# Properties

 Label 4650.2.a.bz.1.1 Level $4650$ Weight $2$ Character 4650.1 Self dual yes Analytic conductor $37.130$ 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] = [4650,2,Mod(1,4650)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4650, 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("4650.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4650.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: $$37.1304369399$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{65})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 16$$ x^2 - x - 16 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 930) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-3.53113$$ of defining polynomial Character $$\chi$$ $$=$$ 4650.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -3.53113 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -3.53113 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.53113 q^{11} -1.00000 q^{12} -6.00000 q^{13} +3.53113 q^{14} +1.00000 q^{16} +4.00000 q^{17} -1.00000 q^{18} -3.53113 q^{19} +3.53113 q^{21} +1.53113 q^{22} +1.53113 q^{23} +1.00000 q^{24} +6.00000 q^{26} -1.00000 q^{27} -3.53113 q^{28} +1.00000 q^{31} -1.00000 q^{32} +1.53113 q^{33} -4.00000 q^{34} +1.00000 q^{36} -9.06226 q^{37} +3.53113 q^{38} +6.00000 q^{39} -9.06226 q^{41} -3.53113 q^{42} +0.468871 q^{43} -1.53113 q^{44} -1.53113 q^{46} -11.0623 q^{47} -1.00000 q^{48} +5.46887 q^{49} -4.00000 q^{51} -6.00000 q^{52} -5.53113 q^{53} +1.00000 q^{54} +3.53113 q^{56} +3.53113 q^{57} +7.06226 q^{59} -11.0623 q^{61} -1.00000 q^{62} -3.53113 q^{63} +1.00000 q^{64} -1.53113 q^{66} +11.0623 q^{67} +4.00000 q^{68} -1.53113 q^{69} -4.46887 q^{71} -1.00000 q^{72} +0.468871 q^{73} +9.06226 q^{74} -3.53113 q^{76} +5.40661 q^{77} -6.00000 q^{78} -0.468871 q^{79} +1.00000 q^{81} +9.06226 q^{82} +8.00000 q^{83} +3.53113 q^{84} -0.468871 q^{86} +1.53113 q^{88} +1.53113 q^{89} +21.1868 q^{91} +1.53113 q^{92} -1.00000 q^{93} +11.0623 q^{94} +1.00000 q^{96} +16.1245 q^{97} -5.46887 q^{98} -1.53113 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} + q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 + q^7 - 2 * q^8 + 2 * q^9 $$2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} + q^{7} - 2 q^{8} + 2 q^{9} + 5 q^{11} - 2 q^{12} - 12 q^{13} - q^{14} + 2 q^{16} + 8 q^{17} - 2 q^{18} + q^{19} - q^{21} - 5 q^{22} - 5 q^{23} + 2 q^{24} + 12 q^{26} - 2 q^{27} + q^{28} + 2 q^{31} - 2 q^{32} - 5 q^{33} - 8 q^{34} + 2 q^{36} - 2 q^{37} - q^{38} + 12 q^{39} - 2 q^{41} + q^{42} + 9 q^{43} + 5 q^{44} + 5 q^{46} - 6 q^{47} - 2 q^{48} + 19 q^{49} - 8 q^{51} - 12 q^{52} - 3 q^{53} + 2 q^{54} - q^{56} - q^{57} - 2 q^{59} - 6 q^{61} - 2 q^{62} + q^{63} + 2 q^{64} + 5 q^{66} + 6 q^{67} + 8 q^{68} + 5 q^{69} - 17 q^{71} - 2 q^{72} + 9 q^{73} + 2 q^{74} + q^{76} + 35 q^{77} - 12 q^{78} - 9 q^{79} + 2 q^{81} + 2 q^{82} + 16 q^{83} - q^{84} - 9 q^{86} - 5 q^{88} - 5 q^{89} - 6 q^{91} - 5 q^{92} - 2 q^{93} + 6 q^{94} + 2 q^{96} - 19 q^{98} + 5 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 + q^7 - 2 * q^8 + 2 * q^9 + 5 * q^11 - 2 * q^12 - 12 * q^13 - q^14 + 2 * q^16 + 8 * q^17 - 2 * q^18 + q^19 - q^21 - 5 * q^22 - 5 * q^23 + 2 * q^24 + 12 * q^26 - 2 * q^27 + q^28 + 2 * q^31 - 2 * q^32 - 5 * q^33 - 8 * q^34 + 2 * q^36 - 2 * q^37 - q^38 + 12 * q^39 - 2 * q^41 + q^42 + 9 * q^43 + 5 * q^44 + 5 * q^46 - 6 * q^47 - 2 * q^48 + 19 * q^49 - 8 * q^51 - 12 * q^52 - 3 * q^53 + 2 * q^54 - q^56 - q^57 - 2 * q^59 - 6 * q^61 - 2 * q^62 + q^63 + 2 * q^64 + 5 * q^66 + 6 * q^67 + 8 * q^68 + 5 * q^69 - 17 * q^71 - 2 * q^72 + 9 * q^73 + 2 * q^74 + q^76 + 35 * q^77 - 12 * q^78 - 9 * q^79 + 2 * q^81 + 2 * q^82 + 16 * q^83 - q^84 - 9 * q^86 - 5 * q^88 - 5 * q^89 - 6 * q^91 - 5 * q^92 - 2 * q^93 + 6 * q^94 + 2 * q^96 - 19 * q^98 + 5 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ −1.00000 −0.577350
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ 1.00000 0.408248
$$7$$ −3.53113 −1.33464 −0.667321 0.744771i $$-0.732559\pi$$
−0.667321 + 0.744771i $$0.732559\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −1.53113 −0.461653 −0.230826 0.972995i $$-0.574143\pi$$
−0.230826 + 0.972995i $$0.574143\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ −6.00000 −1.66410 −0.832050 0.554700i $$-0.812833\pi$$
−0.832050 + 0.554700i $$0.812833\pi$$
$$14$$ 3.53113 0.943734
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 4.00000 0.970143 0.485071 0.874475i $$-0.338794\pi$$
0.485071 + 0.874475i $$0.338794\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ −3.53113 −0.810097 −0.405048 0.914295i $$-0.632745\pi$$
−0.405048 + 0.914295i $$0.632745\pi$$
$$20$$ 0 0
$$21$$ 3.53113 0.770555
$$22$$ 1.53113 0.326438
$$23$$ 1.53113 0.319262 0.159631 0.987177i $$-0.448969\pi$$
0.159631 + 0.987177i $$0.448969\pi$$
$$24$$ 1.00000 0.204124
$$25$$ 0 0
$$26$$ 6.00000 1.17670
$$27$$ −1.00000 −0.192450
$$28$$ −3.53113 −0.667321
$$29$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$30$$ 0 0
