# Properties

 Label 9702.2.a.cy.1.2 Level $9702$ Weight $2$ Character 9702.1 Self dual yes Analytic conductor $77.471$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [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: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 3234) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 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)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{5} -1.00000 q^{8} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{5} -1.00000 q^{8} -2.00000 q^{10} +1.00000 q^{11} -2.58579 q^{13} +1.00000 q^{16} +2.00000 q^{17} -6.24264 q^{19} +2.00000 q^{20} -1.00000 q^{22} -0.828427 q^{23} -1.00000 q^{25} +2.58579 q^{26} +1.65685 q^{29} +2.24264 q^{31} -1.00000 q^{32} -2.00000 q^{34} -4.82843 q^{37} +6.24264 q^{38} -2.00000 q^{40} +0.343146 q^{41} +0.828427 q^{43} +1.00000 q^{44} +0.828427 q^{46} +11.8995 q^{47} +1.00000 q^{50} -2.58579 q^{52} -6.48528 q^{53} +2.00000 q^{55} -1.65685 q^{58} +1.17157 q^{59} -5.41421 q^{61} -2.24264 q^{62} +1.00000 q^{64} -5.17157 q^{65} -6.82843 q^{67} +2.00000 q^{68} +5.65685 q^{71} -0.343146 q^{73} +4.82843 q^{74} -6.24264 q^{76} +0.485281 q^{79} +2.00000 q^{80} -0.343146 q^{82} -9.07107 q^{83} +4.00000 q^{85} -0.828427 q^{86} -1.00000 q^{88} +12.2426 q^{89} -0.828427 q^{92} -11.8995 q^{94} -12.4853 q^{95} -9.89949 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{4} + 4 q^{5} - 2 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 + 2 * q^4 + 4 * q^5 - 2 * q^8 $$2 q - 2 q^{2} + 2 q^{4} + 4 q^{5} - 2 q^{8} - 4 q^{10} + 2 q^{11} - 8 q^{13} + 2 q^{16} + 4 q^{17} - 4 q^{19} + 4 q^{20} - 2 q^{22} + 4 q^{23} - 2 q^{25} + 8 q^{26} - 8 q^{29} - 4 q^{31} - 2 q^{32} - 4 q^{34} - 4 q^{37} + 4 q^{38} - 4 q^{40} + 12 q^{41} - 4 q^{43} + 2 q^{44} - 4 q^{46} + 4 q^{47} + 2 q^{50} - 8 q^{52} + 4 q^{53} + 4 q^{55} + 8 q^{58} + 8 q^{59} - 8 q^{61} + 4 q^{62} + 2 q^{64} - 16 q^{65} - 8 q^{67} + 4 q^{68} - 12 q^{73} + 4 q^{74} - 4 q^{76} - 16 q^{79} + 4 q^{80} - 12 q^{82} - 4 q^{83} + 8 q^{85} + 4 q^{86} - 2 q^{88} + 16 q^{89} + 4 q^{92} - 4 q^{94} - 8 q^{95}+O(q^{100})$$ 2 * q - 2 * q^2 + 2 * q^4 + 4 * q^5 - 2 * q^8 - 4 * q^10 + 2 * q^11 - 8 * q^13 + 2 * q^16 + 4 * q^17 - 4 * q^19 + 4 * q^20 - 2 * q^22 + 4 * q^23 - 2 * q^25 + 8 * q^26 - 8 * q^29 - 4 * q^31 - 2 * q^32 - 4 * q^34 - 4 * q^37 + 4 * q^38 - 4 * q^40 + 12 * q^41 - 4 * q^43 + 2 * q^44 - 4 * q^46 + 4 * q^47 + 2 * q^50 - 8 * q^52 + 4 * q^53 + 4 * q^55 + 8 * q^58 + 8 * q^59 - 8 * q^61 + 4 * q^62 + 2 * q^64 - 16 * q^65 - 8 * q^67 + 4 * q^68 - 12 * q^73 + 4 * q^74 - 4 * q^76 - 16 * q^79 + 4 * q^80 - 12 * q^82 - 4 * q^83 + 8 * q^85 + 4 * q^86 - 2 * q^88 + 16 * q^89 + 4 * q^92 - 4 * q^94 - 8 * q^95

## 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
$$5$$ 2.00000 0.894427 0.447214 0.894427i $$-0.352416\pi$$
0.447214 + 0.894427i $$0.352416\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ −1.00000 −0.353553
$$9$$ 0 0
$$10$$ −2.00000 −0.632456
$$11$$ 1.00000 0.301511
$$12$$ 0 0
$$13$$ −2.58579 −0.717168 −0.358584 0.933497i $$-0.616740\pi$$
−0.358584 + 0.933497i $$0.616740\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 2.00000 0.485071 0.242536 0.970143i $$-0.422021\pi$$
0.242536 + 0.970143i $$0.422021\pi$$
$$18$$ 0 0
$$19$$ −6.24264 −1.43216 −0.716080 0.698018i $$-0.754065\pi$$
−0.716080 + 0.698018i $$0.754065\pi$$
$$20$$ 2.00000 0.447214
$$21$$ 0 0
$$22$$ −1.00000 −0.213201
$$23$$ −0.828427 −0.172739 −0.0863695 0.996263i $$-0.527527\pi$$
−0.0863695 + 0.996263i $$0.527527\pi$$
$$24$$ 0 0
$$25$$ −1.00000 −0.200000
$$26$$ 2.58579 0.507114
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 1.65685 0.307670 0.153835 0.988097i $$-0.450838\pi$$
0.153835 + 0.988097i $$0.450838\pi$$
$$30$$ 0 0
$$31$$ 2.24264 0.402790 0.201395 0.979510i $$-0.435452\pi$$
0.201395 + 0.979510i $$0.435452\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ −2.00000 −0.342997
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −4.82843 −0.793789 −0.396894 0.917864i $$-0.629912\pi$$
−0.396894 + 0.917864i $$0.629912\pi$$
$$38$$ 6.24264 1.01269
$$39$$ 0 0
$$40$$ −2.00000 −0.316228
$$41$$ 0.343146 0.0535904 0.0267952 0.999641i $$-0.491470\pi$$
0.0267952 + 0.999641i $$0.491470\pi$$
$$42$$ 0 0
$$43$$ 0.828427 0.126334 0.0631670 0.998003i $$-0.479880\pi$$
0.0631670 + 0.998003i $$0.479880\pi$$
$$44$$ 1.00000 0.150756
$$45$$ 0 0
$$46$$ 0.828427 0.122145
$$47$$ 11.8995 1.73572 0.867860 0.496809i $$-0.165495\pi$$
0.867860 + 0.496809i $$0.165495\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 1.00000 0.141421
$$51$$ 0 0
$$52$$ −2.58579 −0.358584
