# Properties

 Label 476.2.a.c.1.1 Level $476$ Weight $2$ Character 476.1 Self dual yes Analytic conductor $3.801$ Analytic rank $0$ 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] = [476,2,Mod(1,476)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(476, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("476.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$476 = 2^{2} \cdot 7 \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 476.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: $$3.80087913621$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 3$$ x^2 - x - 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$2.30278$$ of defining polynomial Character $$\chi$$ $$=$$ 476.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.30278 q^{3} -1.30278 q^{5} -1.00000 q^{7} +2.30278 q^{9} +O(q^{10})$$ $$q-2.30278 q^{3} -1.30278 q^{5} -1.00000 q^{7} +2.30278 q^{9} +4.00000 q^{11} -4.60555 q^{13} +3.00000 q^{15} -1.00000 q^{17} +8.60555 q^{19} +2.30278 q^{21} +4.00000 q^{23} -3.30278 q^{25} +1.60555 q^{27} +9.21110 q^{29} +7.30278 q^{31} -9.21110 q^{33} +1.30278 q^{35} +9.81665 q^{37} +10.6056 q^{39} -11.5139 q^{41} -4.30278 q^{43} -3.00000 q^{45} -2.60555 q^{47} +1.00000 q^{49} +2.30278 q^{51} +0.697224 q^{53} -5.21110 q^{55} -19.8167 q^{57} -8.00000 q^{59} +15.5139 q^{61} -2.30278 q^{63} +6.00000 q^{65} +2.69722 q^{67} -9.21110 q^{69} -3.39445 q^{71} +7.51388 q^{73} +7.60555 q^{75} -4.00000 q^{77} -2.60555 q^{79} -10.6056 q^{81} +3.21110 q^{83} +1.30278 q^{85} -21.2111 q^{87} +7.81665 q^{89} +4.60555 q^{91} -16.8167 q^{93} -11.2111 q^{95} -13.3028 q^{97} +9.21110 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{3} + q^{5} - 2 q^{7} + q^{9}+O(q^{10})$$ 2 * q - q^3 + q^5 - 2 * q^7 + q^9 $$2 q - q^{3} + q^{5} - 2 q^{7} + q^{9} + 8 q^{11} - 2 q^{13} + 6 q^{15} - 2 q^{17} + 10 q^{19} + q^{21} + 8 q^{23} - 3 q^{25} - 4 q^{27} + 4 q^{29} + 11 q^{31} - 4 q^{33} - q^{35} - 2 q^{37} + 14 q^{39} - 5 q^{41} - 5 q^{43} - 6 q^{45} + 2 q^{47} + 2 q^{49} + q^{51} + 5 q^{53} + 4 q^{55} - 18 q^{57} - 16 q^{59} + 13 q^{61} - q^{63} + 12 q^{65} + 9 q^{67} - 4 q^{69} - 14 q^{71} - 3 q^{73} + 8 q^{75} - 8 q^{77} + 2 q^{79} - 14 q^{81} - 8 q^{83} - q^{85} - 28 q^{87} - 6 q^{89} + 2 q^{91} - 12 q^{93} - 8 q^{95} - 23 q^{97} + 4 q^{99}+O(q^{100})$$ 2 * q - q^3 + q^5 - 2 * q^7 + q^9 + 8 * q^11 - 2 * q^13 + 6 * q^15 - 2 * q^17 + 10 * q^19 + q^21 + 8 * q^23 - 3 * q^25 - 4 * q^27 + 4 * q^29 + 11 * q^31 - 4 * q^33 - q^35 - 2 * q^37 + 14 * q^39 - 5 * q^41 - 5 * q^43 - 6 * q^45 + 2 * q^47 + 2 * q^49 + q^51 + 5 * q^53 + 4 * q^55 - 18 * q^57 - 16 * q^59 + 13 * q^61 - q^63 + 12 * q^65 + 9 * q^67 - 4 * q^69 - 14 * q^71 - 3 * q^73 + 8 * q^75 - 8 * q^77 + 2 * q^79 - 14 * q^81 - 8 * q^83 - q^85 - 28 * q^87 - 6 * q^89 + 2 * q^91 - 12 * q^93 - 8 * q^95 - 23 * q^97 + 4 * q^99

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −2.30278 −1.32951 −0.664754 0.747062i $$-0.731464\pi$$
−0.664754 + 0.747062i $$0.731464\pi$$
$$4$$ 0 0
$$5$$ −1.30278 −0.582619 −0.291309 0.956629i $$-0.594091\pi$$
−0.291309 + 0.956629i $$0.594091\pi$$
$$6$$ 0 0
$$7$$ −1.00000 −0.377964
$$8$$ 0 0
$$9$$ 2.30278 0.767592
$$10$$ 0 0
$$11$$ 4.00000 1.20605 0.603023 0.797724i $$-0.293963\pi$$
0.603023 + 0.797724i $$0.293963\pi$$
$$12$$ 0 0
$$13$$ −4.60555 −1.27735 −0.638675 0.769477i $$-0.720517\pi$$
−0.638675 + 0.769477i $$0.720517\pi$$
$$14$$ 0 0
$$15$$ 3.00000 0.774597
$$16$$ 0 0
$$17$$ −1.00000 −0.242536
$$18$$ 0 0
$$19$$ 8.60555 1.97425 0.987124 0.159954i $$-0.0511347\pi$$
0.987124 + 0.159954i $$0.0511347\pi$$
$$20$$ 0 0
$$21$$ 2.30278 0.502507
$$22$$ 0 0
$$23$$ 4.00000 0.834058 0.417029 0.908893i $$-0.363071\pi$$
0.417029 + 0.908893i $$0.363071\pi$$
$$24$$ 0 0
$$25$$ −3.30278 −0.660555
$$26$$ 0 0
$$27$$ 1.60555 0.308988
$$28$$ 0 0
$$29$$ 9.21110 1.71046 0.855229 0.518250i $$-0.173416\pi$$
0.855229 + 0.518250i $$0.173416\pi$$
$$30$$ 0 0
$$31$$ 7.30278 1.31162 0.655809 0.754927i $$-0.272328\pi$$
0.655809 + 0.754927i $$0.272328\pi$$
$$32$$ 0 0
$$33$$ −9.21110 −1.60345
$$34$$ 0 0
$$35$$ 1.30278 0.220209
$$36$$ 0 0
$$37$$ 9.81665 1.61385 0.806924 0.590655i $$-0.201131\pi$$
0.806924 + 0.590655i $$0.201131\pi$$
$$38$$ 0 0
$$39$$ 10.6056 1.69825
$$40$$ 0 0
$$41$$ −11.5139 −1.79817 −0.899083 0.437779i $$-0.855765\pi$$
−0.899083 + 0.437779i $$0.855765\pi$$
$$42$$ 0 0
