# Properties

 Label 1850.2.a.v.1.1 Level $1850$ Weight $2$ Character 1850.1 Self dual yes Analytic conductor $14.772$ 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] = [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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{6})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 6$$ x^2 - 6 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 370) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-2.44949$$ 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)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -2.44949 q^{3} +1.00000 q^{4} -2.44949 q^{6} +4.44949 q^{7} +1.00000 q^{8} +3.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -2.44949 q^{3} +1.00000 q^{4} -2.44949 q^{6} +4.44949 q^{7} +1.00000 q^{8} +3.00000 q^{9} +4.89898 q^{11} -2.44949 q^{12} +4.00000 q^{13} +4.44949 q^{14} +1.00000 q^{16} -4.89898 q^{17} +3.00000 q^{18} +3.55051 q^{19} -10.8990 q^{21} +4.89898 q^{22} -8.89898 q^{23} -2.44949 q^{24} +4.00000 q^{26} +4.44949 q^{28} -1.55051 q^{31} +1.00000 q^{32} -12.0000 q^{33} -4.89898 q^{34} +3.00000 q^{36} +1.00000 q^{37} +3.55051 q^{38} -9.79796 q^{39} +2.00000 q^{41} -10.8990 q^{42} +4.00000 q^{43} +4.89898 q^{44} -8.89898 q^{46} +4.44949 q^{47} -2.44949 q^{48} +12.7980 q^{49} +12.0000 q^{51} +4.00000 q^{52} -11.7980 q^{53} +4.44949 q^{56} -8.69694 q^{57} -3.55051 q^{59} +12.0000 q^{61} -1.55051 q^{62} +13.3485 q^{63} +1.00000 q^{64} -12.0000 q^{66} -5.55051 q^{67} -4.89898 q^{68} +21.7980 q^{69} +4.89898 q^{71} +3.00000 q^{72} +4.00000 q^{73} +1.00000 q^{74} +3.55051 q^{76} +21.7980 q^{77} -9.79796 q^{78} +6.44949 q^{79} -9.00000 q^{81} +2.00000 q^{82} -9.55051 q^{83} -10.8990 q^{84} +4.00000 q^{86} +4.89898 q^{88} +15.7980 q^{89} +17.7980 q^{91} -8.89898 q^{92} +3.79796 q^{93} +4.44949 q^{94} -2.44949 q^{96} -2.00000 q^{97} +12.7980 q^{98} +14.6969 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{4} + 4 q^{7} + 2 q^{8} + 6 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^4 + 4 * q^7 + 2 * q^8 + 6 * q^9 $$2 q + 2 q^{2} + 2 q^{4} + 4 q^{7} + 2 q^{8} + 6 q^{9} + 8 q^{13} + 4 q^{14} + 2 q^{16} + 6 q^{18} + 12 q^{19} - 12 q^{21} - 8 q^{23} + 8 q^{26} + 4 q^{28} - 8 q^{31} + 2 q^{32} - 24 q^{33} + 6 q^{36} + 2 q^{37} + 12 q^{38} + 4 q^{41} - 12 q^{42} + 8 q^{43} - 8 q^{46} + 4 q^{47} + 6 q^{49} + 24 q^{51} + 8 q^{52} - 4 q^{53} + 4 q^{56} + 12 q^{57} - 12 q^{59} + 24 q^{61} - 8 q^{62} + 12 q^{63} + 2 q^{64} - 24 q^{66} - 16 q^{67} + 24 q^{69} + 6 q^{72} + 8 q^{73} + 2 q^{74} + 12 q^{76} + 24 q^{77} + 8 q^{79} - 18 q^{81} + 4 q^{82} - 24 q^{83} - 12 q^{84} + 8 q^{86} + 12 q^{89} + 16 q^{91} - 8 q^{92} - 12 q^{93} + 4 q^{94} - 4 q^{97} + 6 q^{98}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^4 + 4 * q^7 + 2 * q^8 + 6 * q^9 + 8 * q^13 + 4 * q^14 + 2 * q^16 + 6 * q^18 + 12 * q^19 - 12 * q^21 - 8 * q^23 + 8 * q^26 + 4 * q^28 - 8 * q^31 + 2 * q^32 - 24 * q^33 + 6 * q^36 + 2 * q^37 + 12 * q^38 + 4 * q^41 - 12 * q^42 + 8 * q^43 - 8 * q^46 + 4 * q^47 + 6 * q^49 + 24 * q^51 + 8 * q^52 - 4 * q^53 + 4 * q^56 + 12 * q^57 - 12 * q^59 + 24 * q^61 - 8 * q^62 + 12 * q^63 + 2 * q^64 - 24 * q^66 - 16 * q^67 + 24 * q^69 + 6 * q^72 + 8 * q^73 + 2 * q^74 + 12 * q^76 + 24 * q^77 + 8 * q^79 - 18 * q^81 + 4 * q^82 - 24 * q^83 - 12 * q^84 + 8 * q^86 + 12 * q^89 + 16 * q^91 - 8 * q^92 - 12 * q^93 + 4 * q^94 - 4 * q^97 + 6 * q^98

