$ 0 0
$$15$$ −0.618034 −0.159576
$$16$$ 0 0
$$17$$ −3.61803 −0.877502 −0.438751 0.898609i $$-0.644579\pi$$
−0.438751 + 0.898609i $$0.644579\pi$$
$$18$$ 0 0
$$19$$ −2.76393 −0.634089 −0.317045 0.948411i $$-0.602691\pi$$
−0.317045 + 0.948411i $$0.602691\pi$$
$$20$$ 0 0
$$21$$ 1.76393 0.384922
$$22$$ 0 0
$$23$$ 5.85410 1.22066 0.610332 0.792145i $$-0.291036\pi$$
0.610332 + 0.792145i $$0.291036\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 3.47214 0.668213
$$28$$ 0 0
$$29$$ −1.00000 −0.185695
$$30$$ 0 0
$$31$$ −1.61803 −0.290607 −0.145304 0.989387i $$-0.546416\pi$$
−0.145304 + 0.989387i $$0.546416\pi$$
$$32$$ 0 0
$$33$$ −2.00000 −0.348155
$$34$$ 0 0
$$35$$ −2.85410 −0.482431
$$36$$ 0 0
$$37$$ −9.23607 −1.51840 −0.759200 0.650857i $$-0.774410\pi$$
−0.759200 + 0.650857i $$0.774410\pi$$
$$38$$ 0 0
$$39$$ −3.14590 −0.503747
$$40$$ 0 0
$$41$$ −3.70820 −0.579124 −0.289562 0.957159i $$-0.593510\pi$$
−0.289562 + 0.957159i $$0.593510\pi$$
$$42$$ 0 0
$$43$$ 7.61803 1.16174 0.580870 0.813997i $$-0.302713\pi$$
0.580870 + 0.813997i $$0.302713\pi$$
$$44$$ 0 0
$$45$$ −2.61803 −0.390273
$$46$$ 0 0
$$47$$ −8.00000 −1.16692 −0.583460 0.812142i $$-0.698301\pi$$
−0.583460 + 0.812142i $$0.698301\pi$$
$$48$$ 0 0
$$49$$ 1.14590 0.163700
$$50$$ 0 0
$$51$$ 2.23607 0.313112
$$52$$ 0 0
$$53$$ −9.32624 −1.28106 −0.640529 0.767934i $$-0.721285\pi$$
−0.640529 + 0.767934i $$0.721285\pi$$
$$54$$ 0 0
$$55$$ 3.23607 0.436351
$$56$$ 0 0
$$57$$ 1.70820 0.226257
$$58$$ 0 0
$$59$$ 9.38197 1.22143 0.610714 0.791851i $$-0.290883\pi$$
0.610714 + 0.791851i $$0.290883\pi$$
$$60$$ 0 0
$$61$$ −2.14590 −0.274754 −0.137377 0.990519i $$-0.543867\pi$$
−0.137377 + 0.990519i $$0.543867\pi$$
$$62$$ 0 0
$$63$$ 7.47214 0.941401
$$64$$ 0 0
$$65$$ 5.09017 0.631358
$$66$$ 0 0
$$67$$ 2.47214 0.302019 0.151010 0.988532i $$-0.451748\pi$$
0.151010 + 0.988532i $$0.451748\pi$$
$$68$$ 0 0
$$69$$ −3.61803 −0.435560
$$70$$ 0 0
$$71$$ 12.9443 1.53620 0.768101 0.640328i $$-0.221202\pi$$
0.768101 + 0.640328i $$0.221202\pi$$
$$72$$ 0 0
$$73$$ 1.90983 0.223529 0.111764 0.993735i $$-0.464350\pi$$
0.111764 + 0.993735i $$0.464350\pi$$
$$74$$ 0 0
$$75$$ −0.618034 −0.0713644
$$76$$ 0 0
$$77$$ −9.23607 −1.05255
$$78$$ 0 0
$$79$$ −2.90983 −0.327381 −0.163691 0.986512i $$-0.552340\pi$$
−0.163691 + 0.986512i $$0.552340\pi$$
$$80$$ 0 0
$$81$$ 5.70820 0.634245
$$82$$ 0 0
$$83$$ −15.7082 −1.72420 −0.862100 0.506739i $$-0.830851\pi$$
−0.862100 + 0.506739i $$0.830851\pi$$
$$84$$ 0 0
$$85$$ −3.61803 −0.392431
$$86$$ 0 0
$$87$$ 0.618034 0.0662602
$$88$$ 0 0
$$89$$ −1.70820 −0.181069 −0.0905346 0.995893i $$-0.528858\pi$$
−0.0905346 + 0.995893i $$0.528858\pi$$
$$90$$ 0 0
$$91$$ −14.5279 −1.52293
$$92$$ 0 0
$$93$$ 1.00000 0.103695
$$94$$ 0 0
$$95$$ −2.76393 −0.283573
$$96$$ 0 0
$$97$$ 8.79837 0.893340 0.446670 0.894699i $$-0.352610\pi$$
0.446670 + 0.894699i $$0.352610\pi$$
$$98$$ 0 0
$$99$$ −8.47214 −0.851482
$$100$$ 0 0
$$101$$ 7.09017 0.705498 0.352749 0.935718i $$-0.385247\pi$$
0.352749 + 0.935718i $$0.385247\pi$$
$$102$$ 0 0
$$103$$ 8.94427 0.881305 0.440653 0.897678i $$-0.354747\pi$$
0.440653 + 0.897678i $$0.354747\pi$$
$$104$$ 0 0
$$105$$ 1.76393 0.172142
$$106$$ 0 0
$$107$$ −0.291796 −0.0282090 −0.0141045 0.999901i $$-0.504490\pi$$
−0.0141045 + 0.999901i $$0.504490\pi$$
$$108$$ 0 0
$$109$$ −11.7082 −1.12144 −0.560721 0.828005i $$-0.689476\pi$$
−0.560721 + 0.828005i $$0.689476\pi$$
$$110$$ 0 0
$$111$$ 5.70820 0.541799
$$112$$ 0 0
$$113$$ 8.79837 0.827681 0.413841 0.910349i $$-0.364187\pi$$
0.413841 + 0.910349i $$0.364187\pi$$
$$114$$ 0 0
$$115$$ 5.85410 0.545898
$$116$$ 0 0
$$117$$ −13.3262 −1.23201
$$118$$ 0 0
$$119$$ 10.3262 0.946605
$$120$$ 0 0
$$121$$ −0.527864 −0.0479876
$$122$$ 0 0
$$123$$ 2.29180 0.206644
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ −0.944272 −0.0837906 −0.0418953 0.999122i $$-0.513340\pi$$
−0.0418953 + 0.999122i $$0.513340\pi$$
$$128$$ 0 0