$$53$$ −6.48528 −0.890822 −0.445411 0.895326i $$-0.646942\pi$$
−0.445411 + 0.895326i $$0.646942\pi$$
$$54$$ 0 0
$$55$$ 2.00000 0.269680
$$56$$ 0 0
$$57$$ 0 0
$$58$$ −1.65685 −0.217556
$$59$$ 1.17157 0.152526 0.0762629 0.997088i $$-0.475701\pi$$
0.0762629 + 0.997088i $$0.475701\pi$$
$$60$$ 0 0
$$61$$ −5.41421 −0.693219 −0.346610 0.938010i $$-0.612667\pi$$
−0.346610 + 0.938010i $$0.612667\pi$$
$$62$$ −2.24264 −0.284816
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ −5.17157 −0.641455
$$66$$ 0 0
$$67$$ −6.82843 −0.834225 −0.417113 0.908855i $$-0.636958\pi$$
−0.417113 + 0.908855i $$0.636958\pi$$
$$68$$ 2.00000 0.242536
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 5.65685 0.671345 0.335673 0.941979i $$-0.391036\pi$$
0.335673 + 0.941979i $$0.391036\pi$$
$$72$$ 0 0
$$73$$ −0.343146 −0.0401622 −0.0200811 0.999798i $$-0.506392\pi$$
−0.0200811 + 0.999798i $$0.506392\pi$$
$$74$$ 4.82843 0.561293
$$75$$ 0 0
$$76$$ −6.24264 −0.716080
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 0.485281 0.0545984 0.0272992 0.999627i $$-0.491309\pi$$
0.0272992 + 0.999627i $$0.491309\pi$$
$$80$$ 2.00000 0.223607
$$81$$ 0 0
$$82$$ −0.343146 −0.0378941
$$83$$ −9.07107 −0.995679 −0.497840 0.867269i $$-0.665873\pi$$
−0.497840 + 0.867269i $$0.665873\pi$$
$$84$$ 0 0
$$85$$ 4.00000 0.433861
$$86$$ −0.828427 −0.0893316
$$87$$ 0 0
$$88$$ −1.00000 −0.106600
$$89$$ 12.2426 1.29772 0.648859 0.760909i $$-0.275247\pi$$
0.648859 + 0.760909i $$0.275247\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −0.828427 −0.0863695
$$93$$ 0 0
$$94$$ −11.8995 −1.22734
$$95$$ −12.4853 −1.28096
$$96$$ 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
$$100$$ −1.00000 −0.100000
$$101$$ −0.928932 −0.0924322 −0.0462161 0.998931i $$-0.514716\pi$$
−0.0462161 + 0.998931i $$0.514716\pi$$
$$102$$ 0 0
$$103$$ 2.24264 0.220974 0.110487 0.993878i $$-0.464759\pi$$
0.110487 + 0.993878i $$0.464759\pi$$
$$104$$ 2.58579 0.253557
$$105$$ 0 0
$$106$$ 6.48528 0.629906
$$107$$ −7.17157 −0.693302 −0.346651 0.937994i $$-0.612681\pi$$
−0.346651 + 0.937994i $$0.612681\pi$$
$$108$$ 0 0
$$109$$ 15.6569 1.49965 0.749827 0.661634i $$-0.230137\pi$$
0.749827 + 0.661634i $$0.230137\pi$$
$$110$$ −2.00000 −0.190693
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 5.65685 0.532152 0.266076 0.963952i $$-0.414273\pi$$
0.266076 + 0.963952i $$0.414273\pi$$
$$114$$ 0 0
$$115$$ −1.65685 −0.154502
$$116$$ 1.65685 0.153835
$$117$$ 0 0
$$118$$ −1.17157 −0.107852
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 5.41421 0.490180
$$123$$ 0 0
$$124$$ 2.24264 0.201395
$$125$$ −12.0000 −1.07331
$$126$$ 0 0
$$127$$ −14.1421 −1.25491 −0.627456 0.778652i $$-0.715904\pi$$
−0.627456 + 0.778652i $$0.715904\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 0 0
$$130$$ 5.17157 0.453577
$$131$$ −13.5563 −1.18442 −0.592212 0.805782i $$-0.701745\pi$$
−0.592212 + 0.805782i $$0.701745\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 6.82843 0.589886
$$135$$ 0 0
$$136$$ −2.00000 −0.171499
$$137$$ −17.3137 −1.47921 −0.739605 0.673041i $$-0.764988\pi$$
−0.739605 + 0.673041i $$0.764988\pi$$
$$138$$ 0 0
$$139$$ −19.8995 −1.68785 −0.843927 0.536459i $$-0.819762\pi$$
−0.843927 + 0.536459i $$0.819762\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −5.65685 −0.474713
$$143$$ −2.58579 −0.216234
$$144$$ 0 0
$$145$$ 3.31371 0.275189
$$146$$ 0.343146 0.0283989
$$147$$ 0 0
$$148$$ −4.82843 −0.396894
$$149$$ −4.00000 −0.327693 −0.163846 0.986486i $$-0.552390\pi$$
−0.163846 + 0.986486i $$0.552390\pi$$
$$150$$ 0 0
$$151$$ 4.00000 0.325515 0.162758 0.986666i $$-0.447961\pi$$
0.162758 + 0.986666i $$0.447961\pi$$
$$152$$ 6.24264 0.506345
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 4.48528 0.360266
$$156$$ 0 0
$$157$$ −7.17157 −0.572354 −0.286177 0.958177i $$-0.592385\pi$$
−0.286177 + 0.958177i $$0.592385\pi$$
$$158$$ −0.485281 −0.0386069
$$159$$ 0 0
$$160$$ −2.00000 −0.158114
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −13.6569 −1.06969 −0.534844 0.844951i $$-0.679630\pi$$
−0.534844 + 0.844951i $$0.679630\pi$$
$$164$$ 0.343146 0.0267952
$$165$$ 0 0
$$166$$ 9.07107 0.704051
$$167$$ −10.8284 −0.837929 −0.418964 0.908003i $$-0.637607\pi$$
−0.418964 + 0.908003i $$0.637607\pi$$
$$168$$ 0 0
$$169$$ −6.31371 −0.485670
$$170$$ −4.00000 −0.306786
$$171$$ 0 0
$$172$$ 0.828427 0.0631670
$$173$$ 13.8995 1.05676 0.528380 0.849008i $$-0.322800\pi$$
0.528380 + 0.849008i $$0.322800\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 1.00000 0.0753778
$$177$$ 0 0
$$178$$ −12.2426 −0.917625
$$179$$ 3.51472 0.262702 0.131351 0.991336i $$-0.458068\pi$$
0.131351 + 0.991336i $$0.458068\pi$$
$$180$$ 0 0