## 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
$$3$$ −2.44949 −1.41421 −0.707107 0.707107i $$-0.750000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ −2.44949 −1.00000
$$7$$ 4.44949 1.68175 0.840875 0.541230i $$-0.182041\pi$$
0.840875 + 0.541230i $$0.182041\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 3.00000 1.00000
$$10$$ 0 0
$$11$$ 4.89898 1.47710 0.738549 0.674200i $$-0.235511\pi$$
0.738549 + 0.674200i $$0.235511\pi$$
$$12$$ −2.44949 −0.707107
$$13$$ 4.00000 1.10940 0.554700 0.832050i $$-0.312833\pi$$
0.554700 + 0.832050i $$0.312833\pi$$
$$14$$ 4.44949 1.18918
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −4.89898 −1.18818 −0.594089 0.804400i $$-0.702487\pi$$
−0.594089 + 0.804400i $$0.702487\pi$$
$$18$$ 3.00000 0.707107
$$19$$ 3.55051 0.814543 0.407271 0.913307i $$-0.366480\pi$$
0.407271 + 0.913307i $$0.366480\pi$$
$$20$$ 0 0
$$21$$ −10.8990 −2.37835
$$22$$ 4.89898 1.04447
$$23$$ −8.89898 −1.85557 −0.927783 0.373121i $$-0.878288\pi$$
−0.927783 + 0.373121i $$0.878288\pi$$
$$24$$ −2.44949 −0.500000
$$25$$ 0 0
$$26$$ 4.00000 0.784465
$$27$$ 0 0
$$28$$ 4.44949 0.840875
$$29$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$30$$ 0 0
$$31$$ −1.55051 −0.278480 −0.139240 0.990259i $$-0.544466\pi$$
−0.139240 + 0.990259i $$0.544466\pi$$
$$32$$ 1.00000 0.176777
$$33$$ −12.0000 −2.08893
$$34$$ −4.89898 −0.840168
$$35$$ 0 0
$$36$$ 3.00000 0.500000
$$37$$ 1.00000 0.164399
$$38$$ 3.55051 0.575969
$$39$$ −9.79796 −1.56893
$$40$$ 0 0
$$41$$ 2.00000 0.312348 0.156174 0.987730i $$-0.450084\pi$$
0.156174 + 0.987730i $$0.450084\pi$$
$$42$$ −10.8990 −1.68175
$$43$$ 4.00000 0.609994 0.304997 0.952353i $$-0.401344\pi$$
0.304997 + 0.952353i $$0.401344\pi$$
$$44$$ 4.89898 0.738549
$$45$$ 0 0
$$46$$ −8.89898 −1.31208
$$47$$ 4.44949 0.649025 0.324512 0.945881i $$-0.394800\pi$$
0.324512 + 0.945881i $$0.394800\pi$$
$$48$$ −2.44949 −0.353553
$$49$$ 12.7980 1.82828
$$50$$ 0 0
$$51$$ 12.0000 1.68034
$$52$$ 4.00000 0.554700
$$53$$ −11.7980 −1.62057 −0.810287 0.586033i $$-0.800689\pi$$
−0.810287 + 0.586033i $$0.800689\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 4.44949 0.594588
$$57$$ −8.69694 −1.15194
$$58$$ 0 0
$$59$$ −3.55051 −0.462237 −0.231119 0.972926i $$-0.574239\pi$$
−0.231119 + 0.972926i $$0.574239\pi$$
$$60$$ 0 0
$$61$$ 12.0000 1.53644 0.768221 0.640184i $$-0.221142\pi$$
0.768221 + 0.640184i $$0.221142\pi$$
$$62$$ −1.55051 −0.196915
$$63$$ 13.3485 1.68175
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ −12.0000 −1.47710
$$67$$ −5.55051 −0.678103 −0.339051 0.940768i $$-0.610106\pi$$