$$129$$ −4.70820 −0.414534
$$130$$ 0 0
$$131$$ 18.9443 1.65517 0.827584 0.561341i $$-0.189715\pi$$
0.827584 + 0.561341i $$0.189715\pi$$
$$132$$ 0 0
$$133$$ 7.88854 0.684023
$$134$$ 0 0
$$135$$ 3.47214 0.298834
$$136$$ 0 0
$$137$$ −0.145898 −0.0124649 −0.00623246 0.999981i $$-0.501984\pi$$
−0.00623246 + 0.999981i $$0.501984\pi$$
$$138$$ 0 0
$$139$$ −5.61803 −0.476515 −0.238258 0.971202i $$-0.576576\pi$$
−0.238258 + 0.971202i $$0.576576\pi$$
$$140$$ 0 0
$$141$$ 4.94427 0.416383
$$142$$ 0 0
$$143$$ 16.4721 1.37747
$$144$$ 0 0
$$145$$ −1.00000 −0.0830455
$$146$$ 0 0
$$147$$ −0.708204 −0.0584117
$$148$$ 0 0
$$149$$ −16.1803 −1.32555 −0.662773 0.748821i $$-0.730620\pi$$
−0.662773 + 0.748821i $$0.730620\pi$$
$$150$$ 0 0
$$151$$ 1.23607 0.100590 0.0502949 0.998734i $$-0.483984\pi$$
0.0502949 + 0.998734i $$0.483984\pi$$
$$152$$ 0 0
$$153$$ 9.47214 0.765777
$$154$$ 0 0
$$155$$ −1.61803 −0.129964
$$156$$ 0 0
$$157$$ −5.05573 −0.403491 −0.201746 0.979438i $$-0.564661\pi$$
−0.201746 + 0.979438i $$0.564661\pi$$
$$158$$ 0 0
$$159$$ 5.76393 0.457110
$$160$$ 0 0
$$161$$ −16.7082 −1.31679
$$162$$ 0 0
$$163$$ 13.5279 1.05958 0.529792 0.848128i $$-0.322270\pi$$
0.529792 + 0.848128i $$0.322270\pi$$
$$164$$ 0 0
$$165$$ −2.00000 −0.155700
$$166$$ 0 0
$$167$$ −4.38197 −0.339087 −0.169543 0.985523i $$-0.554229\pi$$
−0.169543 + 0.985523i $$0.554229\pi$$
$$168$$ 0 0
$$169$$ 12.9098 0.993064
$$170$$ 0 0
$$171$$ 7.23607 0.553356
$$172$$ 0 0
$$173$$ 4.79837 0.364814 0.182407 0.983223i $$-0.441611\pi$$
0.182407 + 0.983223i $$0.441611\pi$$
$$174$$ 0 0
$$175$$ −2.85410 −0.215750
$$176$$ 0 0
$$177$$ −5.79837 −0.435832
$$178$$ 0 0
$$179$$ 1.09017 0.0814831 0.0407416 0.999170i $$-0.487028\pi$$
0.0407416 + 0.999170i $$0.487028\pi$$
$$180$$ 0 0
$$181$$ −0.291796 −0.0216890 −0.0108445 0.999941i $$-0.503452\pi$$
−0.0108445 + 0.999941i $$0.503452\pi$$
$$182$$ 0 0
$$183$$ 1.32624 0.0980383
$$184$$ 0 0
$$185$$ −9.23607 −0.679049
$$186$$ 0 0
$$187$$ −11.7082 −0.856189
$$188$$ 0 0
$$189$$ −9.90983 −0.720834
$$190$$ 0 0
$$191$$ −20.8541 −1.50895 −0.754475 0.656329i $$-0.772108\pi$$
−0.754475 + 0.656329i $$0.772108\pi$$
$$192$$ 0 0
$$193$$ 0.145898 0.0105020 0.00525099 0.999986i $$-0.498329\pi$$
0.00525099 + 0.999986i $$0.498329\pi$$
$$194$$ 0 0
$$195$$ −3.14590 −0.225282
$$196$$ 0 0
$$197$$ −18.8541 −1.34330 −0.671650 0.740869i $$-0.734414\pi$$
−0.671650 + 0.740869i $$0.734414\pi$$
$$198$$ 0 0
$$199$$ −8.76393 −0.621259 −0.310629 0.950531i $$-0.600540\pi$$
−0.310629 + 0.950531i $$0.600540\pi$$
$$200$$ 0 0
$$201$$ −1.52786 −0.107767
$$202$$ 0 0
$$203$$ 2.85410 0.200319
$$204$$ 0 0
$$205$$ −3.70820 −0.258992
$$206$$ 0 0
$$207$$ −15.3262 −1.06525
$$208$$ 0 0
$$209$$ −8.94427 −0.618688
$$210$$ 0 0
$$211$$ −15.2361 −1.04889 −0.524447 0.851443i $$-0.675728\pi$$
−0.524447 + 0.851443i $$0.675728\pi$$
$$212$$ 0 0
$$213$$ −8.00000 −0.548151
$$214$$ 0 0
$$215$$ 7.61803 0.519546
$$216$$ 0 0
$$217$$ 4.61803 0.313493
$$218$$ 0 0
$$219$$ −1.18034 −0.0797600
$$220$$ 0 0
$$221$$ −18.4164 −1.23882
$$222$$ 0 0
$$223$$ −24.9787 −1.67270 −0.836349 0.548197i $$-0.815314\pi$$
−0.836349 + 0.548197i $$0.815314\pi$$
$$224$$ 0 0
$$225$$ −2.61803 −0.174536
$$226$$ 0 0
$$227$$ −2.00000 −0.132745 −0.0663723 0.997795i $$-0.521143\pi$$
−0.0663723 + 0.997795i $$0.521143\pi$$
$$228$$ 0 0
$$229$$ −4.09017 −0.270286 −0.135143 0.990826i $$-0.543149\pi$$
−0.135143 + 0.990826i $$0.543149\pi$$
$$230$$ 0 0
$$231$$ 5.70820 0.375572
$$232$$ 0 0
$$233$$ 16.4721 1.07913 0.539563 0.841945i $$-0.318590\pi$$
0.539563 + 0.841945i $$0.318590\pi$$
$$234$$ 0 0
$$235$$ −8.00000 −0.521862
$$236$$ 0 0
$$237$$ 1.79837 0.116817
$$238$$ 0 0
$$239$$ 11.5279 0.745676 0.372838 0.927897i $$-0.378385\pi$$
0.372838 + 0.927897i $$0.378385\pi$$
$$240$$ 0 0
$$241$$ −1.20163 −0.0774035 −0.0387018 0.999251i $$-0.512322\pi$$
−0.0387018 + 0.999251i $$0.512322\pi$$
$$242$$ 0 0
$$243$$ −13.9443 −0.894525
