# Properties

 Label 1248.2.a.l.1.1 Level $1248$ Weight $2$ Character 1248.1 Self dual yes Analytic conductor $9.965$ 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] = [1248,2,Mod(1,1248)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1248, 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("1248.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1248 = 2^{5} \cdot 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1248.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: $$9.96533017226$$ 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-0.618034$$ of defining polynomial Character $$\chi$$ $$=$$ 1248.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} -1.23607 q^{5} +1.23607 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} -1.23607 q^{5} +1.23607 q^{7} +1.00000 q^{9} -2.00000 q^{11} -1.00000 q^{13} +1.23607 q^{15} +4.47214 q^{17} -5.23607 q^{19} -1.23607 q^{21} +2.47214 q^{23} -3.47214 q^{25} -1.00000 q^{27} +4.47214 q^{29} -7.70820 q^{31} +2.00000 q^{33} -1.52786 q^{35} +0.472136 q^{37} +1.00000 q^{39} +1.23607 q^{41} -6.47214 q^{43} -1.23607 q^{45} -6.94427 q^{47} -5.47214 q^{49} -4.47214 q^{51} +8.47214 q^{53} +2.47214 q^{55} +5.23607 q^{57} -8.47214 q^{59} -12.4721 q^{61} +1.23607 q^{63} +1.23607 q^{65} -6.76393 q^{67} -2.47214 q^{69} -4.47214 q^{71} +0.472136 q^{73} +3.47214 q^{75} -2.47214 q^{77} +8.94427 q^{79} +1.00000 q^{81} -7.52786 q^{83} -5.52786 q^{85} -4.47214 q^{87} -10.1803 q^{89} -1.23607 q^{91} +7.70820 q^{93} +6.47214 q^{95} -4.47214 q^{97} -2.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 + 2 * q^5 - 2 * q^7 + 2 * q^9 $$2 q - 2 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9} - 4 q^{11} - 2 q^{13} - 2 q^{15} - 6 q^{19} + 2 q^{21} - 4 q^{23} + 2 q^{25} - 2 q^{27} - 2 q^{31} + 4 q^{33} - 12 q^{35} - 8 q^{37} + 2 q^{39} - 2 q^{41} - 4 q^{43} + 2 q^{45} + 4 q^{47} - 2 q^{49} + 8 q^{53} - 4 q^{55} + 6 q^{57} - 8 q^{59} - 16 q^{61} - 2 q^{63} - 2 q^{65} - 18 q^{67} + 4 q^{69} - 8 q^{73} - 2 q^{75} + 4 q^{77} + 2 q^{81} - 24 q^{83} - 20 q^{85} + 2 q^{89} + 2 q^{91} + 2 q^{93} + 4 q^{95} - 4 q^{99}+O(q^{100})$$ 2 * q - 2 * q^3 + 2 * q^5 - 2 * q^7 + 2 * q^9 - 4 * q^11 - 2 * q^13 - 2 * q^15 - 6 * q^19 + 2 * q^21 - 4 * q^23 + 2 * q^25 - 2 * q^27 - 2 * q^31 + 4 * q^33 - 12 * q^35 - 8 * q^37 + 2 * q^39 - 2 * q^41 - 4 * q^43 + 2 * q^45 + 4 * q^47 - 2 * q^49 + 8 * q^53 - 4 * q^55 + 6 * q^57 - 8 * q^59 - 16 * q^61 - 2 * q^63 - 2 * q^65 - 18 * q^67 + 4 * q^69 - 8 * q^73 - 2 * q^75 + 4 * q^77 + 2 * q^81 - 24 * q^83 - 20 * q^85 + 2 * q^89 + 2 * q^91 + 2 * q^93 + 4 * q^95 - 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
$$3$$ −1.00000 −0.577350
$$4$$ 0 0
$$5$$ −1.23607 −0.552786 −0.276393 0.961045i $$-0.589139\pi$$
−0.276393 + 0.961045i $$0.589139\pi$$
$$6$$ 0 0
$$7$$ 1.23607 0.467190 0.233595 0.972334i $$-0.424951\pi$$
0.233595 + 0.972334i $$0.424951\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −2.00000 −0.603023 −0.301511 0.953463i $$-0.597491\pi$$
−0.301511 + 0.953463i $$0.597491\pi$$
$$12$$ 0 0
$$13$$ −1.00000 −0.277350
$$14$$ 0 0
$$15$$ 1.23607 0.319151
$$16$$ 0 0
$$17$$ 4.47214 1.08465 0.542326 0.840168i $$-0.317544\pi$$
0.542326 + 0.840168i $$0.317544\pi$$
$$18$$ 0 0
$$19$$ −5.23607 −1.20124 −0.600618 0.799536i $$-0.705079\pi$$
−0.600618 + 0.799536i $$0.705079\pi$$
$$20$$ 0 0
$$21$$ −1.23607 −0.269732
$$22$$ 0 0
$$23$$ 2.47214 0.515476 0.257738 0.966215i $$-0.417023\pi$$
0.257738 + 0.966215i $$0.417023\pi$$
$$24$$ 0 0
$$25$$ −3.47214 −0.694427
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ 4.47214 0.830455 0.415227 0.909718i $$-0.363702\pi$$
0.415227 + 0.909718i $$0.363702\pi$$
$$30$$ 0 0
$$31$$ −7.70820 −1.38443 −0.692217 0.721689i $$-0.743366\pi$$
−0.692217 + 0.721689i $$0.743366\pi$$
$$32$$ 0 0
$$33$$ 2.00000 0.348155
$$34$$ 0 0
$$35$$ −1.52786 −0.258256
$$36$$ 0 0
$$37$$ 0.472136 0.0776187 0.0388093 0.999247i $$-0.487644\pi$$
0.0388093 + 0.999247i $$0.487644\pi$$
$$38$$ 0 0
$$39$$ 1.00000 0.160128
$$40$$ 0 0
$$41$$ 1.23607 0.193041 0.0965207 0.995331i $$-0.469229\pi$$
0.0965207 + 0.995331i $$0.469229\pi$$
$$42$$ 0 0
$$43$$ −6.47214 −0.986991 −0.493496 0.869748i $$-0.664281\pi$$
−0.493496 + 0.869748i $$0.664281\pi$$
$$44$$ 0 0
$$45$$ −1.23607 −0.184262
$$46$$ 0 0
$$47$$ −6.94427 −1.01293 −0.506463 0.862262i $$-0.669047\pi$$