−0.339051 + 0.940768i $$0.610106\pi$$
$$68$$ −4.89898 −0.594089
$$69$$ 21.7980 2.62417
$$70$$ 0 0
$$71$$ 4.89898 0.581402 0.290701 0.956814i $$-0.406112\pi$$
0.290701 + 0.956814i $$0.406112\pi$$
$$72$$ 3.00000 0.353553
$$73$$ 4.00000 0.468165 0.234082 0.972217i $$-0.424791\pi$$
0.234082 + 0.972217i $$0.424791\pi$$
$$74$$ 1.00000 0.116248
$$75$$ 0 0
$$76$$ 3.55051 0.407271
$$77$$ 21.7980 2.48411
$$78$$ −9.79796 −1.10940
$$79$$ 6.44949 0.725624 0.362812 0.931862i $$-0.381817\pi$$
0.362812 + 0.931862i $$0.381817\pi$$
$$80$$ 0 0
$$81$$ −9.00000 −1.00000
$$82$$ 2.00000 0.220863
$$83$$ −9.55051 −1.04830 −0.524152 0.851625i $$-0.675618\pi$$
−0.524152 + 0.851625i $$0.675618\pi$$
$$84$$ −10.8990 −1.18918
$$85$$ 0 0
$$86$$ 4.00000 0.431331
$$87$$ 0 0
$$88$$ 4.89898 0.522233
$$89$$ 15.7980 1.67458 0.837290 0.546759i $$-0.184139\pi$$
0.837290 + 0.546759i $$0.184139\pi$$
$$90$$ 0 0
$$91$$ 17.7980 1.86573
$$92$$ −8.89898 −0.927783
$$93$$ 3.79796 0.393830
$$94$$ 4.44949 0.458930
$$95$$ 0 0
$$96$$ −2.44949 −0.250000
$$97$$ −2.00000 −0.203069 −0.101535 0.994832i $$-0.532375\pi$$
−0.101535 + 0.994832i $$0.532375\pi$$
$$98$$ 12.7980 1.29279
$$99$$ 14.6969 1.47710
$$100$$ 0 0
$$101$$ 10.6969 1.06439 0.532193 0.846623i $$-0.321368\pi$$
0.532193 + 0.846623i $$0.321368\pi$$
$$102$$ 12.0000 1.18818
$$103$$ 9.79796 0.965422 0.482711 0.875780i $$-0.339652\pi$$
0.482711 + 0.875780i $$0.339652\pi$$
$$104$$ 4.00000 0.392232
$$105$$ 0 0
$$106$$ −11.7980 −1.14592
$$107$$ −5.55051 −0.536588 −0.268294 0.963337i $$-0.586460\pi$$
−0.268294 + 0.963337i $$0.586460\pi$$
$$108$$ 0 0
$$109$$ −5.79796 −0.555344 −0.277672 0.960676i $$-0.589563\pi$$
−0.277672 + 0.960676i $$0.589563\pi$$
$$110$$ 0 0
$$111$$ −2.44949 −0.232495
$$112$$ 4.44949 0.420437
$$113$$ −3.10102 −0.291719 −0.145860 0.989305i $$-0.546595\pi$$
−0.145860 + 0.989305i $$0.546595\pi$$
$$114$$ −8.69694 −0.814543
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 12.0000 1.10940
$$118$$ −3.55051 −0.326851
$$119$$ −21.7980 −1.99822
$$120$$ 0 0
$$121$$ 13.0000 1.18182
$$122$$ 12.0000 1.08643
$$123$$ −4.89898 −0.441726
$$124$$ −1.55051 −0.139240
$$125$$ 0 0
$$126$$ 13.3485 1.18918
$$127$$ −2.65153 −0.235285 −0.117643 0.993056i $$-0.537534\pi$$
−0.117643 + 0.993056i $$0.537534\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ −9.79796 −0.862662
$$130$$ 0 0
$$131$$ −10.2474 −0.895324 −0.447662 0.894203i $$-0.647743\pi$$
−0.447662 + 0.894203i $$0.647743\pi$$
$$132$$ −12.0000 −1.04447
$$133$$ 15.7980 1.36986
$$134$$ −5.55051 −0.479491
$$135$$ 0 0
$$136$$ −4.89898 −0.420084
$$137$$ 19.5959 1.67419 0.837096 0.547056i $$-0.184251\pi$$
0.837096 + 0.547056i $$0.184251\pi$$
$$138$$ 21.7980 1.85557
$$139$$ −5.79796 −0.491776 −0.245888 0.969298i $$-0.579080\pi$$
−0.245888 + 0.969298i $$0.579080\pi$$
$$140$$ 0 0
$$141$$ −10.8990 −0.917860
$$142$$ 4.89898 0.411113
$$143$$ 19.5959 1.63869
$$144$$ 3.00000 0.250000
$$145$$ 0 0
$$146$$ 4.00000 0.331042
$$147$$ −31.3485 −2.58558
$$148$$ 1.00000 0.0821995