$$181$$ −13.3137 −0.989600 −0.494800 0.869007i $$-0.664759\pi$$
−0.494800 + 0.869007i $$0.664759\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0.828427 0.0610725
$$185$$ −9.65685 −0.709986
$$186$$ 0 0
$$187$$ 2.00000 0.146254
$$188$$ 11.8995 0.867860
$$189$$ 0 0
$$190$$ 12.4853 0.905778
$$191$$ 17.7990 1.28789 0.643945 0.765072i $$-0.277297\pi$$
0.643945 + 0.765072i $$0.277297\pi$$
$$192$$ 0 0
$$193$$ 18.9706 1.36553 0.682765 0.730638i $$-0.260777\pi$$
0.682765 + 0.730638i $$0.260777\pi$$
$$194$$ 9.89949 0.710742
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −4.34315 −0.309436 −0.154718 0.987959i $$-0.549447\pi$$
−0.154718 + 0.987959i $$0.549447\pi$$
$$198$$ 0 0
$$199$$ 27.2132 1.92909 0.964546 0.263913i $$-0.0850133\pi$$
0.964546 + 0.263913i $$0.0850133\pi$$
$$200$$ 1.00000 0.0707107
$$201$$ 0 0
$$202$$ 0.928932 0.0653594
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0.686292 0.0479327
$$206$$ −2.24264 −0.156252
$$207$$ 0 0
$$208$$ −2.58579 −0.179292
$$209$$ −6.24264 −0.431812
$$210$$ 0 0
$$211$$ −28.8284 −1.98463 −0.992315 0.123734i $$-0.960513\pi$$
−0.992315 + 0.123734i $$0.960513\pi$$
$$212$$ −6.48528 −0.445411
$$213$$ 0 0
$$214$$ 7.17157 0.490239
$$215$$ 1.65685 0.112997
$$216$$ 0 0
$$217$$ 0 0
$$218$$ −15.6569 −1.06042
$$219$$ 0 0
$$220$$ 2.00000 0.134840
$$221$$ −5.17157 −0.347878
$$222$$ 0 0
$$223$$ 2.24264 0.150178 0.0750892 0.997177i $$-0.476076\pi$$
0.0750892 + 0.997177i $$0.476076\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −5.65685 −0.376288
$$227$$ 7.89949 0.524308 0.262154 0.965026i $$-0.415567\pi$$
0.262154 + 0.965026i $$0.415567\pi$$
$$228$$ 0 0
$$229$$ −14.0000 −0.925146 −0.462573 0.886581i $$-0.653074\pi$$
−0.462573 + 0.886581i $$0.653074\pi$$
$$230$$ 1.65685 0.109250
$$231$$ 0 0
$$232$$ −1.65685 −0.108778
$$233$$ −3.65685 −0.239568 −0.119784 0.992800i $$-0.538220\pi$$
−0.119784 + 0.992800i $$0.538220\pi$$
$$234$$ 0 0
$$235$$ 23.7990 1.55247
$$236$$ 1.17157 0.0762629
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −23.3137 −1.50804 −0.754019 0.656852i $$-0.771887\pi$$
−0.754019 + 0.656852i $$0.771887\pi$$
$$240$$ 0 0
$$241$$ −3.65685 −0.235559 −0.117779 0.993040i $$-0.537578\pi$$
−0.117779 + 0.993040i $$0.537578\pi$$
$$242$$ −1.00000 −0.0642824
$$243$$ 0 0
$$244$$ −5.41421 −0.346610
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 16.1421 1.02710
$$248$$ −2.24264 −0.142408
$$249$$ 0 0
$$250$$ 12.0000 0.758947
$$251$$ −8.48528 −0.535586 −0.267793 0.963476i $$-0.586294\pi$$
−0.267793 + 0.963476i $$0.586294\pi$$
$$252$$ 0 0
$$253$$ −0.828427 −0.0520828
$$254$$ 14.1421 0.887357
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 13.2132 0.824217 0.412108 0.911135i $$-0.364792\pi$$
0.412108 + 0.911135i $$0.364792\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ −5.17157 −0.320727
$$261$$ 0 0
$$262$$ 13.5563 0.837514
$$263$$ −14.8284 −0.914360 −0.457180 0.889374i $$-0.651140\pi$$
−0.457180 + 0.889374i $$0.651140\pi$$
$$264$$ 0 0
$$265$$ −12.9706 −0.796775
$$266$$ 0 0
$$267$$ 0 0
$$268$$ −6.82843 −0.417113
$$269$$ −17.3137 −1.05564 −0.527818 0.849358i $$-0.676990\pi$$
−0.527818 + 0.849358i $$0.676990\pi$$
$$270$$ 0 0
$$271$$ 6.14214 0.373108 0.186554 0.982445i $$-0.440268\pi$$
0.186554 + 0.982445i $$0.440268\pi$$
$$272$$ 2.00000 0.121268
$$273$$ 0 0
$$274$$ 17.3137 1.04596
$$275$$ −1.00000 −0.0603023
$$276$$ 0 0
$$277$$ −16.0000 −0.961347 −0.480673 0.876900i $$-0.659608\pi$$
−0.480673 + 0.876900i $$0.659608\pi$$
$$278$$ 19.8995 1.19349
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −21.3137 −1.27147 −0.635735 0.771908i $$-0.719303\pi$$
−0.635735 + 0.771908i $$0.719303\pi$$
$$282$$ 0 0
$$283$$ −6.24264 −0.371086 −0.185543 0.982636i $$-0.559404\pi$$
−0.185543 + 0.982636i $$0.559404\pi$$
$$284$$ 5.65685 0.335673
$$285$$ 0 0
$$286$$ 2.58579 0.152901
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −13.0000 −0.764706
$$290$$ −3.31371 −0.194588
$$291$$ 0 0
$$292$$ −0.343146 −0.0200811
$$293$$ −19.0711 −1.11414 −0.557072 0.830464i $$-0.688075\pi$$
−0.557072 + 0.830464i $$0.688075\pi$$
$$294$$ 0 0
$$295$$ 2.34315 0.136423
$$296$$ 4.82843 0.280647
$$297$$ 0 0
$$298$$ 4.00000 0.231714
$$299$$ 2.14214 0.123883
$$300$$ 0 0
$$301$$ 0 0
$$302$$ −4.00000 −0.230174
$$303$$ 0 0
$$304$$ −6.24264 −0.358040
$$305$$ −10.8284 −0.620034
$$306$$ 0 0
$$307$$ 10.9289 0.623747 0.311874 0.950124i $$-0.399043\pi$$
0.311874 + 0.950124i $$0.399043\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ −4.48528 −0.254747
$$311$$ 27.8995 1.58204 0.791018 0.611793i $$-0.209552\pi$$
0.791018 + 0.611793i $$0.209552\pi$$
$$312$$ 0 0
$$313$$ −0.242641 −0.0137149 −0.00685743 0.999976i $$-0.502183\pi$$