−0.506463 + 0.862262i $$0.669047\pi$$
$$48$$ 0 0
$$49$$ −5.47214 −0.781734
$$50$$ 0 0
$$51$$ −4.47214 −0.626224
$$52$$ 0 0
$$53$$ 8.47214 1.16374 0.581869 0.813283i $$-0.302322\pi$$
0.581869 + 0.813283i $$0.302322\pi$$
$$54$$ 0 0
$$55$$ 2.47214 0.333343
$$56$$ 0 0
$$57$$ 5.23607 0.693534
$$58$$ 0 0
$$59$$ −8.47214 −1.10298 −0.551489 0.834182i $$-0.685940\pi$$
−0.551489 + 0.834182i $$0.685940\pi$$
$$60$$ 0 0
$$61$$ −12.4721 −1.59689 −0.798447 0.602066i $$-0.794345\pi$$
−0.798447 + 0.602066i $$0.794345\pi$$
$$62$$ 0 0
$$63$$ 1.23607 0.155730
$$64$$ 0 0
$$65$$ 1.23607 0.153315
$$66$$ 0 0
$$67$$ −6.76393 −0.826346 −0.413173 0.910653i $$-0.635579\pi$$
−0.413173 + 0.910653i $$0.635579\pi$$
$$68$$ 0 0
$$69$$ −2.47214 −0.297610
$$70$$ 0 0
$$71$$ −4.47214 −0.530745 −0.265372 0.964146i $$-0.585495\pi$$
−0.265372 + 0.964146i $$0.585495\pi$$
$$72$$ 0 0
$$73$$ 0.472136 0.0552593 0.0276297 0.999618i $$-0.491204\pi$$
0.0276297 + 0.999618i $$0.491204\pi$$
$$74$$ 0 0
$$75$$ 3.47214 0.400928
$$76$$ 0 0
$$77$$ −2.47214 −0.281726
$$78$$ 0 0
$$79$$ 8.94427 1.00631 0.503155 0.864196i $$-0.332173\pi$$
0.503155 + 0.864196i $$0.332173\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −7.52786 −0.826290 −0.413145 0.910665i $$-0.635570\pi$$
−0.413145 + 0.910665i $$0.635570\pi$$
$$84$$ 0 0
$$85$$ −5.52786 −0.599581
$$86$$ 0 0
$$87$$ −4.47214 −0.479463
$$88$$ 0 0
$$89$$ −10.1803 −1.07911 −0.539557 0.841949i $$-0.681408\pi$$
−0.539557 + 0.841949i $$0.681408\pi$$
$$90$$ 0 0
$$91$$ −1.23607 −0.129575
$$92$$ 0 0
$$93$$ 7.70820 0.799304
$$94$$ 0 0
$$95$$ 6.47214 0.664027
$$96$$ 0 0
$$97$$ −4.47214 −0.454077 −0.227038 0.973886i $$-0.572904\pi$$
−0.227038 + 0.973886i $$0.572904\pi$$
$$98$$ 0 0
$$99$$ −2.00000 −0.201008
$$100$$ 0 0
$$101$$ −2.94427 −0.292966 −0.146483 0.989213i $$-0.546795\pi$$
−0.146483 + 0.989213i $$0.546795\pi$$
$$102$$ 0 0
$$103$$ 1.52786 0.150545 0.0752725 0.997163i $$-0.476017\pi$$
0.0752725 + 0.997163i $$0.476017\pi$$
$$104$$ 0 0
$$105$$ 1.52786 0.149104
$$106$$ 0 0
$$107$$ −14.4721 −1.39907 −0.699537 0.714596i $$-0.746610\pi$$
−0.699537 + 0.714596i $$0.746610\pi$$
$$108$$ 0 0
$$109$$ −1.05573 −0.101120 −0.0505602 0.998721i $$-0.516101\pi$$
−0.0505602 + 0.998721i $$0.516101\pi$$
$$110$$ 0 0
$$111$$ −0.472136 −0.0448132
$$112$$ 0 0
$$113$$ −2.00000 −0.188144 −0.0940721 0.995565i $$-0.529988\pi$$
−0.0940721 + 0.995565i $$0.529988\pi$$
$$114$$ 0 0
$$115$$ −3.05573 −0.284948
$$116$$ 0 0
$$117$$ −1.00000 −0.0924500
$$118$$ 0 0
$$119$$ 5.52786 0.506738
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ 0 0
$$123$$ −1.23607 −0.111452
$$124$$ 0 0
$$125$$ 10.4721 0.936656
$$126$$ 0 0
$$127$$ −6.47214 −0.574309 −0.287155 0.957884i $$-0.592709\pi$$
−0.287155 + 0.957884i $$0.592709\pi$$
$$128$$ 0 0
$$129$$ 6.47214 0.569840
$$130$$ 0 0
$$131$$ 11.4164 0.997456 0.498728 0.866758i $$-0.333801\pi$$
0.498728 + 0.866758i $$0.333801\pi$$
$$132$$ 0 0
$$133$$ −6.47214 −0.561205
$$134$$ 0 0
$$135$$ 1.23607 0.106384
$$136$$ 0 0
$$137$$ −6.76393 −0.577882 −0.288941 0.957347i $$-0.593303\pi$$
−0.288941 + 0.957347i $$0.593303\pi$$
$$138$$ 0 0
$$139$$ 13.8885 1.17801 0.589005 0.808129i $$-0.299520\pi$$
0.589005 + 0.808129i $$0.299520\pi$$
$$140$$ 0 0
$$141$$ 6.94427 0.584813
$$142$$ 0 0
$$143$$ 2.00000 0.167248
$$144$$ 0 0
$$145$$ −5.52786 −0.459064
$$146$$ 0 0
$$147$$ 5.47214 0.451334
$$148$$ 0 0
$$149$$ 8.29180 0.679290 0.339645 0.940554i $$-0.389693\pi$$
0.339645 + 0.940554i $$0.389693\pi$$
$$150$$ 0 0
$$151$$ 18.7639 1.52699 0.763494 0.645815i $$-0.223482\pi$$
0.763494 + 0.645815i $$0.223482\pi$$
$$152$$ 0 0
$$153$$ 4.47214 0.361551
$$154$$ 0 0
$$155$$ 9.52786 0.765296
$$156$$ 0 0
$$157$$ −13.4164 −1.07075 −0.535373 0.844616i $$-0.679829\pi$$
−0.535373 + 0.844616i $$0.679829\pi$$
$$158$$ 0 0
$$159$$ −8.47214 −0.671884
$$160$$ 0 0
$$161$$ 3.05573 0.240825
$$162$$ 0 0
$$163$$ −2.76393 −0.216488 −0.108244 0.994124i $$-0.534523\pi$$
−0.108244 + 0.994124i $$0.534523\pi$$
$$164$$ 0 0
$$165$$ −2.47214 −0.192456
$$166$$ 0 0
$$167$$ 15.8885 1.22949 0.614746 0.788725i $$-0.289258\pi$$
0.614746 + 0.788725i $$0.289258\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 0 0
$$171$$ −5.23607 −0.400412
$$172$$ 0 0
$$173$$ 2.00000 0.152057 0.0760286 0.997106i $$-0.475776\pi$$
0.0760286 + 0.997106i $$0.475776\pi$$
$$174$$ 0 0
$$175$$ −4.29180 −0.324429
$$176$$ 0 0