$$244$$ 0 0
$$245$$ 1.14590 0.0732087
$$246$$ 0 0
$$247$$ −14.0689 −0.895182
$$248$$ 0 0
$$249$$ 9.70820 0.615232
$$250$$ 0 0
$$251$$ 18.1803 1.14753 0.573766 0.819019i $$-0.305482\pi$$
0.573766 + 0.819019i $$0.305482\pi$$
$$252$$ 0 0
$$253$$ 18.9443 1.19102
$$254$$ 0 0
$$255$$ 2.23607 0.140028
$$256$$ 0 0
$$257$$ −28.0689 −1.75089 −0.875444 0.483319i $$-0.839431\pi$$
−0.875444 + 0.483319i $$0.839431\pi$$
$$258$$ 0 0
$$259$$ 26.3607 1.63797
$$260$$ 0 0
$$261$$ 2.61803 0.162052
$$262$$ 0 0
$$263$$ −17.5279 −1.08081 −0.540407 0.841404i $$-0.681730\pi$$
−0.540407 + 0.841404i $$0.681730\pi$$
$$264$$ 0 0
$$265$$ −9.32624 −0.572906
$$266$$ 0 0
$$267$$ 1.05573 0.0646095
$$268$$ 0 0
$$269$$ −26.5623 −1.61953 −0.809766 0.586753i $$-0.800406\pi$$
−0.809766 + 0.586753i $$0.800406\pi$$
$$270$$ 0 0
$$271$$ −15.4164 −0.936480 −0.468240 0.883601i $$-0.655112\pi$$
−0.468240 + 0.883601i $$0.655112\pi$$
$$272$$ 0 0
$$273$$ 8.97871 0.543416
$$274$$ 0 0
$$275$$ 3.23607 0.195142
$$276$$ 0 0
$$277$$ −30.0000 −1.80253 −0.901263 0.433273i $$-0.857359\pi$$
−0.901263 + 0.433273i $$0.857359\pi$$
$$278$$ 0 0
$$279$$ 4.23607 0.253607
$$280$$ 0 0
$$281$$ −4.85410 −0.289571 −0.144786 0.989463i $$-0.546249\pi$$
−0.144786 + 0.989463i $$0.546249\pi$$
$$282$$ 0 0
$$283$$ −22.1803 −1.31848 −0.659242 0.751931i $$-0.729123\pi$$
−0.659242 + 0.751931i $$0.729123\pi$$
$$284$$ 0 0
$$285$$ 1.70820 0.101185
$$286$$ 0 0
$$287$$ 10.5836 0.624730
$$288$$ 0 0
$$289$$ −3.90983 −0.229990
$$290$$ 0 0
$$291$$ −5.43769 −0.318763
$$292$$ 0 0
$$293$$ −14.1803 −0.828424 −0.414212 0.910180i $$-0.635943\pi$$
−0.414212 + 0.910180i $$0.635943\pi$$
$$294$$ 0 0
$$295$$ 9.38197 0.546239
$$296$$ 0 0
$$297$$ 11.2361 0.651983
$$298$$ 0 0
$$299$$ 29.7984 1.72328
$$300$$ 0 0
$$301$$ −21.7426 −1.25323
$$302$$ 0 0
$$303$$ −4.38197 −0.251737
$$304$$ 0 0
$$305$$ −2.14590 −0.122874
$$306$$ 0 0
$$307$$ 9.52786 0.543784 0.271892 0.962328i $$-0.412351\pi$$
0.271892 + 0.962328i $$0.412351\pi$$
$$308$$ 0 0
$$309$$ −5.52786 −0.314469
$$310$$ 0 0
$$311$$ 6.32624 0.358728 0.179364 0.983783i $$-0.442596\pi$$
0.179364 + 0.983783i $$0.442596\pi$$
$$312$$ 0 0
$$313$$ −3.88854 −0.219793 −0.109897 0.993943i $$-0.535052\pi$$
−0.109897 + 0.993943i $$0.535052\pi$$
$$314$$ 0 0
$$315$$ 7.47214 0.421007
$$316$$ 0 0
$$317$$ 4.76393 0.267569 0.133785 0.991010i $$-0.457287\pi$$
0.133785 + 0.991010i $$0.457287\pi$$
$$318$$ 0 0
$$319$$ −3.23607 −0.181185
$$320$$ 0 0
$$321$$ 0.180340 0.0100656
$$322$$ 0 0
$$323$$ 10.0000 0.556415
$$324$$ 0 0
$$325$$ 5.09017 0.282352
$$326$$ 0 0
$$327$$ 7.23607 0.400155
$$328$$ 0 0
$$329$$ 22.8328 1.25881
$$330$$ 0 0
$$331$$ 10.2918 0.565688 0.282844 0.959166i $$-0.408722\pi$$
0.282844 + 0.959166i $$0.408722\pi$$
$$332$$ 0 0
$$333$$ 24.1803 1.32507
$$334$$ 0 0
$$335$$ 2.47214 0.135067
$$336$$ 0 0
$$337$$ 17.1459 0.933997 0.466998 0.884258i $$-0.345335\pi$$
0.466998 + 0.884258i $$0.345335\pi$$
$$338$$ 0 0
$$339$$ −5.43769 −0.295335
$$340$$ 0 0
$$341$$ −5.23607 −0.283549
$$342$$ 0 0
$$343$$ 16.7082 0.902158
$$344$$ 0 0
$$345$$ −3.61803 −0.194788
$$346$$ 0 0
$$347$$ −30.7639 −1.65149 −0.825747 0.564040i $$-0.809246\pi$$
−0.825747 + 0.564040i $$0.809246\pi$$
$$348$$ 0 0
$$349$$ 12.9443 0.692891 0.346445 0.938070i $$-0.387389\pi$$
0.346445 + 0.938070i $$0.387389\pi$$
$$350$$ 0 0
$$351$$ 17.6738 0.943356
$$352$$ 0 0
$$353$$ −18.4721 −0.983173 −0.491586 0.870829i $$-0.663583\pi$$
−0.491586 + 0.870829i $$0.663583\pi$$
$$354$$ 0 0
$$355$$ 12.9443 0.687011
$$356$$ 0 0
$$357$$ −6.38197 −0.337769
$$358$$ 0 0
$$359$$ 14.9098 0.786911 0.393455 0.919344i $$-0.371280\pi$$
0.393455 + 0.919344i $$0.371280\pi$$
$$360$$ 0 0
$$361$$ −11.3607 −0.597931
$$362$$ 0 0
$$363$$ 0.326238 0.0171231
$$364$$ 0 0
$$365$$ 1.90983 0.0999651
$$366$$ 0 0
$$367$$ −18.1803 −0.949006 −0.474503 0.880254i $$-0.657372\pi$$
−0.474503 + 0.880254i $$0.657372\pi$$
$$368$$ 0 0
$$369$$ 9.70820 0.505389
$$370$$ 0 0
