# Properties

 Label 224.2.a.d.1.2 Level $224$ Weight $2$ Character 224.1 Self dual yes Analytic conductor $1.789$ 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] = [224,2,Mod(1,224)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(224, 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("224.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$224 = 2^{5} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 224.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: $$1.78864900528$$ 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.61803$$ of defining polynomial Character $$\chi$$ $$=$$ 224.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.23607 q^{3} -1.23607 q^{5} -1.00000 q^{7} +7.47214 q^{9} +O(q^{10})$$ $$q+3.23607 q^{3} -1.23607 q^{5} -1.00000 q^{7} +7.47214 q^{9} -2.47214 q^{11} +5.23607 q^{13} -4.00000 q^{15} -4.47214 q^{17} -3.23607 q^{19} -3.23607 q^{21} -4.00000 q^{23} -3.47214 q^{25} +14.4721 q^{27} +4.47214 q^{29} -6.47214 q^{31} -8.00000 q^{33} +1.23607 q^{35} +4.47214 q^{37} +16.9443 q^{39} +0.472136 q^{41} +2.47214 q^{43} -9.23607 q^{45} -1.52786 q^{47} +1.00000 q^{49} -14.4721 q^{51} -10.0000 q^{53} +3.05573 q^{55} -10.4721 q^{57} +4.76393 q^{59} +6.76393 q^{61} -7.47214 q^{63} -6.47214 q^{65} -4.00000 q^{67} -12.9443 q^{69} -12.9443 q^{71} +14.9443 q^{73} -11.2361 q^{75} +2.47214 q^{77} +4.94427 q^{79} +24.4164 q^{81} +4.76393 q^{83} +5.52786 q^{85} +14.4721 q^{87} -6.00000 q^{89} -5.23607 q^{91} -20.9443 q^{93} +4.00000 q^{95} +3.52786 q^{97} -18.4721 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} + 2 q^{5} - 2 q^{7} + 6 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 + 2 * q^5 - 2 * q^7 + 6 * q^9 $$2 q + 2 q^{3} + 2 q^{5} - 2 q^{7} + 6 q^{9} + 4 q^{11} + 6 q^{13} - 8 q^{15} - 2 q^{19} - 2 q^{21} - 8 q^{23} + 2 q^{25} + 20 q^{27} - 4 q^{31} - 16 q^{33} - 2 q^{35} + 16 q^{39} - 8 q^{41} - 4 q^{43} - 14 q^{45} - 12 q^{47} + 2 q^{49} - 20 q^{51} - 20 q^{53} + 24 q^{55} - 12 q^{57} + 14 q^{59} + 18 q^{61} - 6 q^{63} - 4 q^{65} - 8 q^{67} - 8 q^{69} - 8 q^{71} + 12 q^{73} - 18 q^{75} - 4 q^{77} - 8 q^{79} + 22 q^{81} + 14 q^{83} + 20 q^{85} + 20 q^{87} - 12 q^{89} - 6 q^{91} - 24 q^{93} + 8 q^{95} + 16 q^{97} - 28 q^{99}+O(q^{100})$$ 2 * q + 2 * q^3 + 2 * q^5 - 2 * q^7 + 6 * q^9 + 4 * q^11 + 6 * q^13 - 8 * q^15 - 2 * q^19 - 2 * q^21 - 8 * q^23 + 2 * q^25 + 20 * q^27 - 4 * q^31 - 16 * q^33 - 2 * q^35 + 16 * q^39 - 8 * q^41 - 4 * q^43 - 14 * q^45 - 12 * q^47 + 2 * q^49 - 20 * q^51 - 20 * q^53 + 24 * q^55 - 12 * q^57 + 14 * q^59 + 18 * q^61 - 6 * q^63 - 4 * q^65 - 8 * q^67 - 8 * q^69 - 8 * q^71 + 12 * q^73 - 18 * q^75 - 4 * q^77 - 8 * q^79 + 22 * q^81 + 14 * q^83 + 20 * q^85 + 20 * q^87 - 12 * q^89 - 6 * q^91 - 24 * q^93 + 8 * q^95 + 16 * q^97 - 28 * 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
$$3$$ 3.23607 1.86834 0.934172 0.356822i $$-0.116140\pi$$
0.934172 + 0.356822i $$0.116140\pi$$
$$4$$ 0 0
$$5$$ −1.23607 −0.552786 −0.276393 0.961045i $$-0.589139\pi$$
−0.276393 + 0.961045i $$0.589139\pi$$
$$6$$ 0 0
$$7$$ −1.00000 −0.377964
$$8$$ 0 0
$$9$$ 7.47214 2.49071
$$10$$ 0 0
$$11$$ −2.47214 −0.745377 −0.372689 0.927957i $$-0.621564\pi$$
−0.372689 + 0.927957i $$0.621564\pi$$
$$12$$ 0 0
$$13$$ 5.23607 1.45222 0.726112 0.687576i $$-0.241325\pi$$
0.726112 + 0.687576i $$0.241325\pi$$
$$14$$ 0 0
$$15$$ −4.00000 −1.03280
$$16$$ 0 0
$$17$$ −4.47214 −1.08465 −0.542326 0.840168i $$-0.682456\pi$$
−0.542326 + 0.840168i $$0.682456\pi$$
$$18$$ 0 0
$$19$$ −3.23607 −0.742405 −0.371202 0.928552i $$-0.621054\pi$$
−0.371202 + 0.928552i $$0.621054\pi$$
$$20$$ 0 0
$$21$$ −3.23607 −0.706168
$$22$$ 0 0
$$23$$ −4.00000 −0.834058 −0.417029 0.908893i $$-0.636929\pi$$
−0.417029 + 0.908893i $$0.636929\pi$$
$$24$$ 0 0
$$25$$ −3.47214 −0.694427
$$26$$ 0 0
$$27$$ 14.4721 2.78516
$$28$$ 0 0
$$29$$ 4.47214 0.830455 0.415227 0.909718i $$-0.363702\pi$$
0.415227 + 0.909718i $$0.363702\pi$$
$$30$$ 0 0
$$31$$ −6.47214 −1.16243 −0.581215 0.813750i $$-0.697422\pi$$
−0.581215 + 0.813750i $$0.697422\pi$$
$$32$$ 0 0
$$33$$ −8.00000 −1.39262
$$34$$ 0 0
$$35$$ 1.23607 0.208934
$$36$$ 0 0
$$37$$ 4.47214 0.735215 0.367607 0.929981i $$-0.380177\pi$$
0.367607 + 0.929981i $$0.380177\pi$$
$$38$$ 0 0
$$39$$ 16.9443 2.71325
$$40$$ 0 0
$$41$$ 0.472136 0.0737352 0.0368676 0.999320i $$-0.488262\pi$$
0.0368676 + 0.999320i $$0.488262\pi$$
$$42$$ 0 0
$$43$$ 2.47214 0.376997 0.188499 0.982073i $$-0.439638\pi$$
0.188499 + 0.982073i $$0.439638\pi$$
$$44$$ 0 0