$$177$$ 8.47214 0.636805
$$178$$ 0 0
$$179$$ −7.41641 −0.554328 −0.277164 0.960823i $$-0.589395\pi$$
−0.277164 + 0.960823i $$0.589395\pi$$
$$180$$ 0 0
$$181$$ 9.41641 0.699916 0.349958 0.936765i $$-0.386196\pi$$
0.349958 + 0.936765i $$0.386196\pi$$
$$182$$ 0 0
$$183$$ 12.4721 0.921967
$$184$$ 0 0
$$185$$ −0.583592 −0.0429065
$$186$$ 0 0
$$187$$ −8.94427 −0.654070
$$188$$ 0 0
$$189$$ −1.23607 −0.0899107
$$190$$ 0 0
$$191$$ 1.52786 0.110552 0.0552762 0.998471i $$-0.482396\pi$$
0.0552762 + 0.998471i $$0.482396\pi$$
$$192$$ 0 0
$$193$$ 19.8885 1.43161 0.715804 0.698301i $$-0.246060\pi$$
0.715804 + 0.698301i $$0.246060\pi$$
$$194$$ 0 0
$$195$$ −1.23607 −0.0885167
$$196$$ 0 0
$$197$$ 22.7639 1.62186 0.810932 0.585141i $$-0.198961\pi$$
0.810932 + 0.585141i $$0.198961\pi$$
$$198$$ 0 0
$$199$$ 9.52786 0.675412 0.337706 0.941252i $$-0.390349\pi$$
0.337706 + 0.941252i $$0.390349\pi$$
$$200$$ 0 0
$$201$$ 6.76393 0.477091
$$202$$ 0 0
$$203$$ 5.52786 0.387980
$$204$$ 0 0
$$205$$ −1.52786 −0.106711
$$206$$ 0 0
$$207$$ 2.47214 0.171825
$$208$$ 0 0
$$209$$ 10.4721 0.724373
$$210$$ 0 0
$$211$$ −16.9443 −1.16649 −0.583246 0.812296i $$-0.698218\pi$$
−0.583246 + 0.812296i $$0.698218\pi$$
$$212$$ 0 0
$$213$$ 4.47214 0.306426
$$214$$ 0 0
$$215$$ 8.00000 0.545595
$$216$$ 0 0
$$217$$ −9.52786 −0.646794
$$218$$ 0 0
$$219$$ −0.472136 −0.0319040
$$220$$ 0 0
$$221$$ −4.47214 −0.300828
$$222$$ 0 0
$$223$$ −4.65248 −0.311553 −0.155776 0.987792i $$-0.549788\pi$$
−0.155776 + 0.987792i $$0.549788\pi$$
$$224$$ 0 0
$$225$$ −3.47214 −0.231476
$$226$$ 0 0
$$227$$ −10.9443 −0.726397 −0.363198 0.931712i $$-0.618315\pi$$
−0.363198 + 0.931712i $$0.618315\pi$$
$$228$$ 0 0
$$229$$ −6.00000 −0.396491 −0.198246 0.980152i $$-0.563524\pi$$
−0.198246 + 0.980152i $$0.563524\pi$$
$$230$$ 0 0
$$231$$ 2.47214 0.162655
$$232$$ 0 0
$$233$$ −17.4164 −1.14099 −0.570493 0.821302i $$-0.693248\pi$$
−0.570493 + 0.821302i $$0.693248\pi$$
$$234$$ 0 0
$$235$$ 8.58359 0.559932
$$236$$ 0 0
$$237$$ −8.94427 −0.580993
$$238$$ 0 0
$$239$$ −12.4721 −0.806755 −0.403378 0.915034i $$-0.632164\pi$$
−0.403378 + 0.915034i $$0.632164\pi$$
$$240$$ 0 0
$$241$$ −22.3607 −1.44038 −0.720189 0.693778i $$-0.755945\pi$$
−0.720189 + 0.693778i $$0.755945\pi$$
$$242$$ 0 0
$$243$$ −1.00000 −0.0641500
$$244$$ 0 0
$$245$$ 6.76393 0.432132
$$246$$ 0 0
$$247$$ 5.23607 0.333163
$$248$$ 0 0
$$249$$ 7.52786 0.477059
$$250$$ 0 0
$$251$$ 28.9443 1.82695 0.913473 0.406899i $$-0.133390\pi$$
0.913473 + 0.406899i $$0.133390\pi$$
$$252$$ 0 0
$$253$$ −4.94427 −0.310844
$$254$$ 0 0
$$255$$ 5.52786 0.346168
$$256$$ 0 0
$$257$$ −12.4721 −0.777990 −0.388995 0.921240i $$-0.627178\pi$$
−0.388995 + 0.921240i $$0.627178\pi$$
$$258$$ 0 0
$$259$$ 0.583592 0.0362627
$$260$$ 0 0
$$261$$ 4.47214 0.276818
$$262$$ 0 0
$$263$$ 8.94427 0.551527 0.275764 0.961225i $$-0.411069\pi$$
0.275764 + 0.961225i $$0.411069\pi$$
$$264$$ 0 0
$$265$$ −10.4721 −0.643298
$$266$$ 0 0
$$267$$ 10.1803 0.623027
$$268$$ 0 0
$$269$$ 20.4721 1.24821 0.624104 0.781341i $$-0.285464\pi$$
0.624104 + 0.781341i $$0.285464\pi$$
$$270$$ 0 0
$$271$$ 19.1246 1.16174 0.580869 0.813997i $$-0.302713\pi$$
0.580869 + 0.813997i $$0.302713\pi$$
$$272$$ 0 0
$$273$$ 1.23607 0.0748102
$$274$$ 0 0
$$275$$ 6.94427 0.418755
$$276$$ 0 0
$$277$$ −14.9443 −0.897914 −0.448957 0.893553i $$-0.648204\pi$$
−0.448957 + 0.893553i $$0.648204\pi$$
$$278$$ 0 0
$$279$$ −7.70820 −0.461478
$$280$$ 0 0
$$281$$ −3.70820 −0.221213 −0.110606 0.993864i $$-0.535279\pi$$
−0.110606 + 0.993864i $$0.535279\pi$$
$$282$$ 0 0
$$283$$ 22.4721 1.33583 0.667915 0.744238i $$-0.267187\pi$$
0.667915 + 0.744238i $$0.267187\pi$$
$$284$$ 0 0
$$285$$ −6.47214 −0.383376
$$286$$ 0 0
$$287$$ 1.52786 0.0901870
$$288$$ 0 0
$$289$$ 3.00000 0.176471
$$290$$ 0 0
$$291$$ 4.47214 0.262161
$$292$$ 0 0
$$293$$ 10.1803 0.594742 0.297371 0.954762i $$-0.403890\pi$$
0.297371 + 0.954762i $$0.403890\pi$$
$$294$$ 0 0
$$295$$ 10.4721 0.609711
$$296$$ 0 0
$$297$$ 2.00000 0.116052
$$298$$ 0 0
$$299$$ −2.47214 −0.142967
$$300$$ 0 0
$$301$$ −8.00000 −0.461112
$$302$$ 0 0
$$303$$ 2.94427 0.169144
$$304$$ 0 0
$$305$$ 15.4164 0.882741
$$306$$ 0 0
$$307$$ 6.76393 0.386038 0.193019 0.981195i $$-0.438172\pi$$
0.193019 + 0.981195i $$0.438172\pi$$
$$308$$ 0 0
$$309$$ −1.52786 −0.0869171
$$310$$ 0 0
$$311$$ 0.583592 0.0330925 0.0165462 0.999863i $$-0.494733\pi$$