−0.00685743 + 0.999976i $$0.502183\pi$$
$$314$$ 7.17157 0.404715
$$315$$ 0 0
$$316$$ 0.485281 0.0272992
$$317$$ −7.17157 −0.402796 −0.201398 0.979510i $$-0.564548\pi$$
−0.201398 + 0.979510i $$0.564548\pi$$
$$318$$ 0 0
$$319$$ 1.65685 0.0927660
$$320$$ 2.00000 0.111803
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −12.4853 −0.694700
$$324$$ 0 0
$$325$$ 2.58579 0.143434
$$326$$ 13.6569 0.756383
$$327$$ 0 0
$$328$$ −0.343146 −0.0189471
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 11.7990 0.648531 0.324266 0.945966i $$-0.394883\pi$$
0.324266 + 0.945966i $$0.394883\pi$$
$$332$$ −9.07107 −0.497840
$$333$$ 0 0
$$334$$ 10.8284 0.592505
$$335$$ −13.6569 −0.746154
$$336$$ 0 0
$$337$$ 7.17157 0.390660 0.195330 0.980738i $$-0.437422\pi$$
0.195330 + 0.980738i $$0.437422\pi$$
$$338$$ 6.31371 0.343420
$$339$$ 0 0
$$340$$ 4.00000 0.216930
$$341$$ 2.24264 0.121446
$$342$$ 0 0
$$343$$ 0 0
$$344$$ −0.828427 −0.0446658
$$345$$ 0 0
$$346$$ −13.8995 −0.747241
$$347$$ 31.4558 1.68864 0.844319 0.535841i $$-0.180005\pi$$
0.844319 + 0.535841i $$0.180005\pi$$
$$348$$ 0 0
$$349$$ 14.5858 0.780759 0.390380 0.920654i $$-0.372344\pi$$
0.390380 + 0.920654i $$0.372344\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −1.00000 −0.0533002
$$353$$ −21.2132 −1.12906 −0.564532 0.825411i $$-0.690943\pi$$
−0.564532 + 0.825411i $$0.690943\pi$$
$$354$$ 0 0
$$355$$ 11.3137 0.600469
$$356$$ 12.2426 0.648859
$$357$$ 0 0
$$358$$ −3.51472 −0.185759
$$359$$ −10.8284 −0.571503 −0.285751 0.958304i $$-0.592243\pi$$
−0.285751 + 0.958304i $$0.592243\pi$$
$$360$$ 0 0
$$361$$ 19.9706 1.05108
$$362$$ 13.3137 0.699753
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −0.686292 −0.0359221
$$366$$ 0 0
$$367$$ 35.4142 1.84861 0.924303 0.381658i $$-0.124647\pi$$
0.924303 + 0.381658i $$0.124647\pi$$
$$368$$ −0.828427 −0.0431847
$$369$$ 0 0
$$370$$ 9.65685 0.502036
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −31.9411 −1.65385 −0.826924 0.562313i $$-0.809912\pi$$
−0.826924 + 0.562313i $$0.809912\pi$$
$$374$$ −2.00000 −0.103418
$$375$$ 0 0
$$376$$ −11.8995 −0.613670
$$377$$ −4.28427 −0.220651
$$378$$ 0 0
$$379$$ −24.2843 −1.24740 −0.623700 0.781664i $$-0.714371\pi$$
−0.623700 + 0.781664i $$0.714371\pi$$
$$380$$ −12.4853 −0.640481
$$381$$ 0 0
$$382$$ −17.7990 −0.910676
$$383$$ 2.72792 0.139390 0.0696952 0.997568i $$-0.477797\pi$$
0.0696952 + 0.997568i $$0.477797\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −18.9706 −0.965576
$$387$$ 0 0
$$388$$ −9.89949 −0.502571
$$389$$ −22.2843 −1.12986 −0.564929 0.825140i $$-0.691096\pi$$
−0.564929 + 0.825140i $$0.691096\pi$$
$$390$$ 0 0
$$391$$ −1.65685 −0.0837907
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 4.34315 0.218805
$$395$$ 0.970563 0.0488343
$$396$$ 0 0
$$397$$ 16.8284 0.844595 0.422297 0.906457i $$-0.361224\pi$$
0.422297 + 0.906457i $$0.361224\pi$$
$$398$$ −27.2132 −1.36407
$$399$$ 0 0
$$400$$ −1.00000 −0.0500000
$$401$$ −33.3137 −1.66361 −0.831804 0.555070i $$-0.812691\pi$$
−0.831804 + 0.555070i $$0.812691\pi$$
$$402$$ 0 0
$$403$$ −5.79899 −0.288868
$$404$$ −0.928932 −0.0462161
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −4.82843 −0.239336
$$408$$ 0 0
$$409$$ 18.4853 0.914038 0.457019 0.889457i $$-0.348917\pi$$
0.457019 + 0.889457i $$0.348917\pi$$
$$410$$ −0.686292 −0.0338935
$$411$$ 0 0
$$412$$ 2.24264 0.110487
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −18.1421 −0.890562
$$416$$ 2.58579 0.126779
$$417$$ 0 0
$$418$$ 6.24264 0.305338
$$419$$ 16.0000 0.781651 0.390826 0.920465i $$-0.372190\pi$$
0.390826 + 0.920465i $$0.372190\pi$$
$$420$$ 0 0
$$421$$ −16.3431 −0.796516 −0.398258 0.917273i $$-0.630385\pi$$
−0.398258 + 0.917273i $$0.630385\pi$$
$$422$$ 28.8284 1.40335
$$423$$ 0 0
$$424$$ 6.48528 0.314953
$$425$$ −2.00000 −0.0970143
$$426$$ 0 0
$$427$$ 0 0
$$428$$ −7.17157 −0.346651
$$429$$ 0 0
$$430$$ −1.65685 −0.0799006
$$431$$ 26.1421 1.25922 0.629611 0.776910i $$-0.283214\pi$$
0.629611 + 0.776910i $$0.283214\pi$$
$$432$$ 0 0
$$433$$ −2.38478 −0.114605 −0.0573025 0.998357i $$-0.518250\pi$$
−0.0573025 + 0.998357i $$0.518250\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 15.6569 0.749827
$$437$$ 5.17157 0.247390
$$438$$ 0 0
$$439$$ 34.8284 1.66227 0.831135 0.556071i $$-0.187692\pi$$
0.831135 + 0.556071i $$0.187692\pi$$
$$440$$ −2.00000 −0.0953463
$$441$$ 0 0
$$442$$ 5.17157 0.245987
$$443$$ 4.48528 0.213102 0.106551 0.994307i $$-0.466019\pi$$
0.106551 + 0.994307i $$0.466019\pi$$
$$444$$ 0 0
$$445$$ 24.4853 1.16071
$$446$$ −2.24264 −0.106192
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 16.9706 0.800890 0.400445 0.916321i $$-0.368855\pi$$