$$45$$ −9.23607 −1.37683
$$46$$ 0 0
$$47$$ −1.52786 −0.222862 −0.111431 0.993772i $$-0.535543\pi$$
−0.111431 + 0.993772i $$0.535543\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ −14.4721 −2.02650
$$52$$ 0 0
$$53$$ −10.0000 −1.37361 −0.686803 0.726844i $$-0.740986\pi$$
−0.686803 + 0.726844i $$0.740986\pi$$
$$54$$ 0 0
$$55$$ 3.05573 0.412034
$$56$$ 0 0
$$57$$ −10.4721 −1.38707
$$58$$ 0 0
$$59$$ 4.76393 0.620211 0.310106 0.950702i $$-0.399636\pi$$
0.310106 + 0.950702i $$0.399636\pi$$
$$60$$ 0 0
$$61$$ 6.76393 0.866033 0.433016 0.901386i $$-0.357449\pi$$
0.433016 + 0.901386i $$0.357449\pi$$
$$62$$ 0 0
$$63$$ −7.47214 −0.941401
$$64$$ 0 0
$$65$$ −6.47214 −0.802770
$$66$$ 0 0
$$67$$ −4.00000 −0.488678 −0.244339 0.969690i $$-0.578571\pi$$
−0.244339 + 0.969690i $$0.578571\pi$$
$$68$$ 0 0
$$69$$ −12.9443 −1.55831
$$70$$ 0 0
$$71$$ −12.9443 −1.53620 −0.768101 0.640328i $$-0.778798\pi$$
−0.768101 + 0.640328i $$0.778798\pi$$
$$72$$ 0 0
$$73$$ 14.9443 1.74909 0.874547 0.484940i $$-0.161159\pi$$
0.874547 + 0.484940i $$0.161159\pi$$
$$74$$ 0 0
$$75$$ −11.2361 −1.29743
$$76$$ 0 0
$$77$$ 2.47214 0.281726
$$78$$ 0 0
$$79$$ 4.94427 0.556274 0.278137 0.960541i $$-0.410283\pi$$
0.278137 + 0.960541i $$0.410283\pi$$
$$80$$ 0 0
$$81$$ 24.4164 2.71293
$$82$$ 0 0
$$83$$ 4.76393 0.522909 0.261455 0.965216i $$-0.415798\pi$$
0.261455 + 0.965216i $$0.415798\pi$$
$$84$$ 0 0
$$85$$ 5.52786 0.599581
$$86$$ 0 0
$$87$$ 14.4721 1.55158
$$88$$ 0 0
$$89$$ −6.00000 −0.635999 −0.317999 0.948091i $$-0.603011\pi$$
−0.317999 + 0.948091i $$0.603011\pi$$
$$90$$ 0 0
$$91$$ −5.23607 −0.548889
$$92$$ 0 0
$$93$$ −20.9443 −2.17182
$$94$$ 0 0
$$95$$ 4.00000 0.410391
$$96$$ 0 0
$$97$$ 3.52786 0.358200 0.179100 0.983831i $$-0.442681\pi$$
0.179100 + 0.983831i $$0.442681\pi$$
$$98$$ 0 0
$$99$$ −18.4721 −1.85652
$$100$$ 0 0
$$101$$ 11.7082 1.16501 0.582505 0.812827i $$-0.302073\pi$$
0.582505 + 0.812827i $$0.302073\pi$$
$$102$$ 0 0
$$103$$ 14.4721 1.42598 0.712991 0.701173i $$-0.247340\pi$$
0.712991 + 0.701173i $$0.247340\pi$$
$$104$$ 0 0
$$105$$ 4.00000 0.390360
$$106$$ 0 0
$$107$$ 8.94427 0.864675 0.432338 0.901712i $$-0.357689\pi$$
0.432338 + 0.901712i $$0.357689\pi$$
$$108$$ 0 0
$$109$$ −0.472136 −0.0452224 −0.0226112 0.999744i $$-0.507198\pi$$
−0.0226112 + 0.999744i $$0.507198\pi$$
$$110$$ 0 0
$$111$$ 14.4721 1.37363
$$112$$ 0 0
$$113$$ −3.52786 −0.331874 −0.165937 0.986136i $$-0.553065\pi$$
−0.165937 + 0.986136i $$0.553065\pi$$
$$114$$ 0 0
$$115$$ 4.94427 0.461056
$$116$$ 0 0
$$117$$ 39.1246 3.61707
$$118$$ 0 0
$$119$$ 4.47214 0.409960
$$120$$ 0 0
$$121$$ −4.88854 −0.444413
$$122$$ 0 0
$$123$$ 1.52786 0.137763
$$124$$ 0 0
$$125$$ 10.4721 0.936656
$$126$$ 0 0
$$127$$ 8.94427 0.793676 0.396838 0.917889i $$-0.370108\pi$$
0.396838 + 0.917889i $$0.370108\pi$$
$$128$$ 0 0
$$129$$ 8.00000 0.704361
$$130$$ 0 0
$$131$$ 1.70820 0.149246 0.0746232 0.997212i $$-0.476225\pi$$
0.0746232 + 0.997212i $$0.476225\pi$$
$$132$$ 0 0
$$133$$ 3.23607 0.280603
$$134$$ 0 0
$$135$$ −17.8885 −1.53960
$$136$$ 0 0
$$137$$ −2.94427 −0.251546 −0.125773 0.992059i $$-0.540141\pi$$
−0.125773 + 0.992059i $$0.540141\pi$$
$$138$$ 0 0
$$139$$ 3.23607 0.274480 0.137240 0.990538i $$-0.456177\pi$$
0.137240 + 0.990538i $$0.456177\pi$$
$$140$$ 0 0
$$141$$ −4.94427 −0.416383
$$142$$ 0 0
$$143$$ −12.9443 −1.08245
$$144$$ 0 0
$$145$$ −5.52786 −0.459064
$$146$$ 0 0
$$147$$ 3.23607 0.266906
$$148$$ 0 0
$$149$$ −14.9443 −1.22428 −0.612141 0.790748i $$-0.709692\pi$$
−0.612141 + 0.790748i $$0.709692\pi$$
$$150$$ 0 0
$$151$$ −8.94427 −0.727875 −0.363937 0.931423i $$-0.618568\pi$$
−0.363937 + 0.931423i $$0.618568\pi$$
$$152$$ 0 0
$$153$$ −33.4164 −2.70156
$$154$$ 0 0
$$155$$ 8.00000 0.642575
$$156$$ 0 0
$$157$$ 5.23607 0.417884 0.208942 0.977928i $$-0.432998\pi$$
0.208942 + 0.977928i $$0.432998\pi$$
$$158$$ 0 0
$$159$$ −32.3607 −2.56637
$$160$$ 0 0
$$161$$ 4.00000 0.315244
$$162$$ 0 0
$$163$$ −23.4164 −1.83411 −0.917057 0.398755i $$-0.869442\pi$$
−0.917057 + 0.398755i $$0.869442\pi$$
$$164$$ 0 0
$$165$$ 9.88854 0.769822
$$166$$ 0 0
$$167$$ 3.41641 0.264370 0.132185 0.991225i $$-0.457801\pi$$
0.132185 + 0.991225i $$0.457801\pi$$
$$168$$ 0 0
$$169$$ 14.4164 1.10895
$$170$$ 0 0
$$171$$ −24.1803 −1.84912
$$172$$ 0 0
$$173$$ −7.70820 −0.586044 −0.293022 0.956106i $$-0.594661\pi$$
−0.293022 + 0.956106i $$0.594661\pi$$
$$174$$ 0 0
$$175$$ 3.47214 0.262469
$$176$$ 0 0
$$177$$ 15.4164 1.15877
$$178$$ 0 0
$$179$$ 24.9443 1.86442 0.932211 0.361915i $$-0.117877\pi$$