0.0165462 + 0.999863i $$0.494733\pi$$
$$312$$ 0 0
$$313$$ 31.8885 1.80245 0.901224 0.433355i $$-0.142670\pi$$
0.901224 + 0.433355i $$0.142670\pi$$
$$314$$ 0 0
$$315$$ −1.52786 −0.0860854
$$316$$ 0 0
$$317$$ −9.23607 −0.518749 −0.259375 0.965777i $$-0.583516\pi$$
−0.259375 + 0.965777i $$0.583516\pi$$
$$318$$ 0 0
$$319$$ −8.94427 −0.500783
$$320$$ 0 0
$$321$$ 14.4721 0.807756
$$322$$ 0 0
$$323$$ −23.4164 −1.30292
$$324$$ 0 0
$$325$$ 3.47214 0.192599
$$326$$ 0 0
$$327$$ 1.05573 0.0583819
$$328$$ 0 0
$$329$$ −8.58359 −0.473229
$$330$$ 0 0
$$331$$ 14.1803 0.779422 0.389711 0.920937i $$-0.372575\pi$$
0.389711 + 0.920937i $$0.372575\pi$$
$$332$$ 0 0
$$333$$ 0.472136 0.0258729
$$334$$ 0 0
$$335$$ 8.36068 0.456793
$$336$$ 0 0
$$337$$ −6.94427 −0.378279 −0.189139 0.981950i $$-0.560570\pi$$
−0.189139 + 0.981950i $$0.560570\pi$$
$$338$$ 0 0
$$339$$ 2.00000 0.108625
$$340$$ 0 0
$$341$$ 15.4164 0.834845
$$342$$ 0 0
$$343$$ −15.4164 −0.832408
$$344$$ 0 0
$$345$$ 3.05573 0.164515
$$346$$ 0 0
$$347$$ 20.0000 1.07366 0.536828 0.843692i $$-0.319622\pi$$
0.536828 + 0.843692i $$0.319622\pi$$
$$348$$ 0 0
$$349$$ −14.3607 −0.768710 −0.384355 0.923185i $$-0.625576\pi$$
−0.384355 + 0.923185i $$0.625576\pi$$
$$350$$ 0 0
$$351$$ 1.00000 0.0533761
$$352$$ 0 0
$$353$$ −21.5967 −1.14948 −0.574739 0.818336i $$-0.694897\pi$$
−0.574739 + 0.818336i $$0.694897\pi$$
$$354$$ 0 0
$$355$$ 5.52786 0.293389
$$356$$ 0 0
$$357$$ −5.52786 −0.292566
$$358$$ 0 0
$$359$$ 36.8328 1.94396 0.971981 0.235060i $$-0.0755287\pi$$
0.971981 + 0.235060i $$0.0755287\pi$$
$$360$$ 0 0
$$361$$ 8.41641 0.442969
$$362$$ 0 0
$$363$$ 7.00000 0.367405
$$364$$ 0 0
$$365$$ −0.583592 −0.0305466
$$366$$ 0 0
$$367$$ −28.0000 −1.46159 −0.730794 0.682598i $$-0.760850\pi$$
−0.730794 + 0.682598i $$0.760850\pi$$
$$368$$ 0 0
$$369$$ 1.23607 0.0643471
$$370$$ 0 0
$$371$$ 10.4721 0.543686
$$372$$ 0 0
$$373$$ −17.0557 −0.883112 −0.441556 0.897234i $$-0.645573\pi$$
−0.441556 + 0.897234i $$0.645573\pi$$
$$374$$ 0 0
$$375$$ −10.4721 −0.540779
$$376$$ 0 0
$$377$$ −4.47214 −0.230327
$$378$$ 0 0
$$379$$ 25.5967 1.31482 0.657408 0.753535i $$-0.271653\pi$$
0.657408 + 0.753535i $$0.271653\pi$$
$$380$$ 0 0
$$381$$ 6.47214 0.331578
$$382$$ 0 0
$$383$$ −30.0000 −1.53293 −0.766464 0.642287i $$-0.777986\pi$$
−0.766464 + 0.642287i $$0.777986\pi$$
$$384$$ 0 0
$$385$$ 3.05573 0.155734
$$386$$ 0 0
$$387$$ −6.47214 −0.328997
$$388$$ 0 0
$$389$$ 23.8885 1.21120 0.605599 0.795770i $$-0.292934\pi$$
0.605599 + 0.795770i $$0.292934\pi$$
$$390$$ 0 0
$$391$$ 11.0557 0.559112
$$392$$ 0 0
$$393$$ −11.4164 −0.575882
$$394$$ 0 0
$$395$$ −11.0557 −0.556274
$$396$$ 0 0
$$397$$ −7.52786 −0.377813 −0.188906 0.981995i $$-0.560494\pi$$
−0.188906 + 0.981995i $$0.560494\pi$$
$$398$$ 0 0
$$399$$ 6.47214 0.324012
$$400$$ 0 0
$$401$$ −18.1803 −0.907883 −0.453941 0.891032i $$-0.649982\pi$$
−0.453941 + 0.891032i $$0.649982\pi$$
$$402$$ 0 0
$$403$$ 7.70820 0.383973
$$404$$ 0 0
$$405$$ −1.23607 −0.0614207
$$406$$ 0 0
$$407$$ −0.944272 −0.0468058
$$408$$ 0 0
$$409$$ 34.3607 1.69903 0.849513 0.527567i $$-0.176896\pi$$
0.849513 + 0.527567i $$0.176896\pi$$
$$410$$ 0 0
$$411$$ 6.76393 0.333640
$$412$$ 0 0
$$413$$ −10.4721 −0.515300
$$414$$ 0 0
$$415$$ 9.30495 0.456762
$$416$$ 0 0
$$417$$ −13.8885 −0.680125
$$418$$ 0 0
$$419$$ −38.4721 −1.87949 −0.939743 0.341881i $$-0.888936\pi$$
−0.939743 + 0.341881i $$0.888936\pi$$
$$420$$ 0 0
$$421$$ −26.9443 −1.31318 −0.656592 0.754246i $$-0.728003\pi$$
−0.656592 + 0.754246i $$0.728003\pi$$
$$422$$ 0 0
$$423$$ −6.94427 −0.337642
$$424$$ 0 0
$$425$$ −15.5279 −0.753212
$$426$$ 0 0
$$427$$ −15.4164 −0.746052
$$428$$ 0 0
$$429$$ −2.00000 −0.0965609
$$430$$ 0 0
$$431$$ 10.0000 0.481683 0.240842 0.970564i $$-0.422577\pi$$
0.240842 + 0.970564i $$0.422577\pi$$
$$432$$ 0 0
$$433$$ −8.47214 −0.407145 −0.203572 0.979060i $$-0.565255\pi$$
−0.203572 + 0.979060i $$0.565255\pi$$
$$434$$ 0 0
$$435$$ 5.52786 0.265041
$$436$$ 0 0
$$437$$ −12.9443 −0.619208
$$438$$ 0 0
$$439$$ −22.4721 −1.07254 −0.536268 0.844048i $$-0.680166\pi$$
−0.536268 + 0.844048i $$0.680166\pi$$
$$440$$ 0 0
$$441$$ −5.47214 −0.260578
$$442$$ 0 0
$$443$$ 30.4721 1.44777 0.723887 0.689918i $$-0.242353\pi$$
0.723887 + 0.689918i $$0.242353\pi$$
$$444$$ 0 0
$$445$$ 12.5836 0.596519
$$446$$ 0 0
$$447$$ −8.29180 −0.392188
$$448$$ 0 0