$$149$$ −18.6969 −1.53171 −0.765856 0.643012i $$-0.777685\pi$$
−0.765856 + 0.643012i $$0.777685\pi$$
$$150$$ 0 0
$$151$$ −8.00000 −0.651031 −0.325515 0.945537i $$-0.605538\pi$$
−0.325515 + 0.945537i $$0.605538\pi$$
$$152$$ 3.55051 0.287984
$$153$$ −14.6969 −1.18818
$$154$$ 21.7980 1.75653
$$155$$ 0 0
$$156$$ −9.79796 −0.784465
$$157$$ −7.79796 −0.622345 −0.311172 0.950353i $$-0.600722\pi$$
−0.311172 + 0.950353i $$0.600722\pi$$
$$158$$ 6.44949 0.513094
$$159$$ 28.8990 2.29184
$$160$$ 0 0
$$161$$ −39.5959 −3.12060
$$162$$ −9.00000 −0.707107
$$163$$ −21.7980 −1.70735 −0.853674 0.520808i $$-0.825631\pi$$
−0.853674 + 0.520808i $$0.825631\pi$$
$$164$$ 2.00000 0.156174
$$165$$ 0 0
$$166$$ −9.55051 −0.741263
$$167$$ 8.00000 0.619059 0.309529 0.950890i $$-0.399829\pi$$
0.309529 + 0.950890i $$0.399829\pi$$
$$168$$ −10.8990 −0.840875
$$169$$ 3.00000 0.230769
$$170$$ 0 0
$$171$$ 10.6515 0.814543
$$172$$ 4.00000 0.304997
$$173$$ 19.7980 1.50521 0.752605 0.658472i $$-0.228797\pi$$
0.752605 + 0.658472i $$0.228797\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 4.89898 0.369274
$$177$$ 8.69694 0.653702
$$178$$ 15.7980 1.18411
$$179$$ −9.34847 −0.698737 −0.349369 0.936985i $$-0.613604\pi$$
−0.349369 + 0.936985i $$0.613604\pi$$
$$180$$ 0 0
$$181$$ 10.6969 0.795097 0.397549 0.917581i $$-0.369861\pi$$
0.397549 + 0.917581i $$0.369861\pi$$
$$182$$ 17.7980 1.31927
$$183$$ −29.3939 −2.17286
$$184$$ −8.89898 −0.656041
$$185$$ 0 0
$$186$$ 3.79796 0.278480
$$187$$ −24.0000 −1.75505
$$188$$ 4.44949 0.324512
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 11.3485 0.821146 0.410573 0.911828i $$-0.365329\pi$$
0.410573 + 0.911828i $$0.365329\pi$$
$$192$$ −2.44949 −0.176777
$$193$$ 14.0000 1.00774 0.503871 0.863779i $$-0.331909\pi$$
0.503871 + 0.863779i $$0.331909\pi$$
$$194$$ −2.00000 −0.143592
$$195$$ 0 0
$$196$$ 12.7980 0.914140
$$197$$ −2.00000 −0.142494 −0.0712470 0.997459i $$-0.522698\pi$$
−0.0712470 + 0.997459i $$0.522698\pi$$
$$198$$ 14.6969 1.04447
$$199$$ 6.44949 0.457192 0.228596 0.973521i $$-0.426586\pi$$
0.228596 + 0.973521i $$0.426586\pi$$
$$200$$ 0 0
$$201$$ 13.5959 0.958982
$$202$$ 10.6969 0.752634
$$203$$ 0 0
$$204$$ 12.0000 0.840168
$$205$$ 0 0
$$206$$ 9.79796 0.682656
$$207$$ −26.6969 −1.85557
$$208$$ 4.00000 0.277350
$$209$$ 17.3939 1.20316
$$210$$ 0 0
$$211$$ −6.69694 −0.461036 −0.230518 0.973068i $$-0.574042\pi$$
−0.230518 + 0.973068i $$0.574042\pi$$
$$212$$ −11.7980 −0.810287
$$213$$ −12.0000 −0.822226
$$214$$ −5.55051 −0.379425
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −6.89898 −0.468333
$$218$$ −5.79796 −0.392687
$$219$$ −9.79796 −0.662085
$$220$$ 0 0
$$221$$ −19.5959 −1.31816
$$222$$ −2.44949 −0.164399
$$223$$ 0.449490 0.0301001 0.0150500 0.999887i $$-0.495209\pi$$
0.0150500 + 0.999887i $$0.495209\pi$$
$$224$$ 4.44949 0.297294
$$225$$ 0 0
$$226$$ −3.10102 −0.206277
$$227$$ −24.8990 −1.65260 −0.826302 0.563228i $$-0.809559\pi$$
−0.826302 + 0.563228i $$0.809559\pi$$
$$228$$ −8.69694 −0.575969