$$31$$ 1.00000 0.179605
$$32$$ −1.00000 −0.176777
$$33$$ 1.53113 0.266535
$$34$$ −4.00000 −0.685994
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ −9.06226 −1.48983 −0.744913 0.667162i $$-0.767509\pi$$
−0.744913 + 0.667162i $$0.767509\pi$$
$$38$$ 3.53113 0.572825
$$39$$ 6.00000 0.960769
$$40$$ 0 0
$$41$$ −9.06226 −1.41529 −0.707643 0.706570i $$-0.750242\pi$$
−0.707643 + 0.706570i $$0.750242\pi$$
$$42$$ −3.53113 −0.544865
$$43$$ 0.468871 0.0715022 0.0357511 0.999361i $$-0.488618\pi$$
0.0357511 + 0.999361i $$0.488618\pi$$
$$44$$ −1.53113 −0.230826
$$45$$ 0 0
$$46$$ −1.53113 −0.225753
$$47$$ −11.0623 −1.61360 −0.806798 0.590827i $$-0.798801\pi$$
−0.806798 + 0.590827i $$0.798801\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ 5.46887 0.781267
$$50$$ 0 0
$$51$$ −4.00000 −0.560112
$$52$$ −6.00000 −0.832050
$$53$$ −5.53113 −0.759759 −0.379879 0.925036i $$-0.624035\pi$$
−0.379879 + 0.925036i $$0.624035\pi$$
$$54$$ 1.00000 0.136083
$$55$$ 0 0
$$56$$ 3.53113 0.471867
$$57$$ 3.53113 0.467709
$$58$$ 0 0
$$59$$ 7.06226 0.919428 0.459714 0.888067i $$-0.347952\pi$$
0.459714 + 0.888067i $$0.347952\pi$$
$$60$$ 0 0
$$61$$ −11.0623 −1.41638 −0.708188 0.706023i $$-0.750487\pi$$
−0.708188 + 0.706023i $$0.750487\pi$$
$$62$$ −1.00000 −0.127000
$$63$$ −3.53113 −0.444880
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ −1.53113 −0.188469
$$67$$ 11.0623 1.35147 0.675735 0.737145i $$-0.263826\pi$$
0.675735 + 0.737145i $$0.263826\pi$$
$$68$$ 4.00000 0.485071
$$69$$ −1.53113 −0.184326
$$70$$ 0 0
$$71$$ −4.46887 −0.530357 −0.265179 0.964199i $$-0.585431\pi$$
−0.265179 + 0.964199i $$0.585431\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ 0.468871 0.0548772 0.0274386 0.999623i $$-0.491265\pi$$
0.0274386 + 0.999623i $$0.491265\pi$$
$$74$$ 9.06226 1.05347
$$75$$ 0 0
$$76$$ −3.53113 −0.405048
$$77$$ 5.40661 0.616141
$$78$$ −6.00000 −0.679366
$$79$$ −0.468871 −0.0527521 −0.0263761 0.999652i $$-0.508397\pi$$
−0.0263761 + 0.999652i $$0.508397\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 9.06226 1.00076
$$83$$ 8.00000 0.878114 0.439057 0.898459i $$-0.355313\pi$$
0.439057 + 0.898459i $$0.355313\pi$$
$$84$$ 3.53113 0.385278
$$85$$ 0 0
$$86$$ −0.468871 −0.0505597
$$87$$ 0 0
$$88$$ 1.53113 0.163219
$$89$$ 1.53113 0.162299 0.0811497 0.996702i $$-0.474141\pi$$
0.0811497 + 0.996702i $$0.474141\pi$$
$$90$$ 0 0
$$91$$ 21.1868 2.22098
$$92$$ 1.53113 0.159631
$$93$$ −1.00000 −0.103695
$$94$$ 11.0623 1.14098
$$95$$ 0 0
$$96$$ 1.00000 0.102062
$$97$$ 16.1245 1.63720 0.818598 0.574367i $$-0.194752\pi$$
0.818598 + 0.574367i $$0.194752\pi$$
$$98$$ −5.46887 −0.552439
$$99$$ −1.53113 −0.153884
$$100$$ 0 0
$$101$$ −17.5311 −1.74441 −0.872206 0.489138i $$-0.837311\pi$$
−0.872206 + 0.489138i $$0.837311\pi$$
$$102$$ 4.00000 0.396059
$$103$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$104$$ 6.00000 0.588348
$$105$$ 0 0
$$106$$ 5.53113 0.537231
$$107$$ 14.5934 1.41080 0.705398 0.708811i $$-0.250768\pi$$
0.705398 + 0.708811i $$0.250768\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ −1.06226 −0.101746 −0.0508729 0.998705i $$-0.516200\pi$$
−0.0508729 + 0.998705i $$0.516200\pi$$
$$110$$ 0 0
$$111$$ 9.06226 0.860151
$$112$$ −3.53113 −0.333660
$$113$$ −16.5934 −1.56097 −0.780487 0.625172i $$-0.785029\pi$$
−0.780487 + 0.625172i $$0.785029\pi$$
$$114$$ −3.53113 −0.330721
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −6.00000 −0.554700
$$118$$ −7.06226 −0.650134
$$119$$ −14.1245 −1.29479
$$120$$ 0 0
$$121$$ −8.65564 −0.786877
$$122$$ 11.0623 1.00153
$$123$$ 9.06226 0.817116
$$124$$ 1.00000 0.0898027
$$125$$ 0 0
$$126$$ 3.53113 0.314578
$$127$$ 9.06226 0.804145 0.402073 0.915608i $$-0.368290\pi$$
0.402073 + 0.915608i $$0.368290\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ −0.468871 −0.0412818
$$130$$ 0 0
$$131$$ 7.06226 0.617032 0.308516 0.951219i $$-0.400168\pi$$
0.308516 + 0.951219i $$0.400168\pi$$
$$132$$ 1.53113 0.133268
$$133$$ 12.4689 1.08119
$$134$$ −11.0623 −0.955634
$$135$$ 0 0
$$136$$ −4.00000 −0.342997
$$137$$ −0.937742 −0.0801167 −0.0400584 0.999197i $$-0.512754\pi$$
−0.0400584 + 0.999197i $$0.512754\pi$$
$$138$$ 1.53113 0.130338
$$139$$ 6.00000 0.508913 0.254457 0.967084i $$-0.418103\pi$$
0.254457 + 0.967084i $$0.418103\pi$$
$$140$$ 0 0
$$141$$ 11.0623 0.931610
$$142$$ 4.46887 0.375019
$$143$$ 9.18677 0.768237
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ −0.468871 −0.0388041
$$147$$ −5.46887 −0.451065
$$148$$ −9.06226 −0.744913
$$149$$ −10.4689 −0.857643 −0.428822 0.903389i $$-0.641071\pi$$
−0.428822 + 0.903389i $$0.641071\pi$$
$$150$$ 0 0
$$151$$ −18.1245 −1.47495 −0.737476 0.675373i $$-0.763983\pi$$
−0.737476 + 0.675373i $$0.763983\pi$$
$$152$$ 3.53113 0.286412
$$153$$ 4.00000 0.323381
$$154$$ −5.40661 −0.435677
$$155$$ 0 0
$$156$$ 6.00000 0.480384
$$157$$ 14.4689 1.15474 0.577371 0.816482i $$-0.304079\pi$$
0.577371 + 0.816482i $$0.304079\pi$$
$$158$$ 0.468871 0.0373014
$$159$$ 5.53113 0.438647
$$160$$ 0 0