$$449$$ 30.5410 1.44132 0.720660 0.693289i $$-0.243839\pi$$
0.720660 + 0.693289i $$0.243839\pi$$
$$450$$ 0 0
$$451$$ −2.47214 −0.116408
$$452$$ 0 0
$$453$$ −18.7639 −0.881606
$$454$$ 0 0
$$455$$ 1.52786 0.0716274
$$456$$ 0 0
$$457$$ 10.3607 0.484652 0.242326 0.970195i $$-0.422090\pi$$
0.242326 + 0.970195i $$0.422090\pi$$
$$458$$ 0 0
$$459$$ −4.47214 −0.208741
$$460$$ 0 0
$$461$$ 8.65248 0.402986 0.201493 0.979490i $$-0.435421\pi$$
0.201493 + 0.979490i $$0.435421\pi$$
$$462$$ 0 0
$$463$$ 4.29180 0.199457 0.0997283 0.995015i $$-0.468203\pi$$
0.0997283 + 0.995015i $$0.468203\pi$$
$$464$$ 0 0
$$465$$ −9.52786 −0.441844
$$466$$ 0 0
$$467$$ 33.8885 1.56817 0.784087 0.620650i $$-0.213131\pi$$
0.784087 + 0.620650i $$0.213131\pi$$
$$468$$ 0 0
$$469$$ −8.36068 −0.386060
$$470$$ 0 0
$$471$$ 13.4164 0.618195
$$472$$ 0 0
$$473$$ 12.9443 0.595178
$$474$$ 0 0
$$475$$ 18.1803 0.834171
$$476$$ 0 0
$$477$$ 8.47214 0.387912
$$478$$ 0 0
$$479$$ −39.3050 −1.79589 −0.897945 0.440109i $$-0.854940\pi$$
−0.897945 + 0.440109i $$0.854940\pi$$
$$480$$ 0 0
$$481$$ −0.472136 −0.0215275
$$482$$ 0 0
$$483$$ −3.05573 −0.139040
$$484$$ 0 0
$$485$$ 5.52786 0.251007
$$486$$ 0 0
$$487$$ −39.7082 −1.79935 −0.899675 0.436560i $$-0.856197\pi$$
−0.899675 + 0.436560i $$0.856197\pi$$
$$488$$ 0 0
$$489$$ 2.76393 0.124989
$$490$$ 0 0
$$491$$ 28.9443 1.30624 0.653118 0.757256i $$-0.273460\pi$$
0.653118 + 0.757256i $$0.273460\pi$$
$$492$$ 0 0
$$493$$ 20.0000 0.900755
$$494$$ 0 0
$$495$$ 2.47214 0.111114
$$496$$ 0 0
$$497$$ −5.52786 −0.247959
$$498$$ 0 0
$$499$$ −17.8197 −0.797718 −0.398859 0.917012i $$-0.630594\pi$$
−0.398859 + 0.917012i $$0.630594\pi$$
$$500$$ 0 0
$$501$$ −15.8885 −0.709848
$$502$$ 0 0
$$503$$ −31.7771 −1.41687 −0.708435 0.705776i $$-0.750599\pi$$
−0.708435 + 0.705776i $$0.750599\pi$$
$$504$$ 0 0
$$505$$ 3.63932 0.161948
$$506$$ 0 0
$$507$$ −1.00000 −0.0444116
$$508$$ 0 0
$$509$$ −6.18034 −0.273939 −0.136969 0.990575i $$-0.543736\pi$$
−0.136969 + 0.990575i $$0.543736\pi$$
$$510$$ 0 0
$$511$$ 0.583592 0.0258166
$$512$$ 0 0
$$513$$ 5.23607 0.231178
$$514$$ 0 0
$$515$$ −1.88854 −0.0832192
$$516$$ 0 0
$$517$$ 13.8885 0.610817
$$518$$ 0 0
$$519$$ −2.00000 −0.0877903
$$520$$ 0 0
$$521$$ 1.05573 0.0462523 0.0231261 0.999733i $$-0.492638\pi$$
0.0231261 + 0.999733i $$0.492638\pi$$
$$522$$ 0 0
$$523$$ −40.9443 −1.79037 −0.895184 0.445697i $$-0.852956\pi$$
−0.895184 + 0.445697i $$0.852956\pi$$
$$524$$ 0 0
$$525$$ 4.29180 0.187309
$$526$$ 0 0
$$527$$ −34.4721 −1.50163
$$528$$ 0 0
$$529$$ −16.8885 −0.734285
$$530$$ 0 0
$$531$$ −8.47214 −0.367659
$$532$$ 0 0
$$533$$ −1.23607 −0.0535400
$$534$$ 0 0
$$535$$ 17.8885 0.773389
$$536$$ 0 0
$$537$$ 7.41641 0.320042
$$538$$ 0 0
$$539$$ 10.9443 0.471403
$$540$$ 0 0
$$541$$ 2.00000 0.0859867 0.0429934 0.999075i $$-0.486311\pi$$
0.0429934 + 0.999075i $$0.486311\pi$$
$$542$$ 0 0
$$543$$ −9.41641 −0.404097
$$544$$ 0 0
$$545$$ 1.30495 0.0558980
$$546$$ 0 0
$$547$$ 16.9443 0.724485 0.362242 0.932084i $$-0.382011\pi$$
0.362242 + 0.932084i $$0.382011\pi$$
$$548$$ 0 0
$$549$$ −12.4721 −0.532298
$$550$$ 0 0
$$551$$ −23.4164 −0.997573
$$552$$ 0 0
$$553$$ 11.0557 0.470137
$$554$$ 0 0
$$555$$ 0.583592 0.0247721
$$556$$ 0 0
$$557$$ 0.291796 0.0123638 0.00618190 0.999981i $$-0.498032\pi$$
0.00618190 + 0.999981i $$0.498032\pi$$
$$558$$ 0 0
$$559$$ 6.47214 0.273742
$$560$$ 0 0
$$561$$ 8.94427 0.377627
$$562$$ 0 0
$$563$$ −10.4721 −0.441348 −0.220674 0.975348i $$-0.570826\pi$$
−0.220674 + 0.975348i $$0.570826\pi$$
$$564$$ 0 0
$$565$$ 2.47214 0.104004
$$566$$ 0 0
$$567$$ 1.23607 0.0519100
$$568$$ 0 0
$$569$$ −14.3607 −0.602031 −0.301016 0.953619i $$-0.597326\pi$$
−0.301016 + 0.953619i $$0.597326\pi$$
$$570$$ 0 0
$$571$$ 25.5279 1.06831 0.534154 0.845387i $$-0.320630\pi$$
0.534154 + 0.845387i $$0.320630\pi$$
$$572$$ 0 0
$$573$$ −1.52786 −0.0638274
$$574$$ 0 0
$$575$$ −8.58359 −0.357961
$$576$$ 0 0
$$577$$ −1.05573 −0.0439505 −0.0219753 0.999759i $$-0.506996\pi$$
−0.0219753 + 0.999759i $$0.506996\pi$$
$$578$$ 0 0
$$579$$ −19.8885 −0.826540
$$580$$ 0 0
$$581$$ −9.30495 −0.386034
$$582$$ 0 0
$$583$$ −16.9443 −0.701760
$$584$$ 0 0
$$585$$ 1.23607 0.0511051
$$586$$ 0 0
$$587$$ −33.4164 −1.37924 −0.689621 0.724170i $$-0.742223\pi$$
−0.689621 + 0.724170i $$0.742223\pi$$
$$588$$ 0 0
$$589$$ 40.3607 1.66303
$$590$$ 0 0
$$591$$ −22.7639 −0.936383
