# Properties

 Label 1850.2.a.y.1.1 Level $1850$ Weight $2$ Character 1850.1 Self dual yes Analytic conductor $14.772$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1850,2,Mod(1,1850)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1850, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1850.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1850 = 2 \cdot 5^{2} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1850.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: $$14.7723243739$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.1524.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 7x + 1$$ x^3 - x^2 - 7*x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$3.13264$$ of defining polynomial Character $$\chi$$ $$=$$ 1850.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -3.13264 q^{3} +1.00000 q^{4} +3.13264 q^{6} -4.13264 q^{7} -1.00000 q^{8} +6.81342 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} -3.13264 q^{3} +1.00000 q^{4} +3.13264 q^{6} -4.13264 q^{7} -1.00000 q^{8} +6.81342 q^{9} -3.68078 q^{11} -3.13264 q^{12} +0.132637 q^{13} +4.13264 q^{14} +1.00000 q^{16} +5.13264 q^{17} -6.81342 q^{18} +0.451857 q^{19} +12.9461 q^{21} +3.68078 q^{22} +4.26527 q^{23} +3.13264 q^{24} -0.132637 q^{26} -11.9461 q^{27} -4.13264 q^{28} -10.2653 q^{29} +5.58449 q^{31} -1.00000 q^{32} +11.5305 q^{33} -5.13264 q^{34} +6.81342 q^{36} -1.00000 q^{37} -0.451857 q^{38} -0.415505 q^{39} +10.3616 q^{41} -12.9461 q^{42} +5.07869 q^{43} -3.68078 q^{44} -4.26527 q^{46} -3.13264 q^{48} +10.0787 q^{49} -16.0787 q^{51} +0.132637 q^{52} +11.6268 q^{53} +11.9461 q^{54} +4.13264 q^{56} -1.41551 q^{57} +10.2653 q^{58} -13.0787 q^{59} -4.39791 q^{61} -5.58449 q^{62} -28.1574 q^{63} +1.00000 q^{64} -11.5305 q^{66} +0.228923 q^{67} +5.13264 q^{68} -13.3616 q^{69} +9.49420 q^{71} -6.81342 q^{72} -3.54814 q^{73} +1.00000 q^{74} +0.451857 q^{76} +15.2113 q^{77} +0.415505 q^{78} -0.903715 q^{79} +16.9824 q^{81} -10.3616 q^{82} +13.9461 q^{83} +12.9461 q^{84} -5.07869 q^{86} +32.1574 q^{87} +3.68078 q^{88} +0.777066 q^{89} -0.548143 q^{91} +4.26527 q^{92} -17.4942 q^{93} +3.13264 q^{96} -0.638440 q^{97} -10.0787 q^{98} -25.0787 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{2} - q^{3} + 3 q^{4} + q^{6} - 4 q^{7} - 3 q^{8} + 6 q^{9}+O(q^{10})$$ 3 * q - 3 * q^2 - q^3 + 3 * q^4 + q^6 - 4 * q^7 - 3 * q^8 + 6 * q^9 $$3 q - 3 q^{2} - q^{3} + 3 q^{4} + q^{6} - 4 q^{7} - 3 q^{8} + 6 q^{9} - 5 q^{11} - q^{12} - 8 q^{13} + 4 q^{14} + 3 q^{16} + 7 q^{17} - 6 q^{18} - q^{19} + 16 q^{21} + 5 q^{22} - 4 q^{23} + q^{24} + 8 q^{26} - 13 q^{27} - 4 q^{28} - 14 q^{29} + 6 q^{31} - 3 q^{32} + q^{33} - 7 q^{34} + 6 q^{36} - 3 q^{37} + q^{38} - 12 q^{39} + 19 q^{41} - 16 q^{42} - 16 q^{43} - 5 q^{44} + 4 q^{46} - q^{48} - q^{49} - 17 q^{51} - 8 q^{52} + 6 q^{53} + 13 q^{54} + 4 q^{56} - 15 q^{57} + 14 q^{58} - 8 q^{59} + 12 q^{61} - 6 q^{62} - 22 q^{63} + 3 q^{64} - q^{66} - 3 q^{67} + 7 q^{68} - 28 q^{69} + 8 q^{71} - 6 q^{72} - 13 q^{73} + 3 q^{74} - q^{76} + 6 q^{77} + 12 q^{78} + 2 q^{79} + 15 q^{81} - 19 q^{82} + 19 q^{83} + 16 q^{84} + 16 q^{86} + 34 q^{87} + 5 q^{88} + q^{89} - 4 q^{91} - 4 q^{92} - 32 q^{93} + q^{96} - 14 q^{97} + q^{98} - 44 q^{99}+O(q^{100})$$ 3 * q - 3 * q^2 - q^3 + 3 * q^4 + q^6 - 4 * q^7 - 3 * q^8 + 6 * q^9 - 5 * q^11 - q^12 - 8 * q^13 + 4 * q^14 + 3 * q^16 + 7 * q^17 - 6 * q^18 - q^19 + 16 * q^21 + 5 * q^22 - 4 * q^23 + q^24 + 8 * q^26 - 13 * q^27 - 4 * q^28 - 14 * q^29 + 6 * q^31 - 3 * q^32 + q^33 - 7 * q^34 + 6 * q^36 - 3 * q^37 + q^38 - 12 * q^39 + 19 * q^41 - 16 * q^42 - 16 * q^43 - 5 * q^44 + 4 * q^46 - q^48 - q^49 - 17 * q^51 - 8 * q^52 + 6 * q^53 + 13 * q^54 + 4 * q^56 - 15 * q^57 + 14 * q^58 - 8 * q^59 + 12 * q^61 - 6 * q^62 - 22 * q^63 + 3 * q^64 - q^66 - 3 * q^67 + 7 * q^68 - 28 * q^69 + 8 * q^71 - 6 * q^72 - 13 * q^73 + 3 * q^74 - q^76 + 6 * q^77 + 12 * q^78 + 2 * q^79 + 15 * q^81 - 19 * q^82 + 19 * q^83 + 16 * q^84 + 16 * q^86 + 34 * q^87 + 5 * q^88 + q^89 - 4 * q^91 - 4 * q^92 - 32 * q^93 + q^96 - 14 * q^97 + q^98 - 44 * 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
$$3$$ −3.13264 −1.80863 −0.904315 0.426867i $$-0.859617\pi$$
−0.904315 + 0.426867i $$0.859617\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ 3.13264 1.27889
$$7$$ −4.13264 −1.56199 −0.780995 0.624537i $$-0.785288\pi$$
−0.780995 + 0.624537i $$0.785288\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 6.81342 2.27114
$$10$$ 0 0
$$11$$ −3.68078 −1.10980 −0.554898 0.831918i $$-0.687243\pi$$
−0.554898 + 0.831918i $$0.687243\pi$$
$$12$$ −3.13264 −0.904315
$$13$$ 0.132637 0.0367870 0.0183935 0.999831i $$-0.494145\pi$$
0.0183935 + 0.999831i $$0.494145\pi$$
$$14$$ 4.13264 1.10449
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 5.13264 1.24485 0.622424 0.782680i $$-0.286148\pi$$
0.622424 + 0.782680i $$0.286148\pi$$
$$18$$ −6.81342 −1.60594
$$19$$ 0.451857 0.103663 0.0518316 0.998656i $$-0.483494\pi$$
0.0518316 + 0.998656i $$0.483494\pi$$
$$20$$ 0 0
