Properties

 Label 1248.2.a.n.1.1 Level $1248$ Weight $2$ Character 1248.1 Self dual yes Analytic conductor $9.965$ Analytic rank $0$ 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: $$0$$ 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 $$-0.575049\pi$$
−0.233595 + 0.972334i $$0.575049\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 2.00000 0.603023 0.301511 0.953463i $$-0.402509\pi$$
0.301511 + 0.953463i $$0.402509\pi$$
$$12$$ 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 $$-0.294921\pi$$
0.600618 + 0.799536i $$0.294921\pi$$
$$20$$ 0 0
$$21$$ −1.23607 −0.269732
$$22$$ 0 0
$$23$$ −2.47214 −0.515476 −0.257738 0.966215i $$-0.582977\pi$$
−0.257738 + 0.966215i $$0.582977\pi$$
$$24$$ 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 $$-0.256634\pi$$
0.692217 + 0.721689i $$0.256634\pi$$
$$32$$ 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 $$-0.335719\pi$$
0.493496 + 0.869748i $$0.335719\pi$$
$$44$$ 0 0
$$45$$ −1.23607 −0.184262
$$46$$ 0 0
$$47$$ 6.94427 1.01293 0.506463 0.862262i $$-0.330953\pi$$
0.506463 + 0.862262i $$0.330953\pi$$
$$48$$ 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 $$-0.314060\pi$$
0.551489 + 0.834182i $$0.314060\pi$$
$$60$$ 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 $$-0.364421\pi$$
0.413173 + 0.910653i $$0.364421\pi$$
$$68$$ 0 0
$$69$$ −2.47214 −0.297610
$$70$$ 0 0
$$71$$ 4.47214 0.530745 0.265372 0.964146i $$-0.414505\pi$$
0.265372 + 0.964146i $$0.414505\pi$$
$$72$$ 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 $$-0.667827\pi$$
−0.503155 + 0.864196i $$0.667827\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 7.52786 0.826290 0.413145 0.910665i $$-0.364430\pi$$
0.413145 + 0.910665i $$0.364430\pi$$
$$84$$ 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 $$-0.523983\pi$$
−0.0752725 + 0.997163i $$0.523983\pi$$
$$104$$ 0 0
$$105$$ 1.52786 0.149104
$$106$$ 0 0
$$107$$ 14.4721 1.39907 0.699537 0.714596i $$-0.253390\pi$$
0.699537 + 0.714596i $$0.253390\pi$$
$$108$$ 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 $$-0.407291\pi$$
0.287155 + 0.957884i $$0.407291\pi$$
$$128$$ 0 0
$$129$$ 6.47214 0.569840
$$130$$ 0 0
$$131$$ −11.4164 −0.997456 −0.498728 0.866758i $$-0.666199\pi$$
−0.498728 + 0.866758i $$0.666199\pi$$
$$132$$ 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 $$-0.700480\pi$$
−0.589005 + 0.808129i $$0.700480\pi$$
$$140$$ 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 $$-0.776518\pi$$
−0.763494 + 0.645815i $$0.776518\pi$$
$$152$$ 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 $$-0.465477\pi$$
0.108244 + 0.994124i $$0.465477\pi$$
$$164$$ 0 0
$$165$$ −2.47214 −0.192456
$$166$$ 0 0
$$167$$ −15.8885 −1.22949 −0.614746 0.788725i $$-0.710742\pi$$
−0.614746 + 0.788725i $$0.710742\pi$$
$$168$$ 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 $$-0.410605\pi$$
0.277164 + 0.960823i $$0.410605\pi$$
$$180$$ 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 $$-0.517604\pi$$
−0.0552762 + 0.998471i $$0.517604\pi$$
$$192$$ 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 $$-0.609651\pi$$
−0.337706 + 0.941252i $$0.609651\pi$$
$$200$$ 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 $$-0.301782\pi$$
0.583246 + 0.812296i $$0.301782\pi$$
$$212$$ 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 $$-0.450212\pi$$
0.155776 + 0.987792i $$0.450212\pi$$
$$224$$ 0 0
$$225$$ −3.47214 −0.231476
$$226$$ 0 0
$$227$$ 10.9443 0.726397 0.363198 0.931712i $$-0.381685\pi$$
0.363198 + 0.931712i $$0.381685\pi$$
$$228$$ 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 $$-0.367836\pi$$
0.403378 + 0.915034i $$0.367836\pi$$
$$240$$ 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 $$-0.866610\pi$$
−0.913473 + 0.406899i $$0.866610\pi$$
$$252$$ 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 $$-0.588931\pi$$