$$371$$ 26.6180 1.38194
$$372$$ 0 0
$$373$$ −26.2705 −1.36024 −0.680118 0.733103i $$-0.738071\pi$$
−0.680118 + 0.733103i $$0.738071\pi$$
$$374$$ 0 0
$$375$$ −0.618034 −0.0319151
$$376$$ 0 0
$$377$$ −5.09017 −0.262157
$$378$$ 0 0
$$379$$ −38.0689 −1.95547 −0.977734 0.209850i $$-0.932702\pi$$
−0.977734 + 0.209850i $$0.932702\pi$$
$$380$$ 0 0
$$381$$ 0.583592 0.0298983
$$382$$ 0 0
$$383$$ −26.2705 −1.34236 −0.671180 0.741294i $$-0.734212\pi$$
−0.671180 + 0.741294i $$0.734212\pi$$
$$384$$ 0 0
$$385$$ −9.23607 −0.470714
$$386$$ 0 0
$$387$$ −19.9443 −1.01382
$$388$$ 0 0
$$389$$ 18.0000 0.912636 0.456318 0.889817i $$-0.349168\pi$$
0.456318 + 0.889817i $$0.349168\pi$$
$$390$$ 0 0
$$391$$ −21.1803 −1.07114
$$392$$ 0 0
$$393$$ −11.7082 −0.590601
$$394$$ 0 0
$$395$$ −2.90983 −0.146409
$$396$$ 0 0
$$397$$ −5.14590 −0.258265 −0.129133 0.991627i $$-0.541219\pi$$
−0.129133 + 0.991627i $$0.541219\pi$$
$$398$$ 0 0
$$399$$ −4.87539 −0.244075
$$400$$ 0 0
$$401$$ 8.32624 0.415792 0.207896 0.978151i $$-0.433338\pi$$
0.207896 + 0.978151i $$0.433338\pi$$
$$402$$ 0 0
$$403$$ −8.23607 −0.410268
$$404$$ 0 0
$$405$$ 5.70820 0.283643
$$406$$ 0 0
$$407$$ −29.8885 −1.48152
$$408$$ 0 0
$$409$$ 7.41641 0.366718 0.183359 0.983046i $$-0.441303\pi$$
0.183359 + 0.983046i $$0.441303\pi$$
$$410$$ 0 0
$$411$$ 0.0901699 0.00444776
$$412$$ 0 0
$$413$$ −26.7771 −1.31761
$$414$$ 0 0
$$415$$ −15.7082 −0.771085
$$416$$ 0 0
$$417$$ 3.47214 0.170031
$$418$$ 0 0
$$419$$ −3.90983 −0.191008 −0.0955038 0.995429i $$-0.530446\pi$$
−0.0955038 + 0.995429i $$0.530446\pi$$
$$420$$ 0 0
$$421$$ −7.52786 −0.366886 −0.183443 0.983030i $$-0.558724\pi$$
−0.183443 + 0.983030i $$0.558724\pi$$
$$422$$ 0 0
$$423$$ 20.9443 1.01835
$$424$$ 0 0
$$425$$ −3.61803 −0.175500
$$426$$ 0 0
$$427$$ 6.12461 0.296391
$$428$$ 0 0
$$429$$ −10.1803 −0.491511
$$430$$ 0 0
$$431$$ 23.7082 1.14198 0.570992 0.820956i $$-0.306559\pi$$
0.570992 + 0.820956i $$0.306559\pi$$
$$432$$ 0 0
$$433$$ 4.47214 0.214917 0.107459 0.994210i $$-0.465729\pi$$
0.107459 + 0.994210i $$0.465729\pi$$
$$434$$ 0 0
$$435$$ 0.618034 0.0296325
$$436$$ 0 0
$$437$$ −16.1803 −0.774011
$$438$$ 0 0
$$439$$ 6.00000 0.286364 0.143182 0.989696i $$-0.454267\pi$$
0.143182 + 0.989696i $$0.454267\pi$$
$$440$$ 0 0
$$441$$ −3.00000 −0.142857
$$442$$ 0 0
$$443$$ −25.2705 −1.20064 −0.600319 0.799761i $$-0.704960\pi$$
−0.600319 + 0.799761i $$0.704960\pi$$
$$444$$ 0 0
$$445$$ −1.70820 −0.0809766
$$446$$ 0 0
$$447$$ 10.0000 0.472984
$$448$$ 0 0
$$449$$ −24.0689 −1.13588 −0.567940 0.823070i $$-0.692260\pi$$
−0.567940 + 0.823070i $$0.692260\pi$$
$$450$$ 0 0
$$451$$ −12.0000 −0.565058
$$452$$ 0 0
$$453$$ −0.763932 −0.0358927
$$454$$ 0 0
$$455$$ −14.5279 −0.681077
$$456$$ 0 0
$$457$$ 22.0000 1.02912 0.514558 0.857455i $$-0.327956\pi$$
0.514558 + 0.857455i $$0.327956\pi$$
$$458$$ 0 0
$$459$$ −12.5623 −0.586358
$$460$$ 0 0
$$461$$ 19.3262 0.900113 0.450056 0.893000i $$-0.351404\pi$$
0.450056 + 0.893000i $$0.351404\pi$$
$$462$$ 0 0
$$463$$ −38.8328 −1.80471 −0.902357 0.430989i $$-0.858165\pi$$
−0.902357 + 0.430989i $$0.858165\pi$$
$$464$$ 0 0
$$465$$ 1.00000 0.0463739
$$466$$ 0 0
$$467$$ 22.6738 1.04922 0.524608 0.851344i $$-0.324212\pi$$
0.524608 + 0.851344i $$0.324212\pi$$
$$468$$ 0 0
$$469$$ −7.05573 −0.325803
$$470$$ 0 0
$$471$$ 3.12461 0.143975
$$472$$ 0 0
$$473$$ 24.6525 1.13352
$$474$$ 0 0
$$475$$ −2.76393 −0.126818
$$476$$ 0 0
$$477$$ 24.4164 1.11795
$$478$$ 0 0
$$479$$ 0.673762 0.0307850 0.0153925 0.999882i $$-0.495100\pi$$
0.0153925 + 0.999882i $$0.495100\pi$$
$$480$$ 0 0
$$481$$ −47.0132 −2.14362
$$482$$ 0 0
$$483$$ 10.3262 0.469860
$$484$$ 0 0
$$485$$ 8.79837 0.399514
$$486$$ 0 0
$$487$$ −21.2705 −0.963859 −0.481929 0.876210i $$-0.660064\pi$$
−0.481929 + 0.876210i $$0.660064\pi$$
$$488$$ 0 0
$$489$$ −8.36068 −0.378083
$$490$$ 0 0
$$491$$ 41.3050 1.86407 0.932033 0.362373i $$-0.118033\pi$$
0.932033 + 0.362373i $$0.118033\pi$$