$$161$$ −5.40661 −0.426101
$$162$$ −1.00000 −0.0785674
$$163$$ 11.0623 0.866463 0.433231 0.901283i $$-0.357373\pi$$
0.433231 + 0.901283i $$0.357373\pi$$
$$164$$ −9.06226 −0.707643
$$165$$ 0 0
$$166$$ −8.00000 −0.620920
$$167$$ −0.593387 −0.0459176 −0.0229588 0.999736i $$-0.507309\pi$$
−0.0229588 + 0.999736i $$0.507309\pi$$
$$168$$ −3.53113 −0.272433
$$169$$ 23.0000 1.76923
$$170$$ 0 0
$$171$$ −3.53113 −0.270032
$$172$$ 0.468871 0.0357511
$$173$$ −14.0000 −1.06440 −0.532200 0.846619i $$-0.678635\pi$$
−0.532200 + 0.846619i $$0.678635\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −1.53113 −0.115413
$$177$$ −7.06226 −0.530832
$$178$$ −1.53113 −0.114763
$$179$$ 13.0623 0.976319 0.488159 0.872754i $$-0.337668\pi$$
0.488159 + 0.872754i $$0.337668\pi$$
$$180$$ 0 0
$$181$$ 26.5934 1.97667 0.988335 0.152293i $$-0.0486657\pi$$
0.988335 + 0.152293i $$0.0486657\pi$$
$$182$$ −21.1868 −1.57047
$$183$$ 11.0623 0.817746
$$184$$ −1.53113 −0.112876
$$185$$ 0 0
$$186$$ 1.00000 0.0733236
$$187$$ −6.12452 −0.447869
$$188$$ −11.0623 −0.806798
$$189$$ 3.53113 0.256852
$$190$$ 0 0
$$191$$ −8.00000 −0.578860 −0.289430 0.957199i $$-0.593466\pi$$
−0.289430 + 0.957199i $$0.593466\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ −14.0000 −1.00774 −0.503871 0.863779i $$-0.668091\pi$$
−0.503871 + 0.863779i $$0.668091\pi$$
$$194$$ −16.1245 −1.15767
$$195$$ 0 0
$$196$$ 5.46887 0.390634
$$197$$ −6.00000 −0.427482 −0.213741 0.976890i $$-0.568565\pi$$
−0.213741 + 0.976890i $$0.568565\pi$$
$$198$$ 1.53113 0.108813
$$199$$ −0.468871 −0.0332374 −0.0166187 0.999862i $$-0.505290\pi$$
−0.0166187 + 0.999862i $$0.505290\pi$$
$$200$$ 0 0
$$201$$ −11.0623 −0.780272
$$202$$ 17.5311 1.23349
$$203$$ 0 0
$$204$$ −4.00000 −0.280056
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 1.53113 0.106421
$$208$$ −6.00000 −0.416025
$$209$$ 5.40661 0.373983
$$210$$ 0 0
$$211$$ 22.5934 1.55539 0.777696 0.628640i $$-0.216388\pi$$
0.777696 + 0.628640i $$0.216388\pi$$
$$212$$ −5.53113 −0.379879
$$213$$ 4.46887 0.306202
$$214$$ −14.5934 −0.997583
$$215$$ 0 0
$$216$$ 1.00000 0.0680414
$$217$$ −3.53113 −0.239709
$$218$$ 1.06226 0.0719452
$$219$$ −0.468871 −0.0316834
$$220$$ 0 0
$$221$$ −24.0000 −1.61441
$$222$$ −9.06226 −0.608219
$$223$$ −2.00000 −0.133930 −0.0669650 0.997755i $$-0.521332\pi$$
−0.0669650 + 0.997755i $$0.521332\pi$$
$$224$$ 3.53113 0.235933
$$225$$ 0 0
$$226$$ 16.5934 1.10378
$$227$$ 16.4689 1.09308 0.546539 0.837434i $$-0.315945\pi$$
0.546539 + 0.837434i $$0.315945\pi$$
$$228$$ 3.53113 0.233855
$$229$$ 18.5934 1.22869 0.614343 0.789039i $$-0.289421\pi$$
0.614343 + 0.789039i $$0.289421\pi$$
$$230$$ 0 0
$$231$$ −5.40661 −0.355729
$$232$$ 0 0
$$233$$ −9.53113 −0.624405 −0.312203 0.950016i $$-0.601067\pi$$
−0.312203 + 0.950016i $$0.601067\pi$$
$$234$$ 6.00000 0.392232
$$235$$ 0 0
$$236$$ 7.06226 0.459714
$$237$$ 0.468871 0.0304565
$$238$$ 14.1245 0.915556
$$239$$ −8.00000 −0.517477 −0.258738 0.965947i $$-0.583307\pi$$
−0.258738 + 0.965947i $$0.583307\pi$$
$$240$$ 0 0
$$241$$ 2.00000 0.128831 0.0644157 0.997923i $$-0.479482\pi$$
0.0644157 + 0.997923i $$0.479482\pi$$
$$242$$ 8.65564 0.556406
$$243$$ −1.00000 −0.0641500
$$244$$ −11.0623 −0.708188
$$245$$ 0 0
$$246$$ −9.06226 −0.577788
$$247$$ 21.1868 1.34808
$$248$$ −1.00000 −0.0635001
$$249$$ −8.00000 −0.506979
$$250$$ 0 0
$$251$$ −13.0623 −0.824482 −0.412241 0.911075i $$-0.635254\pi$$
−0.412241 + 0.911075i $$0.635254\pi$$
$$252$$ −3.53113 −0.222440
$$253$$ −2.34436 −0.147388
$$254$$ −9.06226 −0.568617
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 27.6556 1.72511 0.862556 0.505962i $$-0.168862\pi$$
0.862556 + 0.505962i $$0.168862\pi$$
$$258$$ 0.468871 0.0291906
$$259$$ 32.0000 1.98838
$$260$$ 0 0
$$261$$ 0 0
$$262$$ −7.06226 −0.436308
$$263$$ 6.93774 0.427800 0.213900 0.976856i $$-0.431383\pi$$
0.213900 + 0.976856i $$0.431383\pi$$
$$264$$ −1.53113 −0.0942345
$$265$$ 0 0
$$266$$ −12.4689 −0.764516
$$267$$ −1.53113 −0.0937036
$$268$$ 11.0623 0.675735
$$269$$ 29.1868 1.77955 0.889774 0.456400i $$-0.150862\pi$$
0.889774 + 0.456400i $$0.150862\pi$$
$$270$$ 0 0
$$271$$ −19.5311 −1.18643 −0.593216 0.805043i $$-0.702142\pi$$
−0.593216 + 0.805043i $$0.702142\pi$$
$$272$$ 4.00000 0.242536
$$273$$ −21.1868 −1.28228
$$274$$ 0.937742 0.0566511
$$275$$ 0 0
$$276$$ −1.53113 −0.0921631
$$277$$ −1.06226 −0.0638249 −0.0319124 0.999491i $$-0.510160\pi$$
−0.0319124 + 0.999491i $$0.510160\pi$$
$$278$$ −6.00000 −0.359856
$$279$$ 1.00000 0.0598684
$$280$$ 0 0
$$281$$ −1.06226 −0.0633690 −0.0316845 0.999498i $$-0.510087\pi$$
−0.0316845 + 0.999498i $$0.510087\pi$$
$$282$$ −11.0623 −0.658748
$$283$$ 12.0000 0.713326 0.356663 0.934233i $$-0.383914\pi$$
0.356663 + 0.934233i $$0.383914\pi$$
$$284$$ −4.46887 −0.265179
$$285$$ 0 0
$$286$$ −9.18677 −0.543225
$$287$$ 32.0000 1.88890
$$288$$ −1.00000 −0.0589256
$$289$$ −1.00000 −0.0588235
$$290$$ 0 0
$$291$$ −16.1245 −0.945236
$$292$$ 0.468871 0.0274386
$$293$$ 14.0000 0.817889 0.408944 0.912559i $$-0.365897\pi$$