$$43$$ −4.30278 −0.656167 −0.328084 0.944649i $$-0.606403\pi$$
−0.328084 + 0.944649i $$0.606403\pi$$
$$44$$ 0 0
$$45$$ −3.00000 −0.447214
$$46$$ 0 0
$$47$$ −2.60555 −0.380059 −0.190029 0.981778i $$-0.560858\pi$$
−0.190029 + 0.981778i $$0.560858\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 2.30278 0.322453
$$52$$ 0 0
$$53$$ 0.697224 0.0957711 0.0478856 0.998853i $$-0.484752\pi$$
0.0478856 + 0.998853i $$0.484752\pi$$
$$54$$ 0 0
$$55$$ −5.21110 −0.702665
$$56$$ 0 0
$$57$$ −19.8167 −2.62478
$$58$$ 0 0
$$59$$ −8.00000 −1.04151 −0.520756 0.853706i $$-0.674350\pi$$
−0.520756 + 0.853706i $$0.674350\pi$$
$$60$$ 0 0
$$61$$ 15.5139 1.98635 0.993174 0.116640i $$-0.0372123\pi$$
0.993174 + 0.116640i $$0.0372123\pi$$
$$62$$ 0 0
$$63$$ −2.30278 −0.290122
$$64$$ 0 0
$$65$$ 6.00000 0.744208
$$66$$ 0 0
$$67$$ 2.69722 0.329518 0.164759 0.986334i $$-0.447315\pi$$
0.164759 + 0.986334i $$0.447315\pi$$
$$68$$ 0 0
$$69$$ −9.21110 −1.10889
$$70$$ 0 0
$$71$$ −3.39445 −0.402847 −0.201423 0.979504i $$-0.564557\pi$$
−0.201423 + 0.979504i $$0.564557\pi$$
$$72$$ 0 0
$$73$$ 7.51388 0.879433 0.439716 0.898137i $$-0.355079\pi$$
0.439716 + 0.898137i $$0.355079\pi$$
$$74$$ 0 0
$$75$$ 7.60555 0.878213
$$76$$ 0 0
$$77$$ −4.00000 −0.455842
$$78$$ 0 0
$$79$$ −2.60555 −0.293147 −0.146574 0.989200i $$-0.546825\pi$$
−0.146574 + 0.989200i $$0.546825\pi$$
$$80$$ 0 0
$$81$$ −10.6056 −1.17839
$$82$$ 0 0
$$83$$ 3.21110 0.352464 0.176232 0.984349i $$-0.443609\pi$$
0.176232 + 0.984349i $$0.443609\pi$$
$$84$$ 0 0
$$85$$ 1.30278 0.141306
$$86$$ 0 0
$$87$$ −21.2111 −2.27407
$$88$$ 0 0
$$89$$ 7.81665 0.828564 0.414282 0.910149i $$-0.364033\pi$$
0.414282 + 0.910149i $$0.364033\pi$$
$$90$$ 0 0
$$91$$ 4.60555 0.482793
$$92$$ 0 0
$$93$$ −16.8167 −1.74381
$$94$$ 0 0
$$95$$ −11.2111 −1.15023
$$96$$ 0 0
$$97$$ −13.3028 −1.35069 −0.675346 0.737501i $$-0.736006\pi$$
−0.675346 + 0.737501i $$0.736006\pi$$
$$98$$ 0 0
$$99$$ 9.21110 0.925751
$$100$$ 0 0
$$101$$ −12.6056 −1.25430 −0.627150 0.778899i $$-0.715779\pi$$
−0.627150 + 0.778899i $$0.715779\pi$$
$$102$$ 0 0
$$103$$ 14.0000 1.37946 0.689730 0.724066i $$-0.257729\pi$$
0.689730 + 0.724066i $$0.257729\pi$$
$$104$$ 0 0
$$105$$ −3.00000 −0.292770
$$106$$ 0 0
$$107$$ −4.60555 −0.445235 −0.222618 0.974906i $$-0.571460\pi$$
−0.222618 + 0.974906i $$0.571460\pi$$
$$108$$ 0 0
$$109$$ 4.78890 0.458693 0.229347 0.973345i $$-0.426341\pi$$
0.229347 + 0.973345i $$0.426341\pi$$
$$110$$ 0 0
$$111$$ −22.6056 −2.14562
$$112$$ 0 0
$$113$$ −9.39445 −0.883755 −0.441878 0.897075i $$-0.645687\pi$$
−0.441878 + 0.897075i $$0.645687\pi$$
$$114$$ 0 0
$$115$$ −5.21110 −0.485938
$$116$$ 0 0
$$117$$ −10.6056 −0.980484
$$118$$ 0 0
$$119$$ 1.00000 0.0916698
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ 0 0
$$123$$ 26.5139 2.39068
$$124$$ 0 0
$$125$$ 10.8167 0.967471
$$126$$ 0 0
$$127$$ 14.5139 1.28790 0.643949 0.765068i $$-0.277295\pi$$
0.643949 + 0.765068i $$0.277295\pi$$
$$128$$ 0 0
$$129$$ 9.90833 0.872380
$$130$$ 0 0
$$131$$ −9.21110 −0.804778 −0.402389 0.915469i $$-0.631820\pi$$
−0.402389 + 0.915469i $$0.631820\pi$$
$$132$$ 0 0
$$133$$ −8.60555 −0.746196
$$134$$ 0 0
$$135$$ −2.09167 −0.180023
$$136$$ 0 0
$$137$$ 18.1194 1.54805 0.774024 0.633157i $$-0.218241\pi$$
0.774024 + 0.633157i $$0.218241\pi$$
$$138$$ 0 0
$$139$$ 8.51388 0.722138 0.361069 0.932539i $$-0.382412\pi$$
0.361069 + 0.932539i $$0.382412\pi$$
$$140$$ 0 0
$$141$$ 6.00000 0.505291
$$142$$ 0 0
$$143$$ −18.4222 −1.54054
$$144$$ 0 0
$$145$$ −12.0000 −0.996546
$$146$$ 0 0
$$147$$ −2.30278 −0.189930
$$148$$ 0 0
$$149$$ −4.30278 −0.352497 −0.176249 0.984346i $$-0.556396\pi$$
−0.176249 + 0.984346i $$0.556396\pi$$
$$150$$ 0 0
$$151$$ 11.6972 0.951907 0.475953 0.879471i $$-0.342103\pi$$
0.475953 + 0.879471i $$0.342103\pi$$
$$152$$ 0 0
$$153$$ −2.30278 −0.186168
$$154$$ 0 0
$$155$$ −9.51388 −0.764173
$$156$$ 0 0
$$157$$ 2.00000 0.159617 0.0798087 0.996810i $$-0.474569\pi$$
0.0798087 + 0.996810i $$0.474569\pi$$
$$158$$ 0 0
$$159$$ −1.60555 −0.127328
$$160$$ 0 0
$$161$$ −4.00000 −0.315244
$$162$$ 0 0
$$163$$ 1.39445 0.109222 0.0546108 0.998508i $$-0.482608\pi$$
0.0546108 + 0.998508i $$0.482608\pi$$
$$164$$ 0 0
$$165$$ 12.0000 0.934199
$$166$$ 0 0
$$167$$ 12.1194 0.937830 0.468915 0.883243i $$-0.344645\pi$$
0.468915 + 0.883243i $$0.344645\pi$$
$$168$$ 0 0
$$169$$ 8.21110 0.631623
