# Properties

 Label 9702.2.a.cp.1.1 Level $9702$ Weight $2$ Character 9702.1 Self dual yes Analytic conductor $77.471$ Analytic rank $1$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9702,2,Mod(1,9702)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9702, 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("9702.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9702 = 2 \cdot 3^{2} \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9702.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: $$77.4708600410$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1078) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 9702.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{4} -4.24264 q^{5} -1.00000 q^{8} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{4} -4.24264 q^{5} -1.00000 q^{8} +4.24264 q^{10} +1.00000 q^{11} +1.00000 q^{16} +5.65685 q^{17} -4.24264 q^{20} -1.00000 q^{22} -6.00000 q^{23} +13.0000 q^{25} -2.00000 q^{29} -1.41421 q^{31} -1.00000 q^{32} -5.65685 q^{34} -10.0000 q^{37} +4.24264 q^{40} +11.3137 q^{41} -8.00000 q^{43} +1.00000 q^{44} +6.00000 q^{46} -4.24264 q^{47} -13.0000 q^{50} -8.00000 q^{53} -4.24264 q^{55} +2.00000 q^{58} -1.41421 q^{59} -2.82843 q^{61} +1.41421 q^{62} +1.00000 q^{64} +2.00000 q^{67} +5.65685 q^{68} +2.00000 q^{71} +8.48528 q^{73} +10.0000 q^{74} +16.0000 q^{79} -4.24264 q^{80} -11.3137 q^{82} +16.9706 q^{83} -24.0000 q^{85} +8.00000 q^{86} -1.00000 q^{88} +7.07107 q^{89} -6.00000 q^{92} +4.24264 q^{94} -9.89949 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{4} - 2 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 $$2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} + 2 q^{11} + 2 q^{16} - 2 q^{22} - 12 q^{23} + 26 q^{25} - 4 q^{29} - 2 q^{32} - 20 q^{37} - 16 q^{43} + 2 q^{44} + 12 q^{46} - 26 q^{50} - 16 q^{53} + 4 q^{58} + 2 q^{64} + 4 q^{67} + 4 q^{71} + 20 q^{74} + 32 q^{79} - 48 q^{85} + 16 q^{86} - 2 q^{88} - 12 q^{92}+O(q^{100})$$ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 + 2 * q^11 + 2 * q^16 - 2 * q^22 - 12 * q^23 + 26 * q^25 - 4 * q^29 - 2 * q^32 - 20 * q^37 - 16 * q^43 + 2 * q^44 + 12 * q^46 - 26 * q^50 - 16 * q^53 + 4 * q^58 + 2 * q^64 + 4 * q^67 + 4 * q^71 + 20 * q^74 + 32 * q^79 - 48 * q^85 + 16 * q^86 - 2 * q^88 - 12 * q^92

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ 0 0
$$4$$ 1.00000 0.500000
$$5$$ −4.24264 −1.89737 −0.948683 0.316228i $$-0.897584\pi$$
−0.948683 + 0.316228i $$0.897584\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ −1.00000 −0.353553
$$9$$ 0 0
$$10$$ 4.24264 1.34164
$$11$$ 1.00000 0.301511
$$12$$ 0 0
$$13$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 5.65685 1.37199 0.685994 0.727607i $$-0.259367\pi$$
0.685994 + 0.727607i $$0.259367\pi$$
$$18$$ 0 0
$$19$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$20$$ −4.24264 −0.948683
$$21$$ 0 0
$$22$$ −1.00000 −0.213201
$$23$$ −6.00000 −1.25109 −0.625543 0.780189i $$-0.715123\pi$$
−0.625543 + 0.780189i $$0.715123\pi$$
$$24$$ 0 0
$$25$$ 13.0000 2.60000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −2.00000 −0.371391 −0.185695 0.982607i $$-0.559454\pi$$
−0.185695 + 0.982607i $$0.559454\pi$$
$$30$$ 0 0
$$31$$ −1.41421 −0.254000 −0.127000 0.991903i $$-0.540535\pi$$
−0.127000 + 0.991903i $$0.540535\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ −5.65685 −0.970143
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −10.0000 −1.64399 −0.821995 0.569495i $$-0.807139\pi$$
−0.821995 + 0.569495i $$0.807139\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 4.24264 0.670820
$$41$$ 11.3137 1.76690 0.883452 0.468521i $$-0.155213\pi$$
0.883452 + 0.468521i $$0.155213\pi$$
$$42$$ 0 0
$$43$$ −8.00000 −1.21999 −0.609994 0.792406i $$-0.708828\pi$$
−0.609994 + 0.792406i $$0.708828\pi$$
$$44$$ 1.00000 0.150756
$$45$$ 0 0
$$46$$ 6.00000 0.884652
$$47$$ −4.24264 −0.618853 −0.309426 0.950923i $$-0.600137\pi$$
−0.309426 + 0.950923i $$0.600137\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ −13.0000 −1.83848
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −8.00000 −1.09888 −0.549442 0.835532i $$-0.685160\pi$$
−0.549442 + 0.835532i $$0.685160\pi$$
$$54$$ 0 0
$$55$$ −4.24264 −0.572078
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 2.00000 0.262613
$$59$$ −1.41421 −0.184115 −0.0920575 0.995754i $$-0.529344\pi$$
−0.0920575 + 0.995754i $$0.529344\pi$$
$$60$$ 0 0
$$61$$ −2.82843 −0.362143 −0.181071 0.983470i $$-0.557957\pi$$
−0.181071 + 0.983470i $$0.557957\pi$$
$$62$$ 1.41421 0.179605
