# Properties

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

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [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 1.2 Root $$1.61803$$ 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)

 $$f(q)$$ $$=$$ $$q+1.61803 q^{3} +1.00000 q^{5} +3.85410 q^{7} -0.381966 q^{9} +O(q^{10})$$ $$q+1.61803 q^{3} +1.00000 q^{5} +3.85410 q^{7} -0.381966 q^{9} -1.23607 q^{11} -6.09017 q^{13} +1.61803 q^{15} -1.38197 q^{17} -7.23607 q^{19} +6.23607 q^{21} -0.854102 q^{23} +1.00000 q^{25} -5.47214 q^{27} -1.00000 q^{29} +0.618034 q^{31} -2.00000 q^{33} +3.85410 q^{35} -4.76393 q^{37} -9.85410 q^{39} +9.70820 q^{41} +5.38197 q^{43} -0.381966 q^{45} -8.00000 q^{47} +7.85410 q^{49} -2.23607 q^{51} +6.32624 q^{53} -1.23607 q^{55} -11.7082 q^{57} +11.6180 q^{59} -8.85410 q^{61} -1.47214 q^{63} -6.09017 q^{65} -6.47214 q^{67} -1.38197 q^{69} -4.94427 q^{71} +13.0902 q^{73} +1.61803 q^{75} -4.76393 q^{77} -14.0902 q^{79} -7.70820 q^{81} -2.29180 q^{83} -1.38197 q^{85} -1.61803 q^{87} +11.7082 q^{89} -23.4721 q^{91} +1.00000 q^{93} -7.23607 q^{95} -15.7984 q^{97} +0.472136 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} + 2 q^{5} + q^{7} - 3 q^{9}+O(q^{10})$$ 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
$$3$$ 1.61803 0.934172 0.467086 0.884212i $$-0.345304\pi$$
0.467086 + 0.884212i $$0.345304\pi$$
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ 3.85410 1.45671 0.728357 0.685198i $$-0.240284\pi$$
0.728357 + 0.685198i $$0.240284\pi$$
$$8$$ 0 0
$$9$$ −0.381966 −0.127322
$$10$$ 0 0
$$11$$ −1.23607 −0.372689 −0.186344 0.982485i $$-0.559664\pi$$
−0.186344 + 0.982485i $$0.559664\pi$$
$$12$$ 0 0
$$13$$ −6.09017 −1.68911 −0.844555 0.535469i $$-0.820135\pi$$
−0.844555 + 0.535469i $$0.820135\pi$$
$$14$$ 0 0
$$15$$ 1.61803 0.417775
$$16$$ 0 0
$$17$$ −1.38197 −0.335176 −0.167588 0.985857i $$-0.553598\pi$$
−0.167588 + 0.985857i $$0.553598\pi$$
$$18$$ 0 0
$$19$$ −7.23607 −1.66007 −0.830034 0.557713i $$-0.811679\pi$$
−0.830034 + 0.557713i $$0.811679\pi$$
$$20$$ 0 0
$$21$$ 6.23607 1.36082
$$22$$ 0 0
$$23$$ −0.854102 −0.178093 −0.0890463 0.996027i $$-0.528382\pi$$
−0.0890463 + 0.996027i $$0.528382\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −5.47214 −1.05311
$$28$$ 0 0
$$29$$ −1.00000 −0.185695
$$30$$ 0 0
$$31$$ 0.618034 0.111002 0.0555011 0.998459i $$-0.482324\pi$$
0.0555011 + 0.998459i $$0.482324\pi$$
$$32$$ 0 0
$$33$$ −2.00000 −0.348155
$$34$$ 0 0
$$35$$ 3.85410 0.651462
$$36$$ 0 0
$$37$$ −4.76393 −0.783186 −0.391593 0.920139i $$-0.628076\pi$$
−0.391593 + 0.920139i $$0.628076\pi$$
$$38$$ 0 0
$$39$$ −9.85410 −1.57792
$$40$$ 0 0
$$41$$ 9.70820 1.51617 0.758083 0.652158i $$-0.226136\pi$$
0.758083 + 0.652158i $$0.226136\pi$$
$$42$$ 0 0
$$43$$ 5.38197 0.820742 0.410371 0.911919i $$-0.365399\pi$$
0.410371 + 0.911919i $$0.365399\pi$$
$$44$$ 0 0
$$45$$ −0.381966 −0.0569401
$$46$$ 0 0
$$47$$ −8.00000 −1.16692 −0.583460 0.812142i $$-0.698301\pi$$
−0.583460 + 0.812142i $$0.698301\pi$$
$$48$$ 0 0
$$49$$ 7.85410 1.12201
$$50$$ 0 0
$$51$$ −2.23607 −0.313112
$$52$$ 0 0
$$53$$ 6.32624 0.868976 0.434488 0.900678i $$-0.356929\pi$$
0.434488 + 0.900678i $$0.356929\pi$$
$$54$$ 0 0
$$55$$ −1.23607 −0.166671
$$56$$ 0 0
$$57$$ −11.7082 −1.55079
$$58$$ 0 0
$$59$$ 11.6180 1.51254 0.756270 0.654260i $$-0.227020\pi$$
0.756270 + 0.654260i $$0.227020\pi$$
$$60$$ 0 0
$$61$$ −8.85410 −1.13365 −0.566826 0.823838i $$-0.691829\pi$$
−0.566826 + 0.823838i $$0.691829\pi$$
$$62$$ 0 0
$$63$$ −1.47214 −0.185472
$$64$$ 0 0
$$65$$ −6.09017 −0.755393
$$66$$ 0 0
$$67$$ −6.47214 −0.790697 −0.395349 0.918531i $$-0.629376\pi$$
−0.395349 + 0.918531i $$0.629376\pi$$
$$68$$ 0 0
$$69$$ −1.38197 −0.166369
$$70$$ 0 0
$$71$$ −4.94427 −0.586777 −0.293389 0.955993i $$-0.594783\pi$$
−0.293389 + 0.955993i $$0.594783\pi$$
$$72$$ 0 0
$$73$$ 13.0902 1.53209 0.766044 0.642788i $$-0.222222\pi$$
0.766044 + 0.642788i $$0.222222\pi$$
$$74$$ 0 0
$$75$$ 1.61803 0.186834
$$76$$ 0 0
$$77$$ −4.76393 −0.542900
$$78$$ 0 0
$$79$$ −14.0902 −1.58527 −0.792634 0.609698i $$-0.791291\pi$$