$$492$$ 0 0
$$493$$ 3.61803 0.162948
$$494$$ 0 0
$$495$$ −8.47214 −0.380794
$$496$$ 0 0
$$497$$ −36.9443 −1.65718
$$498$$ 0 0
$$499$$ 1.67376 0.0749279 0.0374639 0.999298i $$-0.488072\pi$$
0.0374639 + 0.999298i $$0.488072\pi$$
$$500$$ 0 0
$$501$$ 2.70820 0.120994
$$502$$ 0 0
$$503$$ 20.0000 0.891756 0.445878 0.895094i $$-0.352892\pi$$
0.445878 + 0.895094i $$0.352892\pi$$
$$504$$ 0 0
$$505$$ 7.09017 0.315508
$$506$$ 0 0
$$507$$ −7.97871 −0.354347
$$508$$ 0 0
$$509$$ 3.70820 0.164363 0.0821816 0.996617i $$-0.473811\pi$$
0.0821816 + 0.996617i $$0.473811\pi$$
$$510$$ 0 0
$$511$$ −5.45085 −0.241131
$$512$$ 0 0
$$513$$ −9.59675 −0.423707
$$514$$ 0 0
$$515$$ 8.94427 0.394132
$$516$$ 0 0
$$517$$ −25.8885 −1.13858
$$518$$ 0 0
$$519$$ −2.96556 −0.130174
$$520$$ 0 0
$$521$$ −35.3262 −1.54767 −0.773835 0.633387i $$-0.781664\pi$$
−0.773835 + 0.633387i $$0.781664\pi$$
$$522$$ 0 0
$$523$$ 10.1803 0.445155 0.222578 0.974915i $$-0.428553\pi$$
0.222578 + 0.974915i $$0.428553\pi$$
$$524$$ 0 0
$$525$$ 1.76393 0.0769843
$$526$$ 0 0
$$527$$ 5.85410 0.255009
$$528$$ 0 0
$$529$$ 11.2705 0.490022
$$530$$ 0 0
$$531$$ −24.5623 −1.06591
$$532$$ 0 0
$$533$$ −18.8754 −0.817584
$$534$$ 0 0
$$535$$ −0.291796 −0.0126154
$$536$$ 0 0
$$537$$ −0.673762 −0.0290750
$$538$$ 0 0
$$539$$ 3.70820 0.159724
$$540$$ 0 0
$$541$$ −5.03444 −0.216448 −0.108224 0.994127i $$-0.534516\pi$$
−0.108224 + 0.994127i $$0.534516\pi$$
$$542$$ 0 0
$$543$$ 0.180340 0.00773913
$$544$$ 0 0
$$545$$ −11.7082 −0.501524
$$546$$ 0 0
$$547$$ −19.1246 −0.817709 −0.408855 0.912600i $$-0.634072\pi$$
−0.408855 + 0.912600i $$0.634072\pi$$
$$548$$ 0 0
$$549$$ 5.61803 0.239772
$$550$$ 0 0
$$551$$ 2.76393 0.117747
$$552$$ 0 0
$$553$$ 8.30495 0.353162
$$554$$ 0 0
$$555$$ 5.70820 0.242300
$$556$$ 0 0
$$557$$ −24.1459 −1.02309 −0.511547 0.859255i $$-0.670927\pi$$
−0.511547 + 0.859255i $$0.670927\pi$$
$$558$$ 0 0
$$559$$ 38.7771 1.64010
$$560$$ 0 0
$$561$$ 7.23607 0.305507
$$562$$ 0 0
$$563$$ −28.3951 −1.19671 −0.598356 0.801230i $$-0.704179\pi$$
−0.598356 + 0.801230i $$0.704179\pi$$
$$564$$ 0 0
$$565$$ 8.79837 0.370150
$$566$$ 0 0
$$567$$ −16.2918 −0.684191
$$568$$ 0 0
$$569$$ 32.3607 1.35663 0.678315 0.734771i $$-0.262710\pi$$
0.678315 + 0.734771i $$0.262710\pi$$
$$570$$ 0 0
$$571$$ 20.1459 0.843080 0.421540 0.906810i $$-0.361490\pi$$
0.421540 + 0.906810i $$0.361490\pi$$
$$572$$ 0 0
$$573$$ 12.8885 0.538427
$$574$$ 0 0
$$575$$ 5.85410 0.244133
$$576$$ 0 0
$$577$$ 12.7426 0.530483 0.265242 0.964182i $$-0.414548\pi$$
0.265242 + 0.964182i $$0.414548\pi$$
$$578$$ 0 0
$$579$$ −0.0901699 −0.00374733
$$580$$ 0 0
$$581$$ 44.8328 1.85998
$$582$$ 0 0
$$583$$ −30.1803 −1.24994
$$584$$ 0 0
$$585$$ −13.3262 −0.550972
$$586$$ 0 0
$$587$$ 8.83282 0.364569 0.182285 0.983246i $$-0.441651\pi$$
0.182285 + 0.983246i $$0.441651\pi$$
$$588$$ 0 0
$$589$$ 4.47214 0.184271
$$590$$ 0 0
$$591$$ 11.6525 0.479319
$$592$$ 0 0
$$593$$ −6.65248 −0.273184 −0.136592 0.990627i $$-0.543615\pi$$
−0.136592 + 0.990627i $$0.543615\pi$$
$$594$$ 0 0
$$595$$ 10.3262 0.423334
$$596$$ 0 0
$$597$$ 5.41641 0.221679
$$598$$ 0 0
$$599$$ −0.673762 −0.0275292 −0.0137646 0.999905i $$-0.504382\pi$$
−0.0137646 + 0.999905i $$0.504382\pi$$
$$600$$ 0 0
$$601$$ 20.5410 0.837886 0.418943 0.908013i $$-0.362401\pi$$
0.418943 + 0.908013i $$0.362401\pi$$
$$602$$ 0 0
$$603$$ −6.47214 −0.263566
$$604$$ 0 0
$$605$$ −0.527864 −0.0214607
$$606$$ 0 0
$$607$$ −2.47214 −0.100341 −0.0501705 0.998741i $$-0.515976\pi$$
−0.0501705 + 0.998741i $$0.515976\pi$$
$$608$$ 0 0
$$609$$ −1.76393 −0.0714781
$$610$$ 0 0
$$611$$ −40.7214 −1.64741
$$612$$ 0 0
$$613$$ −32.7426 −1.32246 −0.661232 0.750182i $$-0.729966\pi$$
−0.661232 + 0.750182i $$0.729966\pi$$
$$614$$ 0 0
$$615$$ 2.29180 0.0924141
$$616$$ 0 0
$$617$$ −31.7984 −1.28015 −0.640077 0.768311i $$-0.721098\pi$$
−0.640077 + 0.768311i $$0.721098\pi$$
$$618$$ 0 0