$$229$$ −10.0000 −0.660819 −0.330409 0.943838i $$-0.607187\pi$$
−0.330409 + 0.943838i $$0.607187\pi$$
$$230$$ 0 0
$$231$$ −53.3939 −3.51306
$$232$$ 0 0
$$233$$ 9.79796 0.641886 0.320943 0.947099i $$-0.396000\pi$$
0.320943 + 0.947099i $$0.396000\pi$$
$$234$$ 12.0000 0.784465
$$235$$ 0 0
$$236$$ −3.55051 −0.231119
$$237$$ −15.7980 −1.02619
$$238$$ −21.7980 −1.41295
$$239$$ −18.0454 −1.16726 −0.583630 0.812020i $$-0.698368\pi$$
−0.583630 + 0.812020i $$0.698368\pi$$
$$240$$ 0 0
$$241$$ 7.79796 0.502311 0.251155 0.967947i $$-0.419189\pi$$
0.251155 + 0.967947i $$0.419189\pi$$
$$242$$ 13.0000 0.835672
$$243$$ 22.0454 1.41421
$$244$$ 12.0000 0.768221
$$245$$ 0 0
$$246$$ −4.89898 −0.312348
$$247$$ 14.2020 0.903654
$$248$$ −1.55051 −0.0984575
$$249$$ 23.3939 1.48253
$$250$$ 0 0
$$251$$ 21.3485 1.34750 0.673752 0.738958i $$-0.264682\pi$$
0.673752 + 0.738958i $$0.264682\pi$$
$$252$$ 13.3485 0.840875
$$253$$ −43.5959 −2.74085
$$254$$ −2.65153 −0.166372
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −4.89898 −0.305590 −0.152795 0.988258i $$-0.548827\pi$$
−0.152795 + 0.988258i $$0.548827\pi$$
$$258$$ −9.79796 −0.609994
$$259$$ 4.44949 0.276478
$$260$$ 0 0
$$261$$ 0 0
$$262$$ −10.2474 −0.633089
$$263$$ 1.75255 0.108067 0.0540335 0.998539i $$-0.482792\pi$$
0.0540335 + 0.998539i $$0.482792\pi$$
$$264$$ −12.0000 −0.738549
$$265$$ 0 0
$$266$$ 15.7980 0.968635
$$267$$ −38.6969 −2.36821
$$268$$ −5.55051 −0.339051
$$269$$ −18.6969 −1.13997 −0.569986 0.821654i $$-0.693051\pi$$
−0.569986 + 0.821654i $$0.693051\pi$$
$$270$$ 0 0
$$271$$ −32.4949 −1.97392 −0.986962 0.160952i $$-0.948544\pi$$
−0.986962 + 0.160952i $$0.948544\pi$$
$$272$$ −4.89898 −0.297044
$$273$$ −43.5959 −2.63854
$$274$$ 19.5959 1.18383
$$275$$ 0 0
$$276$$ 21.7980 1.31208
$$277$$ 19.5959 1.17740 0.588702 0.808350i $$-0.299639\pi$$
0.588702 + 0.808350i $$0.299639\pi$$
$$278$$ −5.79796 −0.347738
$$279$$ −4.65153 −0.278480
$$280$$ 0 0
$$281$$ 27.7980 1.65829 0.829144 0.559036i $$-0.188829\pi$$
0.829144 + 0.559036i $$0.188829\pi$$
$$282$$ −10.8990 −0.649025
$$283$$ 15.5959 0.927081 0.463541 0.886076i $$-0.346579\pi$$
0.463541 + 0.886076i $$0.346579\pi$$
$$284$$ 4.89898 0.290701
$$285$$ 0 0
$$286$$ 19.5959 1.15873
$$287$$ 8.89898 0.525290
$$288$$ 3.00000 0.176777
$$289$$ 7.00000 0.411765
$$290$$ 0 0
$$291$$ 4.89898 0.287183
$$292$$ 4.00000 0.234082
$$293$$ 25.5959 1.49533 0.747665 0.664076i $$-0.231175\pi$$
0.747665 + 0.664076i $$0.231175\pi$$
$$294$$ −31.3485 −1.82828
$$295$$ 0 0
$$296$$ 1.00000 0.0581238
$$297$$ 0 0
$$298$$ −18.6969 −1.08308
$$299$$ −35.5959 −2.05857
$$300$$ 0 0
$$301$$ 17.7980 1.02586
$$302$$ −8.00000 −0.460348
$$303$$ −26.2020 −1.50527
$$304$$ 3.55051 0.203636
$$305$$ 0 0
$$306$$ −14.6969 −0.840168
$$307$$ 13.1464 0.750306 0.375153 0.926963i $$-0.377590\pi$$
0.375153 + 0.926963i $$0.377590\pi$$
$$308$$ 21.7980 1.24205
$$309$$ −24.0000 −1.36531
$$310$$ 0 0
$$311$$ −7.34847 −0.416693 −0.208347 0.978055i $$-0.566808\pi$$
−0.208347 + 0.978055i $$0.566808\pi$$