−0.275764 + 0.961225i $$0.588931\pi$$
$$264$$ 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 $$-0.697287\pi$$
−0.580869 + 0.813997i $$0.697287\pi$$
$$272$$ 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 $$-0.732813\pi$$
−0.667915 + 0.744238i $$0.732813\pi$$
$$284$$ 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 $$-0.561828\pi$$
−0.193019 + 0.981195i $$0.561828\pi$$
$$308$$ 0 0
$$309$$ −1.52786 −0.0869171
$$310$$ 0 0
$$311$$ −0.583592 −0.0330925 −0.0165462 0.999863i $$-0.505267\pi$$
−0.0165462 + 0.999863i $$0.505267\pi$$
$$312$$ 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 $$-0.627425\pi$$
−0.389711 + 0.920937i $$0.627425\pi$$
$$332$$ 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 $$-0.680378\pi$$
−0.536828 + 0.843692i $$0.680378\pi$$
$$348$$ 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 $$-0.924471\pi$$
−0.971981 + 0.235060i $$0.924471\pi$$
$$360$$ 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 $$-0.239150\pi$$
0.730794 + 0.682598i $$0.239150\pi$$
$$368$$ 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 $$-0.728347\pi$$
−0.657408 + 0.753535i $$0.728347\pi$$
$$380$$ 0 0
$$381$$ 6.47214 0.331578
$$382$$ 0 0
$$383$$ 30.0000 1.53293 0.766464 0.642287i $$-0.222014\pi$$
0.766464 + 0.642287i $$0.222014\pi$$
$$384$$ 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 $$-0.111064\pi$$
0.939743 + 0.341881i $$0.111064\pi$$
$$420$$ 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 $$-0.577423\pi$$
−0.240842 + 0.970564i $$0.577423\pi$$
$$432$$ 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 $$-0.319834\pi$$
0.536268 + 0.844048i $$0.319834\pi$$
$$440$$ 0 0
$$441$$ −5.47214 −0.260578
$$442$$ 0 0
$$443$$ −30.4721 −1.44777 −0.723887 0.689918i $$-0.757647\pi$$
−0.723887 + 0.689918i $$0.757647\pi$$
$$444$$ 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 $$-0.531797\pi$$
−0.0997283 + 0.995015i $$0.531797\pi$$
$$464$$ 0 0
$$465$$ −9.52786 −0.441844
$$466$$ 0 0
$$467$$ −33.8885 −1.56817 −0.784087 0.620650i $$-0.786869\pi$$
−0.784087 + 0.620650i $$0.786869\pi$$
$$468$$ 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 $$-0.145060\pi$$
0.897945 + 0.440109i $$0.145060\pi$$
$$480$$ 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 $$-0.143803\pi$$
0.899675 + 0.436560i $$0.143803\pi$$
$$488$$ 0 0
$$489$$ 2.76393 0.124989
$$490$$ 0 0
$$491$$ −28.9443 −1.30624 −0.653118 0.757256i $$-0.726540\pi$$
−0.653118 + 0.757256i $$0.726540\pi$$
$$492$$ 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 $$-0.369406\pi$$
0.398859 + 0.917012i $$0.369406\pi$$
$$500$$ 0 0
$$501$$ −15.8885 −0.709848
$$502$$ 0 0
$$503$$ 31.7771 1.41687 0.708435 0.705776i $$-0.249401\pi$$
0.708435 + 0.705776i $$0.249401\pi$$
$$504$$ 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 $$-0.147044\pi$$
0.895184 + 0.445697i $$0.147044\pi$$
$$524$$ 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 $$-0.617989\pi$$
−0.362242 + 0.932084i $$0.617989\pi$$
$$548$$ 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 $$-0.429174\pi$$
0.220674 + 0.975348i $$0.429174\pi$$
$$564$$ 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 $$-0.679370\pi$$
−0.534154 + 0.845387i $$0.679370\pi$$
$$572$$ 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 $$-0.257777\pi$$
0.689621 + 0.724170i $$0.257777\pi$$
$$588$$ 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 $$-0.546043\pi$$
−0.144145 + 0.989557i $$0.546043\pi$$
$$600$$ 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 $$-0.307624\pi$$
0.568242 + 0.822861i $$0.307624\pi$$
$$608$$ 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 $$-0.452826\pi$$
0.147661 + 0.989038i $$0.452826\pi$$
$$620$$ 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 $$-0.581037\pi$$