$$619$$ −41.7771 −1.67916 −0.839581 0.543234i $$-0.817200\pi$$
−0.839581 + 0.543234i $$0.817200\pi$$
$$620$$ 0 0
$$621$$ 20.3262 0.815664
$$622$$ 0 0
$$623$$ 4.87539 0.195328
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 5.52786 0.220762
$$628$$ 0 0
$$629$$ 33.4164 1.33240
$$630$$ 0 0
$$631$$ 38.3607 1.52711 0.763557 0.645740i $$-0.223451\pi$$
0.763557 + 0.645740i $$0.223451\pi$$
$$632$$ 0 0
$$633$$ 9.41641 0.374269
$$634$$ 0 0
$$635$$ −0.944272 −0.0374723
$$636$$ 0 0
$$637$$ 5.83282 0.231105
$$638$$ 0 0
$$639$$ −33.8885 −1.34061
$$640$$ 0 0
$$641$$ −38.7639 −1.53108 −0.765542 0.643386i $$-0.777529\pi$$
−0.765542 + 0.643386i $$0.777529\pi$$
$$642$$ 0 0
$$643$$ 25.7082 1.01383 0.506916 0.861995i $$-0.330785\pi$$
0.506916 + 0.861995i $$0.330785\pi$$
$$644$$ 0 0
$$645$$ −4.70820 −0.185385
$$646$$ 0 0
$$647$$ 37.3050 1.46661 0.733304 0.679900i $$-0.237977\pi$$
0.733304 + 0.679900i $$0.237977\pi$$
$$648$$ 0 0
$$649$$ 30.3607 1.19176
$$650$$ 0 0
$$651$$ −2.85410 −0.111861
$$652$$ 0 0
$$653$$ −39.5967 −1.54954 −0.774770 0.632243i $$-0.782134\pi$$
−0.774770 + 0.632243i $$0.782134\pi$$
$$654$$ 0 0
$$655$$ 18.9443 0.740214
$$656$$ 0 0
$$657$$ −5.00000 −0.195069
$$658$$ 0 0
$$659$$ 15.8197 0.616246 0.308123 0.951346i $$-0.400299\pi$$
0.308123 + 0.951346i $$0.400299\pi$$
$$660$$ 0 0
$$661$$ −27.5967 −1.07339 −0.536695 0.843777i $$-0.680327\pi$$
−0.536695 + 0.843777i $$0.680327\pi$$
$$662$$ 0 0
$$663$$ 11.3820 0.442039
$$664$$ 0 0
$$665$$ 7.88854 0.305905
$$666$$ 0 0
$$667$$ −5.85410 −0.226672
$$668$$ 0 0
$$669$$ 15.4377 0.596856
$$670$$ 0 0
$$671$$ −6.94427 −0.268081
$$672$$ 0 0
$$673$$ 18.7639 0.723296 0.361648 0.932315i $$-0.382214\pi$$
0.361648 + 0.932315i $$0.382214\pi$$
$$674$$ 0 0
$$675$$ 3.47214 0.133643
$$676$$ 0 0
$$677$$ 13.8885 0.533780 0.266890 0.963727i $$-0.414004\pi$$
0.266890 + 0.963727i $$0.414004\pi$$
$$678$$ 0 0
$$679$$ −25.1115 −0.963689
$$680$$ 0 0
$$681$$ 1.23607 0.0473662
$$682$$ 0 0
$$683$$ −15.3475 −0.587257 −0.293628 0.955920i $$-0.594863\pi$$
−0.293628 + 0.955920i $$0.594863\pi$$
$$684$$ 0 0
$$685$$ −0.145898 −0.00557448
$$686$$ 0 0
$$687$$ 2.52786 0.0964440
$$688$$ 0 0
$$689$$ −47.4721 −1.80854
$$690$$ 0 0
$$691$$ −13.0344 −0.495854 −0.247927 0.968779i $$-0.579749\pi$$
−0.247927 + 0.968779i $$0.579749\pi$$
$$692$$ 0 0
$$693$$ 24.1803 0.918535
$$694$$ 0 0
$$695$$ −5.61803 −0.213104
$$696$$ 0 0
$$697$$ 13.4164 0.508183
$$698$$ 0 0
$$699$$ −10.1803 −0.385056
$$700$$ 0 0
$$701$$ 13.1246 0.495710 0.247855 0.968797i $$-0.420274\pi$$
0.247855 + 0.968797i $$0.420274\pi$$
$$702$$ 0 0
$$703$$ 25.5279 0.962802
$$704$$ 0 0
$$705$$ 4.94427 0.186212
$$706$$ 0 0
$$707$$ −20.2361 −0.761056
$$708$$ 0 0
$$709$$ 18.2918 0.686963 0.343481 0.939159i $$-0.388394\pi$$
0.343481 + 0.939159i $$0.388394\pi$$
$$710$$ 0 0
$$711$$ 7.61803 0.285699
$$712$$ 0 0
$$713$$ −9.47214 −0.354734
$$714$$ 0 0
$$715$$ 16.4721 0.616023
$$716$$ 0 0
$$717$$ −7.12461 −0.266074
$$718$$ 0 0
$$719$$ 20.6525 0.770207 0.385104 0.922873i $$-0.374166\pi$$
0.385104 + 0.922873i $$0.374166\pi$$
$$720$$ 0 0
$$721$$ −25.5279 −0.950707
$$722$$ 0 0
$$723$$ 0.742646 0.0276193
$$724$$ 0 0
$$725$$ −1.00000 −0.0371391
$$726$$ 0 0
$$727$$ 44.7639 1.66020 0.830101 0.557613i $$-0.188283\pi$$
0.830101 + 0.557613i $$0.188283\pi$$
$$728$$ 0 0
$$729$$ −8.50658 −0.315058
$$730$$ 0 0
$$731$$ −27.5623 −1.01943
$$732$$ 0 0
$$733$$ −35.7771 −1.32146 −0.660728 0.750625i $$-0.729753\pi$$
−0.660728 + 0.750625i $$0.729753\pi$$
$$734$$ 0 0
$$735$$ −0.708204 −0.0261225
$$736$$ 0 0
$$737$$ 8.00000 0.294684
$$738$$ 0 0
$$739$$ 3.41641 0.125675 0.0628373 0.998024i $$-0.479985\pi$$
0.0628373 + 0.998024i $$0.479985\pi$$
$$740$$ 0 0
$$741$$ 8.69505 0.319421
$$742$$ 0 0
$$743$$ −7.41641 −0.272082 −0.136041 0.990703i $$-0.543438\pi$$
−0.136041 + 0.990703i $$0.543438\pi$$
$$744$$ 0 0
$$745$$ −16.1803 −0.592802
$$746$$ 0 0
$$747$$ 41.1246 1.50467
$$748$$ 0 0
$$749$$ 0.832816 0.0304304