$$21$$ 12.9461 2.82506
$$22$$ 3.68078 0.784745
$$23$$ 4.26527 0.889371 0.444686 0.895687i $$-0.353315\pi$$
0.444686 + 0.895687i $$0.353315\pi$$
$$24$$ 3.13264 0.639447
$$25$$ 0 0
$$26$$ −0.132637 −0.0260124
$$27$$ −11.9461 −2.29902
$$28$$ −4.13264 −0.780995
$$29$$ −10.2653 −1.90621 −0.953107 0.302634i $$-0.902134\pi$$
−0.953107 + 0.302634i $$0.902134\pi$$
$$30$$ 0 0
$$31$$ 5.58449 1.00300 0.501502 0.865156i $$-0.332781\pi$$
0.501502 + 0.865156i $$0.332781\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 11.5305 2.00721
$$34$$ −5.13264 −0.880240
$$35$$ 0 0
$$36$$ 6.81342 1.13557
$$37$$ −1.00000 −0.164399
$$38$$ −0.451857 −0.0733009
$$39$$ −0.415505 −0.0665341
$$40$$ 0 0
$$41$$ 10.3616 1.61820 0.809102 0.587668i $$-0.199954\pi$$
0.809102 + 0.587668i $$0.199954\pi$$
$$42$$ −12.9461 −1.99762
$$43$$ 5.07869 0.774493 0.387247 0.921976i $$-0.373426\pi$$
0.387247 + 0.921976i $$0.373426\pi$$
$$44$$ −3.68078 −0.554898
$$45$$ 0 0
$$46$$ −4.26527 −0.628880
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ −3.13264 −0.452157
$$49$$ 10.0787 1.43981
$$50$$ 0 0
$$51$$ −16.0787 −2.25147
$$52$$ 0.132637 0.0183935
$$53$$ 11.6268 1.59707 0.798534 0.601949i $$-0.205609\pi$$
0.798534 + 0.601949i $$0.205609\pi$$
$$54$$ 11.9461 1.62565
$$55$$ 0 0
$$56$$ 4.13264 0.552247
$$57$$ −1.41551 −0.187488
$$58$$ 10.2653 1.34790
$$59$$ −13.0787 −1.70270 −0.851350 0.524597i $$-0.824216\pi$$
−0.851350 + 0.524597i $$0.824216\pi$$
$$60$$ 0 0
$$61$$ −4.39791 −0.563095 −0.281547 0.959547i $$-0.590848\pi$$
−0.281547 + 0.959547i $$0.590848\pi$$
$$62$$ −5.58449 −0.709232
$$63$$ −28.1574 −3.54750
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ −11.5305 −1.41931
$$67$$ 0.228923 0.0279674 0.0139837 0.999902i $$-0.495549\pi$$
0.0139837 + 0.999902i $$0.495549\pi$$
$$68$$ 5.13264 0.622424
$$69$$ −13.3616 −1.60854
$$70$$ 0 0
$$71$$ 9.49420 1.12675 0.563377 0.826200i $$-0.309502\pi$$
0.563377 + 0.826200i $$0.309502\pi$$
$$72$$ −6.81342 −0.802969
$$73$$ −3.54814 −0.415279 −0.207639 0.978205i $$-0.566578\pi$$
−0.207639 + 0.978205i $$0.566578\pi$$
$$74$$ 1.00000 0.116248
$$75$$ 0 0
$$76$$ 0.451857 0.0518316
$$77$$ 15.2113 1.73349
$$78$$ 0.415505 0.0470467
$$79$$ −0.903715 −0.101676 −0.0508379 0.998707i $$-0.516189\pi$$
−0.0508379 + 0.998707i $$0.516189\pi$$
$$80$$ 0 0
$$81$$ 16.9824 1.88693
$$82$$ −10.3616 −1.14424
$$83$$ 13.9461 1.53078 0.765389 0.643568i $$-0.222546\pi$$
0.765389 + 0.643568i $$0.222546\pi$$
$$84$$ 12.9461 1.41253
$$85$$ 0 0
$$86$$ −5.07869 −0.547650
$$87$$ 32.1574 3.44763
$$88$$ 3.68078 0.392372
$$89$$ 0.777066 0.0823688 0.0411844 0.999152i $$-0.486887\pi$$
0.0411844 + 0.999152i $$0.486887\pi$$
$$90$$ 0 0
$$91$$ −0.548143 −0.0574610
$$92$$ 4.26527 0.444686
$$93$$ −17.4942 −1.81406
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 3.13264 0.319723
$$97$$ −0.638440 −0.0648237 −0.0324119 0.999475i $$-0.510319\pi$$
−0.0324119 + 0.999475i $$0.510319\pi$$
$$98$$ −10.0787 −1.01810
$$99$$ −25.0787 −2.52050
$$100$$ 0 0
$$101$$ −5.62684 −0.559891 −0.279946 0.960016i $$-0.590316\pi$$
−0.279946 + 0.960016i $$0.590316\pi$$
$$102$$ 16.0787 1.59203
$$103$$ −10.0423 −0.989501 −0.494751 0.869035i $$-0.664741\pi$$
−0.494751 + 0.869035i $$0.664741\pi$$
$$104$$ −0.132637 −0.0130062
$$105$$ 0 0
$$106$$ −11.6268 −1.12930
$$107$$ −19.6632 −1.90091 −0.950456 0.310859i $$-0.899383\pi$$
−0.950456 + 0.310859i $$0.899383\pi$$
$$108$$ −11.9461 −1.14951
$$109$$ 13.6268 1.30521 0.652607 0.757697i $$-0.273675\pi$$
0.652607 + 0.757697i $$0.273675\pi$$
$$110$$ 0 0
$$111$$ 3.13264 0.297337
$$112$$ −4.13264 −0.390498
$$113$$ −7.94606 −0.747502 −0.373751 0.927529i $$-0.621929\pi$$
−0.373751 + 0.927529i $$0.621929\pi$$
$$114$$ 1.41551 0.132574
$$115$$ 0 0
$$116$$ −10.2653 −0.953107
$$117$$ 0.903715 0.0835484
$$118$$ 13.0787 1.20399
$$119$$ −21.2113 −1.94444
$$120$$ 0 0
$$121$$ 2.54814 0.231649
$$122$$ 4.39791 0.398168
$$123$$ −32.4590 −2.92673
$$124$$ 5.58449 0.501502
$$125$$ 0 0
$$126$$ 28.1574 2.50846
$$127$$ −17.4942 −1.55236 −0.776180 0.630512i $$-0.782845\pi$$
−0.776180 + 0.630512i $$0.782845\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ −15.9097 −1.40077
$$130$$ 0 0
$$131$$ 8.81342 0.770032 0.385016 0.922910i $$-0.374196\pi$$
0.385016 + 0.922910i $$0.374196\pi$$
$$132$$ 11.5305 1.00361
$$133$$ −1.86736 −0.161921
$$134$$ −0.228923 −0.0197759
$$135$$ 0 0
$$136$$ −5.13264 −0.440120
$$137$$ −18.8921 −1.61406 −0.807031 0.590509i $$-0.798927\pi$$
−0.807031 + 0.590509i $$0.798927\pi$$
$$138$$ 13.3616 1.13741
$$139$$ −14.8498 −1.25954 −0.629771 0.776781i $$-0.716851\pi$$
−0.629771 + 0.776781i $$0.716851\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −9.49420 −0.796735
$$143$$ −0.488209 −0.0408261
$$144$$ 6.81342 0.567785
$$145$$ 0 0
$$146$$ 3.54814 0.293646
$$147$$ −31.5729 −2.60409
$$148$$ −1.00000 −0.0821995
$$149$$ −2.41551 −0.197886 −0.0989429 0.995093i $$-0.531546\pi$$
−0.0989429 + 0.995093i $$0.531546\pi$$
$$150$$ 0 0
$$151$$ 10.7231 0.872635 0.436318 0.899793i $$-0.356282\pi$$
0.436318 + 0.899793i $$0.356282\pi$$