−0.792634 + 0.609698i $$0.791291\pi$$
$$80$$ 0 0
$$81$$ −7.70820 −0.856467
$$82$$ 0 0
$$83$$ −2.29180 −0.251557 −0.125779 0.992058i $$-0.540143\pi$$
−0.125779 + 0.992058i $$0.540143\pi$$
$$84$$ 0 0
$$85$$ −1.38197 −0.149895
$$86$$ 0 0
$$87$$ −1.61803 −0.173471
$$88$$ 0 0
$$89$$ 11.7082 1.24107 0.620534 0.784180i $$-0.286916\pi$$
0.620534 + 0.784180i $$0.286916\pi$$
$$90$$ 0 0
$$91$$ −23.4721 −2.46055
$$92$$ 0 0
$$93$$ 1.00000 0.103695
$$94$$ 0 0
$$95$$ −7.23607 −0.742405
$$96$$ 0 0
$$97$$ −15.7984 −1.60408 −0.802041 0.597269i $$-0.796252\pi$$
−0.802041 + 0.597269i $$0.796252\pi$$
$$98$$ 0 0
$$99$$ 0.472136 0.0474514
$$100$$ 0 0
$$101$$ −4.09017 −0.406987 −0.203494 0.979076i $$-0.565230\pi$$
−0.203494 + 0.979076i $$0.565230\pi$$
$$102$$ 0 0
$$103$$ −8.94427 −0.881305 −0.440653 0.897678i $$-0.645253\pi$$
−0.440653 + 0.897678i $$0.645253\pi$$
$$104$$ 0 0
$$105$$ 6.23607 0.608578
$$106$$ 0 0
$$107$$ −13.7082 −1.32522 −0.662611 0.748964i $$-0.730552\pi$$
−0.662611 + 0.748964i $$0.730552\pi$$
$$108$$ 0 0
$$109$$ 1.70820 0.163616 0.0818081 0.996648i $$-0.473931\pi$$
0.0818081 + 0.996648i $$0.473931\pi$$
$$110$$ 0 0
$$111$$ −7.70820 −0.731630
$$112$$ 0 0
$$113$$ −15.7984 −1.48619 −0.743093 0.669188i $$-0.766642\pi$$
−0.743093 + 0.669188i $$0.766642\pi$$
$$114$$ 0 0
$$115$$ −0.854102 −0.0796454
$$116$$ 0 0
$$117$$ 2.32624 0.215061
$$118$$ 0 0
$$119$$ −5.32624 −0.488255
$$120$$ 0 0
$$121$$ −9.47214 −0.861103
$$122$$ 0 0
$$123$$ 15.7082 1.41636
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ 16.9443 1.50356 0.751780 0.659413i $$-0.229195\pi$$
0.751780 + 0.659413i $$0.229195\pi$$
$$128$$ 0 0
$$129$$ 8.70820 0.766715
$$130$$ 0 0
$$131$$ 1.05573 0.0922394 0.0461197 0.998936i $$-0.485314\pi$$
0.0461197 + 0.998936i $$0.485314\pi$$
$$132$$ 0 0
$$133$$ −27.8885 −2.41824
$$134$$ 0 0
$$135$$ −5.47214 −0.470966
$$136$$ 0 0
$$137$$ −6.85410 −0.585585 −0.292793 0.956176i $$-0.594585\pi$$
−0.292793 + 0.956176i $$0.594585\pi$$
$$138$$ 0 0
$$139$$ −3.38197 −0.286855 −0.143427 0.989661i $$-0.545812\pi$$
−0.143427 + 0.989661i $$0.545812\pi$$
$$140$$ 0 0
$$141$$ −12.9443 −1.09010
$$142$$ 0 0
$$143$$ 7.52786 0.629512
$$144$$ 0 0
$$145$$ −1.00000 −0.0830455
$$146$$ 0 0
$$147$$ 12.7082 1.04815
$$148$$ 0 0
$$149$$ 6.18034 0.506313 0.253157 0.967425i $$-0.418531\pi$$
0.253157 + 0.967425i $$0.418531\pi$$
$$150$$ 0 0
$$151$$ −3.23607 −0.263347 −0.131674 0.991293i $$-0.542035\pi$$
−0.131674 + 0.991293i $$0.542035\pi$$
$$152$$ 0 0
$$153$$ 0.527864 0.0426753
$$154$$ 0 0
$$155$$ 0.618034 0.0496417
$$156$$ 0 0
$$157$$ −22.9443 −1.83115 −0.915576 0.402145i $$-0.868265\pi$$
−0.915576 + 0.402145i $$0.868265\pi$$
$$158$$ 0 0
$$159$$ 10.2361 0.811773
$$160$$ 0 0
$$161$$ −3.29180 −0.259430
$$162$$ 0 0
$$163$$ 22.4721 1.76015 0.880077 0.474831i $$-0.157491\pi$$
0.880077 + 0.474831i $$0.157491\pi$$
$$164$$ 0 0
$$165$$ −2.00000 −0.155700
$$166$$ 0 0
$$167$$ −6.61803 −0.512119 −0.256059 0.966661i $$-0.582424\pi$$
−0.256059 + 0.966661i $$0.582424\pi$$
$$168$$ 0 0
$$169$$ 24.0902 1.85309
$$170$$ 0 0
$$171$$ 2.76393 0.211363
$$172$$ 0 0
$$173$$ −19.7984 −1.50524 −0.752621 0.658454i $$-0.771211\pi$$
−0.752621 + 0.658454i $$0.771211\pi$$
$$174$$ 0 0
$$175$$ 3.85410 0.291343
$$176$$ 0 0
$$177$$ 18.7984 1.41297
$$178$$ 0 0
$$179$$ −10.0902 −0.754175 −0.377087 0.926178i $$-0.623074\pi$$
−0.377087 + 0.926178i $$0.623074\pi$$
$$180$$ 0 0
$$181$$ −13.7082 −1.01892 −0.509461 0.860494i $$-0.670155\pi$$
−0.509461 + 0.860494i $$0.670155\pi$$
$$182$$ 0 0
$$183$$ −14.3262 −1.05903
$$184$$ 0 0
$$185$$ −4.76393 −0.350251
$$186$$ 0 0
$$187$$ 1.70820 0.124916
$$188$$ 0 0
$$189$$ −21.0902 −1.53408
$$190$$ 0 0
$$191$$ −14.1459 −1.02356 −0.511781 0.859116i $$-0.671014\pi$$
−0.511781 + 0.859116i $$0.671014\pi$$
$$192$$ 0 0
$$193$$ 6.85410 0.493369 0.246685 0.969096i $$-0.420659\pi$$
0.246685 + 0.969096i $$0.420659\pi$$
$$194$$ 0 0
$$195$$ −9.85410 −0.705667
$$196$$ 0 0
$$197$$ −12.1459 −0.865359 −0.432680 0.901548i $$-0.642432\pi$$
−0.432680 + 0.901548i $$0.642432\pi$$
$$198$$ 0 0
$$199$$ −13.2361 −0.938280 −0.469140 0.883124i $$-0.655436\pi$$