0.400445 + 0.916321i $$0.368855\pi$$
$$450$$ 0 0
$$451$$ 0.343146 0.0161581
$$452$$ 5.65685 0.266076
$$453$$ 0 0
$$454$$ −7.89949 −0.370742
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −6.68629 −0.312772 −0.156386 0.987696i $$-0.549984\pi$$
−0.156386 + 0.987696i $$0.549984\pi$$
$$458$$ 14.0000 0.654177
$$459$$ 0 0
$$460$$ −1.65685 −0.0772512
$$461$$ −33.4142 −1.55626 −0.778128 0.628106i $$-0.783830\pi$$
−0.778128 + 0.628106i $$0.783830\pi$$
$$462$$ 0 0
$$463$$ 19.3137 0.897584 0.448792 0.893636i $$-0.351854\pi$$
0.448792 + 0.893636i $$0.351854\pi$$
$$464$$ 1.65685 0.0769175
$$465$$ 0 0
$$466$$ 3.65685 0.169401
$$467$$ 12.0000 0.555294 0.277647 0.960683i $$-0.410445\pi$$
0.277647 + 0.960683i $$0.410445\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ −23.7990 −1.09777
$$471$$ 0 0
$$472$$ −1.17157 −0.0539260
$$473$$ 0.828427 0.0380911
$$474$$ 0 0
$$475$$ 6.24264 0.286432
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 23.3137 1.06634
$$479$$ −15.3137 −0.699701 −0.349851 0.936806i $$-0.613768\pi$$
−0.349851 + 0.936806i $$0.613768\pi$$
$$480$$ 0 0
$$481$$ 12.4853 0.569280
$$482$$ 3.65685 0.166565
$$483$$ 0 0
$$484$$ 1.00000 0.0454545
$$485$$ −19.7990 −0.899026
$$486$$ 0 0
$$487$$ −28.1421 −1.27524 −0.637621 0.770350i $$-0.720081\pi$$
−0.637621 + 0.770350i $$0.720081\pi$$
$$488$$ 5.41421 0.245090
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −2.62742 −0.118574 −0.0592868 0.998241i $$-0.518883\pi$$
−0.0592868 + 0.998241i $$0.518883\pi$$
$$492$$ 0 0
$$493$$ 3.31371 0.149242
$$494$$ −16.1421 −0.726269
$$495$$ 0 0
$$496$$ 2.24264 0.100698
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 22.1421 0.991218 0.495609 0.868546i $$-0.334945\pi$$
0.495609 + 0.868546i $$0.334945\pi$$
$$500$$ −12.0000 −0.536656
$$501$$ 0 0
$$502$$ 8.48528 0.378717
$$503$$ 8.48528 0.378340 0.189170 0.981944i $$-0.439420\pi$$
0.189170 + 0.981944i $$0.439420\pi$$
$$504$$ 0 0
$$505$$ −1.85786 −0.0826739
$$506$$ 0.828427 0.0368281
$$507$$ 0 0
$$508$$ −14.1421 −0.627456
$$509$$ 34.0000 1.50702 0.753512 0.657434i $$-0.228358\pi$$
0.753512 + 0.657434i $$0.228358\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −1.00000 −0.0441942
$$513$$ 0 0
$$514$$ −13.2132 −0.582809
$$515$$ 4.48528 0.197645
$$516$$ 0 0
$$517$$ 11.8995 0.523339
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 5.17157 0.226788
$$521$$ −21.2132 −0.929367 −0.464684 0.885477i $$-0.653832\pi$$
−0.464684 + 0.885477i $$0.653832\pi$$
$$522$$ 0 0
$$523$$ −5.07107 −0.221742 −0.110871 0.993835i $$-0.535364\pi$$
−0.110871 + 0.993835i $$0.535364\pi$$
$$524$$ −13.5563 −0.592212
$$525$$ 0 0
$$526$$ 14.8284 0.646550
$$527$$ 4.48528 0.195382
$$528$$ 0 0
$$529$$ −22.3137 −0.970161
$$530$$ 12.9706 0.563405
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −0.887302 −0.0384333
$$534$$ 0 0
$$535$$ −14.3431 −0.620108
$$536$$ 6.82843 0.294943
$$537$$ 0 0
$$538$$ 17.3137 0.746447
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −22.6863 −0.975360 −0.487680 0.873023i $$-0.662157\pi$$
−0.487680 + 0.873023i $$0.662157\pi$$
$$542$$ −6.14214 −0.263827
$$543$$ 0 0
$$544$$ −2.00000 −0.0857493
$$545$$ 31.3137 1.34133
$$546$$ 0 0
$$547$$ 2.62742 0.112340 0.0561701 0.998421i $$-0.482111\pi$$
0.0561701 + 0.998421i $$0.482111\pi$$
$$548$$ −17.3137 −0.739605
$$549$$ 0 0
$$550$$ 1.00000 0.0426401
$$551$$ −10.3431 −0.440633
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 16.0000 0.679775
$$555$$ 0 0
$$556$$ −19.8995 −0.843927
$$557$$ −8.34315 −0.353510 −0.176755 0.984255i $$-0.556560\pi$$
−0.176755 + 0.984255i $$0.556560\pi$$
$$558$$ 0 0
$$559$$ −2.14214 −0.0906027
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 21.3137 0.899065
$$563$$ −6.92893 −0.292020 −0.146010 0.989283i $$-0.546643\pi$$
−0.146010 + 0.989283i $$0.546643\pi$$
$$564$$ 0 0
$$565$$ 11.3137 0.475971
$$566$$ 6.24264 0.262398
$$567$$ 0 0
$$568$$ −5.65685 −0.237356
$$569$$ 33.3137 1.39658 0.698292 0.715813i $$-0.253944\pi$$
0.698292 + 0.715813i $$0.253944\pi$$
$$570$$ 0 0
$$571$$ −20.0000 −0.836974 −0.418487 0.908223i $$-0.637439\pi$$
−0.418487 + 0.908223i $$0.637439\pi$$
$$572$$ −2.58579 −0.108117
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0.828427 0.0345478
$$576$$ 0 0
$$577$$ 35.5563 1.48023 0.740115 0.672480i $$-0.234771\pi$$
0.740115 + 0.672480i $$0.234771\pi$$
$$578$$ 13.0000 0.540729
$$579$$ 0 0
$$580$$ 3.31371 0.137594
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −6.48528 −0.268593
$$584$$ 0.343146 0.0141995
$$585$$ 0 0
$$586$$ 19.0711 0.787819
$$587$$ −27.7990 −1.14739 −0.573694 0.819070i $$-0.694490\pi$$
−0.573694 + 0.819070i $$0.694490\pi$$
$$588$$ 0 0
$$589$$ −14.0000 −0.576860
$$590$$ −2.34315 −0.0964658