$$152$$ −0.451857 −0.0366505
$$153$$ 34.9708 2.82722
$$154$$ −15.2113 −1.22576
$$155$$ 0 0
$$156$$ −0.415505 −0.0332670
$$157$$ 3.31922 0.264903 0.132451 0.991190i $$-0.457715\pi$$
0.132451 + 0.991190i $$0.457715\pi$$
$$158$$ 0.903715 0.0718957
$$159$$ −36.4227 −2.88850
$$160$$ 0 0
$$161$$ −17.6268 −1.38919
$$162$$ −16.9824 −1.33426
$$163$$ −15.5481 −1.21782 −0.608912 0.793238i $$-0.708394\pi$$
−0.608912 + 0.793238i $$0.708394\pi$$
$$164$$ 10.3616 0.809102
$$165$$ 0 0
$$166$$ −13.9461 −1.08242
$$167$$ 3.77707 0.292278 0.146139 0.989264i $$-0.453315\pi$$
0.146139 + 0.989264i $$0.453315\pi$$
$$168$$ −12.9461 −0.998810
$$169$$ −12.9824 −0.998647
$$170$$ 0 0
$$171$$ 3.07869 0.235434
$$172$$ 5.07869 0.387247
$$173$$ −20.1574 −1.53254 −0.766269 0.642520i $$-0.777889\pi$$
−0.766269 + 0.642520i $$0.777889\pi$$
$$174$$ −32.1574 −2.43785
$$175$$ 0 0
$$176$$ −3.68078 −0.277449
$$177$$ 40.9708 3.07955
$$178$$ −0.777066 −0.0582435
$$179$$ 20.2537 1.51383 0.756915 0.653513i $$-0.226706\pi$$
0.756915 + 0.653513i $$0.226706\pi$$
$$180$$ 0 0
$$181$$ −17.1690 −1.27616 −0.638080 0.769970i $$-0.720271\pi$$
−0.638080 + 0.769970i $$0.720271\pi$$
$$182$$ 0.548143 0.0406310
$$183$$ 13.7771 1.01843
$$184$$ −4.26527 −0.314440
$$185$$ 0 0
$$186$$ 17.4942 1.28274
$$187$$ −18.8921 −1.38153
$$188$$ 0 0
$$189$$ 49.3687 3.59105
$$190$$ 0 0
$$191$$ −0.192571 −0.0139339 −0.00696696 0.999976i $$-0.502218\pi$$
−0.00696696 + 0.999976i $$0.502218\pi$$
$$192$$ −3.13264 −0.226079
$$193$$ 14.6692 1.05591 0.527955 0.849272i $$-0.322959\pi$$
0.527955 + 0.849272i $$0.322959\pi$$
$$194$$ 0.638440 0.0458373
$$195$$ 0 0
$$196$$ 10.0787 0.719907
$$197$$ 7.84977 0.559273 0.279636 0.960106i $$-0.409786\pi$$
0.279636 + 0.960106i $$0.409786\pi$$
$$198$$ 25.0787 1.78227
$$199$$ −11.3616 −0.805400 −0.402700 0.915332i $$-0.631928\pi$$
−0.402700 + 0.915332i $$0.631928\pi$$
$$200$$ 0 0
$$201$$ −0.717132 −0.0505826
$$202$$ 5.62684 0.395903
$$203$$ 42.4227 2.97749
$$204$$ −16.0787 −1.12573
$$205$$ 0 0
$$206$$ 10.0423 0.699683
$$207$$ 29.0611 2.01989
$$208$$ 0.132637 0.00919676
$$209$$ −1.66319 −0.115045
$$210$$ 0 0
$$211$$ −12.2289 −0.841874 −0.420937 0.907090i $$-0.638299\pi$$
−0.420937 + 0.907090i $$0.638299\pi$$
$$212$$ 11.6268 0.798534
$$213$$ −29.7419 −2.03788
$$214$$ 19.6632 1.34415
$$215$$ 0 0
$$216$$ 11.9461 0.812826
$$217$$ −23.0787 −1.56668
$$218$$ −13.6268 −0.922926
$$219$$ 11.1150 0.751085
$$220$$ 0 0
$$221$$ 0.680780 0.0457942
$$222$$ −3.13264 −0.210249
$$223$$ −28.1574 −1.88556 −0.942779 0.333418i $$-0.891798\pi$$
−0.942779 + 0.333418i $$0.891798\pi$$
$$224$$ 4.13264 0.276123
$$225$$ 0 0
$$226$$ 7.94606 0.528564
$$227$$ −0.282868 −0.0187746 −0.00938729 0.999956i $$-0.502988\pi$$
−0.00938729 + 0.999956i $$0.502988\pi$$
$$228$$ −1.41551 −0.0937441
$$229$$ −8.45785 −0.558910 −0.279455 0.960159i $$-0.590154\pi$$
−0.279455 + 0.960159i $$0.590154\pi$$
$$230$$ 0 0
$$231$$ −47.6516 −3.13524
$$232$$ 10.2653 0.673948
$$233$$ 14.2477 0.933397 0.466698 0.884417i $$-0.345443\pi$$
0.466698 + 0.884417i $$0.345443\pi$$
$$234$$ −0.903715 −0.0590777
$$235$$ 0 0
$$236$$ −13.0787 −0.851350
$$237$$ 2.83101 0.183894
$$238$$ 21.2113 1.37493
$$239$$ 21.8921 1.41608 0.708041 0.706171i $$-0.249579\pi$$
0.708041 + 0.706171i $$0.249579\pi$$
$$240$$ 0 0
$$241$$ −9.13264 −0.588285 −0.294142 0.955762i $$-0.595034\pi$$
−0.294142 + 0.955762i $$0.595034\pi$$
$$242$$ −2.54814 −0.163801
$$243$$ −17.3616 −1.11374
$$244$$ −4.39791 −0.281547
$$245$$ 0 0
$$246$$ 32.4590 2.06951
$$247$$ 0.0599332 0.00381346
$$248$$ −5.58449 −0.354616
$$249$$ −43.6879 −2.76861
$$250$$ 0 0
$$251$$ −9.00000 −0.568075 −0.284037 0.958813i $$-0.591674\pi$$
−0.284037 + 0.958813i $$0.591674\pi$$
$$252$$ −28.1574 −1.77375
$$253$$ −15.6995 −0.987022
$$254$$ 17.4942 1.09768
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 8.90371 0.555398 0.277699 0.960668i $$-0.410428\pi$$
0.277699 + 0.960668i $$0.410428\pi$$
$$258$$ 15.9097 0.990495
$$259$$ 4.13264 0.256790
$$260$$ 0 0
$$261$$ −69.9416 −4.32928
$$262$$ −8.81342 −0.544495
$$263$$ 14.9037 0.919002 0.459501 0.888177i $$-0.348028\pi$$
0.459501 + 0.888177i $$0.348028\pi$$
$$264$$ −11.5305 −0.709656
$$265$$ 0 0
$$266$$ 1.86736 0.114495
$$267$$ −2.43426 −0.148975
$$268$$ 0.228923 0.0139837
$$269$$ −1.55413 −0.0947570 −0.0473785 0.998877i $$-0.515087\pi$$
−0.0473785 + 0.998877i $$0.515087\pi$$
$$270$$ 0 0
$$271$$ 5.68677 0.345447 0.172723 0.984970i $$-0.444743\pi$$
0.172723 + 0.984970i $$0.444743\pi$$
$$272$$ 5.13264 0.311212
$$273$$ 1.71713 0.103926
$$274$$ 18.8921 1.14131
$$275$$ 0 0
$$276$$ −13.3616 −0.804271
$$277$$ 12.9037 0.775309 0.387655 0.921805i $$-0.373285\pi$$
0.387655 + 0.921805i $$0.373285\pi$$
$$278$$ 14.8498 0.890630
$$279$$ 38.0495 2.27796
$$280$$ 0 0
$$281$$ −12.2829 −0.732734 −0.366367 0.930470i $$-0.619399\pi$$
−0.366367 + 0.930470i $$0.619399\pi$$
$$282$$ 0 0
$$283$$ 26.8745 1.59752 0.798762 0.601647i $$-0.205489\pi$$
0.798762 + 0.601647i $$0.205489\pi$$
$$284$$ 9.49420 0.563377
$$285$$ 0 0
$$286$$ 0.488209 0.0288684