−0.469140 + 0.883124i $$0.655436\pi$$
$$200$$ 0 0
$$201$$ −10.4721 −0.738648
$$202$$ 0 0
$$203$$ −3.85410 −0.270505
$$204$$ 0 0
$$205$$ 9.70820 0.678050
$$206$$ 0 0
$$207$$ 0.326238 0.0226751
$$208$$ 0 0
$$209$$ 8.94427 0.618688
$$210$$ 0 0
$$211$$ −10.7639 −0.741020 −0.370510 0.928829i $$-0.620817\pi$$
−0.370510 + 0.928829i $$0.620817\pi$$
$$212$$ 0 0
$$213$$ −8.00000 −0.548151
$$214$$ 0 0
$$215$$ 5.38197 0.367047
$$216$$ 0 0
$$217$$ 2.38197 0.161698
$$218$$ 0 0
$$219$$ 21.1803 1.43123
$$220$$ 0 0
$$221$$ 8.41641 0.566149
$$222$$ 0 0
$$223$$ 21.9787 1.47180 0.735902 0.677088i $$-0.236759\pi$$
0.735902 + 0.677088i $$0.236759\pi$$
$$224$$ 0 0
$$225$$ −0.381966 −0.0254644
$$226$$ 0 0
$$227$$ −2.00000 −0.132745 −0.0663723 0.997795i $$-0.521143\pi$$
−0.0663723 + 0.997795i $$0.521143\pi$$
$$228$$ 0 0
$$229$$ 7.09017 0.468532 0.234266 0.972173i $$-0.424731\pi$$
0.234266 + 0.972173i $$0.424731\pi$$
$$230$$ 0 0
$$231$$ −7.70820 −0.507163
$$232$$ 0 0
$$233$$ 7.52786 0.493167 0.246583 0.969122i $$-0.420692\pi$$
0.246583 + 0.969122i $$0.420692\pi$$
$$234$$ 0 0
$$235$$ −8.00000 −0.521862
$$236$$ 0 0
$$237$$ −22.7984 −1.48091
$$238$$ 0 0
$$239$$ 20.4721 1.32423 0.662116 0.749401i $$-0.269659\pi$$
0.662116 + 0.749401i $$0.269659\pi$$
$$240$$ 0 0
$$241$$ −25.7984 −1.66182 −0.830910 0.556407i $$-0.812179\pi$$
−0.830910 + 0.556407i $$0.812179\pi$$
$$242$$ 0 0
$$243$$ 3.94427 0.253025
$$244$$ 0 0
$$245$$ 7.85410 0.501780
$$246$$ 0 0
$$247$$ 44.0689 2.80404
$$248$$ 0 0
$$249$$ −3.70820 −0.234998
$$250$$ 0 0
$$251$$ −4.18034 −0.263861 −0.131930 0.991259i $$-0.542118\pi$$
−0.131930 + 0.991259i $$0.542118\pi$$
$$252$$ 0 0
$$253$$ 1.05573 0.0663731
$$254$$ 0 0
$$255$$ −2.23607 −0.140028
$$256$$ 0 0
$$257$$ 30.0689 1.87565 0.937823 0.347115i $$-0.112839\pi$$
0.937823 + 0.347115i $$0.112839\pi$$
$$258$$ 0 0
$$259$$ −18.3607 −1.14088
$$260$$ 0 0
$$261$$ 0.381966 0.0236431
$$262$$ 0 0
$$263$$ −26.4721 −1.63234 −0.816171 0.577811i $$-0.803907\pi$$
−0.816171 + 0.577811i $$0.803907\pi$$
$$264$$ 0 0
$$265$$ 6.32624 0.388618
$$266$$ 0 0
$$267$$ 18.9443 1.15937
$$268$$ 0 0
$$269$$ −6.43769 −0.392513 −0.196257 0.980553i $$-0.562879\pi$$
−0.196257 + 0.980553i $$0.562879\pi$$
$$270$$ 0 0
$$271$$ 11.4164 0.693497 0.346749 0.937958i $$-0.387286\pi$$
0.346749 + 0.937958i $$0.387286\pi$$
$$272$$ 0 0
$$273$$ −37.9787 −2.29858
$$274$$ 0 0
$$275$$ −1.23607 −0.0745377
$$276$$ 0 0
$$277$$ −30.0000 −1.80253 −0.901263 0.433273i $$-0.857359\pi$$
−0.901263 + 0.433273i $$0.857359\pi$$
$$278$$ 0 0
$$279$$ −0.236068 −0.0141330
$$280$$ 0 0
$$281$$ 1.85410 0.110606 0.0553032 0.998470i $$-0.482387\pi$$
0.0553032 + 0.998470i $$0.482387\pi$$
$$282$$ 0 0
$$283$$ 0.180340 0.0107201 0.00536005 0.999986i $$-0.498294\pi$$
0.00536005 + 0.999986i $$0.498294\pi$$
$$284$$ 0 0
$$285$$ −11.7082 −0.693534
$$286$$ 0 0
$$287$$ 37.4164 2.20862
$$288$$ 0 0
$$289$$ −15.0902 −0.887657
$$290$$ 0 0
$$291$$ −25.5623 −1.49849
$$292$$ 0 0
$$293$$ 8.18034 0.477901 0.238950 0.971032i $$-0.423197\pi$$
0.238950 + 0.971032i $$0.423197\pi$$
$$294$$ 0 0
$$295$$ 11.6180 0.676428
$$296$$ 0 0
$$297$$ 6.76393 0.392483
$$298$$ 0 0
$$299$$ 5.20163 0.300818
$$300$$ 0 0
$$301$$ 20.7426 1.19559
$$302$$ 0 0
$$303$$ −6.61803 −0.380196
$$304$$ 0 0
$$305$$ −8.85410 −0.506984
$$306$$ 0 0
$$307$$ 18.4721 1.05426 0.527130 0.849785i $$-0.323268\pi$$
0.527130 + 0.849785i $$0.323268\pi$$
$$308$$ 0 0
$$309$$ −14.4721 −0.823291
$$310$$ 0 0
$$311$$ −9.32624 −0.528842 −0.264421 0.964407i $$-0.585181\pi$$
−0.264421 + 0.964407i $$0.585181\pi$$
$$312$$ 0 0
$$313$$ 31.8885 1.80245 0.901224 0.433355i $$-0.142670\pi$$
0.901224 + 0.433355i $$0.142670\pi$$
$$314$$ 0 0
$$315$$ −1.47214 −0.0829455
$$316$$ 0 0
$$317$$ 9.23607 0.518749 0.259375 0.965777i $$-0.416484\pi$$
0.259375 + 0.965777i $$0.416484\pi$$
$$318$$ 0 0
$$319$$ 1.23607 0.0692065
$$320$$ 0 0
$$321$$ −22.1803 −1.23799
$$322$$ 0 0
$$323$$ 10.0000 0.556415
$$324$$ 0 0
$$325$$ −6.09017 −0.337822
$$326$$ 0 0
$$327$$ 2.76393 0.152846
$$328$$ 0 0
$$329$$ −30.8328 −1.69987
$$330$$ 0 0
$$331$$ 23.7082 1.30312 0.651560 0.758597i $$-0.274115\pi$$