$$591$$ 0 0
$$592$$ −4.82843 −0.198447
$$593$$ −14.6863 −0.603094 −0.301547 0.953451i $$-0.597503\pi$$
−0.301547 + 0.953451i $$0.597503\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −4.00000 −0.163846
$$597$$ 0 0
$$598$$ −2.14214 −0.0875984
$$599$$ −8.00000 −0.326871 −0.163436 0.986554i $$-0.552258\pi$$
−0.163436 + 0.986554i $$0.552258\pi$$
$$600$$ 0 0
$$601$$ −20.8284 −0.849609 −0.424805 0.905285i $$-0.639657\pi$$
−0.424805 + 0.905285i $$0.639657\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 4.00000 0.162758
$$605$$ 2.00000 0.0813116
$$606$$ 0 0
$$607$$ 12.6863 0.514921 0.257460 0.966289i $$-0.417114\pi$$
0.257460 + 0.966289i $$0.417114\pi$$
$$608$$ 6.24264 0.253173
$$609$$ 0 0
$$610$$ 10.8284 0.438430
$$611$$ −30.7696 −1.24480
$$612$$ 0 0
$$613$$ −41.6569 −1.68250 −0.841252 0.540643i $$-0.818181\pi$$
−0.841252 + 0.540643i $$0.818181\pi$$
$$614$$ −10.9289 −0.441056
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 8.00000 0.322068 0.161034 0.986949i $$-0.448517\pi$$
0.161034 + 0.986949i $$0.448517\pi$$
$$618$$ 0 0
$$619$$ 10.3431 0.415726 0.207863 0.978158i $$-0.433349\pi$$
0.207863 + 0.978158i $$0.433349\pi$$
$$620$$ 4.48528 0.180133
$$621$$ 0 0
$$622$$ −27.8995 −1.11867
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −19.0000 −0.760000
$$626$$ 0.242641 0.00969787
$$627$$ 0 0
$$628$$ −7.17157 −0.286177
$$629$$ −9.65685 −0.385044
$$630$$ 0 0
$$631$$ 7.85786 0.312817 0.156408 0.987692i $$-0.450008\pi$$
0.156408 + 0.987692i $$0.450008\pi$$
$$632$$ −0.485281 −0.0193035
$$633$$ 0 0
$$634$$ 7.17157 0.284820
$$635$$ −28.2843 −1.12243
$$636$$ 0 0
$$637$$ 0 0
$$638$$ −1.65685 −0.0655955
$$639$$ 0 0
$$640$$ −2.00000 −0.0790569
$$641$$ 12.6863 0.501078 0.250539 0.968106i $$-0.419392\pi$$
0.250539 + 0.968106i $$0.419392\pi$$
$$642$$ 0 0
$$643$$ 13.4558 0.530647 0.265323 0.964159i $$-0.414521\pi$$
0.265323 + 0.964159i $$0.414521\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 12.4853 0.491227
$$647$$ −13.0711 −0.513877 −0.256938 0.966428i $$-0.582714\pi$$
−0.256938 + 0.966428i $$0.582714\pi$$
$$648$$ 0 0
$$649$$ 1.17157 0.0459883
$$650$$ −2.58579 −0.101423
$$651$$ 0 0
$$652$$ −13.6569 −0.534844
$$653$$ 34.2843 1.34165 0.670824 0.741617i $$-0.265941\pi$$
0.670824 + 0.741617i $$0.265941\pi$$
$$654$$ 0 0
$$655$$ −27.1127 −1.05938
$$656$$ 0.343146 0.0133976
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −13.1127 −0.510798 −0.255399 0.966836i $$-0.582207\pi$$
−0.255399 + 0.966836i $$0.582207\pi$$
$$660$$ 0 0
$$661$$ −27.9411 −1.08678 −0.543392 0.839479i $$-0.682860\pi$$
−0.543392 + 0.839479i $$0.682860\pi$$
$$662$$ −11.7990 −0.458581
$$663$$ 0 0
$$664$$ 9.07107 0.352026
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −1.37258 −0.0531466
$$668$$ −10.8284 −0.418964
$$669$$ 0 0
$$670$$ 13.6569 0.527610
$$671$$ −5.41421 −0.209013
$$672$$ 0 0
$$673$$ −34.4853 −1.32931 −0.664655 0.747150i $$-0.731421\pi$$
−0.664655 + 0.747150i $$0.731421\pi$$
$$674$$ −7.17157 −0.276239
$$675$$ 0 0
$$676$$ −6.31371 −0.242835
$$677$$ 12.0416 0.462797 0.231399 0.972859i $$-0.425670\pi$$
0.231399 + 0.972859i $$0.425670\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ −4.00000 −0.153393
$$681$$ 0 0
$$682$$ −2.24264 −0.0858752
$$683$$ 44.7696 1.71306 0.856530 0.516098i $$-0.172616\pi$$
0.856530 + 0.516098i $$0.172616\pi$$
$$684$$ 0 0
$$685$$ −34.6274 −1.32305
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0.828427 0.0315835
$$689$$ 16.7696 0.638869
$$690$$ 0 0
$$691$$ −14.1421 −0.537992 −0.268996 0.963141i $$-0.586692\pi$$
−0.268996 + 0.963141i $$0.586692\pi$$
$$692$$ 13.8995 0.528380
$$693$$ 0 0
$$694$$ −31.4558 −1.19405
$$695$$ −39.7990 −1.50966
$$696$$ 0 0
$$697$$ 0.686292 0.0259951
$$698$$ −14.5858 −0.552080
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 13.3137 0.502852 0.251426 0.967877i $$-0.419101\pi$$
0.251426 + 0.967877i $$0.419101\pi$$
$$702$$ 0 0
$$703$$ 30.1421 1.13683
$$704$$ 1.00000 0.0376889
$$705$$ 0 0
$$706$$ 21.2132 0.798369
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 38.7696 1.45602 0.728011 0.685566i $$-0.240445\pi$$
0.728011 + 0.685566i $$0.240445\pi$$
$$710$$ −11.3137 −0.424596
$$711$$ 0 0
$$712$$ −12.2426 −0.458812
$$713$$ −1.85786 −0.0695776
$$714$$ 0 0
$$715$$ −5.17157 −0.193406
$$716$$ 3.51472 0.131351
$$717$$ 0 0
$$718$$ 10.8284 0.404113
$$719$$ 32.1838 1.20025 0.600126 0.799906i $$-0.295117\pi$$
0.600126 + 0.799906i $$0.295117\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ −19.9706 −0.743227
$$723$$ 0 0
$$724$$ −13.3137 −0.494800
$$725$$ −1.65685 −0.0615340
$$726$$ 0 0
$$727$$ 11.2132 0.415875 0.207937 0.978142i $$-0.433325\pi$$