$$287$$ −42.8206 −2.52762
$$288$$ −6.81342 −0.401484
$$289$$ 9.34397 0.549645
$$290$$ 0 0
$$291$$ 2.00000 0.117242
$$292$$ −3.54814 −0.207639
$$293$$ 16.7535 0.978749 0.489375 0.872074i $$-0.337225\pi$$
0.489375 + 0.872074i $$0.337225\pi$$
$$294$$ 31.5729 1.84137
$$295$$ 0 0
$$296$$ 1.00000 0.0581238
$$297$$ 43.9708 2.55144
$$298$$ 2.41551 0.139926
$$299$$ 0.565735 0.0327173
$$300$$ 0 0
$$301$$ −20.9884 −1.20975
$$302$$ −10.7231 −0.617046
$$303$$ 17.6268 1.01264
$$304$$ 0.451857 0.0259158
$$305$$ 0 0
$$306$$ −34.9708 −1.99915
$$307$$ 8.47661 0.483785 0.241893 0.970303i $$-0.422232\pi$$
0.241893 + 0.970303i $$0.422232\pi$$
$$308$$ 15.2113 0.866746
$$309$$ 31.4590 1.78964
$$310$$ 0 0
$$311$$ 32.4650 1.84092 0.920461 0.390835i $$-0.127814\pi$$
0.920461 + 0.390835i $$0.127814\pi$$
$$312$$ 0.415505 0.0235233
$$313$$ −28.8134 −1.62863 −0.814315 0.580423i $$-0.802887\pi$$
−0.814315 + 0.580423i $$0.802887\pi$$
$$314$$ −3.31922 −0.187314
$$315$$ 0 0
$$316$$ −0.903715 −0.0508379
$$317$$ 6.87335 0.386046 0.193023 0.981194i $$-0.438171\pi$$
0.193023 + 0.981194i $$0.438171\pi$$
$$318$$ 36.4227 2.04248
$$319$$ 37.7842 2.11551
$$320$$ 0 0
$$321$$ 61.5976 3.43804
$$322$$ 17.6268 0.982305
$$323$$ 2.31922 0.129045
$$324$$ 16.9824 0.943467
$$325$$ 0 0
$$326$$ 15.5481 0.861132
$$327$$ −42.6879 −2.36065
$$328$$ −10.3616 −0.572121
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −1.65603 −0.0910238 −0.0455119 0.998964i $$-0.514492\pi$$
−0.0455119 + 0.998964i $$0.514492\pi$$
$$332$$ 13.9461 0.765389
$$333$$ −6.81342 −0.373373
$$334$$ −3.77707 −0.206672
$$335$$ 0 0
$$336$$ 12.9461 0.706265
$$337$$ −10.2537 −0.558553 −0.279277 0.960211i $$-0.590095\pi$$
−0.279277 + 0.960211i $$0.590095\pi$$
$$338$$ 12.9824 0.706150
$$339$$ 24.8921 1.35195
$$340$$ 0 0
$$341$$ −20.5553 −1.11313
$$342$$ −3.07869 −0.166477
$$343$$ −12.7231 −0.686984
$$344$$ −5.07869 −0.273825
$$345$$ 0 0
$$346$$ 20.1574 1.08367
$$347$$ −25.8805 −1.38934 −0.694669 0.719329i $$-0.744449\pi$$
−0.694669 + 0.719329i $$0.744449\pi$$
$$348$$ 32.1574 1.72382
$$349$$ 12.7535 0.682678 0.341339 0.939940i $$-0.389120\pi$$
0.341339 + 0.939940i $$0.389120\pi$$
$$350$$ 0 0
$$351$$ −1.58449 −0.0845741
$$352$$ 3.68078 0.196186
$$353$$ −8.72312 −0.464285 −0.232142 0.972682i $$-0.574574\pi$$
−0.232142 + 0.972682i $$0.574574\pi$$
$$354$$ −40.9708 −2.17757
$$355$$ 0 0
$$356$$ 0.777066 0.0411844
$$357$$ 66.4474 3.51677
$$358$$ −20.2537 −1.07044
$$359$$ 14.9037 0.786588 0.393294 0.919413i $$-0.371335\pi$$
0.393294 + 0.919413i $$0.371335\pi$$
$$360$$ 0 0
$$361$$ −18.7958 −0.989254
$$362$$ 17.1690 0.902382
$$363$$ −7.98241 −0.418968
$$364$$ −0.548143 −0.0287305
$$365$$ 0 0
$$366$$ −13.7771 −0.720139
$$367$$ 14.8558 0.775464 0.387732 0.921772i $$-0.373259\pi$$
0.387732 + 0.921772i $$0.373259\pi$$
$$368$$ 4.26527 0.222343
$$369$$ 70.5976 3.67517
$$370$$ 0 0
$$371$$ −48.0495 −2.49461
$$372$$ −17.4942 −0.907032
$$373$$ 1.77707 0.0920130 0.0460065 0.998941i $$-0.485351\pi$$
0.0460065 + 0.998941i $$0.485351\pi$$
$$374$$ 18.8921 0.976888
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −1.36156 −0.0701239
$$378$$ −49.3687 −2.53925
$$379$$ −26.3863 −1.35537 −0.677687 0.735351i $$-0.737017\pi$$
−0.677687 + 0.735351i $$0.737017\pi$$
$$380$$ 0 0
$$381$$ 54.8030 2.80764
$$382$$ 0.192571 0.00985277
$$383$$ −4.57289 −0.233664 −0.116832 0.993152i $$-0.537274\pi$$
−0.116832 + 0.993152i $$0.537274\pi$$
$$384$$ 3.13264 0.159862
$$385$$ 0 0
$$386$$ −14.6692 −0.746641
$$387$$ 34.6033 1.75898
$$388$$ −0.638440 −0.0324119
$$389$$ 5.03635 0.255353 0.127677 0.991816i $$-0.459248\pi$$
0.127677 + 0.991816i $$0.459248\pi$$
$$390$$ 0 0
$$391$$ 21.8921 1.10713
$$392$$ −10.0787 −0.509051
$$393$$ −27.6092 −1.39270
$$394$$ −7.84977 −0.395466
$$395$$ 0 0
$$396$$ −25.0787 −1.26025
$$397$$ −34.4650 −1.72975 −0.864874 0.501988i $$-0.832602\pi$$
−0.864874 + 0.501988i $$0.832602\pi$$
$$398$$ 11.3616 0.569504
$$399$$ 5.84977 0.292855
$$400$$ 0 0
$$401$$ −4.47661 −0.223551 −0.111775 0.993733i $$-0.535654\pi$$
−0.111775 + 0.993733i $$0.535654\pi$$
$$402$$ 0.717132 0.0357673
$$403$$ 0.740713 0.0368976
$$404$$ −5.62684 −0.279946
$$405$$ 0 0
$$406$$ −42.4227 −2.10540
$$407$$ 3.68078 0.182449
$$408$$ 16.0787 0.796014
$$409$$ −34.7243 −1.71701 −0.858503 0.512809i $$-0.828605\pi$$
−0.858503 + 0.512809i $$0.828605\pi$$
$$410$$ 0 0
$$411$$ 59.1821 2.91924
$$412$$ −10.0423 −0.494751
$$413$$ 54.0495 2.65960
$$414$$ −29.0611 −1.42828
$$415$$ 0 0
$$416$$ −0.132637 −0.00650309
$$417$$ 46.5189 2.27804
$$418$$ 1.66319 0.0813492
$$419$$ −7.15023 −0.349312 −0.174656 0.984630i $$-0.555881\pi$$
−0.174656 + 0.984630i $$0.555881\pi$$
$$420$$ 0 0
$$421$$ 26.6033 1.29656 0.648282 0.761401i $$-0.275488\pi$$
0.648282 + 0.761401i $$0.275488\pi$$
$$422$$ 12.2289 0.595295
$$423$$ 0 0
$$424$$ −11.6268 −0.564649
$$425$$ 0 0
$$426$$ 29.7419 1.44100
$$427$$ 18.1750 0.879549
$$428$$ −19.6632 −0.950456
$$429$$ 1.52938 0.0738393
$$430$$ 0 0
$$431$$ −1.16899 −0.0563082 −0.0281541 0.999604i $$-0.508963\pi$$