$$170$$ 0 0
$$171$$ 19.8167 1.51542
$$172$$ 0 0
$$173$$ 20.7250 1.57569 0.787846 0.615873i $$-0.211197\pi$$
0.787846 + 0.615873i $$0.211197\pi$$
$$174$$ 0 0
$$175$$ 3.30278 0.249666
$$176$$ 0 0
$$177$$ 18.4222 1.38470
$$178$$ 0 0
$$179$$ 9.51388 0.711101 0.355550 0.934657i $$-0.384293\pi$$
0.355550 + 0.934657i $$0.384293\pi$$
$$180$$ 0 0
$$181$$ −6.00000 −0.445976 −0.222988 0.974821i $$-0.571581\pi$$
−0.222988 + 0.974821i $$0.571581\pi$$
$$182$$ 0 0
$$183$$ −35.7250 −2.64087
$$184$$ 0 0
$$185$$ −12.7889 −0.940258
$$186$$ 0 0
$$187$$ −4.00000 −0.292509
$$188$$ 0 0
$$189$$ −1.60555 −0.116787
$$190$$ 0 0
$$191$$ −11.7250 −0.848390 −0.424195 0.905571i $$-0.639443\pi$$
−0.424195 + 0.905571i $$0.639443\pi$$
$$192$$ 0 0
$$193$$ −15.0278 −1.08172 −0.540861 0.841112i $$-0.681901\pi$$
−0.540861 + 0.841112i $$0.681901\pi$$
$$194$$ 0 0
$$195$$ −13.8167 −0.989431
$$196$$ 0 0
$$197$$ −6.60555 −0.470626 −0.235313 0.971920i $$-0.575612\pi$$
−0.235313 + 0.971920i $$0.575612\pi$$
$$198$$ 0 0
$$199$$ 9.69722 0.687418 0.343709 0.939076i $$-0.388317\pi$$
0.343709 + 0.939076i $$0.388317\pi$$
$$200$$ 0 0
$$201$$ −6.21110 −0.438097
$$202$$ 0 0
$$203$$ −9.21110 −0.646493
$$204$$ 0 0
$$205$$ 15.0000 1.04765
$$206$$ 0 0
$$207$$ 9.21110 0.640216
$$208$$ 0 0
$$209$$ 34.4222 2.38103
$$210$$ 0 0
$$211$$ 6.18335 0.425679 0.212840 0.977087i $$-0.431729\pi$$
0.212840 + 0.977087i $$0.431729\pi$$
$$212$$ 0 0
$$213$$ 7.81665 0.535588
$$214$$ 0 0
$$215$$ 5.60555 0.382295
$$216$$ 0 0
$$217$$ −7.30278 −0.495745
$$218$$ 0 0
$$219$$ −17.3028 −1.16921
$$220$$ 0 0
$$221$$ 4.60555 0.309803
$$222$$ 0 0
$$223$$ 13.8167 0.925232 0.462616 0.886559i $$-0.346911\pi$$
0.462616 + 0.886559i $$0.346911\pi$$
$$224$$ 0 0
$$225$$ −7.60555 −0.507037
$$226$$ 0 0
$$227$$ 2.09167 0.138829 0.0694146 0.997588i $$-0.477887\pi$$
0.0694146 + 0.997588i $$0.477887\pi$$
$$228$$ 0 0
$$229$$ −1.21110 −0.0800319 −0.0400160 0.999199i $$-0.512741\pi$$
−0.0400160 + 0.999199i $$0.512741\pi$$
$$230$$ 0 0
$$231$$ 9.21110 0.606046
$$232$$ 0 0
$$233$$ 26.6056 1.74299 0.871494 0.490407i $$-0.163152\pi$$
0.871494 + 0.490407i $$0.163152\pi$$
$$234$$ 0 0
$$235$$ 3.39445 0.221429
$$236$$ 0 0
$$237$$ 6.00000 0.389742
$$238$$ 0 0
$$239$$ 18.1194 1.17205 0.586024 0.810294i $$-0.300692\pi$$
0.586024 + 0.810294i $$0.300692\pi$$
$$240$$ 0 0
$$241$$ −26.5139 −1.70791 −0.853955 0.520348i $$-0.825802\pi$$
−0.853955 + 0.520348i $$0.825802\pi$$
$$242$$ 0 0
$$243$$ 19.6056 1.25770
$$244$$ 0 0
$$245$$ −1.30278 −0.0832313
$$246$$ 0 0
$$247$$ −39.6333 −2.52181
$$248$$ 0 0
$$249$$ −7.39445 −0.468604
$$250$$ 0 0
$$251$$ −2.18335 −0.137812 −0.0689058 0.997623i $$-0.521951\pi$$
−0.0689058 + 0.997623i $$0.521951\pi$$
$$252$$ 0 0
$$253$$ 16.0000 1.00591
$$254$$ 0 0
$$255$$ −3.00000 −0.187867
$$256$$ 0 0
$$257$$ −4.42221 −0.275850 −0.137925 0.990443i $$-0.544043\pi$$
−0.137925 + 0.990443i $$0.544043\pi$$
$$258$$ 0 0
$$259$$ −9.81665 −0.609977
$$260$$ 0 0
$$261$$ 21.2111 1.31293
$$262$$ 0 0
$$263$$ −18.4222 −1.13596 −0.567981 0.823042i $$-0.692275\pi$$
−0.567981 + 0.823042i $$0.692275\pi$$
$$264$$ 0 0
$$265$$ −0.908327 −0.0557981
$$266$$ 0 0
$$267$$ −18.0000 −1.10158
$$268$$ 0 0
$$269$$ 12.7889 0.779753 0.389876 0.920867i $$-0.372518\pi$$
0.389876 + 0.920867i $$0.372518\pi$$
$$270$$ 0 0
$$271$$ −10.4222 −0.633104 −0.316552 0.948575i $$-0.602525\pi$$
−0.316552 + 0.948575i $$0.602525\pi$$
$$272$$ 0 0
$$273$$ −10.6056 −0.641877
$$274$$ 0 0
$$275$$ −13.2111 −0.796659
$$276$$ 0 0
$$277$$ 1.57779 0.0948005 0.0474003 0.998876i $$-0.484906\pi$$
0.0474003 + 0.998876i $$0.484906\pi$$
$$278$$ 0 0
$$279$$ 16.8167 1.00679
$$280$$ 0 0
$$281$$ −19.1194 −1.14057 −0.570285 0.821447i $$-0.693167\pi$$
−0.570285 + 0.821447i $$0.693167\pi$$
$$282$$ 0 0
$$283$$ 20.3305 1.20852 0.604262 0.796785i $$-0.293468\pi$$
0.604262 + 0.796785i $$0.293468\pi$$
$$284$$ 0 0
$$285$$ 25.8167 1.52925
$$286$$ 0 0
$$287$$ 11.5139 0.679643
$$288$$ 0 0
$$289$$ 1.00000 0.0588235
$$290$$ 0 0
$$291$$ 30.6333 1.79576
$$292$$ 0 0
$$293$$ −11.2111 −0.654960 −0.327480 0.944858i $$-0.606199\pi$$
−0.327480 + 0.944858i $$0.606199\pi$$
$$294$$ 0 0
$$295$$ 10.4222 0.606804
$$296$$ 0 0
$$297$$ 6.42221 0.372654
$$298$$ 0 0
$$299$$ −18.4222 −1.06538
$$300$$ 0 0
$$301$$ 4.30278 0.248008
$$302$$ 0 0
$$303$$ 29.0278 1.66760
$$304$$ 0 0
$$305$$ −20.2111 −1.15728
$$306$$ 0 0