0.408944 + 0.912559i $$0.365897\pi$$
$$294$$ 5.46887 0.318951
$$295$$ 0 0
$$296$$ 9.06226 0.526733
$$297$$ 1.53113 0.0888451
$$298$$ 10.4689 0.606445
$$299$$ −9.18677 −0.531285
$$300$$ 0 0
$$301$$ −1.65564 −0.0954298
$$302$$ 18.1245 1.04295
$$303$$ 17.5311 1.00714
$$304$$ −3.53113 −0.202524
$$305$$ 0 0
$$306$$ −4.00000 −0.228665
$$307$$ −2.12452 −0.121253 −0.0606263 0.998161i $$-0.519310\pi$$
−0.0606263 + 0.998161i $$0.519310\pi$$
$$308$$ 5.40661 0.308070
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −8.00000 −0.453638 −0.226819 0.973937i $$-0.572833\pi$$
−0.226819 + 0.973937i $$0.572833\pi$$
$$312$$ −6.00000 −0.339683
$$313$$ −27.0623 −1.52965 −0.764825 0.644239i $$-0.777174\pi$$
−0.764825 + 0.644239i $$0.777174\pi$$
$$314$$ −14.4689 −0.816526
$$315$$ 0 0
$$316$$ −0.468871 −0.0263761
$$317$$ −29.0623 −1.63230 −0.816150 0.577841i $$-0.803895\pi$$
−0.816150 + 0.577841i $$0.803895\pi$$
$$318$$ −5.53113 −0.310170
$$319$$ 0 0
$$320$$ 0 0
$$321$$ −14.5934 −0.814523
$$322$$ 5.40661 0.301299
$$323$$ −14.1245 −0.785909
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ −11.0623 −0.612682
$$327$$ 1.06226 0.0587430
$$328$$ 9.06226 0.500379
$$329$$ 39.0623 2.15357
$$330$$ 0 0
$$331$$ 29.0623 1.59741 0.798703 0.601725i $$-0.205520\pi$$
0.798703 + 0.601725i $$0.205520\pi$$
$$332$$ 8.00000 0.439057
$$333$$ −9.06226 −0.496609
$$334$$ 0.593387 0.0324687
$$335$$ 0 0
$$336$$ 3.53113 0.192639
$$337$$ −29.1868 −1.58990 −0.794952 0.606672i $$-0.792504\pi$$
−0.794952 + 0.606672i $$0.792504\pi$$
$$338$$ −23.0000 −1.25104
$$339$$ 16.5934 0.901229
$$340$$ 0 0
$$341$$ −1.53113 −0.0829153
$$342$$ 3.53113 0.190942
$$343$$ 5.40661 0.291930
$$344$$ −0.468871 −0.0252798
$$345$$ 0 0
$$346$$ 14.0000 0.752645
$$347$$ 22.1245 1.18771 0.593853 0.804573i $$-0.297606\pi$$
0.593853 + 0.804573i $$0.297606\pi$$
$$348$$ 0 0
$$349$$ 25.0623 1.34155 0.670776 0.741660i $$-0.265961\pi$$
0.670776 + 0.741660i $$0.265961\pi$$
$$350$$ 0 0
$$351$$ 6.00000 0.320256
$$352$$ 1.53113 0.0816094
$$353$$ 23.0623 1.22748 0.613740 0.789508i $$-0.289664\pi$$
0.613740 + 0.789508i $$0.289664\pi$$
$$354$$ 7.06226 0.375355
$$355$$ 0 0
$$356$$ 1.53113 0.0811497
$$357$$ 14.1245 0.747549
$$358$$ −13.0623 −0.690362
$$359$$ 11.5311 0.608590 0.304295 0.952578i $$-0.401579\pi$$
0.304295 + 0.952578i $$0.401579\pi$$
$$360$$ 0 0
$$361$$ −6.53113 −0.343744
$$362$$ −26.5934 −1.39772
$$363$$ 8.65564 0.454304
$$364$$ 21.1868 1.11049
$$365$$ 0 0
$$366$$ −11.0623 −0.578233
$$367$$ 10.0000 0.521996 0.260998 0.965339i $$-0.415948\pi$$
0.260998 + 0.965339i $$0.415948\pi$$
$$368$$ 1.53113 0.0798156
$$369$$ −9.06226 −0.471762
$$370$$ 0 0
$$371$$ 19.5311 1.01401
$$372$$ −1.00000 −0.0518476
$$373$$ 10.4689 0.542058 0.271029 0.962571i $$-0.412636\pi$$
0.271029 + 0.962571i $$0.412636\pi$$
$$374$$ 6.12452 0.316691
$$375$$ 0 0
$$376$$ 11.0623 0.570492
$$377$$ 0 0
$$378$$ −3.53113 −0.181622
$$379$$ −2.59339 −0.133213 −0.0666067 0.997779i $$-0.521217\pi$$
−0.0666067 + 0.997779i $$0.521217\pi$$
$$380$$ 0 0
$$381$$ −9.06226 −0.464274
$$382$$ 8.00000 0.409316
$$383$$ −13.0623 −0.667450 −0.333725 0.942670i $$-0.608306\pi$$
−0.333725 + 0.942670i $$0.608306\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 0 0
$$386$$ 14.0000 0.712581
$$387$$ 0.468871 0.0238341
$$388$$ 16.1245 0.818598
$$389$$ 27.0623 1.37211 0.686055 0.727549i $$-0.259341\pi$$
0.686055 + 0.727549i $$0.259341\pi$$
$$390$$ 0 0
$$391$$ 6.12452 0.309730
$$392$$ −5.46887 −0.276220
$$393$$ −7.06226 −0.356244
$$394$$ 6.00000 0.302276
$$395$$ 0 0
$$396$$ −1.53113 −0.0769421
$$397$$ 18.7179 0.939425 0.469712 0.882820i $$-0.344358\pi$$
0.469712 + 0.882820i $$0.344358\pi$$
$$398$$ 0.468871 0.0235024
$$399$$ −12.4689 −0.624224
$$400$$ 0 0
$$401$$ −34.7179 −1.73373 −0.866865 0.498544i $$-0.833868\pi$$
−0.866865 + 0.498544i $$0.833868\pi$$
$$402$$ 11.0623 0.551735
$$403$$ −6.00000 −0.298881
$$404$$ −17.5311 −0.872206
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 13.8755 0.687782
$$408$$ 4.00000 0.198030
$$409$$ −9.06226 −0.448100 −0.224050 0.974578i $$-0.571928\pi$$
−0.224050 + 0.974578i $$0.571928\pi$$
$$410$$ 0 0
$$411$$ 0.937742 0.0462554
$$412$$ 0 0
$$413$$ −24.9377 −1.22711
$$414$$ −1.53113 −0.0752509
$$415$$ 0 0
$$416$$ 6.00000 0.294174
$$417$$ −6.00000 −0.293821
$$418$$ −5.40661 −0.264446
$$419$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$420$$ 0 0
$$421$$ 38.2490 1.86414 0.932072 0.362273i $$-0.117999\pi$$
0.932072 + 0.362273i $$0.117999\pi$$
$$422$$ −22.5934 −1.09983
$$423$$ −11.0623 −0.537865
$$424$$ 5.53113 0.268615
$$425$$ 0 0
$$426$$ −4.46887 −0.216518
$$427$$ 39.0623 1.89036
$$428$$ 14.5934 0.705398
$$429$$ −9.18677 −0.443542
$$430$$ 0 0
$$431$$ −24.0000 −1.15604 −0.578020 0.816023i $$-0.696174\pi$$
−0.578020 + 0.816023i $$0.696174\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ 1.40661 0.0675975 0.0337988 0.999429i $$-0.489239\pi$$
0.0337988 + 0.999429i $$0.489239\pi$$
$$434$$ 3.53113 0.169500
$$435$$ 0 0
$$436$$ −1.06226 −0.0508729