−0.0281541 + 0.999604i $$0.508963\pi$$
$$432$$ −11.9461 −0.574755
$$433$$ −5.00000 −0.240285 −0.120142 0.992757i $$-0.538335\pi$$
−0.120142 + 0.992757i $$0.538335\pi$$
$$434$$ 23.0787 1.10781
$$435$$ 0 0
$$436$$ 13.6268 0.652607
$$437$$ 1.92730 0.0921951
$$438$$ −11.1150 −0.531097
$$439$$ −32.3148 −1.54230 −0.771150 0.636654i $$-0.780318\pi$$
−0.771150 + 0.636654i $$0.780318\pi$$
$$440$$ 0 0
$$441$$ 68.6703 3.27002
$$442$$ −0.680780 −0.0323814
$$443$$ 17.4155 0.827436 0.413718 0.910405i $$-0.364230\pi$$
0.413718 + 0.910405i $$0.364230\pi$$
$$444$$ 3.13264 0.148668
$$445$$ 0 0
$$446$$ 28.1574 1.33329
$$447$$ 7.56690 0.357902
$$448$$ −4.13264 −0.195249
$$449$$ −7.38032 −0.348299 −0.174149 0.984719i $$-0.555718\pi$$
−0.174149 + 0.984719i $$0.555718\pi$$
$$450$$ 0 0
$$451$$ −38.1386 −1.79588
$$452$$ −7.94606 −0.373751
$$453$$ −33.5916 −1.57827
$$454$$ 0.282868 0.0132756
$$455$$ 0 0
$$456$$ 1.41551 0.0662871
$$457$$ −22.7419 −1.06382 −0.531910 0.846801i $$-0.678526\pi$$
−0.531910 + 0.846801i $$0.678526\pi$$
$$458$$ 8.45785 0.395209
$$459$$ −61.3148 −2.86193
$$460$$ 0 0
$$461$$ 39.7115 1.84955 0.924775 0.380515i $$-0.124253\pi$$
0.924775 + 0.380515i $$0.124253\pi$$
$$462$$ 47.6516 2.21695
$$463$$ 30.6456 1.42422 0.712111 0.702067i $$-0.247739\pi$$
0.712111 + 0.702067i $$0.247739\pi$$
$$464$$ −10.2653 −0.476553
$$465$$ 0 0
$$466$$ −14.2477 −0.660011
$$467$$ −24.7958 −1.14741 −0.573707 0.819061i $$-0.694495\pi$$
−0.573707 + 0.819061i $$0.694495\pi$$
$$468$$ 0.903715 0.0417742
$$469$$ −0.946055 −0.0436848
$$470$$ 0 0
$$471$$ −10.3979 −0.479111
$$472$$ 13.0787 0.601996
$$473$$ −18.6936 −0.859530
$$474$$ −2.83101 −0.130033
$$475$$ 0 0
$$476$$ −21.2113 −0.972220
$$477$$ 79.2185 3.62717
$$478$$ −21.8921 −1.00132
$$479$$ −15.5845 −0.712074 −0.356037 0.934472i $$-0.615872\pi$$
−0.356037 + 0.934472i $$0.615872\pi$$
$$480$$ 0 0
$$481$$ −0.132637 −0.00604775
$$482$$ 9.13264 0.415980
$$483$$ 55.2185 2.51253
$$484$$ 2.54814 0.115825
$$485$$ 0 0
$$486$$ 17.3616 0.787536
$$487$$ −3.65720 −0.165724 −0.0828618 0.996561i $$-0.526406\pi$$
−0.0828618 + 0.996561i $$0.526406\pi$$
$$488$$ 4.39791 0.199084
$$489$$ 48.7067 2.20259
$$490$$ 0 0
$$491$$ 33.3264 1.50400 0.751999 0.659164i $$-0.229090\pi$$
0.751999 + 0.659164i $$0.229090\pi$$
$$492$$ −32.4590 −1.46337
$$493$$ −52.6879 −2.37295
$$494$$ −0.0599332 −0.00269652
$$495$$ 0 0
$$496$$ 5.58449 0.250751
$$497$$ −39.2361 −1.75998
$$498$$ 43.6879 1.95770
$$499$$ 7.53654 0.337382 0.168691 0.985669i $$-0.446046\pi$$
0.168691 + 0.985669i $$0.446046\pi$$
$$500$$ 0 0
$$501$$ −11.8322 −0.528623
$$502$$ 9.00000 0.401690
$$503$$ 24.0000 1.07011 0.535054 0.844818i $$-0.320291\pi$$
0.535054 + 0.844818i $$0.320291\pi$$
$$504$$ 28.1574 1.25423
$$505$$ 0 0
$$506$$ 15.6995 0.697930
$$507$$ 40.6692 1.80618
$$508$$ −17.4942 −0.776180
$$509$$ 24.1150 1.06888 0.534440 0.845206i $$-0.320522\pi$$
0.534440 + 0.845206i $$0.320522\pi$$
$$510$$ 0 0
$$511$$ 14.6632 0.648661
$$512$$ −1.00000 −0.0441942
$$513$$ −5.39791 −0.238324
$$514$$ −8.90371 −0.392726
$$515$$ 0 0
$$516$$ −15.9097 −0.700386
$$517$$ 0 0
$$518$$ −4.13264 −0.181578
$$519$$ 63.1458 2.77179
$$520$$ 0 0
$$521$$ −11.8134 −0.517555 −0.258778 0.965937i $$-0.583320\pi$$
−0.258778 + 0.965937i $$0.583320\pi$$
$$522$$ 69.9416 3.06126
$$523$$ 33.9532 1.48467 0.742335 0.670029i $$-0.233718\pi$$
0.742335 + 0.670029i $$0.233718\pi$$
$$524$$ 8.81342 0.385016
$$525$$ 0 0
$$526$$ −14.9037 −0.649833
$$527$$ 28.6632 1.24859
$$528$$ 11.5305 0.501803
$$529$$ −4.80743 −0.209019
$$530$$ 0 0
$$531$$ −89.1106 −3.86707
$$532$$ −1.86736 −0.0809604
$$533$$ 1.37433 0.0595289
$$534$$ 2.43426 0.105341
$$535$$ 0 0
$$536$$ −0.228923 −0.00988796
$$537$$ −63.4474 −2.73796
$$538$$ 1.55413 0.0670033
$$539$$ −37.0975 −1.59790
$$540$$ 0 0
$$541$$ −33.4590 −1.43852 −0.719258 0.694743i $$-0.755518\pi$$
−0.719258 + 0.694743i $$0.755518\pi$$
$$542$$ −5.68677 −0.244268
$$543$$ 53.7842 2.30810
$$544$$ −5.13264 −0.220060
$$545$$ 0 0
$$546$$ −1.71713 −0.0734865
$$547$$ 42.1514 1.80226 0.901132 0.433545i $$-0.142738\pi$$
0.901132 + 0.433545i $$0.142738\pi$$
$$548$$ −18.8921 −0.807031
$$549$$ −29.9648 −1.27887
$$550$$ 0 0
$$551$$ −4.63844 −0.197604
$$552$$ 13.3616 0.568706
$$553$$ 3.73473 0.158817
$$554$$ −12.9037 −0.548226
$$555$$ 0 0
$$556$$ −14.8498 −0.629771
$$557$$ −12.3732 −0.524268 −0.262134 0.965032i $$-0.584426\pi$$
−0.262134 + 0.965032i $$0.584426\pi$$
$$558$$ −38.0495 −1.61076
$$559$$ 0.673625 0.0284913
$$560$$ 0 0
$$561$$ 59.1821 2.49867
$$562$$ 12.2829 0.518122
$$563$$ −40.7782 −1.71860 −0.859299 0.511474i $$-0.829100\pi$$
−0.859299 + 0.511474i $$0.829100\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −26.8745 −1.12962
$$567$$ −70.1821 −2.94737
$$568$$ −9.49420 −0.398368
$$569$$ −20.7419 −0.869545 −0.434772 0.900540i $$-0.643171\pi$$
−0.434772 + 0.900540i $$0.643171\pi$$
$$570$$ 0 0
$$571$$ −37.1458 −1.55450 −0.777251 0.629190i $$-0.783387\pi$$
−0.777251 + 0.629190i $$0.783387\pi$$
$$572$$ −0.488209 −0.0204131
$$573$$ 0.603254 0.0252013