$$307$$ 4.00000 0.228292 0.114146 0.993464i $$-0.463587\pi$$
0.114146 + 0.993464i $$0.463587\pi$$
$$308$$ 0 0
$$309$$ −32.2389 −1.83400
$$310$$ 0 0
$$311$$ −17.7250 −1.00509 −0.502546 0.864551i $$-0.667603\pi$$
−0.502546 + 0.864551i $$0.667603\pi$$
$$312$$ 0 0
$$313$$ 3.90833 0.220912 0.110456 0.993881i $$-0.464769\pi$$
0.110456 + 0.993881i $$0.464769\pi$$
$$314$$ 0 0
$$315$$ 3.00000 0.169031
$$316$$ 0 0
$$317$$ −33.6333 −1.88903 −0.944517 0.328461i $$-0.893470\pi$$
−0.944517 + 0.328461i $$0.893470\pi$$
$$318$$ 0 0
$$319$$ 36.8444 2.06289
$$320$$ 0 0
$$321$$ 10.6056 0.591944
$$322$$ 0 0
$$323$$ −8.60555 −0.478826
$$324$$ 0 0
$$325$$ 15.2111 0.843760
$$326$$ 0 0
$$327$$ −11.0278 −0.609836
$$328$$ 0 0
$$329$$ 2.60555 0.143649
$$330$$ 0 0
$$331$$ 3.72498 0.204743 0.102372 0.994746i $$-0.467357\pi$$
0.102372 + 0.994746i $$0.467357\pi$$
$$332$$ 0 0
$$333$$ 22.6056 1.23878
$$334$$ 0 0
$$335$$ −3.51388 −0.191984
$$336$$ 0 0
$$337$$ −14.4222 −0.785628 −0.392814 0.919618i $$-0.628498\pi$$
−0.392814 + 0.919618i $$0.628498\pi$$
$$338$$ 0 0
$$339$$ 21.6333 1.17496
$$340$$ 0 0
$$341$$ 29.2111 1.58187
$$342$$ 0 0
$$343$$ −1.00000 −0.0539949
$$344$$ 0 0
$$345$$ 12.0000 0.646058
$$346$$ 0 0
$$347$$ 26.8444 1.44108 0.720542 0.693412i $$-0.243893\pi$$
0.720542 + 0.693412i $$0.243893\pi$$
$$348$$ 0 0
$$349$$ 18.4222 0.986118 0.493059 0.869996i $$-0.335879\pi$$
0.493059 + 0.869996i $$0.335879\pi$$
$$350$$ 0 0
$$351$$ −7.39445 −0.394686
$$352$$ 0 0
$$353$$ −16.0000 −0.851594 −0.425797 0.904819i $$-0.640006\pi$$
−0.425797 + 0.904819i $$0.640006\pi$$
$$354$$ 0 0
$$355$$ 4.42221 0.234706
$$356$$ 0 0
$$357$$ −2.30278 −0.121876
$$358$$ 0 0
$$359$$ −1.48612 −0.0784345 −0.0392173 0.999231i $$-0.512486\pi$$
−0.0392173 + 0.999231i $$0.512486\pi$$
$$360$$ 0 0
$$361$$ 55.0555 2.89766
$$362$$ 0 0
$$363$$ −11.5139 −0.604322
$$364$$ 0 0
$$365$$ −9.78890 −0.512374
$$366$$ 0 0
$$367$$ 6.90833 0.360612 0.180306 0.983611i $$-0.442291\pi$$
0.180306 + 0.983611i $$0.442291\pi$$
$$368$$ 0 0
$$369$$ −26.5139 −1.38026
$$370$$ 0 0
$$371$$ −0.697224 −0.0361981
$$372$$ 0 0
$$373$$ −15.3305 −0.793785 −0.396892 0.917865i $$-0.629911\pi$$
−0.396892 + 0.917865i $$0.629911\pi$$
$$374$$ 0 0
$$375$$ −24.9083 −1.28626
$$376$$ 0 0
$$377$$ −42.4222 −2.18485
$$378$$ 0 0
$$379$$ 9.57779 0.491978 0.245989 0.969273i $$-0.420887\pi$$
0.245989 + 0.969273i $$0.420887\pi$$
$$380$$ 0 0
$$381$$ −33.4222 −1.71227
$$382$$ 0 0
$$383$$ −1.57779 −0.0806216 −0.0403108 0.999187i $$-0.512835\pi$$
−0.0403108 + 0.999187i $$0.512835\pi$$
$$384$$ 0 0
$$385$$ 5.21110 0.265582
$$386$$ 0 0
$$387$$ −9.90833 −0.503669
$$388$$ 0 0
$$389$$ −29.7250 −1.50712 −0.753558 0.657381i $$-0.771664\pi$$
−0.753558 + 0.657381i $$0.771664\pi$$
$$390$$ 0 0
$$391$$ −4.00000 −0.202289
$$392$$ 0 0
$$393$$ 21.2111 1.06996
$$394$$ 0 0
$$395$$ 3.39445 0.170793
$$396$$ 0 0
$$397$$ −12.0917 −0.606864 −0.303432 0.952853i $$-0.598132\pi$$
−0.303432 + 0.952853i $$0.598132\pi$$
$$398$$ 0 0
$$399$$ 19.8167 0.992074
$$400$$ 0 0
$$401$$ 2.60555 0.130115 0.0650575 0.997882i $$-0.479277\pi$$
0.0650575 + 0.997882i $$0.479277\pi$$
$$402$$ 0 0
$$403$$ −33.6333 −1.67539
$$404$$ 0 0
$$405$$ 13.8167 0.686555
$$406$$ 0 0
$$407$$ 39.2666 1.94637
$$408$$ 0 0
$$409$$ −5.81665 −0.287615 −0.143808 0.989606i $$-0.545935\pi$$
−0.143808 + 0.989606i $$0.545935\pi$$
$$410$$ 0 0
$$411$$ −41.7250 −2.05814
$$412$$ 0 0
$$413$$ 8.00000 0.393654
$$414$$ 0 0
$$415$$ −4.18335 −0.205352
$$416$$ 0 0
$$417$$ −19.6056 −0.960088
$$418$$ 0 0
$$419$$ −8.51388 −0.415930 −0.207965 0.978136i $$-0.566684\pi$$
−0.207965 + 0.978136i $$0.566684\pi$$
$$420$$ 0 0
$$421$$ 11.0917 0.540575 0.270288 0.962780i $$-0.412881\pi$$
0.270288 + 0.962780i $$0.412881\pi$$
$$422$$ 0 0
$$423$$ −6.00000 −0.291730
$$424$$ 0 0
$$425$$ 3.30278 0.160208
$$426$$ 0 0
$$427$$ −15.5139 −0.750769
$$428$$ 0 0
$$429$$ 42.4222 2.04816
$$430$$ 0 0
$$431$$ −28.8444 −1.38939 −0.694693 0.719306i $$-0.744460\pi$$
−0.694693 + 0.719306i $$0.744460\pi$$
$$432$$ 0 0
$$433$$ −7.39445 −0.355355 −0.177677 0.984089i $$-0.556858\pi$$
−0.177677 + 0.984089i $$0.556858\pi$$
$$434$$ 0 0
$$435$$ 27.6333 1.32492
$$436$$ 0 0
$$437$$ 34.4222 1.64664
$$438$$ 0 0
$$439$$ 21.0917 1.00665 0.503325 0.864097i $$-0.332110\pi$$
0.503325 + 0.864097i $$0.332110\pi$$
$$440$$ 0 0
$$441$$ 2.30278 0.109656
$$442$$ 0 0