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 2.00000 0.244339 0.122169 0.992509i $$-0.461015\pi$$
0.122169 + 0.992509i $$0.461015\pi$$
$$68$$ 5.65685 0.685994
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 2.00000 0.237356 0.118678 0.992933i $$-0.462134\pi$$
0.118678 + 0.992933i $$0.462134\pi$$
$$72$$ 0 0
$$73$$ 8.48528 0.993127 0.496564 0.868000i $$-0.334595\pi$$
0.496564 + 0.868000i $$0.334595\pi$$
$$74$$ 10.0000 1.16248
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 16.0000 1.80014 0.900070 0.435745i $$-0.143515\pi$$
0.900070 + 0.435745i $$0.143515\pi$$
$$80$$ −4.24264 −0.474342
$$81$$ 0 0
$$82$$ −11.3137 −1.24939
$$83$$ 16.9706 1.86276 0.931381 0.364047i $$-0.118605\pi$$
0.931381 + 0.364047i $$0.118605\pi$$
$$84$$ 0 0
$$85$$ −24.0000 −2.60317
$$86$$ 8.00000 0.862662
$$87$$ 0 0
$$88$$ −1.00000 −0.106600
$$89$$ 7.07107 0.749532 0.374766 0.927119i $$-0.377723\pi$$
0.374766 + 0.927119i $$0.377723\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −6.00000 −0.625543
$$93$$ 0 0
$$94$$ 4.24264 0.437595
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −9.89949 −1.00514 −0.502571 0.864536i $$-0.667612\pi$$
−0.502571 + 0.864536i $$0.667612\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 13.0000 1.30000
$$101$$ −5.65685 −0.562878 −0.281439 0.959579i $$-0.590812\pi$$
−0.281439 + 0.959579i $$0.590812\pi$$
$$102$$ 0 0
$$103$$ 18.3848 1.81151 0.905753 0.423806i $$-0.139306\pi$$
0.905753 + 0.423806i $$0.139306\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 8.00000 0.777029
$$107$$ −16.0000 −1.54678 −0.773389 0.633932i $$-0.781440\pi$$
−0.773389 + 0.633932i $$0.781440\pi$$
$$108$$ 0 0
$$109$$ −2.00000 −0.191565 −0.0957826 0.995402i $$-0.530535\pi$$
−0.0957826 + 0.995402i $$0.530535\pi$$
$$110$$ 4.24264 0.404520
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 2.00000 0.188144 0.0940721 0.995565i $$-0.470012\pi$$
0.0940721 + 0.995565i $$0.470012\pi$$
$$114$$ 0 0
$$115$$ 25.4558 2.37377
$$116$$ −2.00000 −0.185695
$$117$$ 0 0
$$118$$ 1.41421 0.130189
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 2.82843 0.256074
$$123$$ 0 0
$$124$$ −1.41421 −0.127000
$$125$$ −33.9411 −3.03579
$$126$$ 0 0
$$127$$ 16.0000 1.41977 0.709885 0.704317i $$-0.248747\pi$$
0.709885 + 0.704317i $$0.248747\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 19.7990 1.72985 0.864923 0.501905i $$-0.167367\pi$$
0.864923 + 0.501905i $$0.167367\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ −2.00000 −0.172774
$$135$$ 0 0
$$136$$ −5.65685 −0.485071
$$137$$ 18.0000 1.53784 0.768922 0.639343i $$-0.220793\pi$$
0.768922 + 0.639343i $$0.220793\pi$$
$$138$$ 0 0
$$139$$ 11.3137 0.959616 0.479808 0.877373i $$-0.340706\pi$$
0.479808 + 0.877373i $$0.340706\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −2.00000 −0.167836
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 8.48528 0.704664
$$146$$ −8.48528 −0.702247
$$147$$ 0 0
$$148$$ −10.0000 −0.821995
$$149$$ 10.0000 0.819232 0.409616 0.912258i $$-0.365663\pi$$
0.409616 + 0.912258i $$0.365663\pi$$
$$150$$ 0 0
$$151$$ 4.00000 0.325515 0.162758 0.986666i $$-0.447961\pi$$
0.162758 + 0.986666i $$0.447961\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 6.00000 0.481932
$$156$$ 0 0
$$157$$ 4.24264 0.338600 0.169300 0.985565i $$-0.445849\pi$$
0.169300 + 0.985565i $$0.445849\pi$$
$$158$$ −16.0000 −1.27289
$$159$$ 0 0
$$160$$ 4.24264 0.335410
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −10.0000 −0.783260 −0.391630 0.920123i $$-0.628089\pi$$
−0.391630 + 0.920123i $$0.628089\pi$$
$$164$$ 11.3137 0.883452
$$165$$ 0 0
$$166$$ −16.9706 −1.31717
$$167$$ −5.65685 −0.437741 −0.218870 0.975754i $$-0.570237\pi$$
−0.218870 + 0.975754i $$0.570237\pi$$
$$168$$ 0 0
$$169$$ −13.0000 −1.00000
$$170$$ 24.0000 1.84072
$$171$$ 0 0
$$172$$ −8.00000 −0.609994
$$173$$ 11.3137 0.860165 0.430083 0.902790i $$-0.358484\pi$$
0.430083 + 0.902790i $$0.358484\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 1.00000 0.0753778
$$177$$ 0 0
$$178$$ −7.07107 −0.529999
$$179$$ −12.0000 −0.896922 −0.448461 0.893802i $$-0.648028\pi$$
−0.448461 + 0.893802i $$0.648028\pi$$
$$180$$ 0 0
$$181$$ −7.07107 −0.525588 −0.262794 0.964852i $$-0.584644\pi$$
−0.262794 + 0.964852i $$0.584644\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 6.00000 0.442326
$$185$$ 42.4264 3.11925
$$186$$ 0 0
$$187$$ 5.65685 0.413670
$$188$$ −4.24264 −0.309426
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −16.0000 −1.15772 −0.578860 0.815427i $$-0.696502\pi$$