$$574$$ 42.8206 1.78730
$$575$$ 0 0
$$576$$ 6.81342 0.283892
$$577$$ 23.0975 0.961560 0.480780 0.876841i $$-0.340354\pi$$
0.480780 + 0.876841i $$0.340354\pi$$
$$578$$ −9.34397 −0.388658
$$579$$ −45.9532 −1.90975
$$580$$ 0 0
$$581$$ −57.6340 −2.39106
$$582$$ −2.00000 −0.0829027
$$583$$ −42.7958 −1.77242
$$584$$ 3.54814 0.146823
$$585$$ 0 0
$$586$$ −16.7535 −0.692080
$$587$$ −11.0176 −0.454745 −0.227372 0.973808i $$-0.573013\pi$$
−0.227372 + 0.973808i $$0.573013\pi$$
$$588$$ −31.5729 −1.30204
$$589$$ 2.52339 0.103975
$$590$$ 0 0
$$591$$ −24.5905 −1.01152
$$592$$ −1.00000 −0.0410997
$$593$$ 0.469450 0.0192780 0.00963900 0.999954i $$-0.496932\pi$$
0.00963900 + 0.999954i $$0.496932\pi$$
$$594$$ −43.9708 −1.80414
$$595$$ 0 0
$$596$$ −2.41551 −0.0989429
$$597$$ 35.5916 1.45667
$$598$$ −0.565735 −0.0231346
$$599$$ −24.7711 −1.01212 −0.506059 0.862499i $$-0.668898\pi$$
−0.506059 + 0.862499i $$0.668898\pi$$
$$600$$ 0 0
$$601$$ −23.4051 −0.954713 −0.477356 0.878710i $$-0.658405\pi$$
−0.477356 + 0.878710i $$0.658405\pi$$
$$602$$ 20.9884 0.855423
$$603$$ 1.55975 0.0635178
$$604$$ 10.7231 0.436318
$$605$$ 0 0
$$606$$ −17.6268 −0.716041
$$607$$ −0.342801 −0.0139139 −0.00695693 0.999976i $$-0.502214\pi$$
−0.00695693 + 0.999976i $$0.502214\pi$$
$$608$$ −0.451857 −0.0183252
$$609$$ −132.895 −5.38517
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 34.9708 1.41361
$$613$$ −35.8921 −1.44967 −0.724834 0.688923i $$-0.758084\pi$$
−0.724834 + 0.688923i $$0.758084\pi$$
$$614$$ −8.47661 −0.342088
$$615$$ 0 0
$$616$$ −15.2113 −0.612882
$$617$$ −7.64443 −0.307753 −0.153877 0.988090i $$-0.549176\pi$$
−0.153877 + 0.988090i $$0.549176\pi$$
$$618$$ −31.4590 −1.26547
$$619$$ −10.4227 −0.418922 −0.209461 0.977817i $$-0.567171\pi$$
−0.209461 + 0.977817i $$0.567171\pi$$
$$620$$ 0 0
$$621$$ −50.9532 −2.04468
$$622$$ −32.4650 −1.30173
$$623$$ −3.21133 −0.128659
$$624$$ −0.415505 −0.0166335
$$625$$ 0 0
$$626$$ 28.8134 1.15162
$$627$$ 5.21016 0.208074
$$628$$ 3.31922 0.132451
$$629$$ −5.13264 −0.204652
$$630$$ 0 0
$$631$$ 26.4227 1.05187 0.525935 0.850525i $$-0.323716\pi$$
0.525935 + 0.850525i $$0.323716\pi$$
$$632$$ 0.903715 0.0359478
$$633$$ 38.3088 1.52264
$$634$$ −6.87335 −0.272976
$$635$$ 0 0
$$636$$ −36.4227 −1.44425
$$637$$ 1.33681 0.0529664
$$638$$ −37.7842 −1.49589
$$639$$ 64.6879 2.55902
$$640$$ 0 0
$$641$$ 11.8074 0.466365 0.233183 0.972433i $$-0.425086\pi$$
0.233183 + 0.972433i $$0.425086\pi$$
$$642$$ −61.5976 −2.43106
$$643$$ −18.3556 −0.723873 −0.361937 0.932203i $$-0.617884\pi$$
−0.361937 + 0.932203i $$0.617884\pi$$
$$644$$ −17.6268 −0.694595
$$645$$ 0 0
$$646$$ −2.31922 −0.0912485
$$647$$ −33.7771 −1.32791 −0.663957 0.747771i $$-0.731124\pi$$
−0.663957 + 0.747771i $$0.731124\pi$$
$$648$$ −16.9824 −0.667132
$$649$$ 48.1398 1.88965
$$650$$ 0 0
$$651$$ 72.2972 2.83355
$$652$$ −15.5481 −0.608912
$$653$$ −18.0247 −0.705363 −0.352681 0.935743i $$-0.614730\pi$$
−0.352681 + 0.935743i $$0.614730\pi$$
$$654$$ 42.6879 1.66923
$$655$$ 0 0
$$656$$ 10.3616 0.404551
$$657$$ −24.1750 −0.943156
$$658$$ 0 0
$$659$$ 16.5669 0.645355 0.322677 0.946509i $$-0.395417\pi$$
0.322677 + 0.946509i $$0.395417\pi$$
$$660$$ 0 0
$$661$$ −23.7842 −0.925099 −0.462549 0.886593i $$-0.653065\pi$$
−0.462549 + 0.886593i $$0.653065\pi$$
$$662$$ 1.65603 0.0643635
$$663$$ −2.13264 −0.0828248
$$664$$ −13.9461 −0.541212
$$665$$ 0 0
$$666$$ 6.81342 0.264015
$$667$$ −43.7842 −1.69533
$$668$$ 3.77707 0.146139
$$669$$ 88.2069 3.41028
$$670$$ 0 0
$$671$$ 16.1877 0.624921
$$672$$ −12.9461 −0.499405
$$673$$ 12.0551 0.464690 0.232345 0.972633i $$-0.425360\pi$$
0.232345 + 0.972633i $$0.425360\pi$$
$$674$$ 10.2537 0.394957
$$675$$ 0 0
$$676$$ −12.9824 −0.499323
$$677$$ 28.8805 1.10997 0.554984 0.831861i $$-0.312724\pi$$
0.554984 + 0.831861i $$0.312724\pi$$
$$678$$ −24.8921 −0.955976
$$679$$ 2.63844 0.101254
$$680$$ 0 0
$$681$$ 0.886122 0.0339563
$$682$$ 20.5553 0.787103
$$683$$ −17.6328 −0.674701 −0.337351 0.941379i $$-0.609531\pi$$
−0.337351 + 0.941379i $$0.609531\pi$$
$$684$$ 3.07869 0.117717
$$685$$ 0 0
$$686$$ 12.7231 0.485771
$$687$$ 26.4954 1.01086
$$688$$ 5.07869 0.193623
$$689$$ 1.54215 0.0587514
$$690$$ 0 0
$$691$$ −20.6572 −0.785837 −0.392918 0.919573i $$-0.628535\pi$$
−0.392918 + 0.919573i $$0.628535\pi$$
$$692$$ −20.1574 −0.766269
$$693$$ 103.641 3.93700
$$694$$ 25.8805 0.982411
$$695$$ 0 0
$$696$$ −32.1574 −1.21892
$$697$$ 53.1821 2.01442
$$698$$ −12.7535 −0.482727
$$699$$ −44.6328 −1.68817
$$700$$ 0 0
$$701$$ 49.2057 1.85847 0.929237 0.369484i $$-0.120466\pi$$
0.929237 + 0.369484i $$0.120466\pi$$
$$702$$ 1.58449 0.0598029
$$703$$ −0.451857 −0.0170421
$$704$$ −3.68078 −0.138725
$$705$$ 0 0
$$706$$ 8.72312 0.328299
$$707$$ 23.2537 0.874544
$$708$$ 40.9708 1.53978
$$709$$ 3.73473 0.140261 0.0701303 0.997538i $$-0.477659\pi$$
0.0701303 + 0.997538i $$0.477659\pi$$
$$710$$ 0 0
$$711$$ −6.15739 −0.230920
$$712$$ −0.777066 −0.0291218
$$713$$ 23.8194 0.892044
$$714$$ −66.4474 −2.48673
$$715$$ 0 0
$$716$$ 20.2537 0.756915
$$717$$ −68.5800 −2.56117