$$592$$ 0 0
$$593$$ 10.7639 0.442022 0.221011 0.975271i $$-0.429064\pi$$
0.221011 + 0.975271i $$0.429064\pi$$
$$594$$ 0 0
$$595$$ −6.83282 −0.280118
$$596$$ 0 0
$$597$$ −9.52786 −0.389950
$$598$$ 0 0
$$599$$ 7.05573 0.288289 0.144145 0.989557i $$-0.453957\pi$$
0.144145 + 0.989557i $$0.453957\pi$$
$$600$$ 0 0
$$601$$ 12.1115 0.494037 0.247018 0.969011i $$-0.420549\pi$$
0.247018 + 0.969011i $$0.420549\pi$$
$$602$$ 0 0
$$603$$ −6.76393 −0.275449
$$604$$ 0 0
$$605$$ 8.65248 0.351773
$$606$$ 0 0
$$607$$ −28.0000 −1.13648 −0.568242 0.822861i $$-0.692376\pi$$
−0.568242 + 0.822861i $$0.692376\pi$$
$$608$$ 0 0
$$609$$ −5.52786 −0.224000
$$610$$ 0 0
$$611$$ 6.94427 0.280935
$$612$$ 0 0
$$613$$ −14.3607 −0.580022 −0.290011 0.957023i $$-0.593659\pi$$
−0.290011 + 0.957023i $$0.593659\pi$$
$$614$$ 0 0
$$615$$ 1.52786 0.0616094
$$616$$ 0 0
$$617$$ −45.5967 −1.83566 −0.917828 0.396978i $$-0.870059\pi$$
−0.917828 + 0.396978i $$0.870059\pi$$
$$618$$ 0 0
$$619$$ −7.34752 −0.295322 −0.147661 0.989038i $$-0.547174\pi$$
−0.147661 + 0.989038i $$0.547174\pi$$
$$620$$ 0 0
$$621$$ −2.47214 −0.0992034
$$622$$ 0 0
$$623$$ −12.5836 −0.504151
$$624$$ 0 0
$$625$$ 4.41641 0.176656
$$626$$ 0 0
$$627$$ −10.4721 −0.418217
$$628$$ 0 0
$$629$$ 2.11146 0.0841893
$$630$$ 0 0
$$631$$ 12.6525 0.503687 0.251844 0.967768i $$-0.418963\pi$$
0.251844 + 0.967768i $$0.418963\pi$$
$$632$$ 0 0
$$633$$ 16.9443 0.673474
$$634$$ 0 0
$$635$$ 8.00000 0.317470
$$636$$ 0 0
$$637$$ 5.47214 0.216814
$$638$$ 0 0
$$639$$ −4.47214 −0.176915
$$640$$ 0 0
$$641$$ −5.41641 −0.213935 −0.106968 0.994263i $$-0.534114\pi$$
−0.106968 + 0.994263i $$0.534114\pi$$
$$642$$ 0 0
$$643$$ 11.1246 0.438712 0.219356 0.975645i $$-0.429604\pi$$
0.219356 + 0.975645i $$0.429604\pi$$
$$644$$ 0 0
$$645$$ −8.00000 −0.315000
$$646$$ 0 0
$$647$$ −32.0000 −1.25805 −0.629025 0.777385i $$-0.716546\pi$$
−0.629025 + 0.777385i $$0.716546\pi$$
$$648$$ 0 0
$$649$$ 16.9443 0.665121
$$650$$ 0 0
$$651$$ 9.52786 0.373426
$$652$$ 0 0
$$653$$ −34.3607 −1.34464 −0.672319 0.740262i $$-0.734702\pi$$
−0.672319 + 0.740262i $$0.734702\pi$$
$$654$$ 0 0
$$655$$ −14.1115 −0.551380
$$656$$ 0 0
$$657$$ 0.472136 0.0184198
$$658$$ 0 0
$$659$$ 18.8328 0.733622 0.366811 0.930295i $$-0.380450\pi$$
0.366811 + 0.930295i $$0.380450\pi$$
$$660$$ 0 0
$$661$$ 44.2492 1.72110 0.860548 0.509370i $$-0.170121\pi$$
0.860548 + 0.509370i $$0.170121\pi$$
$$662$$ 0 0
$$663$$ 4.47214 0.173683
$$664$$ 0 0
$$665$$ 8.00000 0.310227
$$666$$ 0 0
$$667$$ 11.0557 0.428080
$$668$$ 0 0
$$669$$ 4.65248 0.179875
$$670$$ 0 0
$$671$$ 24.9443 0.962963
$$672$$ 0 0
$$673$$ 45.4164 1.75067 0.875337 0.483513i $$-0.160640\pi$$
0.875337 + 0.483513i $$0.160640\pi$$
$$674$$ 0 0
$$675$$ 3.47214 0.133643
$$676$$ 0 0
$$677$$ 10.3607 0.398193 0.199097 0.979980i $$-0.436199\pi$$
0.199097 + 0.979980i $$0.436199\pi$$
$$678$$ 0 0
$$679$$ −5.52786 −0.212140
$$680$$ 0 0
$$681$$ 10.9443 0.419385
$$682$$ 0 0
$$683$$ −10.0000 −0.382639 −0.191320 0.981528i $$-0.561277\pi$$
−0.191320 + 0.981528i $$0.561277\pi$$
$$684$$ 0 0
$$685$$ 8.36068 0.319445
$$686$$ 0 0
$$687$$ 6.00000 0.228914
$$688$$ 0 0
$$689$$ −8.47214 −0.322763
$$690$$ 0 0
$$691$$ −21.5967 −0.821579 −0.410790 0.911730i $$-0.634747\pi$$
−0.410790 + 0.911730i $$0.634747\pi$$
$$692$$ 0 0
$$693$$ −2.47214 −0.0939087
$$694$$ 0 0
$$695$$ −17.1672 −0.651188
$$696$$ 0 0
$$697$$ 5.52786 0.209383
$$698$$ 0 0
$$699$$ 17.4164 0.658749
$$700$$ 0 0
$$701$$ 6.00000 0.226617 0.113308 0.993560i $$-0.463855\pi$$
0.113308 + 0.993560i $$0.463855\pi$$
$$702$$ 0 0
$$703$$ −2.47214 −0.0932384
$$704$$ 0 0
$$705$$ −8.58359 −0.323277
$$706$$ 0 0
$$707$$ −3.63932 −0.136871
$$708$$ 0 0
$$709$$ −14.0000 −0.525781 −0.262891 0.964826i $$-0.584676\pi$$
−0.262891 + 0.964826i $$0.584676\pi$$
$$710$$ 0 0
$$711$$ 8.94427 0.335436
$$712$$ 0 0
$$713$$ −19.0557 −0.713643
$$714$$ 0 0
$$715$$ −2.47214 −0.0924526
$$716$$ 0 0
$$717$$ 12.4721 0.465780
$$718$$ 0 0
$$719$$ 2.47214 0.0921951 0.0460976 0.998937i $$-0.485321\pi$$
0.0460976 + 0.998937i $$0.485321\pi$$
$$720$$ 0 0
$$721$$ 1.88854 0.0703330
$$722$$ 0 0
$$723$$ 22.3607 0.831603
$$724$$ 0 0
$$725$$ −15.5279 −0.576690
$$726$$ 0 0
$$727$$ −1.52786 −0.0566653 −0.0283327 0.999599i $$-0.509020\pi$$
−0.0283327 + 0.999599i $$0.509020\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −28.9443 −1.07054