0.207937 + 0.978142i $$0.433325\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0.686292 0.0254008
$$731$$ 1.65685 0.0612810
$$732$$ 0 0
$$733$$ 11.5563 0.426843 0.213422 0.976960i $$-0.431539\pi$$
0.213422 + 0.976960i $$0.431539\pi$$
$$734$$ −35.4142 −1.30716
$$735$$ 0 0
$$736$$ 0.828427 0.0305362
$$737$$ −6.82843 −0.251528
$$738$$ 0 0
$$739$$ −42.6274 −1.56807 −0.784037 0.620714i $$-0.786843\pi$$
−0.784037 + 0.620714i $$0.786843\pi$$
$$740$$ −9.65685 −0.354993
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −26.6274 −0.976865 −0.488433 0.872602i $$-0.662431\pi$$
−0.488433 + 0.872602i $$0.662431\pi$$
$$744$$ 0 0
$$745$$ −8.00000 −0.293097
$$746$$ 31.9411 1.16945
$$747$$ 0 0
$$748$$ 2.00000 0.0731272
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 16.1421 0.589035 0.294517 0.955646i $$-0.404841\pi$$
0.294517 + 0.955646i $$0.404841\pi$$
$$752$$ 11.8995 0.433930
$$753$$ 0 0
$$754$$ 4.28427 0.156024
$$755$$ 8.00000 0.291150
$$756$$ 0 0
$$757$$ −19.8579 −0.721746 −0.360873 0.932615i $$-0.617521\pi$$
−0.360873 + 0.932615i $$0.617521\pi$$
$$758$$ 24.2843 0.882044
$$759$$ 0 0
$$760$$ 12.4853 0.452889
$$761$$ −18.4853 −0.670091 −0.335045 0.942202i $$-0.608752\pi$$
−0.335045 + 0.942202i $$0.608752\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 17.7990 0.643945
$$765$$ 0 0
$$766$$ −2.72792 −0.0985638
$$767$$ −3.02944 −0.109387
$$768$$ 0 0
$$769$$ 44.1421 1.59181 0.795903 0.605424i $$-0.206996\pi$$
0.795903 + 0.605424i $$0.206996\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 18.9706 0.682765
$$773$$ 22.4853 0.808739 0.404370 0.914596i $$-0.367491\pi$$
0.404370 + 0.914596i $$0.367491\pi$$
$$774$$ 0 0
$$775$$ −2.24264 −0.0805580
$$776$$ 9.89949 0.355371
$$777$$ 0 0
$$778$$ 22.2843 0.798930
$$779$$ −2.14214 −0.0767500
$$780$$ 0 0
$$781$$ 5.65685 0.202418
$$782$$ 1.65685 0.0592490
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −14.3431 −0.511929
$$786$$ 0 0
$$787$$ 0.585786 0.0208810 0.0104405 0.999945i $$-0.496677\pi$$
0.0104405 + 0.999945i $$0.496677\pi$$
$$788$$ −4.34315 −0.154718
$$789$$ 0 0
$$790$$ −0.970563 −0.0345311
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 14.0000 0.497155
$$794$$ −16.8284 −0.597219
$$795$$ 0 0
$$796$$ 27.2132 0.964546
$$797$$ 34.4853 1.22153 0.610766 0.791811i $$-0.290862\pi$$
0.610766 + 0.791811i $$0.290862\pi$$
$$798$$ 0 0
$$799$$ 23.7990 0.841948
$$800$$ 1.00000 0.0353553
$$801$$ 0 0
$$802$$ 33.3137 1.17635
$$803$$ −0.343146 −0.0121094
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 5.79899 0.204261
$$807$$ 0 0
$$808$$ 0.928932 0.0326797
$$809$$ 46.7696 1.64433 0.822165 0.569249i $$-0.192766\pi$$
0.822165 + 0.569249i $$0.192766\pi$$
$$810$$ 0 0
$$811$$ 0.384776 0.0135113 0.00675566 0.999977i $$-0.497850\pi$$
0.00675566 + 0.999977i $$0.497850\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 4.82843 0.169236
$$815$$ −27.3137 −0.956757
$$816$$ 0 0
$$817$$ −5.17157 −0.180930
$$818$$ −18.4853 −0.646323
$$819$$ 0 0
$$820$$ 0.686292 0.0239663
$$821$$ −24.6863 −0.861558 −0.430779 0.902458i $$-0.641761\pi$$
−0.430779 + 0.902458i $$0.641761\pi$$
$$822$$ 0 0
$$823$$ 0.142136 0.00495454 0.00247727 0.999997i $$-0.499211\pi$$
0.00247727 + 0.999997i $$0.499211\pi$$
$$824$$ −2.24264 −0.0781261
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −16.6863 −0.580239 −0.290120 0.956990i $$-0.593695\pi$$
−0.290120 + 0.956990i $$0.593695\pi$$
$$828$$ 0 0
$$829$$ 6.28427 0.218262 0.109131 0.994027i $$-0.465193\pi$$
0.109131 + 0.994027i $$0.465193\pi$$
$$830$$ 18.1421 0.629723
$$831$$ 0 0
$$832$$ −2.58579 −0.0896460
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −21.6569 −0.749466
$$836$$ −6.24264 −0.215906
$$837$$ 0 0
$$838$$ −16.0000 −0.552711
$$839$$ −2.92893 −0.101118 −0.0505590 0.998721i $$-0.516100\pi$$
−0.0505590 + 0.998721i $$0.516100\pi$$
$$840$$ 0 0
$$841$$ −26.2548 −0.905339
$$842$$ 16.3431 0.563222
$$843$$ 0 0
$$844$$ −28.8284 −0.992315
$$845$$ −12.6274 −0.434396
$$846$$ 0 0
$$847$$ 0 0
$$848$$ −6.48528 −0.222705
$$849$$ 0 0
$$850$$ 2.00000 0.0685994
$$851$$ 4.00000 0.137118
$$852$$ 0 0
$$853$$ −4.44365 −0.152148 −0.0760739 0.997102i $$-0.524238\pi$$
−0.0760739 + 0.997102i $$0.524238\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 7.17157 0.245119
$$857$$ −28.1421 −0.961317 −0.480659 0.876908i $$-0.659602\pi$$
−0.480659 + 0.876908i $$0.659602\pi$$
$$858$$ 0 0
$$859$$ −54.4264 −1.85701 −0.928503 0.371326i $$-0.878903\pi$$
−0.928503 + 0.371326i $$0.878903\pi$$
$$860$$ 1.65685 0.0564983
$$861$$ 0 0
$$862$$ −26.1421 −0.890405
$$863$$ −34.7696 −1.18357 −0.591785 0.806096i $$-0.701576\pi$$
−0.591785 + 0.806096i $$0.701576\pi$$
$$864$$ 0 0
$$865$$ 27.7990 0.945194