$$437$$ −5.40661 −0.258633
$$438$$ 0.468871 0.0224035
$$439$$ −8.93774 −0.426575 −0.213288 0.976989i $$-0.568417\pi$$
−0.213288 + 0.976989i $$0.568417\pi$$
$$440$$ 0 0
$$441$$ 5.46887 0.260422
$$442$$ 24.0000 1.14156
$$443$$ −38.5934 −1.83363 −0.916814 0.399316i $$-0.869248\pi$$
−0.916814 + 0.399316i $$0.869248\pi$$
$$444$$ 9.06226 0.430076
$$445$$ 0 0
$$446$$ 2.00000 0.0947027
$$447$$ 10.4689 0.495161
$$448$$ −3.53113 −0.166830
$$449$$ 28.1245 1.32728 0.663639 0.748053i $$-0.269011\pi$$
0.663639 + 0.748053i $$0.269011\pi$$
$$450$$ 0 0
$$451$$ 13.8755 0.653371
$$452$$ −16.5934 −0.780487
$$453$$ 18.1245 0.851564
$$454$$ −16.4689 −0.772922
$$455$$ 0 0
$$456$$ −3.53113 −0.165360
$$457$$ −31.0623 −1.45303 −0.726516 0.687150i $$-0.758862\pi$$
−0.726516 + 0.687150i $$0.758862\pi$$
$$458$$ −18.5934 −0.868812
$$459$$ −4.00000 −0.186704
$$460$$ 0 0
$$461$$ 24.0000 1.11779 0.558896 0.829238i $$-0.311225\pi$$
0.558896 + 0.829238i $$0.311225\pi$$
$$462$$ 5.40661 0.251538
$$463$$ 35.1868 1.63527 0.817634 0.575738i $$-0.195285\pi$$
0.817634 + 0.575738i $$0.195285\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 9.53113 0.441521
$$467$$ −10.1245 −0.468507 −0.234253 0.972176i $$-0.575265\pi$$
−0.234253 + 0.972176i $$0.575265\pi$$
$$468$$ −6.00000 −0.277350
$$469$$ −39.0623 −1.80373
$$470$$ 0 0
$$471$$ −14.4689 −0.666690
$$472$$ −7.06226 −0.325067
$$473$$ −0.717902 −0.0330092
$$474$$ −0.468871 −0.0215360
$$475$$ 0 0
$$476$$ −14.1245 −0.647396
$$477$$ −5.53113 −0.253253
$$478$$ 8.00000 0.365911
$$479$$ 41.6556 1.90329 0.951647 0.307192i $$-0.0993895\pi$$
0.951647 + 0.307192i $$0.0993895\pi$$
$$480$$ 0 0
$$481$$ 54.3735 2.47922
$$482$$ −2.00000 −0.0910975
$$483$$ 5.40661 0.246009
$$484$$ −8.65564 −0.393438
$$485$$ 0 0
$$486$$ 1.00000 0.0453609
$$487$$ −6.93774 −0.314379 −0.157190 0.987568i $$-0.550243\pi$$
−0.157190 + 0.987568i $$0.550243\pi$$
$$488$$ 11.0623 0.500765
$$489$$ −11.0623 −0.500253
$$490$$ 0 0
$$491$$ 16.5934 0.748849 0.374425 0.927257i $$-0.377840\pi$$
0.374425 + 0.927257i $$0.377840\pi$$
$$492$$ 9.06226 0.408558
$$493$$ 0 0
$$494$$ −21.1868 −0.953238
$$495$$ 0 0
$$496$$ 1.00000 0.0449013
$$497$$ 15.7802 0.707837
$$498$$ 8.00000 0.358489
$$499$$ 9.06226 0.405682 0.202841 0.979212i $$-0.434982\pi$$
0.202841 + 0.979212i $$0.434982\pi$$
$$500$$ 0 0
$$501$$ 0.593387 0.0265106
$$502$$ 13.0623 0.582997
$$503$$ −16.0000 −0.713405 −0.356702 0.934218i $$-0.616099\pi$$
−0.356702 + 0.934218i $$0.616099\pi$$
$$504$$ 3.53113 0.157289
$$505$$ 0 0
$$506$$ 2.34436 0.104219
$$507$$ −23.0000 −1.02147
$$508$$ 9.06226 0.402073
$$509$$ 5.87548 0.260426 0.130213 0.991486i $$-0.458434\pi$$
0.130213 + 0.991486i $$0.458434\pi$$
$$510$$ 0 0
$$511$$ −1.65564 −0.0732414
$$512$$ −1.00000 −0.0441942
$$513$$ 3.53113 0.155903
$$514$$ −27.6556 −1.21984
$$515$$ 0 0
$$516$$ −0.468871 −0.0206409
$$517$$ 16.9377 0.744921
$$518$$ −32.0000 −1.40600
$$519$$ 14.0000 0.614532
$$520$$ 0 0
$$521$$ 20.1245 0.881671 0.440836 0.897588i $$-0.354682\pi$$
0.440836 + 0.897588i $$0.354682\pi$$
$$522$$ 0 0
$$523$$ 14.5934 0.638124 0.319062 0.947734i $$-0.396632\pi$$
0.319062 + 0.947734i $$0.396632\pi$$
$$524$$ 7.06226 0.308516
$$525$$ 0 0
$$526$$ −6.93774 −0.302500
$$527$$ 4.00000 0.174243
$$528$$ 1.53113 0.0666338
$$529$$ −20.6556 −0.898071
$$530$$ 0 0
$$531$$ 7.06226 0.306476
$$532$$ 12.4689 0.540594
$$533$$ 54.3735 2.35518
$$534$$ 1.53113 0.0662584
$$535$$ 0 0
$$536$$ −11.0623 −0.477817
$$537$$ −13.0623 −0.563678
$$538$$ −29.1868 −1.25833
$$539$$ −8.37355 −0.360674
$$540$$ 0 0
$$541$$ 28.1245 1.20917 0.604584 0.796542i $$-0.293340\pi$$
0.604584 + 0.796542i $$0.293340\pi$$
$$542$$ 19.5311 0.838934
$$543$$ −26.5934 −1.14123
$$544$$ −4.00000 −0.171499
$$545$$ 0 0
$$546$$ 21.1868 0.906710
$$547$$ −41.1868 −1.76102 −0.880510 0.474028i $$-0.842799\pi$$
−0.880510 + 0.474028i $$0.842799\pi$$
$$548$$ −0.937742 −0.0400584
$$549$$ −11.0623 −0.472126
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 1.53113 0.0651692
$$553$$ 1.65564 0.0704052
$$554$$ 1.06226 0.0451310
$$555$$ 0 0
$$556$$ 6.00000 0.254457
$$557$$ −45.7802 −1.93977 −0.969884 0.243568i $$-0.921682\pi$$
−0.969884 + 0.243568i $$0.921682\pi$$
$$558$$ −1.00000 −0.0423334
$$559$$ −2.81323 −0.118987
$$560$$ 0 0
$$561$$ 6.12452 0.258577
$$562$$ 1.06226 0.0448086
$$563$$ 4.00000 0.168580 0.0842900 0.996441i $$-0.473138\pi$$
0.0842900 + 0.996441i $$0.473138\pi$$
$$564$$ 11.0623 0.465805
$$565$$ 0 0
$$566$$ −12.0000 −0.504398
$$567$$ −3.53113 −0.148293
$$568$$ 4.46887 0.187510
$$569$$ 17.5311 0.734943 0.367472 0.930035i $$-0.380224\pi$$
0.367472 + 0.930035i $$0.380224\pi$$
$$570$$ 0 0
$$571$$ 3.87548 0.162184 0.0810920 0.996707i $$-0.474159\pi$$
0.0810920 + 0.996707i $$0.474159\pi$$
$$572$$ 9.18677 0.384118
$$573$$ 8.00000 0.334205
$$574$$ −32.0000 −1.33565
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ 11.1868 0.465711 0.232856 0.972511i $$-0.425193\pi$$
0.232856 + 0.972511i $$0.425193\pi$$
$$578$$ 1.00000 0.0415945
$$579$$ 14.0000 0.581820