$$732$$ 0 0
$$733$$ 27.8885 1.03009 0.515043 0.857164i $$-0.327776\pi$$
0.515043 + 0.857164i $$0.327776\pi$$
$$734$$ 0 0
$$735$$ −6.76393 −0.249491
$$736$$ 0 0
$$737$$ 13.5279 0.498305
$$738$$ 0 0
$$739$$ −46.1803 −1.69877 −0.849386 0.527773i $$-0.823027\pi$$
−0.849386 + 0.527773i $$0.823027\pi$$
$$740$$ 0 0
$$741$$ −5.23607 −0.192352
$$742$$ 0 0
$$743$$ 16.4721 0.604304 0.302152 0.953260i $$-0.402295\pi$$
0.302152 + 0.953260i $$0.402295\pi$$
$$744$$ 0 0
$$745$$ −10.2492 −0.375502
$$746$$ 0 0
$$747$$ −7.52786 −0.275430
$$748$$ 0 0
$$749$$ −17.8885 −0.653633
$$750$$ 0 0
$$751$$ −32.3607 −1.18086 −0.590429 0.807090i $$-0.701041\pi$$
−0.590429 + 0.807090i $$0.701041\pi$$
$$752$$ 0 0
$$753$$ −28.9443 −1.05479
$$754$$ 0 0
$$755$$ −23.1935 −0.844098
$$756$$ 0 0
$$757$$ 37.4164 1.35992 0.679961 0.733248i $$-0.261997\pi$$
0.679961 + 0.733248i $$0.261997\pi$$
$$758$$ 0 0
$$759$$ 4.94427 0.179466
$$760$$ 0 0
$$761$$ −9.81966 −0.355962 −0.177981 0.984034i $$-0.556957\pi$$
−0.177981 + 0.984034i $$0.556957\pi$$
$$762$$ 0 0
$$763$$ −1.30495 −0.0472424
$$764$$ 0 0
$$765$$ −5.52786 −0.199860
$$766$$ 0 0
$$767$$ 8.47214 0.305911
$$768$$ 0 0
$$769$$ −52.8328 −1.90520 −0.952600 0.304226i $$-0.901602\pi$$
−0.952600 + 0.304226i $$0.901602\pi$$
$$770$$ 0 0
$$771$$ 12.4721 0.449173
$$772$$ 0 0
$$773$$ 8.29180 0.298235 0.149118 0.988819i $$-0.452357\pi$$
0.149118 + 0.988819i $$0.452357\pi$$
$$774$$ 0 0
$$775$$ 26.7639 0.961389
$$776$$ 0 0
$$777$$ −0.583592 −0.0209363
$$778$$ 0 0
$$779$$ −6.47214 −0.231888
$$780$$ 0 0
$$781$$ 8.94427 0.320051
$$782$$ 0 0
$$783$$ −4.47214 −0.159821
$$784$$ 0 0
$$785$$ 16.5836 0.591894
$$786$$ 0 0
$$787$$ −4.29180 −0.152986 −0.0764930 0.997070i $$-0.524372\pi$$
−0.0764930 + 0.997070i $$0.524372\pi$$
$$788$$ 0 0
$$789$$ −8.94427 −0.318425
$$790$$ 0 0
$$791$$ −2.47214 −0.0878990
$$792$$ 0 0
$$793$$ 12.4721 0.442899
$$794$$ 0 0
$$795$$ 10.4721 0.371408
$$796$$ 0 0
$$797$$ −34.0000 −1.20434 −0.602171 0.798367i $$-0.705697\pi$$
−0.602171 + 0.798367i $$0.705697\pi$$
$$798$$ 0 0
$$799$$ −31.0557 −1.09867
$$800$$ 0 0
$$801$$ −10.1803 −0.359705
$$802$$ 0 0
$$803$$ −0.944272 −0.0333226
$$804$$ 0 0
$$805$$ −3.77709 −0.133125
$$806$$ 0 0
$$807$$ −20.4721 −0.720653
$$808$$ 0 0
$$809$$ −30.9443 −1.08794 −0.543971 0.839104i $$-0.683080\pi$$
−0.543971 + 0.839104i $$0.683080\pi$$
$$810$$ 0 0
$$811$$ −26.1803 −0.919316 −0.459658 0.888096i $$-0.652028\pi$$
−0.459658 + 0.888096i $$0.652028\pi$$
$$812$$ 0 0
$$813$$ −19.1246 −0.670729
$$814$$ 0 0
$$815$$ 3.41641 0.119672
$$816$$ 0 0
$$817$$ 33.8885 1.18561
$$818$$ 0 0
$$819$$ −1.23607 −0.0431917
$$820$$ 0 0
$$821$$ 31.1246 1.08626 0.543128 0.839650i $$-0.317240\pi$$
0.543128 + 0.839650i $$0.317240\pi$$
$$822$$ 0 0
$$823$$ 45.3050 1.57923 0.789616 0.613602i $$-0.210280\pi$$
0.789616 + 0.613602i $$0.210280\pi$$
$$824$$ 0 0
$$825$$ −6.94427 −0.241769
$$826$$ 0 0
$$827$$ 33.7771 1.17454 0.587272 0.809389i $$-0.300202\pi$$
0.587272 + 0.809389i $$0.300202\pi$$
$$828$$ 0 0
$$829$$ 48.4721 1.68351 0.841753 0.539862i $$-0.181524\pi$$
0.841753 + 0.539862i $$0.181524\pi$$
$$830$$ 0 0
$$831$$ 14.9443 0.518411
$$832$$ 0 0
$$833$$ −24.4721 −0.847909
$$834$$ 0 0
$$835$$ −19.6393 −0.679647
$$836$$ 0 0
$$837$$ 7.70820 0.266435
$$838$$ 0 0
$$839$$ 30.3607 1.04817 0.524084 0.851667i $$-0.324408\pi$$
0.524084 + 0.851667i $$0.324408\pi$$
$$840$$ 0 0
$$841$$ −9.00000 −0.310345
$$842$$ 0 0
$$843$$ 3.70820 0.127717
$$844$$ 0 0
$$845$$ −1.23607 −0.0425220
$$846$$ 0 0
$$847$$ −8.65248 −0.297303
$$848$$ 0 0
$$849$$ −22.4721 −0.771242
$$850$$ 0 0
$$851$$ 1.16718 0.0400106
$$852$$ 0 0
$$853$$ −20.4721 −0.700953 −0.350476 0.936572i $$-0.613980\pi$$
−0.350476 + 0.936572i $$0.613980\pi$$
$$854$$ 0 0
$$855$$ 6.47214 0.221342
$$856$$ 0 0
$$857$$ −10.5836 −0.361529 −0.180764 0.983526i $$-0.557857\pi$$
−0.180764 + 0.983526i $$0.557857\pi$$
$$858$$ 0 0
$$859$$ 15.0557 0.513695 0.256847 0.966452i $$-0.417316\pi$$
0.256847 + 0.966452i $$0.417316\pi$$
$$860$$ 0 0
$$861$$ −1.52786 −0.0520695
$$862$$ 0 0
$$863$$ −12.1115 −0.412279 −0.206139 0.978523i $$-0.566090\pi$$
−0.206139 + 0.978523i $$0.566090\pi$$
$$864$$ 0 0
$$865$$ −2.47214 −0.0840551
$$866$$ 0 0
$$867$$ −3.00000 −0.101885
$$868$$ 0 0
$$869$$ −17.8885 −0.606827
$$870$$ 0 0
$$871$$ 6.76393 0.229187
$$872$$ 0 0
$$873$$ −4.47214 −0.151359
$$874$$ 0 0