$$750$$ 0 0
$$751$$ −40.0000 −1.45962 −0.729810 0.683650i $$-0.760392\pi$$
−0.729810 + 0.683650i $$0.760392\pi$$
$$752$$ 0 0
$$753$$ −11.2361 −0.409465
$$754$$ 0 0
$$755$$ 1.23607 0.0449851
$$756$$ 0 0
$$757$$ −22.1803 −0.806158 −0.403079 0.915165i $$-0.632060\pi$$
−0.403079 + 0.915165i $$0.632060\pi$$
$$758$$ 0 0
$$759$$ −11.7082 −0.424981
$$760$$ 0 0
$$761$$ −25.0344 −0.907498 −0.453749 0.891130i $$-0.649914\pi$$
−0.453749 + 0.891130i $$0.649914\pi$$
$$762$$ 0 0
$$763$$ 33.4164 1.20976
$$764$$ 0 0
$$765$$ 9.47214 0.342466
$$766$$ 0 0
$$767$$ 47.7558 1.72436
$$768$$ 0 0
$$769$$ 50.6525 1.82657 0.913287 0.407316i $$-0.133535\pi$$
0.913287 + 0.407316i $$0.133535\pi$$
$$770$$ 0 0
$$771$$ 17.3475 0.624756
$$772$$ 0 0
$$773$$ 52.8328 1.90026 0.950132 0.311848i $$-0.100948\pi$$
0.950132 + 0.311848i $$0.100948\pi$$
$$774$$ 0 0
$$775$$ −1.61803 −0.0581215
$$776$$ 0 0
$$777$$ −16.2918 −0.584465
$$778$$ 0 0
$$779$$ 10.2492 0.367217
$$780$$ 0 0
$$781$$ 41.8885 1.49889
$$782$$ 0 0
$$783$$ −3.47214 −0.124084
$$784$$ 0 0
$$785$$ −5.05573 −0.180447
$$786$$ 0 0
$$787$$ 44.6525 1.59169 0.795844 0.605501i $$-0.207027\pi$$
0.795844 + 0.605501i $$0.207027\pi$$
$$788$$ 0 0
$$789$$ 10.8328 0.385658
$$790$$ 0 0
$$791$$ −25.1115 −0.892861
$$792$$ 0 0
$$793$$ −10.9230 −0.387887
$$794$$ 0 0
$$795$$ 5.76393 0.204426
$$796$$ 0 0
$$797$$ −30.8328 −1.09215 −0.546077 0.837735i $$-0.683879\pi$$
−0.546077 + 0.837735i $$0.683879\pi$$
$$798$$ 0 0
$$799$$ 28.9443 1.02397
$$800$$ 0 0
$$801$$ 4.47214 0.158015
$$802$$ 0 0
$$803$$ 6.18034 0.218099
$$804$$ 0 0
$$805$$ −16.7082 −0.588887
$$806$$ 0 0
$$807$$ 16.4164 0.577885
$$808$$ 0 0
$$809$$ 11.4164 0.401380 0.200690 0.979655i $$-0.435682\pi$$
0.200690 + 0.979655i $$0.435682\pi$$
$$810$$ 0 0
$$811$$ 0.673762 0.0236590 0.0118295 0.999930i $$-0.496234\pi$$
0.0118295 + 0.999930i $$0.496234\pi$$
$$812$$ 0 0
$$813$$ 9.52786 0.334157
$$814$$ 0 0
$$815$$ 13.5279 0.473860
$$816$$ 0 0
$$817$$ −21.0557 −0.736647
$$818$$ 0 0
$$819$$ 38.0344 1.32903
$$820$$ 0 0
$$821$$ 34.0689 1.18901 0.594506 0.804091i $$-0.297348\pi$$
0.594506 + 0.804091i $$0.297348\pi$$
$$822$$ 0 0
$$823$$ −45.7771 −1.59569 −0.797844 0.602863i $$-0.794026\pi$$
−0.797844 + 0.602863i $$0.794026\pi$$
$$824$$ 0 0
$$825$$ −2.00000 −0.0696311
$$826$$ 0 0
$$827$$ 1.90983 0.0664113 0.0332056 0.999449i $$-0.489428\pi$$
0.0332056 + 0.999449i $$0.489428\pi$$
$$828$$ 0 0
$$829$$ 45.7984 1.59064 0.795322 0.606188i $$-0.207302\pi$$
0.795322 + 0.606188i $$0.207302\pi$$
$$830$$ 0 0
$$831$$ 18.5410 0.643181
$$832$$ 0 0
$$833$$ −4.14590 −0.143647
$$834$$ 0 0
$$835$$ −4.38197 −0.151644
$$836$$ 0 0
$$837$$ −5.61803 −0.194188
$$838$$ 0 0
$$839$$ −28.3607 −0.979119 −0.489560 0.871970i $$-0.662842\pi$$
−0.489560 + 0.871970i $$0.662842\pi$$
$$840$$ 0 0
$$841$$ 1.00000 0.0344828
$$842$$ 0 0
$$843$$ 3.00000 0.103325
$$844$$ 0 0
$$845$$ 12.9098 0.444112
$$846$$ 0 0
$$847$$ 1.50658 0.0517666
$$848$$ 0 0
$$849$$ 13.7082 0.470464
$$850$$ 0 0
$$851$$ −54.0689 −1.85346
$$852$$ 0 0
$$853$$ −1.81966 −0.0623040 −0.0311520 0.999515i $$-0.509918\pi$$
−0.0311520 + 0.999515i $$0.509918\pi$$
$$854$$ 0 0
$$855$$ 7.23607 0.247468
$$856$$ 0 0
$$857$$ 33.7082 1.15145 0.575725 0.817643i $$-0.304720\pi$$
0.575725 + 0.817643i $$0.304720\pi$$
$$858$$ 0 0
$$859$$ 16.6525 0.568175 0.284088 0.958798i $$-0.408309\pi$$
0.284088 + 0.958798i $$0.408309\pi$$
$$860$$ 0 0
$$861$$ −6.54102 −0.222917
$$862$$ 0 0
$$863$$ 4.03444 0.137334 0.0686670 0.997640i $$-0.478125\pi$$
0.0686670 + 0.997640i $$0.478125\pi$$
$$864$$ 0 0
$$865$$ 4.79837 0.163150
$$866$$ 0 0
$$867$$ 2.41641 0.0820655
$$868$$ 0 0
$$869$$ −9.41641 −0.319430
$$870$$ 0 0
$$871$$ 12.5836 0.426379
$$872$$ 0 0
$$873$$ −23.0344 −0.779598
$$874$$ 0 0
$$875$$ −2.85410 −0.0964863
$$876$$ 0 0
$$877$$ −50.6180 −1.70925 −0.854625 0.519246i $$-0.826213\pi$$
−0.854625 + 0.519246i $$0.826213\pi$$
$$878$$ 0 0
$$879$$ 8.76393 0.295600