$$443$$ −33.2111 −1.57791 −0.788954 0.614453i $$-0.789377\pi$$
−0.788954 + 0.614453i $$0.789377\pi$$
$$444$$ 0 0
$$445$$ −10.1833 −0.482737
$$446$$ 0 0
$$447$$ 9.90833 0.468648
$$448$$ 0 0
$$449$$ −5.02776 −0.237274 −0.118637 0.992938i $$-0.537853\pi$$
−0.118637 + 0.992938i $$0.537853\pi$$
$$450$$ 0 0
$$451$$ −46.0555 −2.16867
$$452$$ 0 0
$$453$$ −26.9361 −1.26557
$$454$$ 0 0
$$455$$ −6.00000 −0.281284
$$456$$ 0 0
$$457$$ −13.9083 −0.650604 −0.325302 0.945610i $$-0.605466\pi$$
−0.325302 + 0.945610i $$0.605466\pi$$
$$458$$ 0 0
$$459$$ −1.60555 −0.0749407
$$460$$ 0 0
$$461$$ 23.3944 1.08959 0.544794 0.838570i $$-0.316608\pi$$
0.544794 + 0.838570i $$0.316608\pi$$
$$462$$ 0 0
$$463$$ −6.72498 −0.312536 −0.156268 0.987715i $$-0.549946\pi$$
−0.156268 + 0.987715i $$0.549946\pi$$
$$464$$ 0 0
$$465$$ 21.9083 1.01597
$$466$$ 0 0
$$467$$ −16.6056 −0.768413 −0.384207 0.923247i $$-0.625525\pi$$
−0.384207 + 0.923247i $$0.625525\pi$$
$$468$$ 0 0
$$469$$ −2.69722 −0.124546
$$470$$ 0 0
$$471$$ −4.60555 −0.212213
$$472$$ 0 0
$$473$$ −17.2111 −0.791367
$$474$$ 0 0
$$475$$ −28.4222 −1.30410
$$476$$ 0 0
$$477$$ 1.60555 0.0735131
$$478$$ 0 0
$$479$$ 17.0917 0.780938 0.390469 0.920616i $$-0.372313\pi$$
0.390469 + 0.920616i $$0.372313\pi$$
$$480$$ 0 0
$$481$$ −45.2111 −2.06145
$$482$$ 0 0
$$483$$ 9.21110 0.419120
$$484$$ 0 0
$$485$$ 17.3305 0.786939
$$486$$ 0 0
$$487$$ −28.0000 −1.26880 −0.634401 0.773004i $$-0.718753\pi$$
−0.634401 + 0.773004i $$0.718753\pi$$
$$488$$ 0 0
$$489$$ −3.21110 −0.145211
$$490$$ 0 0
$$491$$ −22.5139 −1.01604 −0.508019 0.861346i $$-0.669622\pi$$
−0.508019 + 0.861346i $$0.669622\pi$$
$$492$$ 0 0
$$493$$ −9.21110 −0.414847
$$494$$ 0 0
$$495$$ −12.0000 −0.539360
$$496$$ 0 0
$$497$$ 3.39445 0.152262
$$498$$ 0 0
$$499$$ −29.2111 −1.30767 −0.653834 0.756638i $$-0.726841\pi$$
−0.653834 + 0.756638i $$0.726841\pi$$
$$500$$ 0 0
$$501$$ −27.9083 −1.24685
$$502$$ 0 0
$$503$$ −31.1194 −1.38755 −0.693773 0.720193i $$-0.744053\pi$$
−0.693773 + 0.720193i $$0.744053\pi$$
$$504$$ 0 0
$$505$$ 16.4222 0.730779
$$506$$ 0 0
$$507$$ −18.9083 −0.839748
$$508$$ 0 0
$$509$$ 6.60555 0.292786 0.146393 0.989227i $$-0.453234\pi$$
0.146393 + 0.989227i $$0.453234\pi$$
$$510$$ 0 0
$$511$$ −7.51388 −0.332394
$$512$$ 0 0
$$513$$ 13.8167 0.610020
$$514$$ 0 0
$$515$$ −18.2389 −0.803700
$$516$$ 0 0
$$517$$ −10.4222 −0.458368
$$518$$ 0 0
$$519$$ −47.7250 −2.09489
$$520$$ 0 0
$$521$$ −0.486122 −0.0212974 −0.0106487 0.999943i $$-0.503390\pi$$
−0.0106487 + 0.999943i $$0.503390\pi$$
$$522$$ 0 0
$$523$$ 38.0555 1.66405 0.832026 0.554737i $$-0.187181\pi$$
0.832026 + 0.554737i $$0.187181\pi$$
$$524$$ 0 0
$$525$$ −7.60555 −0.331933
$$526$$ 0 0
$$527$$ −7.30278 −0.318114
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ 0 0
$$531$$ −18.4222 −0.799456
$$532$$ 0 0
$$533$$ 53.0278 2.29689
$$534$$ 0 0
$$535$$ 6.00000 0.259403
$$536$$ 0 0
$$537$$ −21.9083 −0.945414
$$538$$ 0 0
$$539$$ 4.00000 0.172292
$$540$$ 0 0
$$541$$ 4.00000 0.171973 0.0859867 0.996296i $$-0.472596\pi$$
0.0859867 + 0.996296i $$0.472596\pi$$
$$542$$ 0 0
$$543$$ 13.8167 0.592929
$$544$$ 0 0
$$545$$ −6.23886 −0.267243
$$546$$ 0 0
$$547$$ 2.00000 0.0855138 0.0427569 0.999086i $$-0.486386\pi$$
0.0427569 + 0.999086i $$0.486386\pi$$
$$548$$ 0 0
$$549$$ 35.7250 1.52471
$$550$$ 0 0
$$551$$ 79.2666 3.37687
$$552$$ 0 0
$$553$$ 2.60555 0.110799
$$554$$ 0 0
$$555$$ 29.4500 1.25008
$$556$$ 0 0
$$557$$ −20.7889 −0.880854 −0.440427 0.897788i $$-0.645173\pi$$
−0.440427 + 0.897788i $$0.645173\pi$$
$$558$$ 0 0
$$559$$ 19.8167 0.838155
$$560$$ 0 0
$$561$$ 9.21110 0.388893
$$562$$ 0 0
$$563$$ 4.00000 0.168580 0.0842900 0.996441i $$-0.473138\pi$$
0.0842900 + 0.996441i $$0.473138\pi$$
$$564$$ 0 0
$$565$$ 12.2389 0.514893
$$566$$ 0 0
$$567$$ 10.6056 0.445391
$$568$$ 0 0
$$569$$ 22.1194 0.927295 0.463647 0.886020i $$-0.346540\pi$$
0.463647 + 0.886020i $$0.346540\pi$$
$$570$$ 0 0
$$571$$ −45.4500 −1.90202 −0.951011 0.309158i $$-0.899953\pi$$
−0.951011 + 0.309158i $$0.899953\pi$$
$$572$$ 0 0
$$573$$ 27.0000 1.12794
$$574$$ 0 0
$$575$$ −13.2111 −0.550941
$$576$$ 0 0
$$577$$ −8.18335 −0.340677 −0.170339 0.985386i $$-0.554486\pi$$
−0.170339 + 0.985386i $$0.554486\pi$$
$$578$$ 0 0
$$579$$ 34.6056 1.43816
$$580$$ 0 0
$$581$$ −3.21110 −0.133219
$$582$$ 0 0
$$583$$ 2.78890 0.115504
$$584$$ 0 0
$$585$$ 13.8167 0.571248
$$586$$ 0 0