−0.578860 + 0.815427i $$0.696502\pi$$
$$192$$ 0 0
$$193$$ −6.00000 −0.431889 −0.215945 0.976406i $$-0.569283\pi$$
−0.215945 + 0.976406i $$0.569283\pi$$
$$194$$ 9.89949 0.710742
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −22.0000 −1.56744 −0.783718 0.621117i $$-0.786679\pi$$
−0.783718 + 0.621117i $$0.786679\pi$$
$$198$$ 0 0
$$199$$ −1.41421 −0.100251 −0.0501255 0.998743i $$-0.515962\pi$$
−0.0501255 + 0.998743i $$0.515962\pi$$
$$200$$ −13.0000 −0.919239
$$201$$ 0 0
$$202$$ 5.65685 0.398015
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −48.0000 −3.35247
$$206$$ −18.3848 −1.28093
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 8.00000 0.550743 0.275371 0.961338i $$-0.411199\pi$$
0.275371 + 0.961338i $$0.411199\pi$$
$$212$$ −8.00000 −0.549442
$$213$$ 0 0
$$214$$ 16.0000 1.09374
$$215$$ 33.9411 2.31477
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 2.00000 0.135457
$$219$$ 0 0
$$220$$ −4.24264 −0.286039
$$221$$ 0 0
$$222$$ 0 0
$$223$$ −21.2132 −1.42054 −0.710271 0.703929i $$-0.751427\pi$$
−0.710271 + 0.703929i $$0.751427\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −2.00000 −0.133038
$$227$$ 14.1421 0.938647 0.469323 0.883026i $$-0.344498\pi$$
0.469323 + 0.883026i $$0.344498\pi$$
$$228$$ 0 0
$$229$$ 9.89949 0.654177 0.327089 0.944994i $$-0.393932\pi$$
0.327089 + 0.944994i $$0.393932\pi$$
$$230$$ −25.4558 −1.67851
$$231$$ 0 0
$$232$$ 2.00000 0.131306
$$233$$ −14.0000 −0.917170 −0.458585 0.888650i $$-0.651644\pi$$
−0.458585 + 0.888650i $$0.651644\pi$$
$$234$$ 0 0
$$235$$ 18.0000 1.17419
$$236$$ −1.41421 −0.0920575
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 12.0000 0.776215 0.388108 0.921614i $$-0.373129\pi$$
0.388108 + 0.921614i $$0.373129\pi$$
$$240$$ 0 0
$$241$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$242$$ −1.00000 −0.0642824
$$243$$ 0 0
$$244$$ −2.82843 −0.181071
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 1.41421 0.0898027
$$249$$ 0 0
$$250$$ 33.9411 2.14663
$$251$$ −18.3848 −1.16044 −0.580218 0.814461i $$-0.697033\pi$$
−0.580218 + 0.814461i $$0.697033\pi$$
$$252$$ 0 0
$$253$$ −6.00000 −0.377217
$$254$$ −16.0000 −1.00393
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −1.41421 −0.0882162 −0.0441081 0.999027i $$-0.514045\pi$$
−0.0441081 + 0.999027i $$0.514045\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 0 0
$$262$$ −19.7990 −1.22319
$$263$$ 8.00000 0.493301 0.246651 0.969104i $$-0.420670\pi$$
0.246651 + 0.969104i $$0.420670\pi$$
$$264$$ 0 0
$$265$$ 33.9411 2.08499
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 2.00000 0.122169
$$269$$ −18.3848 −1.12094 −0.560470 0.828175i $$-0.689379\pi$$
−0.560470 + 0.828175i $$0.689379\pi$$
$$270$$ 0 0
$$271$$ −8.48528 −0.515444 −0.257722 0.966219i $$-0.582972\pi$$
−0.257722 + 0.966219i $$0.582972\pi$$
$$272$$ 5.65685 0.342997
$$273$$ 0 0
$$274$$ −18.0000 −1.08742
$$275$$ 13.0000 0.783929
$$276$$ 0 0
$$277$$ −2.00000 −0.120168 −0.0600842 0.998193i $$-0.519137\pi$$
−0.0600842 + 0.998193i $$0.519137\pi$$
$$278$$ −11.3137 −0.678551
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 14.0000 0.835170 0.417585 0.908638i $$-0.362877\pi$$
0.417585 + 0.908638i $$0.362877\pi$$
$$282$$ 0 0
$$283$$ −19.7990 −1.17693 −0.588464 0.808523i $$-0.700267\pi$$
−0.588464 + 0.808523i $$0.700267\pi$$
$$284$$ 2.00000 0.118678
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 15.0000 0.882353
$$290$$ −8.48528 −0.498273
$$291$$ 0 0
$$292$$ 8.48528 0.496564
$$293$$ 8.48528 0.495715 0.247858 0.968796i $$-0.420273\pi$$
0.247858 + 0.968796i $$0.420273\pi$$
$$294$$ 0 0
$$295$$ 6.00000 0.349334
$$296$$ 10.0000 0.581238
$$297$$ 0 0
$$298$$ −10.0000 −0.579284
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ −4.00000 −0.230174
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 12.0000 0.687118
$$306$$ 0 0
$$307$$ −25.4558 −1.45284 −0.726421 0.687250i $$-0.758818\pi$$
−0.726421 + 0.687250i $$0.758818\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ −6.00000 −0.340777
$$311$$ −18.3848 −1.04251 −0.521253 0.853402i $$-0.674535\pi$$
−0.521253 + 0.853402i $$0.674535\pi$$
$$312$$ 0 0
$$313$$ −12.7279 −0.719425 −0.359712 0.933063i $$-0.617125\pi$$
−0.359712 + 0.933063i $$0.617125\pi$$
$$314$$ −4.24264 −0.239426
$$315$$ 0 0
$$316$$ 16.0000 0.900070
$$317$$ −30.0000 −1.68497 −0.842484 0.538721i $$-0.818908\pi$$
−0.842484 + 0.538721i $$0.818908\pi$$
$$318$$ 0 0
$$319$$ −2.00000 −0.111979
$$320$$ −4.24264 −0.237171
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 10.0000 0.553849