$$580$$ 0 0
$$581$$ −28.2490 −1.17197
$$582$$ 16.1245 0.668383
$$583$$ 8.46887 0.350745
$$584$$ −0.468871 −0.0194020
$$585$$ 0 0
$$586$$ −14.0000 −0.578335
$$587$$ −20.2490 −0.835767 −0.417883 0.908501i $$-0.637228\pi$$
−0.417883 + 0.908501i $$0.637228\pi$$
$$588$$ −5.46887 −0.225532
$$589$$ −3.53113 −0.145498
$$590$$ 0 0
$$591$$ 6.00000 0.246807
$$592$$ −9.06226 −0.372456
$$593$$ 14.0000 0.574911 0.287456 0.957794i $$-0.407191\pi$$
0.287456 + 0.957794i $$0.407191\pi$$
$$594$$ −1.53113 −0.0628230
$$595$$ 0 0
$$596$$ −10.4689 −0.428822
$$597$$ 0.468871 0.0191896
$$598$$ 9.18677 0.375675
$$599$$ 10.5934 0.432834 0.216417 0.976301i $$-0.430563\pi$$
0.216417 + 0.976301i $$0.430563\pi$$
$$600$$ 0 0
$$601$$ 30.0000 1.22373 0.611863 0.790964i $$-0.290420\pi$$
0.611863 + 0.790964i $$0.290420\pi$$
$$602$$ 1.65564 0.0674790
$$603$$ 11.0623 0.450490
$$604$$ −18.1245 −0.737476
$$605$$ 0 0
$$606$$ −17.5311 −0.712153
$$607$$ 38.5934 1.56646 0.783229 0.621734i $$-0.213571\pi$$
0.783229 + 0.621734i $$0.213571\pi$$
$$608$$ 3.53113 0.143206
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 66.3735 2.68519
$$612$$ 4.00000 0.161690
$$613$$ 30.0000 1.21169 0.605844 0.795583i $$-0.292835\pi$$
0.605844 + 0.795583i $$0.292835\pi$$
$$614$$ 2.12452 0.0857385
$$615$$ 0 0
$$616$$ −5.40661 −0.217839
$$617$$ −20.5934 −0.829059 −0.414529 0.910036i $$-0.636054\pi$$
−0.414529 + 0.910036i $$0.636054\pi$$
$$618$$ 0 0
$$619$$ −34.0000 −1.36658 −0.683288 0.730149i $$-0.739451\pi$$
−0.683288 + 0.730149i $$0.739451\pi$$
$$620$$ 0 0
$$621$$ −1.53113 −0.0614421
$$622$$ 8.00000 0.320771
$$623$$ −5.40661 −0.216611
$$624$$ 6.00000 0.240192
$$625$$ 0 0
$$626$$ 27.0623 1.08163
$$627$$ −5.40661 −0.215919
$$628$$ 14.4689 0.577371
$$629$$ −36.2490 −1.44534
$$630$$ 0 0
$$631$$ −9.65564 −0.384385 −0.192193 0.981357i $$-0.561560\pi$$
−0.192193 + 0.981357i $$0.561560\pi$$
$$632$$ 0.468871 0.0186507
$$633$$ −22.5934 −0.898006
$$634$$ 29.0623 1.15421
$$635$$ 0 0
$$636$$ 5.53113 0.219324
$$637$$ −32.8132 −1.30011
$$638$$ 0 0
$$639$$ −4.46887 −0.176786
$$640$$ 0 0
$$641$$ −22.0000 −0.868948 −0.434474 0.900684i $$-0.643066\pi$$
−0.434474 + 0.900684i $$0.643066\pi$$
$$642$$ 14.5934 0.575955
$$643$$ −6.59339 −0.260018 −0.130009 0.991513i $$-0.541501\pi$$
−0.130009 + 0.991513i $$0.541501\pi$$
$$644$$ −5.40661 −0.213050
$$645$$ 0 0
$$646$$ 14.1245 0.555722
$$647$$ −1.53113 −0.0601949 −0.0300974 0.999547i $$-0.509582\pi$$
−0.0300974 + 0.999547i $$0.509582\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ −10.8132 −0.424456
$$650$$ 0 0
$$651$$ 3.53113 0.138396
$$652$$ 11.0623 0.433231
$$653$$ 26.2490 1.02720 0.513602 0.858029i $$-0.328311\pi$$
0.513602 + 0.858029i $$0.328311\pi$$
$$654$$ −1.06226 −0.0415376
$$655$$ 0 0
$$656$$ −9.06226 −0.353822
$$657$$ 0.468871 0.0182924
$$658$$ −39.0623 −1.52281
$$659$$ 16.0000 0.623272 0.311636 0.950202i $$-0.399123\pi$$
0.311636 + 0.950202i $$0.399123\pi$$
$$660$$ 0 0
$$661$$ −10.0000 −0.388955 −0.194477 0.980907i $$-0.562301\pi$$
−0.194477 + 0.980907i $$0.562301\pi$$
$$662$$ −29.0623 −1.12954
$$663$$ 24.0000 0.932083
$$664$$ −8.00000 −0.310460
$$665$$ 0 0
$$666$$ 9.06226 0.351155
$$667$$ 0 0
$$668$$ −0.593387 −0.0229588
$$669$$ 2.00000 0.0773245
$$670$$ 0 0
$$671$$ 16.9377 0.653874
$$672$$ −3.53113 −0.136216
$$673$$ −7.06226 −0.272230 −0.136115 0.990693i $$-0.543462\pi$$
−0.136115 + 0.990693i $$0.543462\pi$$
$$674$$ 29.1868 1.12423
$$675$$ 0 0
$$676$$ 23.0000 0.884615
$$677$$ −3.40661 −0.130927 −0.0654634 0.997855i $$-0.520853\pi$$
−0.0654634 + 0.997855i $$0.520853\pi$$
$$678$$ −16.5934 −0.637265
$$679$$ −56.9377 −2.18507
$$680$$ 0 0
$$681$$ −16.4689 −0.631089
$$682$$ 1.53113 0.0586300
$$683$$ −15.5311 −0.594282 −0.297141 0.954834i $$-0.596033\pi$$
−0.297141 + 0.954834i $$0.596033\pi$$
$$684$$ −3.53113 −0.135016
$$685$$ 0 0
$$686$$ −5.40661 −0.206425
$$687$$ −18.5934 −0.709382
$$688$$ 0.468871 0.0178755
$$689$$ 33.1868 1.26432
$$690$$ 0 0
$$691$$ −7.53113 −0.286498 −0.143249 0.989687i $$-0.545755\pi$$
−0.143249 + 0.989687i $$0.545755\pi$$
$$692$$ −14.0000 −0.532200
$$693$$ 5.40661 0.205380
$$694$$ −22.1245 −0.839835
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −36.2490 −1.37303
$$698$$ −25.0623 −0.948620
$$699$$ 9.53113 0.360500
$$700$$ 0 0
$$701$$ −21.5311 −0.813220 −0.406610 0.913602i $$-0.633289\pi$$
−0.406610 + 0.913602i $$0.633289\pi$$
$$702$$ −6.00000 −0.226455
$$703$$ 32.0000 1.20690
$$704$$ −1.53113 −0.0577066
$$705$$ 0 0
$$706$$ −23.0623 −0.867960
$$707$$ 61.9047 2.32816
$$708$$ −7.06226 −0.265416
$$709$$ 25.4066 0.954165 0.477083 0.878858i $$-0.341694\pi$$
0.477083 + 0.878858i $$0.341694\pi$$
$$710$$ 0 0
$$711$$ −0.468871 −0.0175840
$$712$$ −1.53113 −0.0573815
$$713$$ 1.53113 0.0573412
$$714$$ −14.1245 −0.528597
$$715$$ 0 0
$$716$$ 13.0623 0.488159
$$717$$ 8.00000 0.298765
$$718$$ −11.5311 −0.430338
$$719$$ −33.1868 −1.23766 −0.618829 0.785526i $$-0.712393\pi$$
−0.618829 + 0.785526i $$0.712393\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 6.53113 0.243063