$$312$$ −9.79796 −0.554700
$$313$$ −17.5959 −0.994580 −0.497290 0.867584i $$-0.665672\pi$$
−0.497290 + 0.867584i $$0.665672\pi$$
$$314$$ −7.79796 −0.440064
$$315$$ 0 0
$$316$$ 6.44949 0.362812
$$317$$ 18.0000 1.01098 0.505490 0.862832i $$-0.331312\pi$$
0.505490 + 0.862832i $$0.331312\pi$$
$$318$$ 28.8990 1.62057
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 13.5959 0.758850
$$322$$ −39.5959 −2.20659
$$323$$ −17.3939 −0.967821
$$324$$ −9.00000 −0.500000
$$325$$ 0 0
$$326$$ −21.7980 −1.20728
$$327$$ 14.2020 0.785375
$$328$$ 2.00000 0.110432
$$329$$ 19.7980 1.09150
$$330$$ 0 0
$$331$$ 14.2474 0.783111 0.391555 0.920155i $$-0.371937\pi$$
0.391555 + 0.920155i $$0.371937\pi$$
$$332$$ −9.55051 −0.524152
$$333$$ 3.00000 0.164399
$$334$$ 8.00000 0.437741
$$335$$ 0 0
$$336$$ −10.8990 −0.594588
$$337$$ −3.59592 −0.195882 −0.0979411 0.995192i $$-0.531226\pi$$
−0.0979411 + 0.995192i $$0.531226\pi$$
$$338$$ 3.00000 0.163178
$$339$$ 7.59592 0.412554
$$340$$ 0 0
$$341$$ −7.59592 −0.411342
$$342$$ 10.6515 0.575969
$$343$$ 25.7980 1.39296
$$344$$ 4.00000 0.215666
$$345$$ 0 0
$$346$$ 19.7980 1.06434
$$347$$ 2.20204 0.118212 0.0591059 0.998252i $$-0.481175\pi$$
0.0591059 + 0.998252i $$0.481175\pi$$
$$348$$ 0 0
$$349$$ 7.10102 0.380109 0.190054 0.981774i $$-0.439134\pi$$
0.190054 + 0.981774i $$0.439134\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 4.89898 0.261116
$$353$$ −6.00000 −0.319348 −0.159674 0.987170i $$-0.551044\pi$$
−0.159674 + 0.987170i $$0.551044\pi$$
$$354$$ 8.69694 0.462237
$$355$$ 0 0
$$356$$ 15.7980 0.837290
$$357$$ 53.3939 2.82590
$$358$$ −9.34847 −0.494082
$$359$$ −12.8990 −0.680782 −0.340391 0.940284i $$-0.610559\pi$$
−0.340391 + 0.940284i $$0.610559\pi$$
$$360$$ 0 0
$$361$$ −6.39388 −0.336520
$$362$$ 10.6969 0.562219
$$363$$ −31.8434 −1.67134
$$364$$ 17.7980 0.932867
$$365$$ 0 0
$$366$$ −29.3939 −1.53644
$$367$$ −2.65153 −0.138409 −0.0692044 0.997603i $$-0.522046\pi$$
−0.0692044 + 0.997603i $$0.522046\pi$$
$$368$$ −8.89898 −0.463891
$$369$$ 6.00000 0.312348
$$370$$ 0 0
$$371$$ −52.4949 −2.72540
$$372$$ 3.79796 0.196915
$$373$$ −11.7980 −0.610875 −0.305438 0.952212i $$-0.598803\pi$$
−0.305438 + 0.952212i $$0.598803\pi$$
$$374$$ −24.0000 −1.24101
$$375$$ 0 0
$$376$$ 4.44949 0.229465
$$377$$ 0 0
$$378$$ 0 0
$$379$$ −31.5959 −1.62297 −0.811487 0.584371i $$-0.801341\pi$$
−0.811487 + 0.584371i $$0.801341\pi$$
$$380$$ 0 0
$$381$$ 6.49490 0.332744
$$382$$ 11.3485 0.580638
$$383$$ −34.6969 −1.77293 −0.886465 0.462795i $$-0.846847\pi$$
−0.886465 + 0.462795i $$0.846847\pi$$
$$384$$ −2.44949 −0.125000
$$385$$ 0 0
$$386$$ 14.0000 0.712581
$$387$$ 12.0000 0.609994
$$388$$ −2.00000 −0.101535
$$389$$ −25.7980 −1.30801 −0.654004 0.756491i $$-0.726912\pi$$
−0.654004 + 0.756491i $$0.726912\pi$$
$$390$$ 0 0
$$391$$ 43.5959 2.20474
$$392$$ 12.7980 0.646395
$$393$$ 25.1010 1.26618
$$394$$ −2.00000 −0.100759
$$395$$ 0 0
$$396$$ 14.6969 0.738549
$$397$$ −16.2020 −0.813157 −0.406579 0.913616i $$-0.633278\pi$$