$$327$$ 0 0
$$328$$ −11.3137 −0.624695
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 20.0000 1.09930 0.549650 0.835395i $$-0.314761\pi$$
0.549650 + 0.835395i $$0.314761\pi$$
$$332$$ 16.9706 0.931381
$$333$$ 0 0
$$334$$ 5.65685 0.309529
$$335$$ −8.48528 −0.463600
$$336$$ 0 0
$$337$$ 2.00000 0.108947 0.0544735 0.998515i $$-0.482652\pi$$
0.0544735 + 0.998515i $$0.482652\pi$$
$$338$$ 13.0000 0.707107
$$339$$ 0 0
$$340$$ −24.0000 −1.30158
$$341$$ −1.41421 −0.0765840
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 8.00000 0.431331
$$345$$ 0 0
$$346$$ −11.3137 −0.608229
$$347$$ −20.0000 −1.07366 −0.536828 0.843692i $$-0.680378\pi$$
−0.536828 + 0.843692i $$0.680378\pi$$
$$348$$ 0 0
$$349$$ 14.1421 0.757011 0.378506 0.925599i $$-0.376438\pi$$
0.378506 + 0.925599i $$0.376438\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −1.00000 −0.0533002
$$353$$ −1.41421 −0.0752710 −0.0376355 0.999292i $$-0.511983\pi$$
−0.0376355 + 0.999292i $$0.511983\pi$$
$$354$$ 0 0
$$355$$ −8.48528 −0.450352
$$356$$ 7.07107 0.374766
$$357$$ 0 0
$$358$$ 12.0000 0.634220
$$359$$ 12.0000 0.633336 0.316668 0.948536i $$-0.397436\pi$$
0.316668 + 0.948536i $$0.397436\pi$$
$$360$$ 0 0
$$361$$ −19.0000 −1.00000
$$362$$ 7.07107 0.371647
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −36.0000 −1.88433
$$366$$ 0 0
$$367$$ −21.2132 −1.10732 −0.553660 0.832743i $$-0.686769\pi$$
−0.553660 + 0.832743i $$0.686769\pi$$
$$368$$ −6.00000 −0.312772
$$369$$ 0 0
$$370$$ −42.4264 −2.20564
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −10.0000 −0.517780 −0.258890 0.965907i $$-0.583357\pi$$
−0.258890 + 0.965907i $$0.583357\pi$$
$$374$$ −5.65685 −0.292509
$$375$$ 0 0
$$376$$ 4.24264 0.218797
$$377$$ 0 0
$$378$$ 0 0
$$379$$ −6.00000 −0.308199 −0.154100 0.988055i $$-0.549248\pi$$
−0.154100 + 0.988055i $$0.549248\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 16.0000 0.818631
$$383$$ −15.5563 −0.794892 −0.397446 0.917625i $$-0.630103\pi$$
−0.397446 + 0.917625i $$0.630103\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 6.00000 0.305392
$$387$$ 0 0
$$388$$ −9.89949 −0.502571
$$389$$ 24.0000 1.21685 0.608424 0.793612i $$-0.291802\pi$$
0.608424 + 0.793612i $$0.291802\pi$$
$$390$$ 0 0
$$391$$ −33.9411 −1.71648
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 22.0000 1.10834
$$395$$ −67.8823 −3.41553
$$396$$ 0 0
$$397$$ 12.7279 0.638796 0.319398 0.947621i $$-0.396519\pi$$
0.319398 + 0.947621i $$0.396519\pi$$
$$398$$ 1.41421 0.0708881
$$399$$ 0 0
$$400$$ 13.0000 0.650000
$$401$$ −12.0000 −0.599251 −0.299626 0.954057i $$-0.596862\pi$$
−0.299626 + 0.954057i $$0.596862\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ −5.65685 −0.281439
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −10.0000 −0.495682
$$408$$ 0 0
$$409$$ −2.82843 −0.139857 −0.0699284 0.997552i $$-0.522277\pi$$
−0.0699284 + 0.997552i $$0.522277\pi$$
$$410$$ 48.0000 2.37055
$$411$$ 0 0
$$412$$ 18.3848 0.905753
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −72.0000 −3.53434
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −24.0416 −1.17451 −0.587255 0.809402i $$-0.699792\pi$$
−0.587255 + 0.809402i $$0.699792\pi$$
$$420$$ 0 0
$$421$$ −20.0000 −0.974740 −0.487370 0.873195i $$-0.662044\pi$$
−0.487370 + 0.873195i $$0.662044\pi$$
$$422$$ −8.00000 −0.389434
$$423$$ 0 0
$$424$$ 8.00000 0.388514
$$425$$ 73.5391 3.56717
$$426$$ 0 0
$$427$$ 0 0
$$428$$ −16.0000 −0.773389
$$429$$ 0 0
$$430$$ −33.9411 −1.63679
$$431$$ −32.0000 −1.54139 −0.770693 0.637207i $$-0.780090\pi$$
−0.770693 + 0.637207i $$0.780090\pi$$
$$432$$ 0 0
$$433$$ −12.7279 −0.611665 −0.305832 0.952085i $$-0.598935\pi$$
−0.305832 + 0.952085i $$0.598935\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −2.00000 −0.0957826
$$437$$ 0 0
$$438$$ 0 0
$$439$$ −25.4558 −1.21494 −0.607471 0.794342i $$-0.707816\pi$$
−0.607471 + 0.794342i $$0.707816\pi$$
$$440$$ 4.24264 0.202260
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −36.0000 −1.71041 −0.855206 0.518289i $$-0.826569\pi$$
−0.855206 + 0.518289i $$0.826569\pi$$
$$444$$ 0 0
$$445$$ −30.0000 −1.42214
$$446$$ 21.2132 1.00447
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 20.0000 0.943858 0.471929 0.881636i $$-0.343558\pi$$
0.471929 + 0.881636i $$0.343558\pi$$
$$450$$ 0 0
$$451$$ 11.3137 0.532742
$$452$$ 2.00000 0.0940721
$$453$$ 0 0
$$454$$ −14.1421 −0.663723
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 14.0000 0.654892 0.327446 0.944870i $$-0.393812\pi$$
0.327446 + 0.944870i $$0.393812\pi$$