$$723$$ −2.00000 −0.0743808
$$724$$ 26.5934 0.988335
$$725$$ 0 0
$$726$$ −8.65564 −0.321241
$$727$$ 50.8424 1.88564 0.942820 0.333301i $$-0.108163\pi$$
0.942820 + 0.333301i $$0.108163\pi$$
$$728$$ −21.1868 −0.785234
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 1.87548 0.0693673
$$732$$ 11.0623 0.408873
$$733$$ −4.12452 −0.152342 −0.0761712 0.997095i $$-0.524270\pi$$
−0.0761712 + 0.997095i $$0.524270\pi$$
$$734$$ −10.0000 −0.369107
$$735$$ 0 0
$$736$$ −1.53113 −0.0564382
$$737$$ −16.9377 −0.623910
$$738$$ 9.06226 0.333586
$$739$$ 27.1868 1.00008 0.500041 0.866002i $$-0.333318\pi$$
0.500041 + 0.866002i $$0.333318\pi$$
$$740$$ 0 0
$$741$$ −21.1868 −0.778316
$$742$$ −19.5311 −0.717010
$$743$$ −11.6556 −0.427604 −0.213802 0.976877i $$-0.568585\pi$$
−0.213802 + 0.976877i $$0.568585\pi$$
$$744$$ 1.00000 0.0366618
$$745$$ 0 0
$$746$$ −10.4689 −0.383293
$$747$$ 8.00000 0.292705
$$748$$ −6.12452 −0.223934
$$749$$ −51.5311 −1.88291
$$750$$ 0 0
$$751$$ 47.0623 1.71733 0.858663 0.512540i $$-0.171296\pi$$
0.858663 + 0.512540i $$0.171296\pi$$
$$752$$ −11.0623 −0.403399
$$753$$ 13.0623 0.476015
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 3.53113 0.128426
$$757$$ −26.0000 −0.944986 −0.472493 0.881334i $$-0.656646\pi$$
−0.472493 + 0.881334i $$0.656646\pi$$
$$758$$ 2.59339 0.0941960
$$759$$ 2.34436 0.0850947
$$760$$ 0 0
$$761$$ −12.5934 −0.456510 −0.228255 0.973601i $$-0.573302\pi$$
−0.228255 + 0.973601i $$0.573302\pi$$
$$762$$ 9.06226 0.328291
$$763$$ 3.75097 0.135794
$$764$$ −8.00000 −0.289430
$$765$$ 0 0
$$766$$ 13.0623 0.471959
$$767$$ −42.3735 −1.53002
$$768$$ −1.00000 −0.0360844
$$769$$ −28.5934 −1.03110 −0.515552 0.856858i $$-0.672413\pi$$
−0.515552 + 0.856858i $$0.672413\pi$$
$$770$$ 0 0
$$771$$ −27.6556 −0.995994
$$772$$ −14.0000 −0.503871
$$773$$ −14.4689 −0.520409 −0.260205 0.965554i $$-0.583790\pi$$
−0.260205 + 0.965554i $$0.583790\pi$$
$$774$$ −0.468871 −0.0168532
$$775$$ 0 0
$$776$$ −16.1245 −0.578836
$$777$$ −32.0000 −1.14799
$$778$$ −27.0623 −0.970229
$$779$$ 32.0000 1.14652
$$780$$ 0 0
$$781$$ 6.84242 0.244841
$$782$$ −6.12452 −0.219012
$$783$$ 0 0
$$784$$ 5.46887 0.195317
$$785$$ 0 0
$$786$$ 7.06226 0.251902
$$787$$ 28.4689 1.01481 0.507403 0.861709i $$-0.330606\pi$$
0.507403 + 0.861709i $$0.330606\pi$$
$$788$$ −6.00000 −0.213741
$$789$$ −6.93774 −0.246990
$$790$$ 0 0
$$791$$ 58.5934 2.08334
$$792$$ 1.53113 0.0544063
$$793$$ 66.3735 2.35699
$$794$$ −18.7179 −0.664273
$$795$$ 0 0
$$796$$ −0.468871 −0.0166187
$$797$$ 20.1245 0.712847 0.356423 0.934325i $$-0.383996\pi$$
0.356423 + 0.934325i $$0.383996\pi$$
$$798$$ 12.4689 0.441393
$$799$$ −44.2490 −1.56542
$$800$$ 0 0
$$801$$ 1.53113 0.0540998
$$802$$ 34.7179 1.22593
$$803$$ −0.717902 −0.0253342
$$804$$ −11.0623 −0.390136
$$805$$ 0 0
$$806$$ 6.00000 0.211341
$$807$$ −29.1868 −1.02742
$$808$$ 17.5311 0.616743
$$809$$ −8.59339 −0.302127 −0.151064 0.988524i $$-0.548270\pi$$
−0.151064 + 0.988524i $$0.548270\pi$$
$$810$$ 0 0
$$811$$ −39.7802 −1.39687 −0.698435 0.715673i $$-0.746120\pi$$
−0.698435 + 0.715673i $$0.746120\pi$$
$$812$$ 0 0
$$813$$ 19.5311 0.684987
$$814$$ −13.8755 −0.486335
$$815$$ 0 0
$$816$$ −4.00000 −0.140028
$$817$$ −1.65564 −0.0579237
$$818$$ 9.06226 0.316854
$$819$$ 21.1868 0.740326
$$820$$ 0 0
$$821$$ 9.87548 0.344657 0.172328 0.985040i $$-0.444871\pi$$
0.172328 + 0.985040i $$0.444871\pi$$
$$822$$ −0.937742 −0.0327075
$$823$$ −7.87548 −0.274522 −0.137261 0.990535i $$-0.543830\pi$$
−0.137261 + 0.990535i $$0.543830\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 24.9377 0.867695
$$827$$ −8.00000 −0.278187 −0.139094 0.990279i $$-0.544419\pi$$
−0.139094 + 0.990279i $$0.544419\pi$$
$$828$$ 1.53113 0.0532104
$$829$$ −9.65564 −0.335354 −0.167677 0.985842i $$-0.553627\pi$$
−0.167677 + 0.985842i $$0.553627\pi$$
$$830$$ 0 0
$$831$$ 1.06226 0.0368493
$$832$$ −6.00000 −0.208013
$$833$$ 21.8755 0.757941
$$834$$ 6.00000 0.207763
$$835$$ 0 0
$$836$$ 5.40661 0.186992
$$837$$ −1.00000 −0.0345651
$$838$$ 0 0
$$839$$ −54.8424 −1.89337 −0.946685 0.322160i $$-0.895591\pi$$
−0.946685 + 0.322160i $$0.895591\pi$$
$$840$$ 0 0
$$841$$ −29.0000 −1.00000
$$842$$ −38.2490 −1.31815
$$843$$ 1.06226 0.0365861
$$844$$ 22.5934 0.777696
$$845$$ 0 0
$$846$$ 11.0623 0.380328
$$847$$ 30.5642 1.05020
$$848$$ −5.53113 −0.189940
$$849$$ −12.0000 −0.411839
$$850$$ 0 0
$$851$$ −13.8755 −0.475645
$$852$$ 4.46887 0.153101
$$853$$ −1.28210 −0.0438982 −0.0219491 0.999759i $$-0.506987\pi$$
−0.0219491 + 0.999759i $$0.506987\pi$$
$$854$$ −39.0623 −1.33668
$$855$$ 0 0
$$856$$ −14.5934 −0.498792
$$857$$ −14.0000 −0.478231 −0.239115 0.970991i $$-0.576857\pi$$
−0.239115 + 0.970991i $$0.576857\pi$$
$$858$$ 9.18677 0.313631
$$859$$ −6.93774 −0.236713 −0.118356 0.992971i $$-0.537763\pi$$
−0.118356 + 0.992971i $$0.537763\pi$$
$$860$$ 0 0
$$861$$ −32.0000 −1.09056
$$862$$ 24.0000 0.817443
$$863$$ −41.5311 −1.41374 −0.706868 0.707345i $$-0.749893\pi$$