−0.406579 + 0.913616i $$0.633278\pi$$
$$398$$ 6.44949 0.323284
$$399$$ −38.6969 −1.93727
$$400$$ 0 0
$$401$$ −23.7980 −1.18841 −0.594207 0.804312i $$-0.702534\pi$$
−0.594207 + 0.804312i $$0.702534\pi$$
$$402$$ 13.5959 0.678103
$$403$$ −6.20204 −0.308946
$$404$$ 10.6969 0.532193
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 4.89898 0.242833
$$408$$ 12.0000 0.594089
$$409$$ −10.0000 −0.494468 −0.247234 0.968956i $$-0.579522\pi$$
−0.247234 + 0.968956i $$0.579522\pi$$
$$410$$ 0 0
$$411$$ −48.0000 −2.36767
$$412$$ 9.79796 0.482711
$$413$$ −15.7980 −0.777367
$$414$$ −26.6969 −1.31208
$$415$$ 0 0
$$416$$ 4.00000 0.196116
$$417$$ 14.2020 0.695477
$$418$$ 17.3939 0.850762
$$419$$ 5.79796 0.283249 0.141624 0.989920i $$-0.454767\pi$$
0.141624 + 0.989920i $$0.454767\pi$$
$$420$$ 0 0
$$421$$ −19.5959 −0.955047 −0.477523 0.878619i $$-0.658465\pi$$
−0.477523 + 0.878619i $$0.658465\pi$$
$$422$$ −6.69694 −0.326002
$$423$$ 13.3485 0.649025
$$424$$ −11.7980 −0.572960
$$425$$ 0 0
$$426$$ −12.0000 −0.581402
$$427$$ 53.3939 2.58391
$$428$$ −5.55051 −0.268294
$$429$$ −48.0000 −2.31746
$$430$$ 0 0
$$431$$ −15.7526 −0.758774 −0.379387 0.925238i $$-0.623865\pi$$
−0.379387 + 0.925238i $$0.623865\pi$$
$$432$$ 0 0
$$433$$ 21.3939 1.02812 0.514062 0.857753i $$-0.328140\pi$$
0.514062 + 0.857753i $$0.328140\pi$$
$$434$$ −6.89898 −0.331162
$$435$$ 0 0
$$436$$ −5.79796 −0.277672
$$437$$ −31.5959 −1.51144
$$438$$ −9.79796 −0.468165
$$439$$ 20.6515 0.985644 0.492822 0.870130i $$-0.335965\pi$$
0.492822 + 0.870130i $$0.335965\pi$$
$$440$$ 0 0
$$441$$ 38.3939 1.82828
$$442$$ −19.5959 −0.932083
$$443$$ 24.6515 1.17123 0.585615 0.810589i $$-0.300853\pi$$
0.585615 + 0.810589i $$0.300853\pi$$
$$444$$ −2.44949 −0.116248
$$445$$ 0 0
$$446$$ 0.449490 0.0212840
$$447$$ 45.7980 2.16617
$$448$$ 4.44949 0.210219
$$449$$ −15.7980 −0.745552 −0.372776 0.927921i $$-0.621594\pi$$
−0.372776 + 0.927921i $$0.621594\pi$$
$$450$$ 0 0
$$451$$ 9.79796 0.461368
$$452$$ −3.10102 −0.145860
$$453$$ 19.5959 0.920697
$$454$$ −24.8990 −1.16857
$$455$$ 0 0
$$456$$ −8.69694 −0.407271
$$457$$ −2.00000 −0.0935561 −0.0467780 0.998905i $$-0.514895\pi$$
−0.0467780 + 0.998905i $$0.514895\pi$$
$$458$$ −10.0000 −0.467269
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −33.7980 −1.57413 −0.787064 0.616871i $$-0.788400\pi$$
−0.787064 + 0.616871i $$0.788400\pi$$
$$462$$ −53.3939 −2.48411
$$463$$ −20.4949 −0.952479 −0.476239 0.879316i $$-0.658000\pi$$
−0.476239 + 0.879316i $$0.658000\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 9.79796 0.453882
$$467$$ −12.0000 −0.555294 −0.277647 0.960683i $$-0.589555\pi$$
−0.277647 + 0.960683i $$0.589555\pi$$
$$468$$ 12.0000 0.554700
$$469$$ −24.6969 −1.14040
$$470$$ 0 0
$$471$$ 19.1010 0.880129
$$472$$ −3.55051 −0.163425
$$473$$ 19.5959 0.901021
$$474$$ −15.7980 −0.725624
$$475$$ 0 0
$$476$$ −21.7980 −0.999108
$$477$$ −35.3939 −1.62057
$$478$$ −18.0454 −0.825378
$$479$$ 5.14643 0.235146 0.117573 0.993064i $$-0.462489\pi$$