$$458$$ −9.89949 −0.462573
$$459$$ 0 0
$$460$$ 25.4558 1.18688
$$461$$ −2.82843 −0.131733 −0.0658665 0.997828i $$-0.520981\pi$$
−0.0658665 + 0.997828i $$0.520981\pi$$
$$462$$ 0 0
$$463$$ 26.0000 1.20832 0.604161 0.796862i $$-0.293508\pi$$
0.604161 + 0.796862i $$0.293508\pi$$
$$464$$ −2.00000 −0.0928477
$$465$$ 0 0
$$466$$ 14.0000 0.648537
$$467$$ 4.24264 0.196326 0.0981630 0.995170i $$-0.468703\pi$$
0.0981630 + 0.995170i $$0.468703\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ −18.0000 −0.830278
$$471$$ 0 0
$$472$$ 1.41421 0.0650945
$$473$$ −8.00000 −0.367840
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 0 0
$$478$$ −12.0000 −0.548867
$$479$$ −2.82843 −0.129234 −0.0646171 0.997910i $$-0.520583\pi$$
−0.0646171 + 0.997910i $$0.520583\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 1.00000 0.0454545
$$485$$ 42.0000 1.90712
$$486$$ 0 0
$$487$$ 2.00000 0.0906287 0.0453143 0.998973i $$-0.485571\pi$$
0.0453143 + 0.998973i $$0.485571\pi$$
$$488$$ 2.82843 0.128037
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 12.0000 0.541552 0.270776 0.962642i $$-0.412720\pi$$
0.270776 + 0.962642i $$0.412720\pi$$
$$492$$ 0 0
$$493$$ −11.3137 −0.509544
$$494$$ 0 0
$$495$$ 0 0
$$496$$ −1.41421 −0.0635001
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 6.00000 0.268597 0.134298 0.990941i $$-0.457122\pi$$
0.134298 + 0.990941i $$0.457122\pi$$
$$500$$ −33.9411 −1.51789
$$501$$ 0 0
$$502$$ 18.3848 0.820553
$$503$$ −31.1127 −1.38725 −0.693623 0.720338i $$-0.743987\pi$$
−0.693623 + 0.720338i $$0.743987\pi$$
$$504$$ 0 0
$$505$$ 24.0000 1.06799
$$506$$ 6.00000 0.266733
$$507$$ 0 0
$$508$$ 16.0000 0.709885
$$509$$ −12.7279 −0.564155 −0.282078 0.959392i $$-0.591024\pi$$
−0.282078 + 0.959392i $$0.591024\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −1.00000 −0.0441942
$$513$$ 0 0
$$514$$ 1.41421 0.0623783
$$515$$ −78.0000 −3.43709
$$516$$ 0 0
$$517$$ −4.24264 −0.186591
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −1.41421 −0.0619578 −0.0309789 0.999520i $$-0.509862\pi$$
−0.0309789 + 0.999520i $$0.509862\pi$$
$$522$$ 0 0
$$523$$ 8.48528 0.371035 0.185518 0.982641i $$-0.440604\pi$$
0.185518 + 0.982641i $$0.440604\pi$$
$$524$$ 19.7990 0.864923
$$525$$ 0 0
$$526$$ −8.00000 −0.348817
$$527$$ −8.00000 −0.348485
$$528$$ 0 0
$$529$$ 13.0000 0.565217
$$530$$ −33.9411 −1.47431
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 67.8823 2.93481
$$536$$ −2.00000 −0.0863868
$$537$$ 0 0
$$538$$ 18.3848 0.792624
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −30.0000 −1.28980 −0.644900 0.764267i $$-0.723101\pi$$
−0.644900 + 0.764267i $$0.723101\pi$$
$$542$$ 8.48528 0.364474
$$543$$ 0 0
$$544$$ −5.65685 −0.242536
$$545$$ 8.48528 0.363470
$$546$$ 0 0
$$547$$ 44.0000 1.88130 0.940652 0.339372i $$-0.110215\pi$$
0.940652 + 0.339372i $$0.110215\pi$$
$$548$$ 18.0000 0.768922
$$549$$ 0 0
$$550$$ −13.0000 −0.554322
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 2.00000 0.0849719
$$555$$ 0 0
$$556$$ 11.3137 0.479808
$$557$$ 30.0000 1.27114 0.635570 0.772043i $$-0.280765\pi$$
0.635570 + 0.772043i $$0.280765\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −14.0000 −0.590554
$$563$$ −22.6274 −0.953632 −0.476816 0.879003i $$-0.658209\pi$$
−0.476816 + 0.879003i $$0.658209\pi$$
$$564$$ 0 0
$$565$$ −8.48528 −0.356978
$$566$$ 19.7990 0.832214
$$567$$ 0 0
$$568$$ −2.00000 −0.0839181
$$569$$ −30.0000 −1.25767 −0.628833 0.777541i $$-0.716467\pi$$
−0.628833 + 0.777541i $$0.716467\pi$$
$$570$$ 0 0
$$571$$ 8.00000 0.334790 0.167395 0.985890i $$-0.446465\pi$$
0.167395 + 0.985890i $$0.446465\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −78.0000 −3.25282
$$576$$ 0 0
$$577$$ −7.07107 −0.294372 −0.147186 0.989109i $$-0.547022\pi$$
−0.147186 + 0.989109i $$0.547022\pi$$
$$578$$ −15.0000 −0.623918
$$579$$ 0 0
$$580$$ 8.48528 0.352332
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −8.00000 −0.331326
$$584$$ −8.48528 −0.351123
$$585$$ 0 0
$$586$$ −8.48528 −0.350524
$$587$$ 26.8701 1.10905 0.554523 0.832168i $$-0.312901\pi$$
0.554523 + 0.832168i $$0.312901\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ −6.00000 −0.247016
$$591$$ 0 0
$$592$$ −10.0000 −0.410997
$$593$$ −42.4264 −1.74224 −0.871122 0.491067i $$-0.836607\pi$$
−0.871122 + 0.491067i $$0.836607\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 10.0000 0.409616
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 6.00000 0.245153 0.122577 0.992459i $$-0.460884\pi$$
0.122577 + 0.992459i $$0.460884\pi$$
$$600$$ 0 0