0.651560 + 0.758597i $$0.274115\pi$$
$$332$$ 0 0
$$333$$ 1.81966 0.0997168
$$334$$ 0 0
$$335$$ −6.47214 −0.353611
$$336$$ 0 0
$$337$$ 23.8541 1.29942 0.649708 0.760184i $$-0.274891\pi$$
0.649708 + 0.760184i $$0.274891\pi$$
$$338$$ 0 0
$$339$$ −25.5623 −1.38835
$$340$$ 0 0
$$341$$ −0.763932 −0.0413692
$$342$$ 0 0
$$343$$ 3.29180 0.177740
$$344$$ 0 0
$$345$$ −1.38197 −0.0744025
$$346$$ 0 0
$$347$$ −35.2361 −1.89157 −0.945786 0.324792i $$-0.894706\pi$$
−0.945786 + 0.324792i $$0.894706\pi$$
$$348$$ 0 0
$$349$$ −4.94427 −0.264661 −0.132330 0.991206i $$-0.542246\pi$$
−0.132330 + 0.991206i $$0.542246\pi$$
$$350$$ 0 0
$$351$$ 33.3262 1.77882
$$352$$ 0 0
$$353$$ −9.52786 −0.507117 −0.253559 0.967320i $$-0.581601\pi$$
−0.253559 + 0.967320i $$0.581601\pi$$
$$354$$ 0 0
$$355$$ −4.94427 −0.262415
$$356$$ 0 0
$$357$$ −8.61803 −0.456115
$$358$$ 0 0
$$359$$ 26.0902 1.37699 0.688493 0.725243i $$-0.258272\pi$$
0.688493 + 0.725243i $$0.258272\pi$$
$$360$$ 0 0
$$361$$ 33.3607 1.75583
$$362$$ 0 0
$$363$$ −15.3262 −0.804419
$$364$$ 0 0
$$365$$ 13.0902 0.685171
$$366$$ 0 0
$$367$$ 4.18034 0.218212 0.109106 0.994030i $$-0.465201\pi$$
0.109106 + 0.994030i $$0.465201\pi$$
$$368$$ 0 0
$$369$$ −3.70820 −0.193041
$$370$$ 0 0
$$371$$ 24.3820 1.26585
$$372$$ 0 0
$$373$$ 7.27051 0.376453 0.188226 0.982126i $$-0.439726\pi$$
0.188226 + 0.982126i $$0.439726\pi$$
$$374$$ 0 0
$$375$$ 1.61803 0.0835549
$$376$$ 0 0
$$377$$ 6.09017 0.313660
$$378$$ 0 0
$$379$$ 20.0689 1.03087 0.515435 0.856929i $$-0.327630\pi$$
0.515435 + 0.856929i $$0.327630\pi$$
$$380$$ 0 0
$$381$$ 27.4164 1.40459
$$382$$ 0 0
$$383$$ 7.27051 0.371506 0.185753 0.982596i $$-0.440528\pi$$
0.185753 + 0.982596i $$0.440528\pi$$
$$384$$ 0 0
$$385$$ −4.76393 −0.242792
$$386$$ 0 0
$$387$$ −2.05573 −0.104499
$$388$$ 0 0
$$389$$ 18.0000 0.912636 0.456318 0.889817i $$-0.349168\pi$$
0.456318 + 0.889817i $$0.349168\pi$$
$$390$$ 0 0
$$391$$ 1.18034 0.0596924
$$392$$ 0 0
$$393$$ 1.70820 0.0861675
$$394$$ 0 0
$$395$$ −14.0902 −0.708953
$$396$$ 0 0
$$397$$ −11.8541 −0.594940 −0.297470 0.954731i $$-0.596143\pi$$
−0.297470 + 0.954731i $$0.596143\pi$$
$$398$$ 0 0
$$399$$ −45.1246 −2.25906
$$400$$ 0 0
$$401$$ −7.32624 −0.365855 −0.182927 0.983126i $$-0.558557\pi$$
−0.182927 + 0.983126i $$0.558557\pi$$
$$402$$ 0 0
$$403$$ −3.76393 −0.187495
$$404$$ 0 0
$$405$$ −7.70820 −0.383024
$$406$$ 0 0
$$407$$ 5.88854 0.291884
$$408$$ 0 0
$$409$$ −19.4164 −0.960080 −0.480040 0.877247i $$-0.659378\pi$$
−0.480040 + 0.877247i $$0.659378\pi$$
$$410$$ 0 0
$$411$$ −11.0902 −0.547038
$$412$$ 0 0
$$413$$ 44.7771 2.20334
$$414$$ 0 0
$$415$$ −2.29180 −0.112500
$$416$$ 0 0
$$417$$ −5.47214 −0.267972
$$418$$ 0 0
$$419$$ −15.0902 −0.737203 −0.368602 0.929587i $$-0.620163\pi$$
−0.368602 + 0.929587i $$0.620163\pi$$
$$420$$ 0 0
$$421$$ −16.4721 −0.802803 −0.401401 0.915902i $$-0.631477\pi$$
−0.401401 + 0.915902i $$0.631477\pi$$
$$422$$ 0 0
$$423$$ 3.05573 0.148575
$$424$$ 0 0
$$425$$ −1.38197 −0.0670352
$$426$$ 0 0
$$427$$ −34.1246 −1.65141
$$428$$ 0 0
$$429$$ 12.1803 0.588072
$$430$$ 0 0
$$431$$ 10.2918 0.495738 0.247869 0.968794i $$-0.420270\pi$$
0.247869 + 0.968794i $$0.420270\pi$$
$$432$$ 0 0
$$433$$ −4.47214 −0.214917 −0.107459 0.994210i $$-0.534271\pi$$
−0.107459 + 0.994210i $$0.534271\pi$$
$$434$$ 0 0
$$435$$ −1.61803 −0.0775788
$$436$$ 0 0
$$437$$ 6.18034 0.295646
$$438$$ 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
$$443$$ 8.27051 0.392944 0.196472 0.980509i $$-0.437052\pi$$
0.196472 + 0.980509i $$0.437052\pi$$
$$444$$ 0 0
$$445$$ 11.7082 0.555022
$$446$$ 0 0
$$447$$ 10.0000 0.472984
$$448$$ 0 0
$$449$$ 34.0689 1.60781 0.803905 0.594758i $$-0.202752\pi$$
0.803905 + 0.594758i $$0.202752\pi$$
$$450$$ 0 0
$$451$$ −12.0000 −0.565058
$$452$$ 0 0
$$453$$ −5.23607 −0.246012
$$454$$ 0 0
$$455$$ −23.4721 −1.10039
$$456$$ 0 0
$$457$$ 22.0000 1.02912 0.514558 0.857455i $$-0.327956\pi$$
0.514558 + 0.857455i $$0.327956\pi$$
$$458$$ 0 0
$$459$$ 7.56231 0.352978
$$460$$ 0 0
$$461$$ 3.67376 0.171104 0.0855521 0.996334i $$-0.472735\pi$$
0.0855521 + 0.996334i $$0.472735\pi$$
$$462$$ 0 0