$$587$$ −11.3944 −0.470299 −0.235150 0.971959i $$-0.575558\pi$$
−0.235150 + 0.971959i $$0.575558\pi$$
$$588$$ 0 0
$$589$$ 62.8444 2.58946
$$590$$ 0 0
$$591$$ 15.2111 0.625701
$$592$$ 0 0
$$593$$ 22.1833 0.910961 0.455480 0.890246i $$-0.349468\pi$$
0.455480 + 0.890246i $$0.349468\pi$$
$$594$$ 0 0
$$595$$ −1.30278 −0.0534086
$$596$$ 0 0
$$597$$ −22.3305 −0.913928
$$598$$ 0 0
$$599$$ 26.7250 1.09195 0.545977 0.837800i $$-0.316159\pi$$
0.545977 + 0.837800i $$0.316159\pi$$
$$600$$ 0 0
$$601$$ 45.2666 1.84646 0.923232 0.384243i $$-0.125538\pi$$
0.923232 + 0.384243i $$0.125538\pi$$
$$602$$ 0 0
$$603$$ 6.21110 0.252936
$$604$$ 0 0
$$605$$ −6.51388 −0.264827
$$606$$ 0 0
$$607$$ 0.330532 0.0134159 0.00670794 0.999978i $$-0.497865\pi$$
0.00670794 + 0.999978i $$0.497865\pi$$
$$608$$ 0 0
$$609$$ 21.2111 0.859517
$$610$$ 0 0
$$611$$ 12.0000 0.485468
$$612$$ 0 0
$$613$$ −21.7250 −0.877464 −0.438732 0.898618i $$-0.644572\pi$$
−0.438732 + 0.898618i $$0.644572\pi$$
$$614$$ 0 0
$$615$$ −34.5416 −1.39285
$$616$$ 0 0
$$617$$ 7.63331 0.307305 0.153653 0.988125i $$-0.450896\pi$$
0.153653 + 0.988125i $$0.450896\pi$$
$$618$$ 0 0
$$619$$ 37.2111 1.49564 0.747820 0.663901i $$-0.231100\pi$$
0.747820 + 0.663901i $$0.231100\pi$$
$$620$$ 0 0
$$621$$ 6.42221 0.257714
$$622$$ 0 0
$$623$$ −7.81665 −0.313168
$$624$$ 0 0
$$625$$ 2.42221 0.0968882
$$626$$ 0 0
$$627$$ −79.2666 −3.16560
$$628$$ 0 0
$$629$$ −9.81665 −0.391416
$$630$$ 0 0
$$631$$ 9.11943 0.363039 0.181519 0.983387i $$-0.441898\pi$$
0.181519 + 0.983387i $$0.441898\pi$$
$$632$$ 0 0
$$633$$ −14.2389 −0.565944
$$634$$ 0 0
$$635$$ −18.9083 −0.750354
$$636$$ 0 0
$$637$$ −4.60555 −0.182479
$$638$$ 0 0
$$639$$ −7.81665 −0.309222
$$640$$ 0 0
$$641$$ 43.6333 1.72341 0.861706 0.507408i $$-0.169396\pi$$
0.861706 + 0.507408i $$0.169396\pi$$
$$642$$ 0 0
$$643$$ 7.88057 0.310779 0.155390 0.987853i $$-0.450337\pi$$
0.155390 + 0.987853i $$0.450337\pi$$
$$644$$ 0 0
$$645$$ −12.9083 −0.508265
$$646$$ 0 0
$$647$$ −35.6333 −1.40089 −0.700445 0.713706i $$-0.747015\pi$$
−0.700445 + 0.713706i $$0.747015\pi$$
$$648$$ 0 0
$$649$$ −32.0000 −1.25611
$$650$$ 0 0
$$651$$ 16.8167 0.659097
$$652$$ 0 0
$$653$$ 40.4222 1.58184 0.790922 0.611918i $$-0.209602\pi$$
0.790922 + 0.611918i $$0.209602\pi$$
$$654$$ 0 0
$$655$$ 12.0000 0.468879
$$656$$ 0 0
$$657$$ 17.3028 0.675046
$$658$$ 0 0
$$659$$ 21.1194 0.822696 0.411348 0.911478i $$-0.365058\pi$$
0.411348 + 0.911478i $$0.365058\pi$$
$$660$$ 0 0
$$661$$ −5.02776 −0.195557 −0.0977785 0.995208i $$-0.531174\pi$$
−0.0977785 + 0.995208i $$0.531174\pi$$
$$662$$ 0 0
$$663$$ −10.6056 −0.411885
$$664$$ 0 0
$$665$$ 11.2111 0.434748
$$666$$ 0 0
$$667$$ 36.8444 1.42662
$$668$$ 0 0
$$669$$ −31.8167 −1.23010
$$670$$ 0 0
$$671$$ 62.0555 2.39563
$$672$$ 0 0
$$673$$ −7.02776 −0.270900 −0.135450 0.990784i $$-0.543248\pi$$
−0.135450 + 0.990784i $$0.543248\pi$$
$$674$$ 0 0
$$675$$ −5.30278 −0.204104
$$676$$ 0 0
$$677$$ −6.00000 −0.230599 −0.115299 0.993331i $$-0.536783\pi$$
−0.115299 + 0.993331i $$0.536783\pi$$
$$678$$ 0 0
$$679$$ 13.3028 0.510514
$$680$$ 0 0
$$681$$ −4.81665 −0.184575
$$682$$ 0 0
$$683$$ 13.3944 0.512524 0.256262 0.966607i $$-0.417509\pi$$
0.256262 + 0.966607i $$0.417509\pi$$
$$684$$ 0 0
$$685$$ −23.6056 −0.901922
$$686$$ 0 0
$$687$$ 2.78890 0.106403
$$688$$ 0 0
$$689$$ −3.21110 −0.122333
$$690$$ 0 0
$$691$$ −1.93608 −0.0736521 −0.0368260 0.999322i $$-0.511725\pi$$
−0.0368260 + 0.999322i $$0.511725\pi$$
$$692$$ 0 0
$$693$$ −9.21110 −0.349901
$$694$$ 0 0
$$695$$ −11.0917 −0.420731
$$696$$ 0 0
$$697$$ 11.5139 0.436119
$$698$$ 0 0
$$699$$ −61.2666 −2.31732
$$700$$ 0 0
$$701$$ −11.5778 −0.437287 −0.218644 0.975805i $$-0.570163\pi$$
−0.218644 + 0.975805i $$0.570163\pi$$
$$702$$ 0 0
$$703$$ 84.4777 3.18614
$$704$$ 0 0
$$705$$ −7.81665 −0.294392
$$706$$ 0 0
$$707$$ 12.6056 0.474081
$$708$$ 0 0
$$709$$ −4.23886 −0.159194 −0.0795968 0.996827i $$-0.525363\pi$$
−0.0795968 + 0.996827i $$0.525363\pi$$
$$710$$ 0 0
$$711$$ −6.00000 −0.225018
$$712$$ 0 0
$$713$$ 29.2111 1.09396
$$714$$ 0 0
$$715$$ 24.0000 0.897549
$$716$$ 0 0
$$717$$ −41.7250 −1.55825
$$718$$ 0 0
$$719$$ 9.69722 0.361645 0.180823 0.983516i $$-0.442124\pi$$
0.180823 + 0.983516i $$0.442124\pi$$
$$720$$ 0 0
$$721$$ −14.0000 −0.521387
$$722$$ 0 0
$$723$$ 61.0555 2.27068
$$724$$ 0 0
$$725$$ −30.4222 −1.12985
$$726$$ 0 0
$$727$$ −14.4222 −0.534890 −0.267445 0.963573i $$-0.586179\pi$$