$$601$$ 5.65685 0.230748 0.115374 0.993322i $$-0.463193\pi$$
0.115374 + 0.993322i $$0.463193\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 4.00000 0.162758
$$605$$ −4.24264 −0.172488
$$606$$ 0 0
$$607$$ 16.9706 0.688814 0.344407 0.938820i $$-0.388080\pi$$
0.344407 + 0.938820i $$0.388080\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ −12.0000 −0.485866
$$611$$ 0 0
$$612$$ 0 0
$$613$$ −10.0000 −0.403896 −0.201948 0.979396i $$-0.564727\pi$$
−0.201948 + 0.979396i $$0.564727\pi$$
$$614$$ 25.4558 1.02731
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −34.0000 −1.36879 −0.684394 0.729112i $$-0.739933\pi$$
−0.684394 + 0.729112i $$0.739933\pi$$
$$618$$ 0 0
$$619$$ 9.89949 0.397894 0.198947 0.980010i $$-0.436248\pi$$
0.198947 + 0.980010i $$0.436248\pi$$
$$620$$ 6.00000 0.240966
$$621$$ 0 0
$$622$$ 18.3848 0.737162
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 79.0000 3.16000
$$626$$ 12.7279 0.508710
$$627$$ 0 0
$$628$$ 4.24264 0.169300
$$629$$ −56.5685 −2.25554
$$630$$ 0 0
$$631$$ 24.0000 0.955425 0.477712 0.878516i $$-0.341466\pi$$
0.477712 + 0.878516i $$0.341466\pi$$
$$632$$ −16.0000 −0.636446
$$633$$ 0 0
$$634$$ 30.0000 1.19145
$$635$$ −67.8823 −2.69382
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 2.00000 0.0791808
$$639$$ 0 0
$$640$$ 4.24264 0.167705
$$641$$ 34.0000 1.34292 0.671460 0.741041i $$-0.265668\pi$$
0.671460 + 0.741041i $$0.265668\pi$$
$$642$$ 0 0
$$643$$ −38.1838 −1.50582 −0.752910 0.658123i $$-0.771351\pi$$
−0.752910 + 0.658123i $$0.771351\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 15.5563 0.611583 0.305792 0.952098i $$-0.401079\pi$$
0.305792 + 0.952098i $$0.401079\pi$$
$$648$$ 0 0
$$649$$ −1.41421 −0.0555127
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −10.0000 −0.391630
$$653$$ −12.0000 −0.469596 −0.234798 0.972044i $$-0.575443\pi$$
−0.234798 + 0.972044i $$0.575443\pi$$
$$654$$ 0 0
$$655$$ −84.0000 −3.28215
$$656$$ 11.3137 0.441726
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$660$$ 0 0
$$661$$ 12.7279 0.495059 0.247529 0.968880i $$-0.420381\pi$$
0.247529 + 0.968880i $$0.420381\pi$$
$$662$$ −20.0000 −0.777322
$$663$$ 0 0
$$664$$ −16.9706 −0.658586
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 12.0000 0.464642
$$668$$ −5.65685 −0.218870
$$669$$ 0 0
$$670$$ 8.48528 0.327815
$$671$$ −2.82843 −0.109190
$$672$$ 0 0
$$673$$ −22.0000 −0.848038 −0.424019 0.905653i $$-0.639381\pi$$
−0.424019 + 0.905653i $$0.639381\pi$$
$$674$$ −2.00000 −0.0770371
$$675$$ 0 0
$$676$$ −13.0000 −0.500000
$$677$$ 39.5980 1.52187 0.760937 0.648826i $$-0.224740\pi$$
0.760937 + 0.648826i $$0.224740\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 24.0000 0.920358
$$681$$ 0 0
$$682$$ 1.41421 0.0541530
$$683$$ 28.0000 1.07139 0.535695 0.844411i $$-0.320050\pi$$
0.535695 + 0.844411i $$0.320050\pi$$
$$684$$ 0 0
$$685$$ −76.3675 −2.91785
$$686$$ 0 0
$$687$$ 0 0
$$688$$ −8.00000 −0.304997
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 15.5563 0.591791 0.295896 0.955220i $$-0.404382\pi$$
0.295896 + 0.955220i $$0.404382\pi$$
$$692$$ 11.3137 0.430083
$$693$$ 0 0
$$694$$ 20.0000 0.759190
$$695$$ −48.0000 −1.82074
$$696$$ 0 0
$$697$$ 64.0000 2.42417
$$698$$ −14.1421 −0.535288
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −50.0000 −1.88847 −0.944237 0.329267i $$-0.893198\pi$$
−0.944237 + 0.329267i $$0.893198\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 1.00000 0.0376889
$$705$$ 0 0
$$706$$ 1.41421 0.0532246
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −20.0000 −0.751116 −0.375558 0.926799i $$-0.622549\pi$$
−0.375558 + 0.926799i $$0.622549\pi$$
$$710$$ 8.48528 0.318447
$$711$$ 0 0
$$712$$ −7.07107 −0.264999
$$713$$ 8.48528 0.317776
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −12.0000 −0.448461
$$717$$ 0 0
$$718$$ −12.0000 −0.447836
$$719$$ −18.3848 −0.685636 −0.342818 0.939402i $$-0.611381\pi$$
−0.342818 + 0.939402i $$0.611381\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 19.0000 0.707107
$$723$$ 0 0
$$724$$ −7.07107 −0.262794
$$725$$ −26.0000 −0.965616
$$726$$ 0 0
$$727$$ 12.7279 0.472052 0.236026 0.971747i $$-0.424155\pi$$
0.236026 + 0.971747i $$0.424155\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 36.0000 1.33242
$$731$$ −45.2548 −1.67381
$$732$$ 0 0
$$733$$ 14.1421 0.522352 0.261176 0.965291i $$-0.415890\pi$$
0.261176 + 0.965291i $$0.415890\pi$$
$$734$$ 21.2132 0.782994
$$735$$ 0 0
$$736$$ 6.00000 0.221163
$$737$$ 2.00000 0.0736709
$$738$$ 0 0