$$463$$ 14.8328 0.689339 0.344670 0.938724i $$-0.387991\pi$$
0.344670 + 0.938724i $$0.387991\pi$$
$$464$$ 0 0
$$465$$ 1.00000 0.0463739
$$466$$ 0 0
$$467$$ 38.3262 1.77353 0.886763 0.462224i $$-0.152948\pi$$
0.886763 + 0.462224i $$0.152948\pi$$
$$468$$ 0 0
$$469$$ −24.9443 −1.15182
$$470$$ 0 0
$$471$$ −37.1246 −1.71061
$$472$$ 0 0
$$473$$ −6.65248 −0.305881
$$474$$ 0 0
$$475$$ −7.23607 −0.332014
$$476$$ 0 0
$$477$$ −2.41641 −0.110640
$$478$$ 0 0
$$479$$ 16.3262 0.745965 0.372982 0.927838i $$-0.378335\pi$$
0.372982 + 0.927838i $$0.378335\pi$$
$$480$$ 0 0
$$481$$ 29.0132 1.32289
$$482$$ 0 0
$$483$$ −5.32624 −0.242352
$$484$$ 0 0
$$485$$ −15.7984 −0.717367
$$486$$ 0 0
$$487$$ 12.2705 0.556030 0.278015 0.960577i $$-0.410324\pi$$
0.278015 + 0.960577i $$0.410324\pi$$
$$488$$ 0 0
$$489$$ 36.3607 1.64429
$$490$$ 0 0
$$491$$ −21.3050 −0.961479 −0.480740 0.876863i $$-0.659632\pi$$
−0.480740 + 0.876863i $$0.659632\pi$$
$$492$$ 0 0
$$493$$ 1.38197 0.0622406
$$494$$ 0 0
$$495$$ 0.472136 0.0212209
$$496$$ 0 0
$$497$$ −19.0557 −0.854766
$$498$$ 0 0
$$499$$ 17.3262 0.775629 0.387814 0.921737i $$-0.373230\pi$$
0.387814 + 0.921737i $$0.373230\pi$$
$$500$$ 0 0
$$501$$ −10.7082 −0.478407
$$502$$ 0 0
$$503$$ 20.0000 0.891756 0.445878 0.895094i $$-0.352892\pi$$
0.445878 + 0.895094i $$0.352892\pi$$
$$504$$ 0 0
$$505$$ −4.09017 −0.182010
$$506$$ 0 0
$$507$$ 38.9787 1.73111
$$508$$ 0 0
$$509$$ −9.70820 −0.430309 −0.215154 0.976580i $$-0.569025\pi$$
−0.215154 + 0.976580i $$0.569025\pi$$
$$510$$ 0 0
$$511$$ 50.4508 2.23181
$$512$$ 0 0
$$513$$ 39.5967 1.74824
$$514$$ 0 0
$$515$$ −8.94427 −0.394132
$$516$$ 0 0
$$517$$ 9.88854 0.434898
$$518$$ 0 0
$$519$$ −32.0344 −1.40616
$$520$$ 0 0
$$521$$ −19.6738 −0.861923 −0.430962 0.902370i $$-0.641826\pi$$
−0.430962 + 0.902370i $$0.641826\pi$$
$$522$$ 0 0
$$523$$ −12.1803 −0.532609 −0.266305 0.963889i $$-0.585803\pi$$
−0.266305 + 0.963889i $$0.585803\pi$$
$$524$$ 0 0
$$525$$ 6.23607 0.272164
$$526$$ 0 0
$$527$$ −0.854102 −0.0372053
$$528$$ 0 0
$$529$$ −22.2705 −0.968283
$$530$$ 0 0
$$531$$ −4.43769 −0.192580
$$532$$ 0 0
$$533$$ −59.1246 −2.56097
$$534$$ 0 0
$$535$$ −13.7082 −0.592657
$$536$$ 0 0
$$537$$ −16.3262 −0.704529
$$538$$ 0 0
$$539$$ −9.70820 −0.418162
$$540$$ 0 0
$$541$$ 24.0344 1.03332 0.516661 0.856190i $$-0.327175\pi$$
0.516661 + 0.856190i $$0.327175\pi$$
$$542$$ 0 0
$$543$$ −22.1803 −0.951849
$$544$$ 0 0
$$545$$ 1.70820 0.0731714
$$546$$ 0 0
$$547$$ 21.1246 0.903223 0.451612 0.892215i $$-0.350849\pi$$
0.451612 + 0.892215i $$0.350849\pi$$
$$548$$ 0 0
$$549$$ 3.38197 0.144339
$$550$$ 0 0
$$551$$ 7.23607 0.308267
$$552$$ 0 0
$$553$$ −54.3050 −2.30928
$$554$$ 0 0
$$555$$ −7.70820 −0.327195
$$556$$ 0 0
$$557$$ −30.8541 −1.30733 −0.653665 0.756784i $$-0.726770\pi$$
−0.653665 + 0.756784i $$0.726770\pi$$
$$558$$ 0 0
$$559$$ −32.7771 −1.38632
$$560$$ 0 0
$$561$$ 2.76393 0.116693
$$562$$ 0 0
$$563$$ 45.3951 1.91318 0.956588 0.291443i $$-0.0941354\pi$$
0.956588 + 0.291443i $$0.0941354\pi$$
$$564$$ 0 0
$$565$$ −15.7984 −0.664643
$$566$$ 0 0
$$567$$ −29.7082 −1.24763
$$568$$ 0 0
$$569$$ −12.3607 −0.518187 −0.259093 0.965852i $$-0.583424\pi$$
−0.259093 + 0.965852i $$0.583424\pi$$
$$570$$ 0 0
$$571$$ 26.8541 1.12381 0.561905 0.827202i $$-0.310069\pi$$
0.561905 + 0.827202i $$0.310069\pi$$
$$572$$ 0 0
$$573$$ −22.8885 −0.956183
$$574$$ 0 0
$$575$$ −0.854102 −0.0356185
$$576$$ 0 0
$$577$$ −29.7426 −1.23820 −0.619101 0.785311i $$-0.712503\pi$$
−0.619101 + 0.785311i $$0.712503\pi$$
$$578$$ 0 0
$$579$$ 11.0902 0.460892
$$580$$ 0 0
$$581$$ −8.83282 −0.366447
$$582$$ 0 0
$$583$$ −7.81966 −0.323857
$$584$$ 0 0
$$585$$ 2.32624 0.0961781
$$586$$ 0 0
$$587$$ −44.8328 −1.85045 −0.925224 0.379421i $$-0.876123\pi$$
−0.925224 + 0.379421i $$0.876123\pi$$
$$588$$ 0 0
$$589$$ −4.47214 −0.184271
$$590$$ 0 0
$$591$$ −19.6525 −0.808395
$$592$$ 0 0
$$593$$ 24.6525 1.01236 0.506178 0.862429i $$-0.331058\pi$$
0.506178 + 0.862429i $$0.331058\pi$$
$$594$$ 0 0
$$595$$ −5.32624 −0.218354
$$596$$ 0 0
$$597$$ −21.4164 −0.876515
$$598$$ 0 0
$$599$$ −16.3262 −0.667072 −0.333536 0.942737i $$-0.608242\pi$$