0.932211 + 0.361915i $$0.117877\pi$$
$$180$$ 0 0
$$181$$ 10.1803 0.756699 0.378349 0.925663i $$-0.376492\pi$$
0.378349 + 0.925663i $$0.376492\pi$$
$$182$$ 0 0
$$183$$ 21.8885 1.61805
$$184$$ 0 0
$$185$$ −5.52786 −0.406417
$$186$$ 0 0
$$187$$ 11.0557 0.808475
$$188$$ 0 0
$$189$$ −14.4721 −1.05269
$$190$$ 0 0
$$191$$ 12.9443 0.936615 0.468307 0.883566i $$-0.344864\pi$$
0.468307 + 0.883566i $$0.344864\pi$$
$$192$$ 0 0
$$193$$ −8.47214 −0.609838 −0.304919 0.952378i $$-0.598629\pi$$
−0.304919 + 0.952378i $$0.598629\pi$$
$$194$$ 0 0
$$195$$ −20.9443 −1.49985
$$196$$ 0 0
$$197$$ −6.94427 −0.494759 −0.247379 0.968919i $$-0.579569\pi$$
−0.247379 + 0.968919i $$0.579569\pi$$
$$198$$ 0 0
$$199$$ 11.4164 0.809288 0.404644 0.914474i $$-0.367395\pi$$
0.404644 + 0.914474i $$0.367395\pi$$
$$200$$ 0 0
$$201$$ −12.9443 −0.913019
$$202$$ 0 0
$$203$$ −4.47214 −0.313882
$$204$$ 0 0
$$205$$ −0.583592 −0.0407598
$$206$$ 0 0
$$207$$ −29.8885 −2.07740
$$208$$ 0 0
$$209$$ 8.00000 0.553372
$$210$$ 0 0
$$211$$ 12.0000 0.826114 0.413057 0.910705i $$-0.364461\pi$$
0.413057 + 0.910705i $$0.364461\pi$$
$$212$$ 0 0
$$213$$ −41.8885 −2.87016
$$214$$ 0 0
$$215$$ −3.05573 −0.208399
$$216$$ 0 0
$$217$$ 6.47214 0.439357
$$218$$ 0 0
$$219$$ 48.3607 3.26791
$$220$$ 0 0
$$221$$ −23.4164 −1.57516
$$222$$ 0 0
$$223$$ 4.94427 0.331093 0.165546 0.986202i $$-0.447061\pi$$
0.165546 + 0.986202i $$0.447061\pi$$
$$224$$ 0 0
$$225$$ −25.9443 −1.72962
$$226$$ 0 0
$$227$$ 12.7639 0.847172 0.423586 0.905856i $$-0.360771\pi$$
0.423586 + 0.905856i $$0.360771\pi$$
$$228$$ 0 0
$$229$$ −25.5967 −1.69148 −0.845740 0.533595i $$-0.820841\pi$$
−0.845740 + 0.533595i $$0.820841\pi$$
$$230$$ 0 0
$$231$$ 8.00000 0.526361
$$232$$ 0 0
$$233$$ 19.8885 1.30294 0.651471 0.758674i $$-0.274152\pi$$
0.651471 + 0.758674i $$0.274152\pi$$
$$234$$ 0 0
$$235$$ 1.88854 0.123195
$$236$$ 0 0
$$237$$ 16.0000 1.03931
$$238$$ 0 0
$$239$$ −21.8885 −1.41585 −0.707926 0.706287i $$-0.750369\pi$$
−0.707926 + 0.706287i $$0.750369\pi$$
$$240$$ 0 0
$$241$$ 3.52786 0.227250 0.113625 0.993524i $$-0.463754\pi$$
0.113625 + 0.993524i $$0.463754\pi$$
$$242$$ 0 0
$$243$$ 35.5967 2.28353
$$244$$ 0 0
$$245$$ −1.23607 −0.0789695
$$246$$ 0 0
$$247$$ −16.9443 −1.07814
$$248$$ 0 0
$$249$$ 15.4164 0.976975
$$250$$ 0 0
$$251$$ −17.7082 −1.11773 −0.558866 0.829258i $$-0.688763\pi$$
−0.558866 + 0.829258i $$0.688763\pi$$
$$252$$ 0 0
$$253$$ 9.88854 0.621687
$$254$$ 0 0
$$255$$ 17.8885 1.12022
$$256$$ 0 0
$$257$$ −14.0000 −0.873296 −0.436648 0.899632i $$-0.643834\pi$$
−0.436648 + 0.899632i $$0.643834\pi$$
$$258$$ 0 0
$$259$$ −4.47214 −0.277885
$$260$$ 0 0
$$261$$ 33.4164 2.06842
$$262$$ 0 0
$$263$$ −28.9443 −1.78478 −0.892390 0.451265i $$-0.850973\pi$$
−0.892390 + 0.451265i $$0.850973\pi$$
$$264$$ 0 0
$$265$$ 12.3607 0.759311
$$266$$ 0 0
$$267$$ −19.4164 −1.18826
$$268$$ 0 0
$$269$$ 18.1803 1.10847 0.554237 0.832359i $$-0.313010\pi$$
0.554237 + 0.832359i $$0.313010\pi$$
$$270$$ 0 0
$$271$$ 24.0000 1.45790 0.728948 0.684569i $$-0.240010\pi$$
0.728948 + 0.684569i $$0.240010\pi$$
$$272$$ 0 0
$$273$$ −16.9443 −1.02551
$$274$$ 0 0
$$275$$ 8.58359 0.517610
$$276$$ 0 0
$$277$$ −27.8885 −1.67566 −0.837830 0.545931i $$-0.816176\pi$$
−0.837830 + 0.545931i $$0.816176\pi$$
$$278$$ 0 0
$$279$$ −48.3607 −2.89528
$$280$$ 0 0
$$281$$ 26.0000 1.55103 0.775515 0.631329i $$-0.217490\pi$$
0.775515 + 0.631329i $$0.217490\pi$$
$$282$$ 0 0
$$283$$ −16.1803 −0.961821 −0.480911 0.876770i $$-0.659694\pi$$
−0.480911 + 0.876770i $$0.659694\pi$$
$$284$$ 0 0
$$285$$ 12.9443 0.766752
$$286$$ 0 0
$$287$$ −0.472136 −0.0278693
$$288$$ 0 0
$$289$$ 3.00000 0.176471
$$290$$ 0 0
$$291$$ 11.4164 0.669242
$$292$$ 0 0
$$293$$ −17.2361 −1.00694 −0.503471 0.864012i $$-0.667944\pi$$
−0.503471 + 0.864012i $$0.667944\pi$$
$$294$$ 0 0
$$295$$ −5.88854 −0.342844
$$296$$ 0 0
$$297$$ −35.7771 −2.07600
$$298$$ 0 0
$$299$$ −20.9443 −1.21124
$$300$$ 0 0
$$301$$ −2.47214 −0.142492
$$302$$ 0 0
$$303$$ 37.8885 2.17664
$$304$$ 0 0
$$305$$ −8.36068 −0.478731
$$306$$ 0 0
$$307$$ −24.1803 −1.38004 −0.690022 0.723788i $$-0.742399\pi$$
−0.690022 + 0.723788i $$0.742399\pi$$
$$308$$ 0 0
$$309$$ 46.8328 2.66423
$$310$$ 0 0
$$311$$ −8.00000 −0.453638 −0.226819 0.973937i $$-0.572833\pi$$
−0.226819 + 0.973937i $$0.572833\pi$$
$$312$$ 0 0
$$313$$ 0.472136 0.0266867 0.0133434 0.999911i $$-0.495753\pi$$
0.0133434 + 0.999911i $$0.495753\pi$$
$$314$$ 0 0
$$315$$ 9.23607 0.520393
$$316$$ 0 0
$$317$$ 26.9443 1.51334 0.756671 0.653796i $$-0.226825\pi$$