$$739$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$740$$ 42.4264 1.55963
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 44.0000 1.61420 0.807102 0.590412i $$-0.201035\pi$$
0.807102 + 0.590412i $$0.201035\pi$$
$$744$$ 0 0
$$745$$ −42.4264 −1.55438
$$746$$ 10.0000 0.366126
$$747$$ 0 0
$$748$$ 5.65685 0.206835
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −14.0000 −0.510867 −0.255434 0.966827i $$-0.582218\pi$$
−0.255434 + 0.966827i $$0.582218\pi$$
$$752$$ −4.24264 −0.154713
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −16.9706 −0.617622
$$756$$ 0 0
$$757$$ −8.00000 −0.290765 −0.145382 0.989376i $$-0.546441\pi$$
−0.145382 + 0.989376i $$0.546441\pi$$
$$758$$ 6.00000 0.217930
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 42.4264 1.53796 0.768978 0.639275i $$-0.220766\pi$$
0.768978 + 0.639275i $$0.220766\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ −16.0000 −0.578860
$$765$$ 0 0
$$766$$ 15.5563 0.562074
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 19.7990 0.713970 0.356985 0.934110i $$-0.383805\pi$$
0.356985 + 0.934110i $$0.383805\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −6.00000 −0.215945
$$773$$ −1.41421 −0.0508657 −0.0254329 0.999677i $$-0.508096\pi$$
−0.0254329 + 0.999677i $$0.508096\pi$$
$$774$$ 0 0
$$775$$ −18.3848 −0.660401
$$776$$ 9.89949 0.355371
$$777$$ 0 0
$$778$$ −24.0000 −0.860442
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 2.00000 0.0715656
$$782$$ 33.9411 1.21373
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −18.0000 −0.642448
$$786$$ 0 0
$$787$$ −5.65685 −0.201645 −0.100823 0.994904i $$-0.532147\pi$$
−0.100823 + 0.994904i $$0.532147\pi$$
$$788$$ −22.0000 −0.783718
$$789$$ 0 0
$$790$$ 67.8823 2.41514
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 0 0
$$794$$ −12.7279 −0.451697
$$795$$ 0 0
$$796$$ −1.41421 −0.0501255
$$797$$ −7.07107 −0.250470 −0.125235 0.992127i $$-0.539968\pi$$
−0.125235 + 0.992127i $$0.539968\pi$$
$$798$$ 0 0
$$799$$ −24.0000 −0.849059
$$800$$ −13.0000 −0.459619
$$801$$ 0 0
$$802$$ 12.0000 0.423735
$$803$$ 8.48528 0.299439
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 5.65685 0.199007
$$809$$ −54.0000 −1.89854 −0.949269 0.314464i $$-0.898175\pi$$
−0.949269 + 0.314464i $$0.898175\pi$$
$$810$$ 0 0
$$811$$ 36.7696 1.29115 0.645577 0.763695i $$-0.276617\pi$$
0.645577 + 0.763695i $$0.276617\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 10.0000 0.350500
$$815$$ 42.4264 1.48613
$$816$$ 0 0
$$817$$ 0 0
$$818$$ 2.82843 0.0988936
$$819$$ 0 0
$$820$$ −48.0000 −1.67623
$$821$$ 38.0000 1.32621 0.663105 0.748527i $$-0.269238\pi$$
0.663105 + 0.748527i $$0.269238\pi$$
$$822$$ 0 0
$$823$$ 40.0000 1.39431 0.697156 0.716919i $$-0.254448\pi$$
0.697156 + 0.716919i $$0.254448\pi$$
$$824$$ −18.3848 −0.640464
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 32.0000 1.11275 0.556375 0.830932i $$-0.312192\pi$$
0.556375 + 0.830932i $$0.312192\pi$$
$$828$$ 0 0
$$829$$ −4.24264 −0.147353 −0.0736765 0.997282i $$-0.523473\pi$$
−0.0736765 + 0.997282i $$0.523473\pi$$
$$830$$ 72.0000 2.49916
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 24.0000 0.830554
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 24.0416 0.830504
$$839$$ −1.41421 −0.0488241 −0.0244120 0.999702i $$-0.507771\pi$$
−0.0244120 + 0.999702i $$0.507771\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ 20.0000 0.689246
$$843$$ 0 0
$$844$$ 8.00000 0.275371
$$845$$ 55.1543 1.89737
$$846$$ 0 0
$$847$$ 0 0
$$848$$ −8.00000 −0.274721
$$849$$ 0 0
$$850$$ −73.5391 −2.52237
$$851$$ 60.0000 2.05677
$$852$$ 0 0
$$853$$ 28.2843 0.968435 0.484218 0.874948i $$-0.339104\pi$$
0.484218 + 0.874948i $$0.339104\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 16.0000 0.546869
$$857$$ 25.4558 0.869555 0.434778 0.900538i $$-0.356827\pi$$
0.434778 + 0.900538i $$0.356827\pi$$
$$858$$ 0 0
$$859$$ 12.7279 0.434271 0.217136 0.976141i $$-0.430329\pi$$
0.217136 + 0.976141i $$0.430329\pi$$
$$860$$ 33.9411 1.15738
$$861$$ 0 0
$$862$$ 32.0000 1.08992
$$863$$ 24.0000 0.816970 0.408485 0.912765i $$-0.366057\pi$$
0.408485 + 0.912765i $$0.366057\pi$$
$$864$$ 0 0
$$865$$ −48.0000 −1.63205
$$866$$ 12.7279 0.432512
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 16.0000 0.542763
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 2.00000 0.0677285
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 34.0000 1.14810 0.574049 0.818821i $$-0.305372\pi$$
0.574049 + 0.818821i $$0.305372\pi$$
$$878$$ 25.4558 0.859093
$$879$$ 0 0
$$880$$ −4.24264 −0.143019