$$718$$ −14.9037 −0.556202
$$719$$ −47.0858 −1.75601 −0.878003 0.478655i $$-0.841124\pi$$
−0.878003 + 0.478655i $$0.841124\pi$$
$$720$$ 0 0
$$721$$ 41.5014 1.54559
$$722$$ 18.7958 0.699508
$$723$$ 28.6092 1.06399
$$724$$ −17.1690 −0.638080
$$725$$ 0 0
$$726$$ 7.98241 0.296255
$$727$$ −27.3616 −1.01478 −0.507392 0.861715i $$-0.669390\pi$$
−0.507392 + 0.861715i $$0.669390\pi$$
$$728$$ 0.548143 0.0203155
$$729$$ 3.44025 0.127417
$$730$$ 0 0
$$731$$ 26.0671 0.964126
$$732$$ 13.7771 0.509215
$$733$$ −47.5070 −1.75471 −0.877355 0.479842i $$-0.840694\pi$$
−0.877355 + 0.479842i $$0.840694\pi$$
$$734$$ −14.8558 −0.548336
$$735$$ 0 0
$$736$$ −4.26527 −0.157220
$$737$$ −0.842615 −0.0310381
$$738$$ −70.5976 −2.59873
$$739$$ 19.6149 0.721544 0.360772 0.932654i $$-0.382513\pi$$
0.360772 + 0.932654i $$0.382513\pi$$
$$740$$ 0 0
$$741$$ −0.187749 −0.00689713
$$742$$ 48.0495 1.76395
$$743$$ −22.3148 −0.818650 −0.409325 0.912389i $$-0.634236\pi$$
−0.409325 + 0.912389i $$0.634236\pi$$
$$744$$ 17.4942 0.641368
$$745$$ 0 0
$$746$$ −1.77707 −0.0650630
$$747$$ 95.0203 3.47661
$$748$$ −18.8921 −0.690764
$$749$$ 81.2608 2.96921
$$750$$ 0 0
$$751$$ 20.7958 0.758850 0.379425 0.925222i $$-0.376122\pi$$
0.379425 + 0.925222i $$0.376122\pi$$
$$752$$ 0 0
$$753$$ 28.1937 1.02744
$$754$$ 1.36156 0.0495851
$$755$$ 0 0
$$756$$ 49.3687 1.79552
$$757$$ 0.530550 0.0192832 0.00964158 0.999954i $$-0.496931\pi$$
0.00964158 + 0.999954i $$0.496931\pi$$
$$758$$ 26.3863 0.958394
$$759$$ 49.1810 1.78516
$$760$$ 0 0
$$761$$ 4.63245 0.167926 0.0839631 0.996469i $$-0.473242\pi$$
0.0839631 + 0.996469i $$0.473242\pi$$
$$762$$ −54.8030 −1.98530
$$763$$ −56.3148 −2.03873
$$764$$ −0.192571 −0.00696696
$$765$$ 0 0
$$766$$ 4.57289 0.165225
$$767$$ −1.73473 −0.0626373
$$768$$ −3.13264 −0.113039
$$769$$ 7.78867 0.280867 0.140433 0.990090i $$-0.455150\pi$$
0.140433 + 0.990090i $$0.455150\pi$$
$$770$$ 0 0
$$771$$ −27.8921 −1.00451
$$772$$ 14.6692 0.527955
$$773$$ 8.04234 0.289263 0.144631 0.989486i $$-0.453800\pi$$
0.144631 + 0.989486i $$0.453800\pi$$
$$774$$ −34.6033 −1.24379
$$775$$ 0 0
$$776$$ 0.638440 0.0229186
$$777$$ −12.9461 −0.464437
$$778$$ −5.03635 −0.180562
$$779$$ 4.68195 0.167748
$$780$$ 0 0
$$781$$ −34.9461 −1.25047
$$782$$ −21.8921 −0.782860
$$783$$ 122.630 4.38242
$$784$$ 10.0787 0.359953
$$785$$ 0 0
$$786$$ 27.6092 0.984789
$$787$$ −25.6268 −0.913498 −0.456749 0.889596i $$-0.650986\pi$$
−0.456749 + 0.889596i $$0.650986\pi$$
$$788$$ 7.84977 0.279636
$$789$$ −46.6879 −1.66213
$$790$$ 0 0
$$791$$ 32.8382 1.16759
$$792$$ 25.0787 0.891133
$$793$$ −0.583328 −0.0207146
$$794$$ 34.4650 1.22312
$$795$$ 0 0
$$796$$ −11.3616 −0.402700
$$797$$ −11.4095 −0.404146 −0.202073 0.979370i $$-0.564768\pi$$
−0.202073 + 0.979370i $$0.564768\pi$$
$$798$$ −5.84977 −0.207080
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 5.29447 0.187071
$$802$$ 4.47661 0.158074
$$803$$ 13.0599 0.460875
$$804$$ −0.717132 −0.0252913
$$805$$ 0 0
$$806$$ −0.740713 −0.0260905
$$807$$ 4.86853 0.171380
$$808$$ 5.62684 0.197951
$$809$$ 21.5189 0.756566 0.378283 0.925690i $$-0.376515\pi$$
0.378283 + 0.925690i $$0.376515\pi$$
$$810$$ 0 0
$$811$$ −44.0847 −1.54802 −0.774011 0.633172i $$-0.781753\pi$$
−0.774011 + 0.633172i $$0.781753\pi$$
$$812$$ 42.4227 1.48874
$$813$$ −17.8146 −0.624785
$$814$$ −3.68078 −0.129011
$$815$$ 0 0
$$816$$ −16.0787 −0.562867
$$817$$ 2.29484 0.0802864
$$818$$ 34.7243 1.21411
$$819$$ −3.73473 −0.130502
$$820$$ 0 0
$$821$$ −9.95766 −0.347525 −0.173762 0.984788i $$-0.555592\pi$$
−0.173762 + 0.984788i $$0.555592\pi$$
$$822$$ −59.1821 −2.06421
$$823$$ −0.663187 −0.0231173 −0.0115586 0.999933i $$-0.503679\pi$$
−0.0115586 + 0.999933i $$0.503679\pi$$
$$824$$ 10.0423 0.349842
$$825$$ 0 0
$$826$$ −54.0495 −1.88062
$$827$$ 41.0671 1.42804 0.714021 0.700124i $$-0.246872\pi$$
0.714021 + 0.700124i $$0.246872\pi$$
$$828$$ 29.0611 1.00994
$$829$$ 6.48259 0.225150 0.112575 0.993643i $$-0.464090\pi$$
0.112575 + 0.993643i $$0.464090\pi$$
$$830$$ 0 0
$$831$$ −40.4227 −1.40225
$$832$$ 0.132637 0.00459838
$$833$$ 51.7303 1.79235
$$834$$ −46.5189 −1.61082
$$835$$ 0 0
$$836$$ −1.66319 −0.0575225
$$837$$ −66.7127 −2.30593
$$838$$ 7.15023 0.247001
$$839$$ 0.771077 0.0266205 0.0133103 0.999911i $$-0.495763\pi$$
0.0133103 + 0.999911i $$0.495763\pi$$
$$840$$ 0 0
$$841$$ 76.3759 2.63365
$$842$$ −26.6033 −0.916809
$$843$$ 38.4778 1.32524
$$844$$ −12.2289 −0.420937
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −10.5305 −0.361834
$$848$$ 11.6268 0.399267
$$849$$ −84.1881 −2.88933
$$850$$ 0 0
$$851$$ −4.26527 −0.146212
$$852$$ −29.7419 −1.01894
$$853$$ −30.0975 −1.03052 −0.515259 0.857035i $$-0.672304\pi$$
−0.515259 + 0.857035i $$0.672304\pi$$
$$854$$ −18.1750 −0.621935
$$855$$ 0 0
$$856$$ 19.6632 0.672074
$$857$$ −8.22892 −0.281095 −0.140547 0.990074i $$-0.544886\pi$$
−0.140547 + 0.990074i $$0.544886\pi$$
$$858$$ −1.52938 −0.0522123
$$859$$ 14.4286 0.492299 0.246150 0.969232i $$-0.420835\pi$$
0.246150 + 0.969232i $$0.420835\pi$$
$$860$$ 0 0