0.756671 + 0.653796i $$0.226825\pi$$
$$318$$ 0 0
$$319$$ −11.0557 −0.619002
$$320$$ 0 0
$$321$$ 28.9443 1.61551
$$322$$ 0 0
$$323$$ 14.4721 0.805251
$$324$$ 0 0
$$325$$ −18.1803 −1.00846
$$326$$ 0 0
$$327$$ −1.52786 −0.0844911
$$328$$ 0 0
$$329$$ 1.52786 0.0842339
$$330$$ 0 0
$$331$$ −13.5279 −0.743559 −0.371779 0.928321i $$-0.621252\pi$$
−0.371779 + 0.928321i $$0.621252\pi$$
$$332$$ 0 0
$$333$$ 33.4164 1.83121
$$334$$ 0 0
$$335$$ 4.94427 0.270134
$$336$$ 0 0
$$337$$ −34.3607 −1.87175 −0.935873 0.352338i $$-0.885387\pi$$
−0.935873 + 0.352338i $$0.885387\pi$$
$$338$$ 0 0
$$339$$ −11.4164 −0.620054
$$340$$ 0 0
$$341$$ 16.0000 0.866449
$$342$$ 0 0
$$343$$ −1.00000 −0.0539949
$$344$$ 0 0
$$345$$ 16.0000 0.861411
$$346$$ 0 0
$$347$$ 2.47214 0.132711 0.0663556 0.997796i $$-0.478863\pi$$
0.0663556 + 0.997796i $$0.478863\pi$$
$$348$$ 0 0
$$349$$ −4.65248 −0.249041 −0.124521 0.992217i $$-0.539739\pi$$
−0.124521 + 0.992217i $$0.539739\pi$$
$$350$$ 0 0
$$351$$ 75.7771 4.04468
$$352$$ 0 0
$$353$$ 19.8885 1.05856 0.529280 0.848447i $$-0.322462\pi$$
0.529280 + 0.848447i $$0.322462\pi$$
$$354$$ 0 0
$$355$$ 16.0000 0.849192
$$356$$ 0 0
$$357$$ 14.4721 0.765947
$$358$$ 0 0
$$359$$ −0.944272 −0.0498368 −0.0249184 0.999689i $$-0.507933\pi$$
−0.0249184 + 0.999689i $$0.507933\pi$$
$$360$$ 0 0
$$361$$ −8.52786 −0.448835
$$362$$ 0 0
$$363$$ −15.8197 −0.830317
$$364$$ 0 0
$$365$$ −18.4721 −0.966876
$$366$$ 0 0
$$367$$ −30.8328 −1.60946 −0.804730 0.593641i $$-0.797690\pi$$
−0.804730 + 0.593641i $$0.797690\pi$$
$$368$$ 0 0
$$369$$ 3.52786 0.183653
$$370$$ 0 0
$$371$$ 10.0000 0.519174
$$372$$ 0 0
$$373$$ −14.9443 −0.773785 −0.386893 0.922125i $$-0.626452\pi$$
−0.386893 + 0.922125i $$0.626452\pi$$
$$374$$ 0 0
$$375$$ 33.8885 1.75000
$$376$$ 0 0
$$377$$ 23.4164 1.20601
$$378$$ 0 0
$$379$$ 31.4164 1.61375 0.806876 0.590721i $$-0.201156\pi$$
0.806876 + 0.590721i $$0.201156\pi$$
$$380$$ 0 0
$$381$$ 28.9443 1.48286
$$382$$ 0 0
$$383$$ −11.4164 −0.583351 −0.291676 0.956517i $$-0.594213\pi$$
−0.291676 + 0.956517i $$0.594213\pi$$
$$384$$ 0 0
$$385$$ −3.05573 −0.155734
$$386$$ 0 0
$$387$$ 18.4721 0.938991
$$388$$ 0 0
$$389$$ 4.47214 0.226746 0.113373 0.993552i $$-0.463834\pi$$
0.113373 + 0.993552i $$0.463834\pi$$
$$390$$ 0 0
$$391$$ 17.8885 0.904663
$$392$$ 0 0
$$393$$ 5.52786 0.278844
$$394$$ 0 0
$$395$$ −6.11146 −0.307501
$$396$$ 0 0
$$397$$ −10.7639 −0.540226 −0.270113 0.962829i $$-0.587061\pi$$
−0.270113 + 0.962829i $$0.587061\pi$$
$$398$$ 0 0
$$399$$ 10.4721 0.524263
$$400$$ 0 0
$$401$$ −32.4721 −1.62158 −0.810791 0.585336i $$-0.800962\pi$$
−0.810791 + 0.585336i $$0.800962\pi$$
$$402$$ 0 0
$$403$$ −33.8885 −1.68811
$$404$$ 0 0
$$405$$ −30.1803 −1.49967
$$406$$ 0 0
$$407$$ −11.0557 −0.548012
$$408$$ 0 0
$$409$$ 5.41641 0.267824 0.133912 0.990993i $$-0.457246\pi$$
0.133912 + 0.990993i $$0.457246\pi$$
$$410$$ 0 0
$$411$$ −9.52786 −0.469975
$$412$$ 0 0
$$413$$ −4.76393 −0.234418
$$414$$ 0 0
$$415$$ −5.88854 −0.289057
$$416$$ 0 0
$$417$$ 10.4721 0.512823
$$418$$ 0 0
$$419$$ 0.180340 0.00881018 0.00440509 0.999990i $$-0.498598\pi$$
0.00440509 + 0.999990i $$0.498598\pi$$
$$420$$ 0 0
$$421$$ −2.00000 −0.0974740 −0.0487370 0.998812i $$-0.515520\pi$$
−0.0487370 + 0.998812i $$0.515520\pi$$
$$422$$ 0 0
$$423$$ −11.4164 −0.555085
$$424$$ 0 0
$$425$$ 15.5279 0.753212
$$426$$ 0 0
$$427$$ −6.76393 −0.327330
$$428$$ 0 0
$$429$$ −41.8885 −2.02240
$$430$$ 0 0
$$431$$ 28.0000 1.34871 0.674356 0.738406i $$-0.264421\pi$$
0.674356 + 0.738406i $$0.264421\pi$$
$$432$$ 0 0
$$433$$ −17.4164 −0.836979 −0.418490 0.908222i $$-0.637440\pi$$
−0.418490 + 0.908222i $$0.637440\pi$$
$$434$$ 0 0
$$435$$ −17.8885 −0.857690
$$436$$ 0 0
$$437$$ 12.9443 0.619208
$$438$$ 0 0
$$439$$ 32.0000 1.52728 0.763638 0.645644i $$-0.223411\pi$$
0.763638 + 0.645644i $$0.223411\pi$$
$$440$$ 0 0
$$441$$ 7.47214 0.355816
$$442$$ 0 0
$$443$$ −21.8885 −1.03996 −0.519978 0.854180i $$-0.674060\pi$$
−0.519978 + 0.854180i $$0.674060\pi$$
$$444$$ 0 0
$$445$$ 7.41641 0.351571
$$446$$ 0 0
$$447$$ −48.3607 −2.28738
$$448$$ 0 0
$$449$$ 27.8885 1.31614 0.658071 0.752956i $$-0.271373\pi$$
0.658071 + 0.752956i $$0.271373\pi$$
$$450$$ 0 0
$$451$$ −1.16718 −0.0549606
$$452$$ 0 0
$$453$$ −28.9443 −1.35992
$$454$$ 0 0
$$455$$ 6.47214 0.303418
$$456$$ 0 0
$$457$$ −16.4721 −0.770534 −0.385267 0.922805i $$-0.625891\pi$$
−0.385267 + 0.922805i $$0.625891\pi$$
$$458$$ 0 0
$$459$$ −64.7214 −3.02093
$$460$$ 0 0