$$866$$ 2.38478 0.0810380
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 0.485281 0.0164620
$$870$$ 0 0
$$871$$ 17.6569 0.598280
$$872$$ −15.6569 −0.530208
$$873$$ 0 0
$$874$$ −5.17157 −0.174931
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −29.3137 −0.989854 −0.494927 0.868935i $$-0.664805\pi$$
−0.494927 + 0.868935i $$0.664805\pi$$
$$878$$ −34.8284 −1.17540
$$879$$ 0 0
$$880$$ 2.00000 0.0674200
$$881$$ 18.1005 0.609822 0.304911 0.952381i $$-0.401373\pi$$
0.304911 + 0.952381i $$0.401373\pi$$
$$882$$ 0 0
$$883$$ 34.6274 1.16531 0.582653 0.812721i $$-0.302015\pi$$
0.582653 + 0.812721i $$0.302015\pi$$
$$884$$ −5.17157 −0.173939
$$885$$ 0 0
$$886$$ −4.48528 −0.150686
$$887$$ −52.2843 −1.75553 −0.877767 0.479088i $$-0.840968\pi$$
−0.877767 + 0.479088i $$0.840968\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ −24.4853 −0.820748
$$891$$ 0 0
$$892$$ 2.24264 0.0750892
$$893$$ −74.2843 −2.48583
$$894$$ 0 0
$$895$$ 7.02944 0.234968
$$896$$ 0 0
$$897$$ 0 0
$$898$$ −16.9706 −0.566315
$$899$$ 3.71573 0.123926
$$900$$ 0 0
$$901$$ −12.9706 −0.432112
$$902$$ −0.343146 −0.0114255
$$903$$ 0 0
$$904$$ −5.65685 −0.188144
$$905$$ −26.6274 −0.885125
$$906$$ 0 0
$$907$$ −47.7990 −1.58714 −0.793570 0.608479i $$-0.791780\pi$$
−0.793570 + 0.608479i $$0.791780\pi$$
$$908$$ 7.89949 0.262154
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −39.4558 −1.30723 −0.653615 0.756827i $$-0.726749\pi$$
−0.653615 + 0.756827i $$0.726749\pi$$
$$912$$ 0 0
$$913$$ −9.07107 −0.300209
$$914$$ 6.68629 0.221163
$$915$$ 0 0
$$916$$ −14.0000 −0.462573
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 32.9706 1.08760 0.543799 0.839215i $$-0.316985\pi$$
0.543799 + 0.839215i $$0.316985\pi$$
$$920$$ 1.65685 0.0546249
$$921$$ 0 0
$$922$$ 33.4142 1.10044
$$923$$ −14.6274 −0.481467
$$924$$ 0 0
$$925$$ 4.82843 0.158758
$$926$$ −19.3137 −0.634688
$$927$$ 0 0
$$928$$ −1.65685 −0.0543889
$$929$$ 53.2132 1.74587 0.872934 0.487838i $$-0.162214\pi$$
0.872934 + 0.487838i $$0.162214\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ −3.65685 −0.119784
$$933$$ 0 0
$$934$$ −12.0000 −0.392652
$$935$$ 4.00000 0.130814
$$936$$ 0 0
$$937$$ −44.9117 −1.46720 −0.733600 0.679581i $$-0.762162\pi$$
−0.733600 + 0.679581i $$0.762162\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 23.7990 0.776237
$$941$$ −3.55635 −0.115934 −0.0579668 0.998319i $$-0.518462\pi$$
−0.0579668 + 0.998319i $$0.518462\pi$$
$$942$$ 0 0
$$943$$ −0.284271 −0.00925715
$$944$$ 1.17157 0.0381314
$$945$$ 0 0
$$946$$ −0.828427 −0.0269345
$$947$$ −60.5685 −1.96821 −0.984107 0.177579i $$-0.943174\pi$$
−0.984107 + 0.177579i $$0.943174\pi$$
$$948$$ 0 0
$$949$$ 0.887302 0.0288030
$$950$$ −6.24264 −0.202538
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −26.4853 −0.857942 −0.428971 0.903318i $$-0.641124\pi$$
−0.428971 + 0.903318i $$0.641124\pi$$
$$954$$ 0 0
$$955$$ 35.5980 1.15192
$$956$$ −23.3137 −0.754019
$$957$$ 0 0
$$958$$ 15.3137 0.494763
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −25.9706 −0.837760
$$962$$ −12.4853 −0.402542
$$963$$ 0 0
$$964$$ −3.65685 −0.117779
$$965$$ 37.9411 1.22137
$$966$$ 0 0
$$967$$ 57.9411 1.86326 0.931630 0.363407i $$-0.118387\pi$$
0.931630 + 0.363407i $$0.118387\pi$$
$$968$$ −1.00000 −0.0321412
$$969$$ 0 0
$$970$$ 19.7990 0.635707
$$971$$ 27.3137 0.876539 0.438269 0.898844i $$-0.355592\pi$$
0.438269 + 0.898844i $$0.355592\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 28.1421 0.901732
$$975$$ 0 0
$$976$$ −5.41421 −0.173305
$$977$$ 18.6274 0.595944 0.297972 0.954575i $$-0.403690\pi$$
0.297972 + 0.954575i $$0.403690\pi$$
$$978$$ 0 0
$$979$$ 12.2426 0.391276
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 2.62742 0.0838442
$$983$$ −52.8701 −1.68629 −0.843146 0.537684i $$-0.819299\pi$$
−0.843146 + 0.537684i $$0.819299\pi$$
$$984$$ 0 0
$$985$$ −8.68629 −0.276768
$$986$$ −3.31371 −0.105530
$$987$$ 0 0
$$988$$ 16.1421 0.513550
$$989$$ −0.686292 −0.0218228
$$990$$ 0 0
$$991$$ −50.9117 −1.61726 −0.808632 0.588315i $$-0.799791\pi$$
−0.808632 + 0.588315i $$0.799791\pi$$
$$992$$ −2.24264 −0.0712039
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 54.4264 1.72543
$$996$$ 0 0
$$997$$ −33.4142 −1.05824 −0.529119 0.848547i $$-0.677478\pi$$
−0.529119 + 0.848547i $$0.677478\pi$$
$$998$$ −22.1421 −0.700897
$$999$$ 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.cy.1.2 2
3.2 odd 2 3234.2.a.bd.1.2 yes 2
7.6 odd 2 9702.2.a.ci.1.1 2
21.20 even 2 3234.2.a.bc.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
3234.2.a.bc.1.1 2 21.20 even 2
3234.2.a.bd.1.2 yes 2 3.2 odd 2
9702.2.a.ci.1.1 2 7.6 odd 2
9702.2.a.cy.1.2 2 1.1 even 1 trivial