−0.267445 + 0.963573i $$0.586179\pi$$
$$728$$ 0 0
$$729$$ −13.3305 −0.493723
$$730$$ 0 0
$$731$$ 4.30278 0.159144
$$732$$ 0 0
$$733$$ 16.2389 0.599796 0.299898 0.953971i $$-0.403047\pi$$
0.299898 + 0.953971i $$0.403047\pi$$
$$734$$ 0 0
$$735$$ 3.00000 0.110657
$$736$$ 0 0
$$737$$ 10.7889 0.397414
$$738$$ 0 0
$$739$$ −23.3305 −0.858227 −0.429114 0.903250i $$-0.641174\pi$$
−0.429114 + 0.903250i $$0.641174\pi$$
$$740$$ 0 0
$$741$$ 91.2666 3.35276
$$742$$ 0 0
$$743$$ −34.6056 −1.26955 −0.634777 0.772695i $$-0.718908\pi$$
−0.634777 + 0.772695i $$0.718908\pi$$
$$744$$ 0 0
$$745$$ 5.60555 0.205372
$$746$$ 0 0
$$747$$ 7.39445 0.270549
$$748$$ 0 0
$$749$$ 4.60555 0.168283
$$750$$ 0 0
$$751$$ 3.39445 0.123865 0.0619326 0.998080i $$-0.480274\pi$$
0.0619326 + 0.998080i $$0.480274\pi$$
$$752$$ 0 0
$$753$$ 5.02776 0.183222
$$754$$ 0 0
$$755$$ −15.2389 −0.554599
$$756$$ 0 0
$$757$$ −7.69722 −0.279760 −0.139880 0.990168i $$-0.544672\pi$$
−0.139880 + 0.990168i $$0.544672\pi$$
$$758$$ 0 0
$$759$$ −36.8444 −1.33737
$$760$$ 0 0
$$761$$ −50.2389 −1.82116 −0.910579 0.413336i $$-0.864364\pi$$
−0.910579 + 0.413336i $$0.864364\pi$$
$$762$$ 0 0
$$763$$ −4.78890 −0.173370
$$764$$ 0 0
$$765$$ 3.00000 0.108465
$$766$$ 0 0
$$767$$ 36.8444 1.33037
$$768$$ 0 0
$$769$$ −36.0555 −1.30020 −0.650098 0.759851i $$-0.725272\pi$$
−0.650098 + 0.759851i $$0.725272\pi$$
$$770$$ 0 0
$$771$$ 10.1833 0.366744
$$772$$ 0 0
$$773$$ 17.0278 0.612446 0.306223 0.951960i $$-0.400935\pi$$
0.306223 + 0.951960i $$0.400935\pi$$
$$774$$ 0 0
$$775$$ −24.1194 −0.866395
$$776$$ 0 0
$$777$$ 22.6056 0.810970
$$778$$ 0 0
$$779$$ −99.0833 −3.55003
$$780$$ 0 0
$$781$$ −13.5778 −0.485852
$$782$$ 0 0
$$783$$ 14.7889 0.528512
$$784$$ 0 0
$$785$$ −2.60555 −0.0929961
$$786$$ 0 0
$$787$$ 50.4222 1.79736 0.898679 0.438607i $$-0.144528\pi$$
0.898679 + 0.438607i $$0.144528\pi$$
$$788$$ 0 0
$$789$$ 42.4222 1.51027
$$790$$ 0 0
$$791$$ 9.39445 0.334028
$$792$$ 0 0
$$793$$ −71.4500 −2.53726
$$794$$ 0 0
$$795$$ 2.09167 0.0741840
$$796$$ 0 0
$$797$$ 48.0555 1.70221 0.851107 0.524993i $$-0.175932\pi$$
0.851107 + 0.524993i $$0.175932\pi$$
$$798$$ 0 0
$$799$$ 2.60555 0.0921778
$$800$$ 0 0
$$801$$ 18.0000 0.635999
$$802$$ 0 0
$$803$$ 30.0555 1.06064
$$804$$ 0 0
$$805$$ 5.21110 0.183667
$$806$$ 0 0
$$807$$ −29.4500 −1.03669
$$808$$ 0 0
$$809$$ −19.8167 −0.696716 −0.348358 0.937361i $$-0.613261\pi$$
−0.348358 + 0.937361i $$0.613261\pi$$
$$810$$ 0 0
$$811$$ −4.48612 −0.157529 −0.0787645 0.996893i $$-0.525098\pi$$
−0.0787645 + 0.996893i $$0.525098\pi$$
$$812$$ 0 0
$$813$$ 24.0000 0.841717
$$814$$ 0 0
$$815$$ −1.81665 −0.0636346
$$816$$ 0 0
$$817$$ −37.0278 −1.29544
$$818$$ 0 0
$$819$$ 10.6056 0.370588
$$820$$ 0 0
$$821$$ −8.23886 −0.287538 −0.143769 0.989611i $$-0.545922\pi$$
−0.143769 + 0.989611i $$0.545922\pi$$
$$822$$ 0 0
$$823$$ −38.6056 −1.34570 −0.672852 0.739777i $$-0.734931\pi$$
−0.672852 + 0.739777i $$0.734931\pi$$
$$824$$ 0 0
$$825$$ 30.4222 1.05917
$$826$$ 0 0
$$827$$ 45.2111 1.57214 0.786072 0.618135i $$-0.212111\pi$$
0.786072 + 0.618135i $$0.212111\pi$$
$$828$$ 0 0
$$829$$ −16.2389 −0.563999 −0.281999 0.959415i $$-0.590998\pi$$
−0.281999 + 0.959415i $$0.590998\pi$$
$$830$$ 0 0
$$831$$ −3.63331 −0.126038
$$832$$ 0 0
$$833$$ −1.00000 −0.0346479
$$834$$ 0 0
$$835$$ −15.7889 −0.546397
$$836$$ 0 0
$$837$$ 11.7250 0.405275
$$838$$ 0 0
$$839$$ 18.4222 0.636005 0.318003 0.948090i $$-0.396988\pi$$
0.318003 + 0.948090i $$0.396988\pi$$
$$840$$ 0 0
$$841$$ 55.8444 1.92567
$$842$$ 0 0
$$843$$ 44.0278 1.51640
$$844$$ 0 0
$$845$$ −10.6972 −0.367996
$$846$$ 0 0
$$847$$ −5.00000 −0.171802
$$848$$ 0 0
$$849$$ −46.8167 −1.60674
$$850$$ 0 0
$$851$$ 39.2666 1.34604
$$852$$ 0 0
$$853$$ −10.0000 −0.342393 −0.171197 0.985237i $$-0.554763\pi$$
−0.171197 + 0.985237i $$0.554763\pi$$
$$854$$ 0 0
$$855$$ −25.8167 −0.882911
$$856$$ 0 0
$$857$$ −45.9638 −1.57009 −0.785047 0.619436i $$-0.787361\pi$$
−0.785047 + 0.619436i $$0.787361\pi$$
$$858$$ 0 0
$$859$$ −54.6056 −1.86312 −0.931559 0.363591i $$-0.881551\pi$$
−0.931559 + 0.363591i $$0.881551\pi$$
$$860$$ 0 0
$$861$$ −26.5139 −0.903591
$$862$$ 0 0
$$863$$ 23.3583 0.795125 0.397563 0.917575i $$-0.369856\pi$$
0.397563 + 0.917575i $$0.369856\pi$$
$$864$$ 0 0
$$865$$ −27.0000 −0.918028
$$866$$ 0 0
$$867$$ −2.30278 −0.0782064
$$868$$ 0 0
$$869$$ −10.4222 −0.353549