−0.333536 + 0.942737i $$0.608242\pi$$
$$600$$ 0 0
$$601$$ −46.5410 −1.89845 −0.949224 0.314601i $$-0.898129\pi$$
−0.949224 + 0.314601i $$0.898129\pi$$
$$602$$ 0 0
$$603$$ 2.47214 0.100673
$$604$$ 0 0
$$605$$ −9.47214 −0.385097
$$606$$ 0 0
$$607$$ 6.47214 0.262696 0.131348 0.991336i $$-0.458069\pi$$
0.131348 + 0.991336i $$0.458069\pi$$
$$608$$ 0 0
$$609$$ −6.23607 −0.252698
$$610$$ 0 0
$$611$$ 48.7214 1.97106
$$612$$ 0 0
$$613$$ 9.74265 0.393502 0.196751 0.980454i $$-0.436961\pi$$
0.196751 + 0.980454i $$0.436961\pi$$
$$614$$ 0 0
$$615$$ 15.7082 0.633416
$$616$$ 0 0
$$617$$ −7.20163 −0.289927 −0.144963 0.989437i $$-0.546306\pi$$
−0.144963 + 0.989437i $$0.546306\pi$$
$$618$$ 0 0
$$619$$ 29.7771 1.19684 0.598421 0.801182i $$-0.295795\pi$$
0.598421 + 0.801182i $$0.295795\pi$$
$$620$$ 0 0
$$621$$ 4.67376 0.187552
$$622$$ 0 0
$$623$$ 45.1246 1.80788
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 14.4721 0.577961
$$628$$ 0 0
$$629$$ 6.58359 0.262505
$$630$$ 0 0
$$631$$ −6.36068 −0.253215 −0.126607 0.991953i $$-0.540409\pi$$
−0.126607 + 0.991953i $$0.540409\pi$$
$$632$$ 0 0
$$633$$ −17.4164 −0.692240
$$634$$ 0 0
$$635$$ 16.9443 0.672413
$$636$$ 0 0
$$637$$ −47.8328 −1.89521
$$638$$ 0 0
$$639$$ 1.88854 0.0747096
$$640$$ 0 0
$$641$$ −43.2361 −1.70772 −0.853861 0.520501i $$-0.825745\pi$$
−0.853861 + 0.520501i $$0.825745\pi$$
$$642$$ 0 0
$$643$$ 12.2918 0.484741 0.242371 0.970184i $$-0.422075\pi$$
0.242371 + 0.970184i $$0.422075\pi$$
$$644$$ 0 0
$$645$$ 8.70820 0.342885
$$646$$ 0 0
$$647$$ −25.3050 −0.994840 −0.497420 0.867510i $$-0.665719\pi$$
−0.497420 + 0.867510i $$0.665719\pi$$
$$648$$ 0 0
$$649$$ −14.3607 −0.563706
$$650$$ 0 0
$$651$$ 3.85410 0.151054
$$652$$ 0 0
$$653$$ 9.59675 0.375550 0.187775 0.982212i $$-0.439872\pi$$
0.187775 + 0.982212i $$0.439872\pi$$
$$654$$ 0 0
$$655$$ 1.05573 0.0412507
$$656$$ 0 0
$$657$$ −5.00000 −0.195069
$$658$$ 0 0
$$659$$ 38.1803 1.48729 0.743647 0.668572i $$-0.233094\pi$$
0.743647 + 0.668572i $$0.233094\pi$$
$$660$$ 0 0
$$661$$ 21.5967 0.840016 0.420008 0.907520i $$-0.362027\pi$$
0.420008 + 0.907520i $$0.362027\pi$$
$$662$$ 0 0
$$663$$ 13.6180 0.528881
$$664$$ 0 0
$$665$$ −27.8885 −1.08147
$$666$$ 0 0
$$667$$ 0.854102 0.0330710
$$668$$ 0 0
$$669$$ 35.5623 1.37492
$$670$$ 0 0
$$671$$ 10.9443 0.422499
$$672$$ 0 0
$$673$$ 23.2361 0.895685 0.447842 0.894113i $$-0.352193\pi$$
0.447842 + 0.894113i $$0.352193\pi$$
$$674$$ 0 0
$$675$$ −5.47214 −0.210623
$$676$$ 0 0
$$677$$ −21.8885 −0.841245 −0.420623 0.907236i $$-0.638188\pi$$
−0.420623 + 0.907236i $$0.638188\pi$$
$$678$$ 0 0
$$679$$ −60.8885 −2.33669
$$680$$ 0 0
$$681$$ −3.23607 −0.124006
$$682$$ 0 0
$$683$$ −46.6525 −1.78511 −0.892554 0.450941i $$-0.851088\pi$$
−0.892554 + 0.450941i $$0.851088\pi$$
$$684$$ 0 0
$$685$$ −6.85410 −0.261882
$$686$$ 0 0
$$687$$ 11.4721 0.437689
$$688$$ 0 0
$$689$$ −38.5279 −1.46779
$$690$$ 0 0
$$691$$ 16.0344 0.609979 0.304989 0.952356i $$-0.401347\pi$$
0.304989 + 0.952356i $$0.401347\pi$$
$$692$$ 0 0
$$693$$ 1.81966 0.0691232
$$694$$ 0 0
$$695$$ −3.38197 −0.128285
$$696$$ 0 0
$$697$$ −13.4164 −0.508183
$$698$$ 0 0
$$699$$ 12.1803 0.460703
$$700$$ 0 0
$$701$$ −27.1246 −1.02448 −0.512241 0.858842i $$-0.671185\pi$$
−0.512241 + 0.858842i $$0.671185\pi$$
$$702$$ 0 0
$$703$$ 34.4721 1.30014
$$704$$ 0 0
$$705$$ −12.9443 −0.487509
$$706$$ 0 0
$$707$$ −15.7639 −0.592864
$$708$$ 0 0
$$709$$ 31.7082 1.19083 0.595413 0.803420i $$-0.296988\pi$$
0.595413 + 0.803420i $$0.296988\pi$$
$$710$$ 0 0
$$711$$ 5.38197 0.201839
$$712$$ 0 0
$$713$$ −0.527864 −0.0197687
$$714$$ 0 0
$$715$$ 7.52786 0.281526
$$716$$ 0 0
$$717$$ 33.1246 1.23706
$$718$$ 0 0
$$719$$ −10.6525 −0.397270 −0.198635 0.980074i $$-0.563651\pi$$
−0.198635 + 0.980074i $$0.563651\pi$$
$$720$$ 0 0
$$721$$ −34.4721 −1.28381
$$722$$ 0 0
$$723$$ −41.7426 −1.55243
$$724$$ 0 0
$$725$$ −1.00000 −0.0371391
$$726$$ 0 0
$$727$$ 49.2361 1.82606 0.913032 0.407887i $$-0.133734\pi$$
0.913032 + 0.407887i $$0.133734\pi$$
$$728$$ 0 0
$$729$$ 29.5066 1.09284
$$730$$ 0 0
$$731$$ −7.43769 −0.275093
$$732$$ 0 0
$$733$$ 35.7771 1.32146 0.660728 0.750625i $$-0.270247\pi$$