$$461$$ 8.29180 0.386187 0.193094 0.981180i $$-0.438148\pi$$
0.193094 + 0.981180i $$0.438148\pi$$
$$462$$ 0 0
$$463$$ −35.7771 −1.66270 −0.831351 0.555748i $$-0.812432\pi$$
−0.831351 + 0.555748i $$0.812432\pi$$
$$464$$ 0 0
$$465$$ 25.8885 1.20055
$$466$$ 0 0
$$467$$ −26.0689 −1.20632 −0.603162 0.797619i $$-0.706093\pi$$
−0.603162 + 0.797619i $$0.706093\pi$$
$$468$$ 0 0
$$469$$ 4.00000 0.184703
$$470$$ 0 0
$$471$$ 16.9443 0.780751
$$472$$ 0 0
$$473$$ −6.11146 −0.281005
$$474$$ 0 0
$$475$$ 11.2361 0.515546
$$476$$ 0 0
$$477$$ −74.7214 −3.42126
$$478$$ 0 0
$$479$$ 35.4164 1.61822 0.809108 0.587659i $$-0.199950\pi$$
0.809108 + 0.587659i $$0.199950\pi$$
$$480$$ 0 0
$$481$$ 23.4164 1.06770
$$482$$ 0 0
$$483$$ 12.9443 0.588985
$$484$$ 0 0
$$485$$ −4.36068 −0.198008
$$486$$ 0 0
$$487$$ −20.0000 −0.906287 −0.453143 0.891438i $$-0.649697\pi$$
−0.453143 + 0.891438i $$0.649697\pi$$
$$488$$ 0 0
$$489$$ −75.7771 −3.42676
$$490$$ 0 0
$$491$$ −2.11146 −0.0952887 −0.0476443 0.998864i $$-0.515171\pi$$
−0.0476443 + 0.998864i $$0.515171\pi$$
$$492$$ 0 0
$$493$$ −20.0000 −0.900755
$$494$$ 0 0
$$495$$ 22.8328 1.02626
$$496$$ 0 0
$$497$$ 12.9443 0.580630
$$498$$ 0 0
$$499$$ 13.8885 0.621737 0.310868 0.950453i $$-0.399380\pi$$
0.310868 + 0.950453i $$0.399380\pi$$
$$500$$ 0 0
$$501$$ 11.0557 0.493934
$$502$$ 0 0
$$503$$ −12.9443 −0.577157 −0.288578 0.957456i $$-0.593183\pi$$
−0.288578 + 0.957456i $$0.593183\pi$$
$$504$$ 0 0
$$505$$ −14.4721 −0.644002
$$506$$ 0 0
$$507$$ 46.6525 2.07191
$$508$$ 0 0
$$509$$ −0.875388 −0.0388009 −0.0194004 0.999812i $$-0.506176\pi$$
−0.0194004 + 0.999812i $$0.506176\pi$$
$$510$$ 0 0
$$511$$ −14.9443 −0.661096
$$512$$ 0 0
$$513$$ −46.8328 −2.06772
$$514$$ 0 0
$$515$$ −17.8885 −0.788263
$$516$$ 0 0
$$517$$ 3.77709 0.166116
$$518$$ 0 0
$$519$$ −24.9443 −1.09493
$$520$$ 0 0
$$521$$ −33.4164 −1.46400 −0.732000 0.681305i $$-0.761413\pi$$
−0.732000 + 0.681305i $$0.761413\pi$$
$$522$$ 0 0
$$523$$ 17.7082 0.774326 0.387163 0.922011i $$-0.373455\pi$$
0.387163 + 0.922011i $$0.373455\pi$$
$$524$$ 0 0
$$525$$ 11.2361 0.490382
$$526$$ 0 0
$$527$$ 28.9443 1.26083
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ 0 0
$$531$$ 35.5967 1.54477
$$532$$ 0 0
$$533$$ 2.47214 0.107080
$$534$$ 0 0
$$535$$ −11.0557 −0.477981
$$536$$ 0 0
$$537$$ 80.7214 3.48338
$$538$$ 0 0
$$539$$ −2.47214 −0.106482
$$540$$ 0 0
$$541$$ −22.9443 −0.986451 −0.493226 0.869901i $$-0.664182\pi$$
−0.493226 + 0.869901i $$0.664182\pi$$
$$542$$ 0 0
$$543$$ 32.9443 1.41377
$$544$$ 0 0
$$545$$ 0.583592 0.0249983
$$546$$ 0 0
$$547$$ 31.4164 1.34327 0.671634 0.740883i $$-0.265593\pi$$
0.671634 + 0.740883i $$0.265593\pi$$
$$548$$ 0 0
$$549$$ 50.5410 2.15704
$$550$$ 0 0
$$551$$ −14.4721 −0.616534
$$552$$ 0 0
$$553$$ −4.94427 −0.210252
$$554$$ 0 0
$$555$$ −17.8885 −0.759326
$$556$$ 0 0
$$557$$ 26.9443 1.14167 0.570833 0.821066i $$-0.306620\pi$$
0.570833 + 0.821066i $$0.306620\pi$$
$$558$$ 0 0
$$559$$ 12.9443 0.547484
$$560$$ 0 0
$$561$$ 35.7771 1.51051
$$562$$ 0 0
$$563$$ 40.1803 1.69340 0.846700 0.532071i $$-0.178586\pi$$
0.846700 + 0.532071i $$0.178586\pi$$
$$564$$ 0 0
$$565$$ 4.36068 0.183455
$$566$$ 0 0
$$567$$ −24.4164 −1.02539
$$568$$ 0 0
$$569$$ −18.3607 −0.769720 −0.384860 0.922975i $$-0.625750\pi$$
−0.384860 + 0.922975i $$0.625750\pi$$
$$570$$ 0 0
$$571$$ 5.52786 0.231334 0.115667 0.993288i $$-0.463099\pi$$
0.115667 + 0.993288i $$0.463099\pi$$
$$572$$ 0 0
$$573$$ 41.8885 1.74992
$$574$$ 0 0
$$575$$ 13.8885 0.579192
$$576$$ 0 0
$$577$$ −6.00000 −0.249783 −0.124892 0.992170i $$-0.539858\pi$$
−0.124892 + 0.992170i $$0.539858\pi$$
$$578$$ 0 0
$$579$$ −27.4164 −1.13939
$$580$$ 0 0
$$581$$ −4.76393 −0.197641
$$582$$ 0 0
$$583$$ 24.7214 1.02385
$$584$$ 0 0
$$585$$ −48.3607 −1.99947
$$586$$ 0 0
$$587$$ −9.70820 −0.400700 −0.200350 0.979724i $$-0.564208\pi$$
−0.200350 + 0.979724i $$0.564208\pi$$
$$588$$ 0 0
$$589$$ 20.9443 0.862994
$$590$$ 0 0
$$591$$ −22.4721 −0.924380
$$592$$ 0 0
$$593$$ −20.8328 −0.855501 −0.427751 0.903897i $$-0.640694\pi$$
−0.427751 + 0.903897i $$0.640694\pi$$
$$594$$ 0 0
$$595$$ −5.52786 −0.226620
$$596$$ 0 0
$$597$$ 36.9443 1.51203
$$598$$ 0 0
$$599$$ 17.8885 0.730906 0.365453 0.930830i $$-0.380914\pi$$
0.365453 + 0.930830i $$0.380914\pi$$
$$600$$ 0 0
$$601$$ −41.7771 −1.70412 −0.852061 0.523442i $$-0.824648\pi$$
−0.852061 + 0.523442i $$0.824648\pi$$
$$602$$ 0 0
$$603$$ −29.8885 −1.21716
$$604$$ 0 0
$$605$$ 6.04257 0.245666
$$606$$ 0 0
$$607$$ 25.8885 1.05078 0.525392 0.850860i $$-0.323919\pi$$