$$881$$ 29.6985 1.00057 0.500284 0.865862i $$-0.333229\pi$$
0.500284 + 0.865862i $$0.333229\pi$$
$$882$$ 0 0
$$883$$ −36.0000 −1.21150 −0.605748 0.795656i $$-0.707126\pi$$
−0.605748 + 0.795656i $$0.707126\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 36.0000 1.20944
$$887$$ −36.7696 −1.23460 −0.617300 0.786728i $$-0.711774\pi$$
−0.617300 + 0.786728i $$0.711774\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 30.0000 1.00560
$$891$$ 0 0
$$892$$ −21.2132 −0.710271
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 50.9117 1.70179
$$896$$ 0 0
$$897$$ 0 0
$$898$$ −20.0000 −0.667409
$$899$$ 2.82843 0.0943333
$$900$$ 0 0
$$901$$ −45.2548 −1.50766
$$902$$ −11.3137 −0.376705
$$903$$ 0 0
$$904$$ −2.00000 −0.0665190
$$905$$ 30.0000 0.997234
$$906$$ 0 0
$$907$$ −14.0000 −0.464862 −0.232431 0.972613i $$-0.574668\pi$$
−0.232431 + 0.972613i $$0.574668\pi$$
$$908$$ 14.1421 0.469323
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −30.0000 −0.993944 −0.496972 0.867766i $$-0.665555\pi$$
−0.496972 + 0.867766i $$0.665555\pi$$
$$912$$ 0 0
$$913$$ 16.9706 0.561644
$$914$$ −14.0000 −0.463079
$$915$$ 0 0
$$916$$ 9.89949 0.327089
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 8.00000 0.263896 0.131948 0.991257i $$-0.457877\pi$$
0.131948 + 0.991257i $$0.457877\pi$$
$$920$$ −25.4558 −0.839254
$$921$$ 0 0
$$922$$ 2.82843 0.0931493
$$923$$ 0 0
$$924$$ 0 0
$$925$$ −130.000 −4.27437
$$926$$ −26.0000 −0.854413
$$927$$ 0 0
$$928$$ 2.00000 0.0656532
$$929$$ 32.5269 1.06717 0.533587 0.845745i $$-0.320844\pi$$
0.533587 + 0.845745i $$0.320844\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ −14.0000 −0.458585
$$933$$ 0 0
$$934$$ −4.24264 −0.138823
$$935$$ −24.0000 −0.784884
$$936$$ 0 0
$$937$$ 25.4558 0.831606 0.415803 0.909455i $$-0.363501\pi$$
0.415803 + 0.909455i $$0.363501\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 18.0000 0.587095
$$941$$ −31.1127 −1.01424 −0.507122 0.861874i $$-0.669291\pi$$
−0.507122 + 0.861874i $$0.669291\pi$$
$$942$$ 0 0
$$943$$ −67.8823 −2.21055
$$944$$ −1.41421 −0.0460287
$$945$$ 0 0
$$946$$ 8.00000 0.260102
$$947$$ −52.0000 −1.68977 −0.844886 0.534946i $$-0.820332\pi$$
−0.844886 + 0.534946i $$0.820332\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −14.0000 −0.453504 −0.226752 0.973952i $$-0.572811\pi$$
−0.226752 + 0.973952i $$0.572811\pi$$
$$954$$ 0 0
$$955$$ 67.8823 2.19662
$$956$$ 12.0000 0.388108
$$957$$ 0 0
$$958$$ 2.82843 0.0913823
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −29.0000 −0.935484
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 25.4558 0.819453
$$966$$ 0 0
$$967$$ 8.00000 0.257263 0.128631 0.991692i $$-0.458942\pi$$
0.128631 + 0.991692i $$0.458942\pi$$
$$968$$ −1.00000 −0.0321412
$$969$$ 0 0
$$970$$ −42.0000 −1.34854
$$971$$ −52.3259 −1.67922 −0.839609 0.543191i $$-0.817216\pi$$
−0.839609 + 0.543191i $$0.817216\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ −2.00000 −0.0640841
$$975$$ 0 0
$$976$$ −2.82843 −0.0905357
$$977$$ −52.0000 −1.66363 −0.831814 0.555055i $$-0.812697\pi$$
−0.831814 + 0.555055i $$0.812697\pi$$
$$978$$ 0 0
$$979$$ 7.07107 0.225992
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −12.0000 −0.382935
$$983$$ −1.41421 −0.0451064 −0.0225532 0.999746i $$-0.507180\pi$$
−0.0225532 + 0.999746i $$0.507180\pi$$
$$984$$ 0 0
$$985$$ 93.3381 2.97400
$$986$$ 11.3137 0.360302
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 48.0000 1.52631
$$990$$ 0 0
$$991$$ −46.0000 −1.46124 −0.730619 0.682785i $$-0.760768\pi$$
−0.730619 + 0.682785i $$0.760768\pi$$
$$992$$ 1.41421 0.0449013
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 6.00000 0.190213
$$996$$ 0 0
$$997$$ 16.9706 0.537463 0.268732 0.963215i $$-0.413396\pi$$
0.268732 + 0.963215i $$0.413396\pi$$
$$998$$ −6.00000 −0.189927
$$999$$ 0 0
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 9702.2.a.cp.1.1 2
3.2 odd 2 1078.2.a.u.1.2 yes 2
7.6 odd 2 inner 9702.2.a.cp.1.2 2
12.11 even 2 8624.2.a.bs.1.1 2
21.2 odd 6 1078.2.e.p.67.1 4
21.5 even 6 1078.2.e.p.67.2 4
21.11 odd 6 1078.2.e.p.177.1 4
21.17 even 6 1078.2.e.p.177.2 4
21.20 even 2 1078.2.a.u.1.1 2
84.83 odd 2 8624.2.a.bs.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
1078.2.a.u.1.1 2 21.20 even 2
1078.2.a.u.1.2 yes 2 3.2 odd 2
1078.2.e.p.67.1 4 21.2 odd 6
1078.2.e.p.67.2 4 21.5 even 6
1078.2.e.p.177.1 4 21.11 odd 6
1078.2.e.p.177.2 4 21.17 even 6
8624.2.a.bs.1.1 2 12.11 even 2
8624.2.a.bs.1.2 2 84.83 odd 2
9702.2.a.cp.1.1 2 1.1 even 1 trivial
9702.2.a.cp.1.2 2 7.6 odd 2 inner