$$870$$ 0 0
$$871$$ −12.4222 −0.420910
$$872$$ 0 0
$$873$$ −30.6333 −1.03678
$$874$$ 0 0
$$875$$ −10.8167 −0.365670
$$876$$ 0 0
$$877$$ −1.39445 −0.0470872 −0.0235436 0.999723i $$-0.507495\pi$$
−0.0235436 + 0.999723i $$0.507495\pi$$
$$878$$ 0 0
$$879$$ 25.8167 0.870774
$$880$$ 0 0
$$881$$ −13.1194 −0.442005 −0.221002 0.975273i $$-0.570933\pi$$
−0.221002 + 0.975273i $$0.570933\pi$$
$$882$$ 0 0
$$883$$ 12.0917 0.406917 0.203459 0.979084i $$-0.434782\pi$$
0.203459 + 0.979084i $$0.434782\pi$$
$$884$$ 0 0
$$885$$ −24.0000 −0.806751
$$886$$ 0 0
$$887$$ 55.7805 1.87293 0.936463 0.350767i $$-0.114079\pi$$
0.936463 + 0.350767i $$0.114079\pi$$
$$888$$ 0 0
$$889$$ −14.5139 −0.486780
$$890$$ 0 0
$$891$$ −42.4222 −1.42120
$$892$$ 0 0
$$893$$ −22.4222 −0.750330
$$894$$ 0 0
$$895$$ −12.3944 −0.414301
$$896$$ 0 0
$$897$$ 42.4222 1.41644
$$898$$ 0 0
$$899$$ 67.2666 2.24347
$$900$$ 0 0
$$901$$ −0.697224 −0.0232279
$$902$$ 0 0
$$903$$ −9.90833 −0.329728
$$904$$ 0 0
$$905$$ 7.81665 0.259834
$$906$$ 0 0
$$907$$ 27.2111 0.903530 0.451765 0.892137i $$-0.350795\pi$$
0.451765 + 0.892137i $$0.350795\pi$$
$$908$$ 0 0
$$909$$ −29.0278 −0.962790
$$910$$ 0 0
$$911$$ −15.8167 −0.524029 −0.262015 0.965064i $$-0.584387\pi$$
−0.262015 + 0.965064i $$0.584387\pi$$
$$912$$ 0 0
$$913$$ 12.8444 0.425088
$$914$$ 0 0
$$915$$ 46.5416 1.53862
$$916$$ 0 0
$$917$$ 9.21110 0.304177
$$918$$ 0 0
$$919$$ 26.3583 0.869480 0.434740 0.900556i $$-0.356840\pi$$
0.434740 + 0.900556i $$0.356840\pi$$
$$920$$ 0 0
$$921$$ −9.21110 −0.303516
$$922$$ 0 0
$$923$$ 15.6333 0.514577
$$924$$ 0 0
$$925$$ −32.4222 −1.06604
$$926$$ 0 0
$$927$$ 32.2389 1.05886
$$928$$ 0 0
$$929$$ 36.0917 1.18413 0.592065 0.805890i $$-0.298313\pi$$
0.592065 + 0.805890i $$0.298313\pi$$
$$930$$ 0 0
$$931$$ 8.60555 0.282036
$$932$$ 0 0
$$933$$ 40.8167 1.33628
$$934$$ 0 0
$$935$$ 5.21110 0.170421
$$936$$ 0 0
$$937$$ −37.2111 −1.21563 −0.607817 0.794077i $$-0.707955\pi$$
−0.607817 + 0.794077i $$0.707955\pi$$
$$938$$ 0 0
$$939$$ −9.00000 −0.293704
$$940$$ 0 0
$$941$$ −33.6972 −1.09850 −0.549249 0.835659i $$-0.685086\pi$$
−0.549249 + 0.835659i $$0.685086\pi$$
$$942$$ 0 0
$$943$$ −46.0555 −1.49977
$$944$$ 0 0
$$945$$ 2.09167 0.0680421
$$946$$ 0 0
$$947$$ 24.8444 0.807335 0.403667 0.914906i $$-0.367735\pi$$
0.403667 + 0.914906i $$0.367735\pi$$
$$948$$ 0 0
$$949$$ −34.6056 −1.12334
$$950$$ 0 0
$$951$$ 77.4500 2.51149
$$952$$ 0 0
$$953$$ −21.3583 −0.691863 −0.345931 0.938260i $$-0.612437\pi$$
−0.345931 + 0.938260i $$0.612437\pi$$
$$954$$ 0 0
$$955$$ 15.2750 0.494288
$$956$$ 0 0
$$957$$ −84.8444 −2.74263
$$958$$ 0 0
$$959$$ −18.1194 −0.585107
$$960$$ 0 0
$$961$$ 22.3305 0.720340
$$962$$ 0 0
$$963$$ −10.6056 −0.341759
$$964$$ 0 0
$$965$$ 19.5778 0.630232
$$966$$ 0 0
$$967$$ −9.51388 −0.305946 −0.152973 0.988230i $$-0.548885\pi$$
−0.152973 + 0.988230i $$0.548885\pi$$
$$968$$ 0 0
$$969$$ 19.8167 0.636603
$$970$$ 0 0
$$971$$ 2.78890 0.0895000 0.0447500 0.998998i $$-0.485751\pi$$
0.0447500 + 0.998998i $$0.485751\pi$$
$$972$$ 0 0
$$973$$ −8.51388 −0.272942
$$974$$ 0 0
$$975$$ −35.0278 −1.12179
$$976$$ 0 0
$$977$$ −8.69722 −0.278249 −0.139124 0.990275i $$-0.544429\pi$$
−0.139124 + 0.990275i $$0.544429\pi$$
$$978$$ 0 0
$$979$$ 31.2666 0.999285
$$980$$ 0 0
$$981$$ 11.0278 0.352089
$$982$$ 0 0
$$983$$ 47.7805 1.52396 0.761981 0.647600i $$-0.224227\pi$$
0.761981 + 0.647600i $$0.224227\pi$$
$$984$$ 0 0
$$985$$ 8.60555 0.274196
$$986$$ 0 0
$$987$$ −6.00000 −0.190982
$$988$$ 0 0
$$989$$ −17.2111 −0.547281
$$990$$ 0 0
$$991$$ −11.2111 −0.356132 −0.178066 0.984019i $$-0.556984\pi$$
−0.178066 + 0.984019i $$0.556984\pi$$
$$992$$ 0 0
$$993$$ −8.57779 −0.272208
$$994$$ 0 0
$$995$$ −12.6333 −0.400503
$$996$$ 0 0
$$997$$ −45.6972 −1.44725 −0.723623 0.690196i $$-0.757524\pi$$
−0.723623 + 0.690196i $$0.757524\pi$$
$$998$$ 0 0
$$999$$ 15.7611 0.498660
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 476.2.a.c.1.1 2
3.2 odd 2 4284.2.a.l.1.2 2
4.3 odd 2 1904.2.a.k.1.2 2
7.6 odd 2 3332.2.a.k.1.2 2
8.3 odd 2 7616.2.a.o.1.1 2
8.5 even 2 7616.2.a.t.1.2 2
17.16 even 2 8092.2.a.l.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
476.2.a.c.1.1 2 1.1 even 1 trivial
1904.2.a.k.1.2 2 4.3 odd 2
3332.2.a.k.1.2 2 7.6 odd 2
4284.2.a.l.1.2 2 3.2 odd 2
7616.2.a.o.1.1 2 8.3 odd 2
7616.2.a.t.1.2 2 8.5 even 2
8092.2.a.l.1.2 2 17.16 even 2