0.525392 + 0.850860i $$0.323919\pi$$
$$608$$ 0 0
$$609$$ −14.4721 −0.586441
$$610$$ 0 0
$$611$$ −8.00000 −0.323645
$$612$$ 0 0
$$613$$ 2.58359 0.104350 0.0521752 0.998638i $$-0.483385\pi$$
0.0521752 + 0.998638i $$0.483385\pi$$
$$614$$ 0 0
$$615$$ −1.88854 −0.0761534
$$616$$ 0 0
$$617$$ −10.3607 −0.417105 −0.208553 0.978011i $$-0.566875\pi$$
−0.208553 + 0.978011i $$0.566875\pi$$
$$618$$ 0 0
$$619$$ −10.0689 −0.404703 −0.202351 0.979313i $$-0.564858\pi$$
−0.202351 + 0.979313i $$0.564858\pi$$
$$620$$ 0 0
$$621$$ −57.8885 −2.32299
$$622$$ 0 0
$$623$$ 6.00000 0.240385
$$624$$ 0 0
$$625$$ 4.41641 0.176656
$$626$$ 0 0
$$627$$ 25.8885 1.03389
$$628$$ 0 0
$$629$$ −20.0000 −0.797452
$$630$$ 0 0
$$631$$ 27.0557 1.07707 0.538536 0.842603i $$-0.318978\pi$$
0.538536 + 0.842603i $$0.318978\pi$$
$$632$$ 0 0
$$633$$ 38.8328 1.54347
$$634$$ 0 0
$$635$$ −11.0557 −0.438733
$$636$$ 0 0
$$637$$ 5.23607 0.207461
$$638$$ 0 0
$$639$$ −96.7214 −3.82624
$$640$$ 0 0
$$641$$ 41.4164 1.63585 0.817925 0.575325i $$-0.195124\pi$$
0.817925 + 0.575325i $$0.195124\pi$$
$$642$$ 0 0
$$643$$ 30.2918 1.19459 0.597296 0.802021i $$-0.296242\pi$$
0.597296 + 0.802021i $$0.296242\pi$$
$$644$$ 0 0
$$645$$ −9.88854 −0.389361
$$646$$ 0 0
$$647$$ −24.3607 −0.957717 −0.478859 0.877892i $$-0.658949\pi$$
−0.478859 + 0.877892i $$0.658949\pi$$
$$648$$ 0 0
$$649$$ −11.7771 −0.462291
$$650$$ 0 0
$$651$$ 20.9443 0.820871
$$652$$ 0 0
$$653$$ 17.4164 0.681557 0.340778 0.940144i $$-0.389309\pi$$
0.340778 + 0.940144i $$0.389309\pi$$
$$654$$ 0 0
$$655$$ −2.11146 −0.0825014
$$656$$ 0 0
$$657$$ 111.666 4.35649
$$658$$ 0 0
$$659$$ 25.3050 0.985741 0.492870 0.870103i $$-0.335948\pi$$
0.492870 + 0.870103i $$0.335948\pi$$
$$660$$ 0 0
$$661$$ 8.65248 0.336542 0.168271 0.985741i $$-0.446182\pi$$
0.168271 + 0.985741i $$0.446182\pi$$
$$662$$ 0 0
$$663$$ −75.7771 −2.94294
$$664$$ 0 0
$$665$$ −4.00000 −0.155113
$$666$$ 0 0
$$667$$ −17.8885 −0.692647
$$668$$ 0 0
$$669$$ 16.0000 0.618596
$$670$$ 0 0
$$671$$ −16.7214 −0.645521
$$672$$ 0 0
$$673$$ 14.9443 0.576059 0.288030 0.957621i $$-0.407000\pi$$
0.288030 + 0.957621i $$0.407000\pi$$
$$674$$ 0 0
$$675$$ −50.2492 −1.93409
$$676$$ 0 0
$$677$$ 42.1803 1.62112 0.810561 0.585654i $$-0.199162\pi$$
0.810561 + 0.585654i $$0.199162\pi$$
$$678$$ 0 0
$$679$$ −3.52786 −0.135387
$$680$$ 0 0
$$681$$ 41.3050 1.58281
$$682$$ 0 0
$$683$$ 47.7771 1.82814 0.914070 0.405557i $$-0.132922\pi$$
0.914070 + 0.405557i $$0.132922\pi$$
$$684$$ 0 0
$$685$$ 3.63932 0.139051
$$686$$ 0 0
$$687$$ −82.8328 −3.16027
$$688$$ 0 0
$$689$$ −52.3607 −1.99478
$$690$$ 0 0
$$691$$ −8.18034 −0.311195 −0.155597 0.987821i $$-0.549730\pi$$
−0.155597 + 0.987821i $$0.549730\pi$$
$$692$$ 0 0
$$693$$ 18.4721 0.701698
$$694$$ 0 0
$$695$$ −4.00000 −0.151729
$$696$$ 0 0
$$697$$ −2.11146 −0.0799771
$$698$$ 0 0
$$699$$ 64.3607 2.43434
$$700$$ 0 0
$$701$$ −14.5836 −0.550815 −0.275407 0.961328i $$-0.588813\pi$$
−0.275407 + 0.961328i $$0.588813\pi$$
$$702$$ 0 0
$$703$$ −14.4721 −0.545827
$$704$$ 0 0
$$705$$ 6.11146 0.230171
$$706$$ 0 0
$$707$$ −11.7082 −0.440332
$$708$$ 0 0
$$709$$ 46.3607 1.74111 0.870556 0.492069i $$-0.163759\pi$$
0.870556 + 0.492069i $$0.163759\pi$$
$$710$$ 0 0
$$711$$ 36.9443 1.38552
$$712$$ 0 0
$$713$$ 25.8885 0.969534
$$714$$ 0 0
$$715$$ 16.0000 0.598366
$$716$$ 0 0
$$717$$ −70.8328 −2.64530
$$718$$ 0 0
$$719$$ 47.1935 1.76002 0.880010 0.474955i $$-0.157536\pi$$
0.880010 + 0.474955i $$0.157536\pi$$
$$720$$ 0 0
$$721$$ −14.4721 −0.538971
$$722$$ 0 0
$$723$$ 11.4164 0.424581
$$724$$ 0 0
$$725$$ −15.5279 −0.576690
$$726$$ 0 0
$$727$$ 32.3607 1.20019 0.600096 0.799928i $$-0.295129\pi$$
0.600096 + 0.799928i $$0.295129\pi$$
$$728$$ 0 0
$$729$$ 41.9443 1.55349
$$730$$ 0 0
$$731$$ −11.0557 −0.408911
$$732$$ 0 0
$$733$$ −9.23607 −0.341142 −0.170571 0.985345i $$-0.554561\pi$$
−0.170571 + 0.985345i $$0.554561\pi$$
$$734$$ 0 0
$$735$$ −4.00000 −0.147542
$$736$$ 0 0
$$737$$ 9.88854 0.364249
$$738$$ 0 0
$$739$$ −23.4164 −0.861386 −0.430693 0.902498i $$-0.641731\pi$$
−0.430693 + 0.902498i $$0.641731\pi$$
$$740$$ 0 0
$$741$$ −54.8328 −2.01433
$$742$$ 0 0
$$743$$ −7.05573 −0.258850 −0.129425 0.991589i $$-0.541313\pi$$
−0.129425 + 0.991589i $$0.541313\pi$$
$$744$$ 0 0
$$745$$ 18.4721 0.676767
$$746$$ 0 0
$$747$$ 35.5967 1.30242
$$748$$ 0 0
$$749$$ −8.94427 −0.326817
$$750$$ 0 0
$$751$$ −36.0000 −1.31366 −0.656829 0.754039i $$-0.728103\pi$$
−0.656829 + 0.754039i $$0.728103\pi$$
$$752$$ 0 0
$$753$$ −57.3050 −2.08831
$$754$$ 0 0