$$880$$ 0 0
$$881$$ 10.1803 0.342984 0.171492 0.985185i $$-0.445141\pi$$
0.171492 + 0.985185i $$0.445141\pi$$
$$882$$ 0 0
$$883$$ −21.5967 −0.726788 −0.363394 0.931635i $$-0.618382\pi$$
−0.363394 + 0.931635i $$0.618382\pi$$
$$884$$ 0 0
$$885$$ −5.79837 −0.194910
$$886$$ 0 0
$$887$$ 31.4164 1.05486 0.527430 0.849599i $$-0.323156\pi$$
0.527430 + 0.849599i $$0.323156\pi$$
$$888$$ 0 0
$$889$$ 2.69505 0.0903890
$$890$$ 0 0
$$891$$ 18.4721 0.618840
$$892$$ 0 0
$$893$$ 22.1115 0.739932
$$894$$ 0 0
$$895$$ 1.09017 0.0364404
$$896$$ 0 0
$$897$$ −18.4164 −0.614906
$$898$$ 0 0
$$899$$ 1.61803 0.0539645
$$900$$ 0 0
$$901$$ 33.7426 1.12413
$$902$$ 0 0
$$903$$ 13.4377 0.447178
$$904$$ 0 0
$$905$$ −0.291796 −0.00969963
$$906$$ 0 0
$$907$$ 25.9098 0.860322 0.430161 0.902752i $$-0.358457\pi$$
0.430161 + 0.902752i $$0.358457\pi$$
$$908$$ 0 0
$$909$$ −18.5623 −0.615673
$$910$$ 0 0
$$911$$ 39.2705 1.30109 0.650545 0.759468i $$-0.274541\pi$$
0.650545 + 0.759468i $$0.274541\pi$$
$$912$$ 0 0
$$913$$ −50.8328 −1.68232
$$914$$ 0 0
$$915$$ 1.32624 0.0438441
$$916$$ 0 0
$$917$$ −54.0689 −1.78551
$$918$$ 0 0
$$919$$ 26.4721 0.873235 0.436618 0.899647i $$-0.356176\pi$$
0.436618 + 0.899647i $$0.356176\pi$$
$$920$$ 0 0
$$921$$ −5.88854 −0.194034
$$922$$ 0 0
$$923$$ 65.8885 2.16875
$$924$$ 0 0
$$925$$ −9.23607 −0.303680
$$926$$ 0 0
$$927$$ −23.4164 −0.769096
$$928$$ 0 0
$$929$$ −44.4508 −1.45839 −0.729193 0.684309i $$-0.760104\pi$$
−0.729193 + 0.684309i $$0.760104\pi$$
$$930$$ 0 0
$$931$$ −3.16718 −0.103800
$$932$$ 0 0
$$933$$ −3.90983 −0.128002
$$934$$ 0 0
$$935$$ −11.7082 −0.382899
$$936$$ 0 0
$$937$$ 43.1246 1.40882 0.704410 0.709793i $$-0.251212\pi$$
0.704410 + 0.709793i $$0.251212\pi$$
$$938$$ 0 0
$$939$$ 2.40325 0.0784272
$$940$$ 0 0
$$941$$ −32.3607 −1.05493 −0.527464 0.849577i $$-0.676857\pi$$
−0.527464 + 0.849577i $$0.676857\pi$$
$$942$$ 0 0
$$943$$ −21.7082 −0.706916
$$944$$ 0 0
$$945$$ −9.90983 −0.322367
$$946$$ 0 0
$$947$$ −55.0902 −1.79019 −0.895095 0.445875i $$-0.852892\pi$$
−0.895095 + 0.445875i $$0.852892\pi$$
$$948$$ 0 0
$$949$$ 9.72136 0.315569
$$950$$ 0 0
$$951$$ −2.94427 −0.0954746
$$952$$ 0 0
$$953$$ 12.8328 0.415696 0.207848 0.978161i $$-0.433354\pi$$
0.207848 + 0.978161i $$0.433354\pi$$
$$954$$ 0 0
$$955$$ −20.8541 −0.674823
$$956$$ 0 0
$$957$$ 2.00000 0.0646508
$$958$$ 0 0
$$959$$ 0.416408 0.0134465
$$960$$ 0 0
$$961$$ −28.3820 −0.915547
$$962$$ 0 0
$$963$$ 0.763932 0.0246174
$$964$$ 0 0
$$965$$ 0.145898 0.00469662
$$966$$ 0 0
$$967$$ −23.5967 −0.758820 −0.379410 0.925229i $$-0.623873\pi$$
−0.379410 + 0.925229i $$0.623873\pi$$
$$968$$ 0 0
$$969$$ −6.18034 −0.198541
$$970$$ 0 0
$$971$$ −52.0689 −1.67097 −0.835485 0.549513i $$-0.814813\pi$$
−0.835485 + 0.549513i $$0.814813\pi$$
$$972$$ 0 0
$$973$$ 16.0344 0.514041
$$974$$ 0 0
$$975$$ −3.14590 −0.100749
$$976$$ 0 0
$$977$$ 10.6525 0.340803 0.170401 0.985375i $$-0.445494\pi$$
0.170401 + 0.985375i $$0.445494\pi$$
$$978$$ 0 0
$$979$$ −5.52786 −0.176671
$$980$$ 0 0
$$981$$ 30.6525 0.978658
$$982$$ 0 0
$$983$$ −13.0557 −0.416413 −0.208207 0.978085i $$-0.566763\pi$$
−0.208207 + 0.978085i $$0.566763\pi$$
$$984$$ 0 0
$$985$$ −18.8541 −0.600742
$$986$$ 0 0
$$987$$ −14.1115 −0.449173
$$988$$ 0 0
$$989$$ 44.5967 1.41809
$$990$$ 0 0
$$991$$ −10.0689 −0.319849 −0.159924 0.987129i $$-0.551125\pi$$
−0.159924 + 0.987129i $$0.551125\pi$$
$$992$$ 0 0
$$993$$ −6.36068 −0.201850
$$994$$ 0 0
$$995$$ −8.76393 −0.277835
$$996$$ 0 0
$$997$$ −44.1803 −1.39921 −0.699603 0.714532i $$-0.746640\pi$$
−0.699603 + 0.714532i $$0.746640\pi$$
$$998$$ 0 0
$$999$$ −32.0689 −1.01461
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.bd.1.1 2
4.3 odd 2 9280.2.a.y.1.2 2
8.3 odd 2 4640.2.a.i.1.1 yes 2
8.5 even 2 4640.2.a.g.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
4640.2.a.g.1.2 2 8.5 even 2
4640.2.a.i.1.1 yes 2 8.3 odd 2
9280.2.a.y.1.2 2 4.3 odd 2
9280.2.a.bd.1.1 2 1.1 even 1 trivial