0.660728 + 0.750625i $$0.270247\pi$$
$$734$$ 0 0
$$735$$ 12.7082 0.468749
$$736$$ 0 0
$$737$$ 8.00000 0.294684
$$738$$ 0 0
$$739$$ −23.4164 −0.861386 −0.430693 0.902498i $$-0.641731\pi$$
−0.430693 + 0.902498i $$0.641731\pi$$
$$740$$ 0 0
$$741$$ 71.3050 2.61945
$$742$$ 0 0
$$743$$ 19.4164 0.712319 0.356159 0.934425i $$-0.384086\pi$$
0.356159 + 0.934425i $$0.384086\pi$$
$$744$$ 0 0
$$745$$ 6.18034 0.226430
$$746$$ 0 0
$$747$$ 0.875388 0.0320288
$$748$$ 0 0
$$749$$ −52.8328 −1.93047
$$750$$ 0 0
$$751$$ −40.0000 −1.45962 −0.729810 0.683650i $$-0.760392\pi$$
−0.729810 + 0.683650i $$0.760392\pi$$
$$752$$ 0 0
$$753$$ −6.76393 −0.246491
$$754$$ 0 0
$$755$$ −3.23607 −0.117773
$$756$$ 0 0
$$757$$ 0.180340 0.00655456 0.00327728 0.999995i $$-0.498957\pi$$
0.00327728 + 0.999995i $$0.498957\pi$$
$$758$$ 0 0
$$759$$ 1.70820 0.0620039
$$760$$ 0 0
$$761$$ 4.03444 0.146248 0.0731242 0.997323i $$-0.476703\pi$$
0.0731242 + 0.997323i $$0.476703\pi$$
$$762$$ 0 0
$$763$$ 6.58359 0.238342
$$764$$ 0 0
$$765$$ 0.527864 0.0190850
$$766$$ 0 0
$$767$$ −70.7558 −2.55484
$$768$$ 0 0
$$769$$ 19.3475 0.697690 0.348845 0.937181i $$-0.386574\pi$$
0.348845 + 0.937181i $$0.386574\pi$$
$$770$$ 0 0
$$771$$ 48.6525 1.75218
$$772$$ 0 0
$$773$$ −0.832816 −0.0299543 −0.0149771 0.999888i $$-0.504768\pi$$
−0.0149771 + 0.999888i $$0.504768\pi$$
$$774$$ 0 0
$$775$$ 0.618034 0.0222004
$$776$$ 0 0
$$777$$ −29.7082 −1.06578
$$778$$ 0 0
$$779$$ −70.2492 −2.51694
$$780$$ 0 0
$$781$$ 6.11146 0.218685
$$782$$ 0 0
$$783$$ 5.47214 0.195558
$$784$$ 0 0
$$785$$ −22.9443 −0.818916
$$786$$ 0 0
$$787$$ 13.3475 0.475788 0.237894 0.971291i $$-0.423543\pi$$
0.237894 + 0.971291i $$0.423543\pi$$
$$788$$ 0 0
$$789$$ −42.8328 −1.52489
$$790$$ 0 0
$$791$$ −60.8885 −2.16495
$$792$$ 0 0
$$793$$ 53.9230 1.91486
$$794$$ 0 0
$$795$$ 10.2361 0.363036
$$796$$ 0 0
$$797$$ 22.8328 0.808780 0.404390 0.914587i $$-0.367484\pi$$
0.404390 + 0.914587i $$0.367484\pi$$
$$798$$ 0 0
$$799$$ 11.0557 0.391124
$$800$$ 0 0
$$801$$ −4.47214 −0.158015
$$802$$ 0 0
$$803$$ −16.1803 −0.570992
$$804$$ 0 0
$$805$$ −3.29180 −0.116021
$$806$$ 0 0
$$807$$ −10.4164 −0.366675
$$808$$ 0 0
$$809$$ −15.4164 −0.542012 −0.271006 0.962578i $$-0.587356\pi$$
−0.271006 + 0.962578i $$0.587356\pi$$
$$810$$ 0 0
$$811$$ 16.3262 0.573292 0.286646 0.958037i $$-0.407460\pi$$
0.286646 + 0.958037i $$0.407460\pi$$
$$812$$ 0 0
$$813$$ 18.4721 0.647846
$$814$$ 0 0
$$815$$ 22.4721 0.787165
$$816$$ 0 0
$$817$$ −38.9443 −1.36249
$$818$$ 0 0
$$819$$ 8.96556 0.313282
$$820$$ 0 0
$$821$$ −24.0689 −0.840010 −0.420005 0.907522i $$-0.637972\pi$$
−0.420005 + 0.907522i $$0.637972\pi$$
$$822$$ 0 0
$$823$$ 25.7771 0.898533 0.449266 0.893398i $$-0.351685\pi$$
0.449266 + 0.893398i $$0.351685\pi$$
$$824$$ 0 0
$$825$$ −2.00000 −0.0696311
$$826$$ 0 0
$$827$$ 13.0902 0.455190 0.227595 0.973756i $$-0.426914\pi$$
0.227595 + 0.973756i $$0.426914\pi$$
$$828$$ 0 0
$$829$$ 21.2016 0.736363 0.368181 0.929754i $$-0.379981\pi$$
0.368181 + 0.929754i $$0.379981\pi$$
$$830$$ 0 0
$$831$$ −48.5410 −1.68387
$$832$$ 0 0
$$833$$ −10.8541 −0.376072
$$834$$ 0 0
$$835$$ −6.61803 −0.229027
$$836$$ 0 0
$$837$$ −3.38197 −0.116898
$$838$$ 0 0
$$839$$ 16.3607 0.564833 0.282417 0.959292i $$-0.408864\pi$$
0.282417 + 0.959292i $$0.408864\pi$$
$$840$$ 0 0
$$841$$ 1.00000 0.0344828
$$842$$ 0 0
$$843$$ 3.00000 0.103325
$$844$$ 0 0
$$845$$ 24.0902 0.828727
$$846$$ 0 0
$$847$$ −36.5066 −1.25438
$$848$$ 0 0
$$849$$ 0.291796 0.0100144
$$850$$ 0 0
$$851$$ 4.06888 0.139480
$$852$$ 0 0
$$853$$ −24.1803 −0.827919 −0.413960 0.910295i $$-0.635855\pi$$
−0.413960 + 0.910295i $$0.635855\pi$$
$$854$$ 0 0
$$855$$ 2.76393 0.0945245
$$856$$ 0 0
$$857$$ 20.2918 0.693155 0.346577 0.938021i $$-0.387344\pi$$
0.346577 + 0.938021i $$0.387344\pi$$
$$858$$ 0 0
$$859$$ −14.6525 −0.499936 −0.249968 0.968254i $$-0.580420\pi$$
−0.249968 + 0.968254i $$0.580420\pi$$
$$860$$ 0 0
$$861$$ 60.5410 2.06323
$$862$$ 0 0
$$863$$ −25.0344 −0.852182 −0.426091 0.904680i $$-0.640110\pi$$
−0.426091 + 0.904680i $$0.640110\pi$$
$$864$$ 0 0
$$865$$ −19.7984 −0.673165
$$866$$ 0 0