$$755$$ 11.0557 0.402359
$$756$$ 0 0
$$757$$ −23.3050 −0.847033 −0.423516 0.905888i $$-0.639204\pi$$
−0.423516 + 0.905888i $$0.639204\pi$$
$$758$$ 0 0
$$759$$ 32.0000 1.16153
$$760$$ 0 0
$$761$$ −12.4721 −0.452115 −0.226057 0.974114i $$-0.572584\pi$$
−0.226057 + 0.974114i $$0.572584\pi$$
$$762$$ 0 0
$$763$$ 0.472136 0.0170925
$$764$$ 0 0
$$765$$ 41.3050 1.49338
$$766$$ 0 0
$$767$$ 24.9443 0.900685
$$768$$ 0 0
$$769$$ 26.3607 0.950590 0.475295 0.879826i $$-0.342341\pi$$
0.475295 + 0.879826i $$0.342341\pi$$
$$770$$ 0 0
$$771$$ −45.3050 −1.63162
$$772$$ 0 0
$$773$$ 17.8197 0.640929 0.320464 0.947261i $$-0.396161\pi$$
0.320464 + 0.947261i $$0.396161\pi$$
$$774$$ 0 0
$$775$$ 22.4721 0.807223
$$776$$ 0 0
$$777$$ −14.4721 −0.519185
$$778$$ 0 0
$$779$$ −1.52786 −0.0547414
$$780$$ 0 0
$$781$$ 32.0000 1.14505
$$782$$ 0 0
$$783$$ 64.7214 2.31295
$$784$$ 0 0
$$785$$ −6.47214 −0.231000
$$786$$ 0 0
$$787$$ 25.7082 0.916399 0.458199 0.888850i $$-0.348495\pi$$
0.458199 + 0.888850i $$0.348495\pi$$
$$788$$ 0 0
$$789$$ −93.6656 −3.33458
$$790$$ 0 0
$$791$$ 3.52786 0.125436
$$792$$ 0 0
$$793$$ 35.4164 1.25767
$$794$$ 0 0
$$795$$ 40.0000 1.41865
$$796$$ 0 0
$$797$$ −29.8197 −1.05627 −0.528133 0.849161i $$-0.677108\pi$$
−0.528133 + 0.849161i $$0.677108\pi$$
$$798$$ 0 0
$$799$$ 6.83282 0.241728
$$800$$ 0 0
$$801$$ −44.8328 −1.58409
$$802$$ 0 0
$$803$$ −36.9443 −1.30374
$$804$$ 0 0
$$805$$ −4.94427 −0.174263
$$806$$ 0 0
$$807$$ 58.8328 2.07101
$$808$$ 0 0
$$809$$ 9.41641 0.331063 0.165532 0.986204i $$-0.447066\pi$$
0.165532 + 0.986204i $$0.447066\pi$$
$$810$$ 0 0
$$811$$ −23.0132 −0.808101 −0.404051 0.914737i $$-0.632398\pi$$
−0.404051 + 0.914737i $$0.632398\pi$$
$$812$$ 0 0
$$813$$ 77.6656 2.72385
$$814$$ 0 0
$$815$$ 28.9443 1.01387
$$816$$ 0 0
$$817$$ −8.00000 −0.279885
$$818$$ 0 0
$$819$$ −39.1246 −1.36712
$$820$$ 0 0
$$821$$ 39.8885 1.39212 0.696060 0.717984i $$-0.254935\pi$$
0.696060 + 0.717984i $$0.254935\pi$$
$$822$$ 0 0
$$823$$ −51.7771 −1.80484 −0.902418 0.430862i $$-0.858210\pi$$
−0.902418 + 0.430862i $$0.858210\pi$$
$$824$$ 0 0
$$825$$ 27.7771 0.967074
$$826$$ 0 0
$$827$$ −32.9443 −1.14558 −0.572792 0.819701i $$-0.694140\pi$$
−0.572792 + 0.819701i $$0.694140\pi$$
$$828$$ 0 0
$$829$$ 6.76393 0.234921 0.117461 0.993078i $$-0.462525\pi$$
0.117461 + 0.993078i $$0.462525\pi$$
$$830$$ 0 0
$$831$$ −90.2492 −3.13071
$$832$$ 0 0
$$833$$ −4.47214 −0.154950
$$834$$ 0 0
$$835$$ −4.22291 −0.146140
$$836$$ 0 0
$$837$$ −93.6656 −3.23756
$$838$$ 0 0
$$839$$ 30.4721 1.05201 0.526007 0.850480i $$-0.323688\pi$$
0.526007 + 0.850480i $$0.323688\pi$$
$$840$$ 0 0
$$841$$ −9.00000 −0.310345
$$842$$ 0 0
$$843$$ 84.1378 2.89786
$$844$$ 0 0
$$845$$ −17.8197 −0.613015
$$846$$ 0 0
$$847$$ 4.88854 0.167972
$$848$$ 0 0
$$849$$ −52.3607 −1.79701
$$850$$ 0 0
$$851$$ −17.8885 −0.613211
$$852$$ 0 0
$$853$$ 0.291796 0.00999091 0.00499545 0.999988i $$-0.498410\pi$$
0.00499545 + 0.999988i $$0.498410\pi$$
$$854$$ 0 0
$$855$$ 29.8885 1.02217
$$856$$ 0 0
$$857$$ −36.4721 −1.24586 −0.622932 0.782276i $$-0.714059\pi$$
−0.622932 + 0.782276i $$0.714059\pi$$
$$858$$ 0 0
$$859$$ 56.1803 1.91685 0.958424 0.285347i $$-0.0921089\pi$$
0.958424 + 0.285347i $$0.0921089\pi$$
$$860$$ 0 0
$$861$$ −1.52786 −0.0520695
$$862$$ 0 0
$$863$$ 3.05573 0.104018 0.0520091 0.998647i $$-0.483438\pi$$
0.0520091 + 0.998647i $$0.483438\pi$$
$$864$$ 0 0
$$865$$ 9.52786 0.323957
$$866$$ 0 0
$$867$$ 9.70820 0.329708
$$868$$ 0 0
$$869$$ −12.2229 −0.414634
$$870$$ 0 0
$$871$$ −20.9443 −0.709670
$$872$$ 0 0
$$873$$ 26.3607 0.892174
$$874$$ 0 0
$$875$$ −10.4721 −0.354023
$$876$$ 0 0
$$877$$ −13.4164 −0.453040 −0.226520 0.974007i $$-0.572735\pi$$
−0.226520 + 0.974007i $$0.572735\pi$$
$$878$$ 0 0
$$879$$ −55.7771 −1.88131
$$880$$ 0 0
$$881$$ −28.8328 −0.971402 −0.485701 0.874125i $$-0.661436\pi$$
−0.485701 + 0.874125i $$0.661436\pi$$
$$882$$ 0 0
$$883$$ −40.9443 −1.37788 −0.688942 0.724816i $$-0.741925\pi$$
−0.688942 + 0.724816i $$0.741925\pi$$
$$884$$ 0 0
$$885$$ −19.0557 −0.640551
$$886$$ 0 0
$$887$$ −29.3050 −0.983964 −0.491982 0.870605i $$-0.663727\pi$$
−0.491982 + 0.870605i $$0.663727\pi$$
$$888$$ 0 0
$$889$$ −8.94427 −0.299981
$$890$$ 0 0
$$891$$ −60.3607 −2.02216
$$892$$ 0 0
$$893$$ 4.94427 0.165454
$$894$$ 0 0
$$895$$ −30.8328 −1.03063
$$896$$ 0 0
$$897$$ −67.7771 −2.26301
$$898$$ 0 0
$$899$$ −28.9443 −0.965346
$$900$$ 0 0
$$901$$ 44.7214 1.48988
$$902$$ 0 0
$$903$$ −8.00000 −0.266223
$$904$$ 0 0
$$905$$ −12.5836 −0.418293
$$906$$ 0 0