$$861$$ 134.141 4.57152
$$862$$ 1.16899 0.0398159
$$863$$ −33.9049 −1.15414 −0.577068 0.816696i $$-0.695803\pi$$
−0.577068 + 0.816696i $$0.695803\pi$$
$$864$$ 11.9461 0.406413
$$865$$ 0 0
$$866$$ 5.00000 0.169907
$$867$$ −29.2713 −0.994104
$$868$$ −23.0787 −0.783342
$$869$$ 3.32637 0.112840
$$870$$ 0 0
$$871$$ 0.0303638 0.00102884
$$872$$ −13.6268 −0.461463
$$873$$ −4.34996 −0.147224
$$874$$ −1.92730 −0.0651918
$$875$$ 0 0
$$876$$ 11.1150 0.375543
$$877$$ −1.36872 −0.0462182 −0.0231091 0.999733i $$-0.507357\pi$$
−0.0231091 + 0.999733i $$0.507357\pi$$
$$878$$ 32.3148 1.09057
$$879$$ −52.4826 −1.77019
$$880$$ 0 0
$$881$$ 47.5741 1.60281 0.801405 0.598122i $$-0.204086\pi$$
0.801405 + 0.598122i $$0.204086\pi$$
$$882$$ −68.6703 −2.31225
$$883$$ −9.88051 −0.332506 −0.166253 0.986083i $$-0.553167\pi$$
−0.166253 + 0.986083i $$0.553167\pi$$
$$884$$ 0.680780 0.0228971
$$885$$ 0 0
$$886$$ −17.4155 −0.585085
$$887$$ −21.8801 −0.734663 −0.367331 0.930090i $$-0.619729\pi$$
−0.367331 + 0.930090i $$0.619729\pi$$
$$888$$ −3.13264 −0.105124
$$889$$ 72.2972 2.42477
$$890$$ 0 0
$$891$$ −62.5085 −2.09411
$$892$$ −28.1574 −0.942779
$$893$$ 0 0
$$894$$ −7.56690 −0.253075
$$895$$ 0 0
$$896$$ 4.13264 0.138062
$$897$$ −1.77224 −0.0591735
$$898$$ 7.38032 0.246284
$$899$$ −57.3264 −1.91194
$$900$$ 0 0
$$901$$ 59.6763 1.98811
$$902$$ 38.1386 1.26988
$$903$$ 65.7490 2.18799
$$904$$ 7.94606 0.264282
$$905$$ 0 0
$$906$$ 33.5916 1.11601
$$907$$ −8.46346 −0.281025 −0.140512 0.990079i $$-0.544875\pi$$
−0.140512 + 0.990079i $$0.544875\pi$$
$$908$$ −0.282868 −0.00938729
$$909$$ −38.3380 −1.27159
$$910$$ 0 0
$$911$$ −34.1997 −1.13309 −0.566544 0.824032i $$-0.691720\pi$$
−0.566544 + 0.824032i $$0.691720\pi$$
$$912$$ −1.41551 −0.0468721
$$913$$ −51.3324 −1.69885
$$914$$ 22.7419 0.752235
$$915$$ 0 0
$$916$$ −8.45785 −0.279455
$$917$$ −36.4227 −1.20278
$$918$$ 61.3148 2.02369
$$919$$ −0.723121 −0.0238536 −0.0119268 0.999929i $$-0.503797\pi$$
−0.0119268 + 0.999929i $$0.503797\pi$$
$$920$$ 0 0
$$921$$ −26.5541 −0.874988
$$922$$ −39.7115 −1.30783
$$923$$ 1.25929 0.0414499
$$924$$ −47.6516 −1.56762
$$925$$ 0 0
$$926$$ −30.6456 −1.00708
$$927$$ −68.4227 −2.24730
$$928$$ 10.2653 0.336974
$$929$$ −16.7902 −0.550869 −0.275434 0.961320i $$-0.588822\pi$$
−0.275434 + 0.961320i $$0.588822\pi$$
$$930$$ 0 0
$$931$$ 4.55413 0.149256
$$932$$ 14.2477 0.466698
$$933$$ −101.701 −3.32954
$$934$$ 24.7958 0.811344
$$935$$ 0 0
$$936$$ −0.903715 −0.0295388
$$937$$ −7.01759 −0.229255 −0.114627 0.993409i $$-0.536567\pi$$
−0.114627 + 0.993409i $$0.536567\pi$$
$$938$$ 0.946055 0.0308898
$$939$$ 90.2620 2.94559
$$940$$ 0 0
$$941$$ 30.7958 1.00392 0.501958 0.864892i $$-0.332613\pi$$
0.501958 + 0.864892i $$0.332613\pi$$
$$942$$ 10.3979 0.338782
$$943$$ 44.1949 1.43918
$$944$$ −13.0787 −0.425675
$$945$$ 0 0
$$946$$ 18.6936 0.607780
$$947$$ −6.19257 −0.201232 −0.100616 0.994925i $$-0.532081\pi$$
−0.100616 + 0.994925i $$0.532081\pi$$
$$948$$ 2.83101 0.0919469
$$949$$ −0.470617 −0.0152769
$$950$$ 0 0
$$951$$ −21.5317 −0.698214
$$952$$ 21.2113 0.687463
$$953$$ 37.1750 1.20422 0.602108 0.798415i $$-0.294328\pi$$
0.602108 + 0.798415i $$0.294328\pi$$
$$954$$ −79.2185 −2.56479
$$955$$ 0 0
$$956$$ 21.8921 0.708041
$$957$$ −118.364 −3.82617
$$958$$ 15.5845 0.503512
$$959$$ 78.0742 2.52115
$$960$$ 0 0
$$961$$ 0.186582 0.00601878
$$962$$ 0.132637 0.00427640
$$963$$ −133.974 −4.31724
$$964$$ −9.13264 −0.294142
$$965$$ 0 0
$$966$$ −55.2185 −1.77663
$$967$$ 34.4722 1.10855 0.554275 0.832334i $$-0.312996\pi$$
0.554275 + 0.832334i $$0.312996\pi$$
$$968$$ −2.54814 −0.0819004
$$969$$ −7.26527 −0.233394
$$970$$ 0 0
$$971$$ −54.2488 −1.74093 −0.870464 0.492232i $$-0.836181\pi$$
−0.870464 + 0.492232i $$0.836181\pi$$
$$972$$ −17.3616 −0.556872
$$973$$ 61.3687 1.96739
$$974$$ 3.65720 0.117184
$$975$$ 0 0
$$976$$ −4.39791 −0.140774
$$977$$ 55.4706 1.77466 0.887331 0.461133i $$-0.152557\pi$$
0.887331 + 0.461133i $$0.152557\pi$$
$$978$$ −48.7067 −1.55747
$$979$$ −2.86021 −0.0914126
$$980$$ 0 0
$$981$$ 92.8453 2.96432
$$982$$ −33.3264 −1.06349
$$983$$ 0.422660 0.0134808 0.00674038 0.999977i $$-0.497854\pi$$
0.00674038 + 0.999977i $$0.497854\pi$$
$$984$$ 32.4590 1.03476
$$985$$ 0 0
$$986$$ 52.6879 1.67793
$$987$$ 0 0
$$988$$ 0.0599332 0.00190673
$$989$$ 21.6620 0.688812
$$990$$ 0 0
$$991$$ 7.93445 0.252046 0.126023 0.992027i $$-0.459779\pi$$
0.126023 + 0.992027i $$0.459779\pi$$
$$992$$ −5.58449 −0.177308
$$993$$ 5.18775 0.164628
$$994$$ 39.2361 1.24449
$$995$$ 0 0
$$996$$ −43.6879 −1.38431
$$997$$ −48.8933 −1.54847 −0.774233 0.632901i $$-0.781864\pi$$
−0.774233 + 0.632901i $$0.781864\pi$$
$$998$$ −7.53654 −0.238565
$$999$$ 11.9461 0.377956
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 1850.2.a.y.1.1 3
5.2 odd 4 1850.2.b.p.149.3 6
5.3 odd 4 1850.2.b.p.149.4 6
5.4 even 2 1850.2.a.bc.1.3 yes 3

By twisted newform
Twist Min Dim Char Parity Ord Type
1850.2.a.y.1.1 3 1.1 even 1 trivial
1850.2.a.bc.1.3 yes 3 5.4 even 2
1850.2.b.p.149.3 6 5.2 odd 4
1850.2.b.p.149.4 6 5.3 odd 4