−0.251844 + 0.967768i $$0.581037\pi$$
$$632$$ 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 $$-0.570396\pi$$
−0.219356 + 0.975645i $$0.570396\pi$$
$$644$$ 0 0
$$645$$ −8.00000 −0.315000
$$646$$ 0 0
$$647$$ 32.0000 1.25805 0.629025 0.777385i $$-0.283454\pi$$
0.629025 + 0.777385i $$0.283454\pi$$
$$648$$ 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 $$-0.619550\pi$$
−0.366811 + 0.930295i $$0.619550\pi$$
$$660$$ 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 $$-0.438723\pi$$
0.191320 + 0.981528i $$0.438723\pi$$
$$684$$ 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 $$-0.365253\pi$$
0.410790 + 0.911730i $$0.365253\pi$$
$$692$$ 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 $$-0.514679\pi$$
−0.0460976 + 0.998937i $$0.514679\pi$$
$$720$$ 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 $$-0.490980\pi$$
0.0283327 + 0.999599i $$0.490980\pi$$
$$728$$ 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 $$-0.176973\pi$$
0.849386 + 0.527773i $$0.176973\pi$$
$$740$$ 0 0
$$741$$ −5.23607 −0.192352
$$742$$ 0 0
$$743$$ −16.4721 −0.604304 −0.302152 0.953260i $$-0.597705\pi$$
−0.302152 + 0.953260i $$0.597705\pi$$
$$744$$ 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 $$-0.298959\pi$$
0.590429 + 0.807090i $$0.298959\pi$$
$$752$$ 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 $$-0.475628\pi$$
0.0764930 + 0.997070i $$0.475628\pi$$
$$788$$ 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 $$-0.347972\pi$$
0.459658 + 0.888096i $$0.347972\pi$$
$$812$$ 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 $$-0.789720\pi$$
−0.789616 + 0.613602i $$0.789720\pi$$
$$824$$ 0 0
$$825$$ −6.94427 −0.241769
$$826$$ 0 0
$$827$$ −33.7771 −1.17454 −0.587272 0.809389i $$-0.699798\pi$$
−0.587272 + 0.809389i $$0.699798\pi$$
$$828$$ 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 $$-0.675592\pi$$
−0.524084 + 0.851667i $$0.675592\pi$$
$$840$$ 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 $$-0.582684\pi$$
−0.256847 + 0.966452i $$0.582684\pi$$
$$860$$ 0 0
$$861$$ −1.52786 −0.0520695
$$862$$ 0 0
$$863$$ 12.1115 0.412279 0.206139 0.978523i $$-0.433910\pi$$
0.206139 + 0.978523i $$0.433910\pi$$
$$864$$ 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 $$-0.802568\pi$$
−0.813733 + 0.581239i $$0.802568\pi$$
$$884$$ 0 0
$$885$$ −10.4721 −0.352017
$$886$$ 0 0
$$887$$ 45.3050 1.52119 0.760596 0.649226i $$-0.224907\pi$$
0.760596 + 0.649226i $$0.224907\pi$$
$$888$$ 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 $$-0.632534\pi$$
−0.404442 + 0.914564i $$0.632534\pi$$
$$908$$ 0 0
$$909$$ −2.94427 −0.0976553
$$910$$ 0 0
$$911$$ −54.4721 −1.80474 −0.902371 0.430960i $$-0.858175\pi$$
−0.902371 + 0.430960i $$0.858175\pi$$
$$912$$ 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 $$-0.382429\pi$$
0.361018 + 0.932559i $$0.382429\pi$$
$$920$$ 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 $$-0.423852\pi$$
0.236952 + 0.971521i $$0.423852\pi$$
$$948$$ 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 $$-0.444631\pi$$
0.173072 + 0.984909i $$0.444631\pi$$
$$968$$ 0 0
$$969$$ 23.4164 0.752243
$$970$$ 0 0
$$971$$ 2.83282 0.0909094 0.0454547 0.998966i $$-0.485526\pi$$
0.0454547 + 0.998966i $$0.485526\pi$$
$$972$$ 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 $$-0.788919\pi$$
−0.788069 + 0.615587i $$0.788919\pi$$
$$984$$ 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 $$-0.619340\pi$$
−0.366195 + 0.930538i $$0.619340\pi$$
$$992$$ 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.n.1.1 yes 2
3.2 odd 2 3744.2.a.s.1.2 2
4.3 odd 2 1248.2.a.l.1.1 2
8.3 odd 2 2496.2.a.bh.1.2 2
8.5 even 2 2496.2.a.be.1.2 2
12.11 even 2 3744.2.a.r.1.2 2
24.5 odd 2 7488.2.a.ct.1.1 2
24.11 even 2 7488.2.a.cs.1.1 2

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