$$907$$ 16.9443 0.562625 0.281313 0.959616i $$-0.409230\pi$$
0.281313 + 0.959616i $$0.409230\pi$$
$$908$$ 0 0
$$909$$ 87.4853 2.90170
$$910$$ 0 0
$$911$$ −18.8328 −0.623959 −0.311980 0.950089i $$-0.600992\pi$$
−0.311980 + 0.950089i $$0.600992\pi$$
$$912$$ 0 0
$$913$$ −11.7771 −0.389765
$$914$$ 0 0
$$915$$ −27.0557 −0.894435
$$916$$ 0 0
$$917$$ −1.70820 −0.0564099
$$918$$ 0 0
$$919$$ −35.7771 −1.18018 −0.590089 0.807338i $$-0.700907\pi$$
−0.590089 + 0.807338i $$0.700907\pi$$
$$920$$ 0 0
$$921$$ −78.2492 −2.57840
$$922$$ 0 0
$$923$$ −67.7771 −2.23091
$$924$$ 0 0
$$925$$ −15.5279 −0.510553
$$926$$ 0 0
$$927$$ 108.138 3.55171
$$928$$ 0 0
$$929$$ 15.3050 0.502139 0.251070 0.967969i $$-0.419218\pi$$
0.251070 + 0.967969i $$0.419218\pi$$
$$930$$ 0 0
$$931$$ −3.23607 −0.106058
$$932$$ 0 0
$$933$$ −25.8885 −0.847553
$$934$$ 0 0
$$935$$ −13.6656 −0.446914
$$936$$ 0 0
$$937$$ −26.9443 −0.880231 −0.440115 0.897941i $$-0.645063\pi$$
−0.440115 + 0.897941i $$0.645063\pi$$
$$938$$ 0 0
$$939$$ 1.52786 0.0498600
$$940$$ 0 0
$$941$$ 13.5967 0.443241 0.221621 0.975133i $$-0.428865\pi$$
0.221621 + 0.975133i $$0.428865\pi$$
$$942$$ 0 0
$$943$$ −1.88854 −0.0614994
$$944$$ 0 0
$$945$$ 17.8885 0.581914
$$946$$ 0 0
$$947$$ 31.4164 1.02090 0.510448 0.859909i $$-0.329480\pi$$
0.510448 + 0.859909i $$0.329480\pi$$
$$948$$ 0 0
$$949$$ 78.2492 2.54008
$$950$$ 0 0
$$951$$ 87.1935 2.82744
$$952$$ 0 0
$$953$$ 16.1115 0.521901 0.260951 0.965352i $$-0.415964\pi$$
0.260951 + 0.965352i $$0.415964\pi$$
$$954$$ 0 0
$$955$$ −16.0000 −0.517748
$$956$$ 0 0
$$957$$ −35.7771 −1.15651
$$958$$ 0 0
$$959$$ 2.94427 0.0950755
$$960$$ 0 0
$$961$$ 10.8885 0.351243
$$962$$ 0 0
$$963$$ 66.8328 2.15366
$$964$$ 0 0
$$965$$ 10.4721 0.337110
$$966$$ 0 0
$$967$$ 5.88854 0.189363 0.0946814 0.995508i $$-0.469817\pi$$
0.0946814 + 0.995508i $$0.469817\pi$$
$$968$$ 0 0
$$969$$ 46.8328 1.50449
$$970$$ 0 0
$$971$$ −27.2361 −0.874047 −0.437024 0.899450i $$-0.643967\pi$$
−0.437024 + 0.899450i $$0.643967\pi$$
$$972$$ 0 0
$$973$$ −3.23607 −0.103744
$$974$$ 0 0
$$975$$ −58.8328 −1.88416
$$976$$ 0 0
$$977$$ 40.8328 1.30636 0.653179 0.757204i $$-0.273435\pi$$
0.653179 + 0.757204i $$0.273435\pi$$
$$978$$ 0 0
$$979$$ 14.8328 0.474059
$$980$$ 0 0
$$981$$ −3.52786 −0.112636
$$982$$ 0 0
$$983$$ 14.4721 0.461589 0.230795 0.973002i $$-0.425867\pi$$
0.230795 + 0.973002i $$0.425867\pi$$
$$984$$ 0 0
$$985$$ 8.58359 0.273496
$$986$$ 0 0
$$987$$ 4.94427 0.157378
$$988$$ 0 0
$$989$$ −9.88854 −0.314437
$$990$$ 0 0
$$991$$ 24.0000 0.762385 0.381193 0.924496i $$-0.375513\pi$$
0.381193 + 0.924496i $$0.375513\pi$$
$$992$$ 0 0
$$993$$ −43.7771 −1.38922
$$994$$ 0 0
$$995$$ −14.1115 −0.447363
$$996$$ 0 0
$$997$$ −40.0689 −1.26899 −0.634497 0.772925i $$-0.718793\pi$$
−0.634497 + 0.772925i $$0.718793\pi$$
$$998$$ 0 0
$$999$$ 64.7214 2.04769
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 224.2.a.d.1.2 yes 2
3.2 odd 2 2016.2.a.o.1.2 2
4.3 odd 2 224.2.a.c.1.1 2
5.4 even 2 5600.2.a.z.1.1 2
7.2 even 3 1568.2.i.m.1537.1 4
7.3 odd 6 1568.2.i.w.961.2 4
7.4 even 3 1568.2.i.m.961.1 4
7.5 odd 6 1568.2.i.w.1537.2 4
7.6 odd 2 1568.2.a.k.1.1 2
8.3 odd 2 448.2.a.j.1.2 2
8.5 even 2 448.2.a.i.1.1 2
12.11 even 2 2016.2.a.r.1.2 2
16.3 odd 4 1792.2.b.k.897.1 4
16.5 even 4 1792.2.b.m.897.1 4
16.11 odd 4 1792.2.b.k.897.4 4
16.13 even 4 1792.2.b.m.897.4 4
20.19 odd 2 5600.2.a.bk.1.2 2
24.5 odd 2 4032.2.a.bv.1.1 2
24.11 even 2 4032.2.a.bw.1.1 2
28.3 even 6 1568.2.i.n.961.1 4
28.11 odd 6 1568.2.i.v.961.2 4
28.19 even 6 1568.2.i.n.1537.1 4
28.23 odd 6 1568.2.i.v.1537.2 4
28.27 even 2 1568.2.a.v.1.2 2
56.13 odd 2 3136.2.a.by.1.2 2
56.27 even 2 3136.2.a.bf.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
224.2.a.c.1.1 2 4.3 odd 2
224.2.a.d.1.2 yes 2 1.1 even 1 trivial
448.2.a.i.1.1 2 8.5 even 2
448.2.a.j.1.2 2 8.3 odd 2
1568.2.a.k.1.1 2 7.6 odd 2
1568.2.a.v.1.2 2 28.27 even 2
1568.2.i.m.961.1 4 7.4 even 3
1568.2.i.m.1537.1 4 7.2 even 3
1568.2.i.n.961.1 4 28.3 even 6
1568.2.i.n.1537.1 4 28.19 even 6
1568.2.i.v.961.2 4 28.11 odd 6
1568.2.i.v.1537.2 4 28.23 odd 6
1568.2.i.w.961.2 4 7.3 odd 6
1568.2.i.w.1537.2 4 7.5 odd 6
1792.2.b.k.897.1 4 16.3 odd 4
1792.2.b.k.897.4 4 16.11 odd 4
1792.2.b.m.897.1 4 16.5 even 4
1792.2.b.m.897.4 4 16.13 even 4
2016.2.a.o.1.2 2 3.2 odd 2
2016.2.a.r.1.2 2 12.11 even 2
3136.2.a.bf.1.1 2 56.27 even 2
3136.2.a.by.1.2 2 56.13 odd 2
4032.2.a.bv.1.1 2 24.5 odd 2
4032.2.a.bw.1.1 2 24.11 even 2
5600.2.a.z.1.1 2 5.4 even 2
5600.2.a.bk.1.2 2 20.19 odd 2