−0.706868 + 0.707345i $$0.749893\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ 0 0
$$866$$ −1.40661 −0.0477987
$$867$$ 1.00000 0.0339618
$$868$$ −3.53113 −0.119854
$$869$$ 0.717902 0.0243532
$$870$$ 0 0
$$871$$ −66.3735 −2.24898
$$872$$ 1.06226 0.0359726
$$873$$ 16.1245 0.545732
$$874$$ 5.40661 0.182881
$$875$$ 0 0
$$876$$ −0.468871 −0.0158417
$$877$$ −44.1245 −1.48998 −0.744990 0.667076i $$-0.767546\pi$$
−0.744990 + 0.667076i $$0.767546\pi$$
$$878$$ 8.93774 0.301634
$$879$$ −14.0000 −0.472208
$$880$$ 0 0
$$881$$ 36.1245 1.21707 0.608533 0.793529i $$-0.291758\pi$$
0.608533 + 0.793529i $$0.291758\pi$$
$$882$$ −5.46887 −0.184146
$$883$$ 53.6556 1.80566 0.902828 0.430002i $$-0.141487\pi$$
0.902828 + 0.430002i $$0.141487\pi$$
$$884$$ −24.0000 −0.807207
$$885$$ 0 0
$$886$$ 38.5934 1.29657
$$887$$ 25.1868 0.845689 0.422845 0.906202i $$-0.361032\pi$$
0.422845 + 0.906202i $$0.361032\pi$$
$$888$$ −9.06226 −0.304109
$$889$$ −32.0000 −1.07325
$$890$$ 0 0
$$891$$ −1.53113 −0.0512947
$$892$$ −2.00000 −0.0669650
$$893$$ 39.0623 1.30717
$$894$$ −10.4689 −0.350131
$$895$$ 0 0
$$896$$ 3.53113 0.117967
$$897$$ 9.18677 0.306737
$$898$$ −28.1245 −0.938527
$$899$$ 0 0
$$900$$ 0 0
$$901$$ −22.1245 −0.737074
$$902$$ −13.8755 −0.462003
$$903$$ 1.65564 0.0550964
$$904$$ 16.5934 0.551888
$$905$$ 0 0
$$906$$ −18.1245 −0.602147
$$907$$ 32.2490 1.07081 0.535406 0.844595i $$-0.320159\pi$$
0.535406 + 0.844595i $$0.320159\pi$$
$$908$$ 16.4689 0.546539
$$909$$ −17.5311 −0.581471
$$910$$ 0 0
$$911$$ −16.0000 −0.530104 −0.265052 0.964234i $$-0.585389\pi$$
−0.265052 + 0.964234i $$0.585389\pi$$
$$912$$ 3.53113 0.116927
$$913$$ −12.2490 −0.405384
$$914$$ 31.0623 1.02745
$$915$$ 0 0
$$916$$ 18.5934 0.614343
$$917$$ −24.9377 −0.823517
$$918$$ 4.00000 0.132020
$$919$$ −8.93774 −0.294829 −0.147414 0.989075i $$-0.547095\pi$$
−0.147414 + 0.989075i $$0.547095\pi$$
$$920$$ 0 0
$$921$$ 2.12452 0.0700052
$$922$$ −24.0000 −0.790398
$$923$$ 26.8132 0.882568
$$924$$ −5.40661 −0.177865
$$925$$ 0 0
$$926$$ −35.1868 −1.15631
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 36.8424 1.20876 0.604380 0.796696i $$-0.293421\pi$$
0.604380 + 0.796696i $$0.293421\pi$$
$$930$$ 0 0
$$931$$ −19.3113 −0.632902
$$932$$ −9.53113 −0.312203
$$933$$ 8.00000 0.261908
$$934$$ 10.1245 0.331284
$$935$$ 0 0
$$936$$ 6.00000 0.196116
$$937$$ 25.0623 0.818748 0.409374 0.912367i $$-0.365747\pi$$
0.409374 + 0.912367i $$0.365747\pi$$
$$938$$ 39.0623 1.27543
$$939$$ 27.0623 0.883143
$$940$$ 0 0
$$941$$ −21.8755 −0.713120 −0.356560 0.934272i $$-0.616051\pi$$
−0.356560 + 0.934272i $$0.616051\pi$$
$$942$$ 14.4689 0.471421
$$943$$ −13.8755 −0.451848
$$944$$ 7.06226 0.229857
$$945$$ 0 0
$$946$$ 0.717902 0.0233410
$$947$$ −35.0623 −1.13937 −0.569685 0.821863i $$-0.692935\pi$$
−0.569685 + 0.821863i $$0.692935\pi$$
$$948$$ 0.468871 0.0152282
$$949$$ −2.81323 −0.0913212
$$950$$ 0 0
$$951$$ 29.0623 0.942408
$$952$$ 14.1245 0.457778
$$953$$ −54.1245 −1.75327 −0.876633 0.481161i $$-0.840215\pi$$
−0.876633 + 0.481161i $$0.840215\pi$$
$$954$$ 5.53113 0.179077
$$955$$ 0 0
$$956$$ −8.00000 −0.258738
$$957$$ 0 0
$$958$$ −41.6556 −1.34583
$$959$$ 3.31129 0.106927
$$960$$ 0 0
$$961$$ 1.00000 0.0322581
$$962$$ −54.3735 −1.75307
$$963$$ 14.5934 0.470265
$$964$$ 2.00000 0.0644157
$$965$$ 0 0
$$966$$ −5.40661 −0.173955
$$967$$ 17.0623 0.548685 0.274343 0.961632i $$-0.411540\pi$$
0.274343 + 0.961632i $$0.411540\pi$$
$$968$$ 8.65564 0.278203
$$969$$ 14.1245 0.453745
$$970$$ 0 0
$$971$$ −33.1868 −1.06501 −0.532507 0.846426i $$-0.678750\pi$$
−0.532507 + 0.846426i $$0.678750\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ −21.1868 −0.679217
$$974$$ 6.93774 0.222300
$$975$$ 0 0
$$976$$ −11.0623 −0.354094
$$977$$ 38.0000 1.21573 0.607864 0.794041i $$-0.292027\pi$$
0.607864 + 0.794041i $$0.292027\pi$$
$$978$$ 11.0623 0.353732
$$979$$ −2.34436 −0.0749259
$$980$$ 0 0
$$981$$ −1.06226 −0.0339153
$$982$$ −16.5934 −0.529516
$$983$$ −17.0623 −0.544202 −0.272101 0.962269i $$-0.587718\pi$$
−0.272101 + 0.962269i $$0.587718\pi$$
$$984$$ −9.06226 −0.288894
$$985$$ 0 0
$$986$$ 0 0
$$987$$ −39.0623 −1.24337
$$988$$ 21.1868 0.674041
$$989$$ 0.717902 0.0228280
$$990$$ 0 0
$$991$$ −24.4689 −0.777279 −0.388640 0.921390i $$-0.627055\pi$$
−0.388640 + 0.921390i $$0.627055\pi$$
$$992$$ −1.00000 −0.0317500
$$993$$ −29.0623 −0.922263
$$994$$ −15.7802 −0.500516
$$995$$ 0 0
$$996$$ −8.00000 −0.253490
$$997$$ −62.2490 −1.97145 −0.985723 0.168373i $$-0.946149\pi$$
−0.985723 + 0.168373i $$0.946149\pi$$
$$998$$ −9.06226 −0.286861
$$999$$ 9.06226 0.286717
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 4650.2.a.bz.1.1 2
5.2 odd 4 4650.2.d.bg.3349.1 4
5.3 odd 4 4650.2.d.bg.3349.4 4
5.4 even 2 930.2.a.q.1.2 2
15.14 odd 2 2790.2.a.bf.1.2 2
20.19 odd 2 7440.2.a.bd.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.a.q.1.2 2 5.4 even 2
2790.2.a.bf.1.2 2 15.14 odd 2
4650.2.a.bz.1.1 2 1.1 even 1 trivial
4650.2.d.bg.3349.1 4 5.2 odd 4
4650.2.d.bg.3349.4 4 5.3 odd 4
7440.2.a.bd.1.1 2 20.19 odd 2