$$875$$ 12.9443 0.437596
$$876$$ 0 0
$$877$$ −35.3050 −1.19216 −0.596082 0.802924i $$-0.703277\pi$$
−0.596082 + 0.802924i $$0.703277\pi$$
$$878$$ 0 0
$$879$$ −10.1803 −0.343374
$$880$$ 0 0
$$881$$ 50.3607 1.69669 0.848347 0.529440i $$-0.177598\pi$$
0.848347 + 0.529440i $$0.177598\pi$$
$$882$$ 0 0
$$883$$ 48.3607 1.62747 0.813733 0.581239i $$-0.197432\pi$$
0.813733 + 0.581239i $$0.197432\pi$$
$$884$$ 0 0
$$885$$ −10.4721 −0.352017
$$886$$ 0 0
$$887$$ −45.3050 −1.52119 −0.760596 0.649226i $$-0.775093\pi$$
−0.760596 + 0.649226i $$0.775093\pi$$
$$888$$ 0 0
$$889$$ −8.00000 −0.268311
$$890$$ 0 0
$$891$$ −2.00000 −0.0670025
$$892$$ 0 0
$$893$$ 36.3607 1.21676
$$894$$ 0 0
$$895$$ 9.16718 0.306425
$$896$$ 0 0
$$897$$ 2.47214 0.0825422
$$898$$ 0 0
$$899$$ −34.4721 −1.14971
$$900$$ 0 0
$$901$$ 37.8885 1.26225
$$902$$ 0 0
$$903$$ 8.00000 0.266223
$$904$$ 0 0
$$905$$ −11.6393 −0.386904
$$906$$ 0 0
$$907$$ 24.3607 0.808883 0.404442 0.914564i $$-0.367466\pi$$
0.404442 + 0.914564i $$0.367466\pi$$
$$908$$ 0 0
$$909$$ −2.94427 −0.0976553
$$910$$ 0 0
$$911$$ 54.4721 1.80474 0.902371 0.430960i $$-0.141825\pi$$
0.902371 + 0.430960i $$0.141825\pi$$
$$912$$ 0 0
$$913$$ 15.0557 0.498272
$$914$$ 0 0
$$915$$ −15.4164 −0.509651
$$916$$ 0 0
$$917$$ 14.1115 0.466001
$$918$$ 0 0
$$919$$ −21.8885 −0.722036 −0.361018 0.932559i $$-0.617571\pi$$
−0.361018 + 0.932559i $$0.617571\pi$$
$$920$$ 0 0
$$921$$ −6.76393 −0.222879
$$922$$ 0 0
$$923$$ 4.47214 0.147202
$$924$$ 0 0
$$925$$ −1.63932 −0.0539005
$$926$$ 0 0
$$927$$ 1.52786 0.0501816
$$928$$ 0 0
$$929$$ −58.5410 −1.92067 −0.960334 0.278851i $$-0.910046\pi$$
−0.960334 + 0.278851i $$0.910046\pi$$
$$930$$ 0 0
$$931$$ 28.6525 0.939047
$$932$$ 0 0
$$933$$ −0.583592 −0.0191059
$$934$$ 0 0
$$935$$ 11.0557 0.361561
$$936$$ 0 0
$$937$$ −31.3050 −1.02269 −0.511344 0.859376i $$-0.670852\pi$$
−0.511344 + 0.859376i $$0.670852\pi$$
$$938$$ 0 0
$$939$$ −31.8885 −1.04064
$$940$$ 0 0
$$941$$ −26.7639 −0.872479 −0.436240 0.899831i $$-0.643690\pi$$
−0.436240 + 0.899831i $$0.643690\pi$$
$$942$$ 0 0
$$943$$ 3.05573 0.0995082
$$944$$ 0 0
$$945$$ 1.52786 0.0497014
$$946$$ 0 0
$$947$$ −14.5836 −0.473903 −0.236952 0.971521i $$-0.576148\pi$$
−0.236952 + 0.971521i $$0.576148\pi$$
$$948$$ 0 0
$$949$$ −0.472136 −0.0153262
$$950$$ 0 0
$$951$$ 9.23607 0.299500
$$952$$ 0 0
$$953$$ 40.4721 1.31102 0.655511 0.755186i $$-0.272453\pi$$
0.655511 + 0.755186i $$0.272453\pi$$
$$954$$ 0 0
$$955$$ −1.88854 −0.0611118
$$956$$ 0 0
$$957$$ 8.94427 0.289127
$$958$$ 0 0
$$959$$ −8.36068 −0.269980
$$960$$ 0 0
$$961$$ 28.4164 0.916658
$$962$$ 0 0
$$963$$ −14.4721 −0.466358
$$964$$ 0 0
$$965$$ −24.5836 −0.791374
$$966$$ 0 0
$$967$$ −10.7639 −0.346145 −0.173072 0.984909i $$-0.555369\pi$$
−0.173072 + 0.984909i $$0.555369\pi$$
$$968$$ 0 0
$$969$$ 23.4164 0.752243
$$970$$ 0 0
$$971$$ −2.83282 −0.0909094 −0.0454547 0.998966i $$-0.514474\pi$$
−0.0454547 + 0.998966i $$0.514474\pi$$
$$972$$ 0 0
$$973$$ 17.1672 0.550355
$$974$$ 0 0
$$975$$ −3.47214 −0.111197
$$976$$ 0 0
$$977$$ −0.291796 −0.00933538 −0.00466769 0.999989i $$-0.501486\pi$$
−0.00466769 + 0.999989i $$0.501486\pi$$
$$978$$ 0 0
$$979$$ 20.3607 0.650730
$$980$$ 0 0
$$981$$ −1.05573 −0.0337068
$$982$$ 0 0
$$983$$ 49.4164 1.57614 0.788069 0.615587i $$-0.211081\pi$$
0.788069 + 0.615587i $$0.211081\pi$$
$$984$$ 0 0
$$985$$ −28.1378 −0.896544
$$986$$ 0 0
$$987$$ 8.58359 0.273219
$$988$$ 0 0
$$989$$ −16.0000 −0.508770
$$990$$ 0 0
$$991$$ 23.0557 0.732389 0.366195 0.930538i $$-0.380660\pi$$
0.366195 + 0.930538i $$0.380660\pi$$
$$992$$ 0 0
$$993$$ −14.1803 −0.449999
$$994$$ 0 0
$$995$$ −11.7771 −0.373359
$$996$$ 0 0
$$997$$ −24.1115 −0.763617 −0.381809 0.924241i $$-0.624699\pi$$
−0.381809 + 0.924241i $$0.624699\pi$$
$$998$$ 0 0
$$999$$ −0.472136 −0.0149377
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 1248.2.a.l.1.1 2
3.2 odd 2 3744.2.a.r.1.2 2
4.3 odd 2 1248.2.a.n.1.1 yes 2
8.3 odd 2 2496.2.a.be.1.2 2
8.5 even 2 2496.2.a.bh.1.2 2
12.11 even 2 3744.2.a.s.1.2 2
24.5 odd 2 7488.2.a.cs.1.1 2
24.11 even 2 7488.2.a.ct.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
1248.2.a.l.1.1 2 1.1 even 1 trivial
1248.2.a.n.1.1 yes 2 4.3 odd 2
2496.2.a.be.1.2 2 8.3 odd 2
2496.2.a.bh.1.2 2 8.5 even 2
3744.2.a.r.1.2 2 3.2 odd 2
3744.2.a.s.1.2 2 12.11 even 2
7488.2.a.cs.1.1 2 24.5 odd 2
7488.2.a.ct.1.1 2 24.11 even 2