$$867$$ −24.4164 −0.829225
$$868$$ 0 0
$$869$$ 17.4164 0.590811
$$870$$ 0 0
$$871$$ 39.4164 1.33557
$$872$$ 0 0
$$873$$ 6.03444 0.204235
$$874$$ 0 0
$$875$$ 3.85410 0.130292
$$876$$ 0 0
$$877$$ −48.3820 −1.63374 −0.816871 0.576820i $$-0.804293\pi$$
−0.816871 + 0.576820i $$0.804293\pi$$
$$878$$ 0 0
$$879$$ 13.2361 0.446441
$$880$$ 0 0
$$881$$ −12.1803 −0.410366 −0.205183 0.978724i $$-0.565779\pi$$
−0.205183 + 0.978724i $$0.565779\pi$$
$$882$$ 0 0
$$883$$ 27.5967 0.928705 0.464352 0.885651i $$-0.346287\pi$$
0.464352 + 0.885651i $$0.346287\pi$$
$$884$$ 0 0
$$885$$ 18.7984 0.631900
$$886$$ 0 0
$$887$$ 4.58359 0.153902 0.0769510 0.997035i $$-0.475482\pi$$
0.0769510 + 0.997035i $$0.475482\pi$$
$$888$$ 0 0
$$889$$ 65.3050 2.19026
$$890$$ 0 0
$$891$$ 9.52786 0.319195
$$892$$ 0 0
$$893$$ 57.8885 1.93717
$$894$$ 0 0
$$895$$ −10.0902 −0.337277
$$896$$ 0 0
$$897$$ 8.41641 0.281016
$$898$$ 0 0
$$899$$ −0.618034 −0.0206126
$$900$$ 0 0
$$901$$ −8.74265 −0.291260
$$902$$ 0 0
$$903$$ 33.5623 1.11688
$$904$$ 0 0
$$905$$ −13.7082 −0.455676
$$906$$ 0 0
$$907$$ 37.0902 1.23156 0.615779 0.787919i $$-0.288841\pi$$
0.615779 + 0.787919i $$0.288841\pi$$
$$908$$ 0 0
$$909$$ 1.56231 0.0518184
$$910$$ 0 0
$$911$$ 5.72949 0.189826 0.0949132 0.995486i $$-0.469743\pi$$
0.0949132 + 0.995486i $$0.469743\pi$$
$$912$$ 0 0
$$913$$ 2.83282 0.0937525
$$914$$ 0 0
$$915$$ −14.3262 −0.473611
$$916$$ 0 0
$$917$$ 4.06888 0.134366
$$918$$ 0 0
$$919$$ 17.5279 0.578191 0.289095 0.957300i $$-0.406646\pi$$
0.289095 + 0.957300i $$0.406646\pi$$
$$920$$ 0 0
$$921$$ 29.8885 0.984861
$$922$$ 0 0
$$923$$ 30.1115 0.991131
$$924$$ 0 0
$$925$$ −4.76393 −0.156637
$$926$$ 0 0
$$927$$ 3.41641 0.112210
$$928$$ 0 0
$$929$$ 11.4508 0.375690 0.187845 0.982199i $$-0.439850\pi$$
0.187845 + 0.982199i $$0.439850\pi$$
$$930$$ 0 0
$$931$$ −56.8328 −1.86262
$$932$$ 0 0
$$933$$ −15.0902 −0.494030
$$934$$ 0 0
$$935$$ 1.70820 0.0558642
$$936$$ 0 0
$$937$$ 2.87539 0.0939348 0.0469674 0.998896i $$-0.485044\pi$$
0.0469674 + 0.998896i $$0.485044\pi$$
$$938$$ 0 0
$$939$$ 51.5967 1.68380
$$940$$ 0 0
$$941$$ 12.3607 0.402947 0.201473 0.979494i $$-0.435427\pi$$
0.201473 + 0.979494i $$0.435427\pi$$
$$942$$ 0 0
$$943$$ −8.29180 −0.270018
$$944$$ 0 0
$$945$$ −21.0902 −0.686063
$$946$$ 0 0
$$947$$ −43.9098 −1.42688 −0.713439 0.700717i $$-0.752863\pi$$
−0.713439 + 0.700717i $$0.752863\pi$$
$$948$$ 0 0
$$949$$ −79.7214 −2.58786
$$950$$ 0 0
$$951$$ 14.9443 0.484601
$$952$$ 0 0
$$953$$ −40.8328 −1.32270 −0.661352 0.750075i $$-0.730017\pi$$
−0.661352 + 0.750075i $$0.730017\pi$$
$$954$$ 0 0
$$955$$ −14.1459 −0.457751
$$956$$ 0 0
$$957$$ 2.00000 0.0646508
$$958$$ 0 0
$$959$$ −26.4164 −0.853030
$$960$$ 0 0
$$961$$ −30.6180 −0.987679
$$962$$ 0 0
$$963$$ 5.23607 0.168730
$$964$$ 0 0
$$965$$ 6.85410 0.220641
$$966$$ 0 0
$$967$$ 25.5967 0.823136 0.411568 0.911379i $$-0.364981\pi$$
0.411568 + 0.911379i $$0.364981\pi$$
$$968$$ 0 0
$$969$$ 16.1803 0.519787
$$970$$ 0 0
$$971$$ 6.06888 0.194760 0.0973799 0.995247i $$-0.468954\pi$$
0.0973799 + 0.995247i $$0.468954\pi$$
$$972$$ 0 0
$$973$$ −13.0344 −0.417865
$$974$$ 0 0
$$975$$ −9.85410 −0.315584
$$976$$ 0 0
$$977$$ −20.6525 −0.660731 −0.330366 0.943853i $$-0.607172\pi$$
−0.330366 + 0.943853i $$0.607172\pi$$
$$978$$ 0 0
$$979$$ −14.4721 −0.462531
$$980$$ 0 0
$$981$$ −0.652476 −0.0208320
$$982$$ 0 0
$$983$$ −30.9443 −0.986969 −0.493484 0.869755i $$-0.664277\pi$$
−0.493484 + 0.869755i $$0.664277\pi$$
$$984$$ 0 0
$$985$$ −12.1459 −0.387000
$$986$$ 0 0
$$987$$ −49.8885 −1.58797
$$988$$ 0 0
$$989$$ −4.59675 −0.146168
$$990$$ 0 0
$$991$$ 48.0689 1.52696 0.763479 0.645832i $$-0.223490\pi$$
0.763479 + 0.645832i $$0.223490\pi$$
$$992$$ 0 0
$$993$$ 38.3607 1.21734
$$994$$ 0 0
$$995$$ −13.2361 −0.419612
$$996$$ 0 0
$$997$$ −21.8197 −0.691036 −0.345518 0.938412i $$-0.612297\pi$$
−0.345518 + 0.938412i $$0.612297\pi$$
$$998$$ 0 0
$$999$$ 26.0689 0.824783
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.1.2 2
4.3 odd 2 9280.2.a.y.1.1 2
8.3 odd 2 4640.2.a.i.1.2 yes 2
8.5 even 2 4640.2.a.g.1.1 2

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