0.117573 + 0.993064i $$0.462489\pi$$
$$480$$ 0 0
$$481$$ 4.00000 0.182384
$$482$$ 7.79796 0.355187
$$483$$ 96.9898 4.41319
$$484$$ 13.0000 0.590909
$$485$$ 0 0
$$486$$ 22.0454 1.00000
$$487$$ −29.3939 −1.33196 −0.665982 0.745968i $$-0.731987\pi$$
−0.665982 + 0.745968i $$0.731987\pi$$
$$488$$ 12.0000 0.543214
$$489$$ 53.3939 2.41455
$$490$$ 0 0
$$491$$ −9.30306 −0.419841 −0.209921 0.977718i $$-0.567321\pi$$
−0.209921 + 0.977718i $$0.567321\pi$$
$$492$$ −4.89898 −0.220863
$$493$$ 0 0
$$494$$ 14.2020 0.638980
$$495$$ 0 0
$$496$$ −1.55051 −0.0696200
$$497$$ 21.7980 0.977772
$$498$$ 23.3939 1.04830
$$499$$ −10.6515 −0.476828 −0.238414 0.971164i $$-0.576628\pi$$
−0.238414 + 0.971164i $$0.576628\pi$$
$$500$$ 0 0
$$501$$ −19.5959 −0.875481
$$502$$ 21.3485 0.952829
$$503$$ −1.79796 −0.0801670 −0.0400835 0.999196i $$-0.512762\pi$$
−0.0400835 + 0.999196i $$0.512762\pi$$
$$504$$ 13.3485 0.594588
$$505$$ 0 0
$$506$$ −43.5959 −1.93807
$$507$$ −7.34847 −0.326357
$$508$$ −2.65153 −0.117643
$$509$$ 30.0000 1.32973 0.664863 0.746965i $$-0.268490\pi$$
0.664863 + 0.746965i $$0.268490\pi$$
$$510$$ 0 0
$$511$$ 17.7980 0.787335
$$512$$ 1.00000 0.0441942
$$513$$ 0 0
$$514$$ −4.89898 −0.216085
$$515$$ 0 0
$$516$$ −9.79796 −0.431331
$$517$$ 21.7980 0.958673
$$518$$ 4.44949 0.195499
$$519$$ −48.4949 −2.12869
$$520$$ 0 0
$$521$$ 17.7980 0.779743 0.389871 0.920869i $$-0.372519\pi$$
0.389871 + 0.920869i $$0.372519\pi$$
$$522$$ 0 0
$$523$$ −8.89898 −0.389125 −0.194563 0.980890i $$-0.562329\pi$$
−0.194563 + 0.980890i $$0.562329\pi$$
$$524$$ −10.2474 −0.447662
$$525$$ 0 0
$$526$$ 1.75255 0.0764149
$$527$$ 7.59592 0.330883
$$528$$ −12.0000 −0.522233
$$529$$ 56.1918 2.44312
$$530$$ 0 0
$$531$$ −10.6515 −0.462237
$$532$$ 15.7980 0.684928
$$533$$ 8.00000 0.346518
$$534$$ −38.6969 −1.67458
$$535$$ 0 0
$$536$$ −5.55051 −0.239746
$$537$$ 22.8990 0.988164
$$538$$ −18.6969 −0.806082
$$539$$ 62.6969 2.70055
$$540$$ 0 0
$$541$$ 0.404082 0.0173728 0.00868642 0.999962i $$-0.497235\pi$$
0.00868642 + 0.999962i $$0.497235\pi$$
$$542$$ −32.4949 −1.39578
$$543$$ −26.2020 −1.12444
$$544$$ −4.89898 −0.210042
$$545$$ 0 0
$$546$$ −43.5959 −1.86573
$$547$$ −10.6969 −0.457368 −0.228684 0.973501i $$-0.573442\pi$$
−0.228684 + 0.973501i $$0.573442\pi$$
$$548$$ 19.5959 0.837096
$$549$$ 36.0000 1.53644
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 21.7980 0.927783
$$553$$ 28.6969 1.22032
$$554$$ 19.5959 0.832551
$$555$$ 0 0
$$556$$ −5.79796 −0.245888
$$557$$ −12.0000 −0.508456 −0.254228 0.967144i $$-0.581821\pi$$
−0.254228 + 0.967144i $$0.581821\pi$$
$$558$$ −4.65153 −0.196915
$$559$$ 16.0000 0.676728
$$560$$ 0 0
$$561$$ 58.7878 2.48202
$$562$$ 27.7980 1.17259
$$563$$ −34.6969 −1.46230 −0.731151 0.682216i $$-0.761016\pi$$
−0.731151 + 0.682216i $$0.761016\pi$$
$$564$$ −10.8990 −0.458930
$$565$$ 0 0
$$566$$ 15.5959 0.655545
$$567$$ −40.0454 −1.68175
$$568$$ 4.89898 0.205557
$$569$$ 21.5959 0.905348 0.452674 0.891676i $$-0.350470\pi$$
0.452674 + 0.891676i $$0.350470\pi$$
$$570$$ 0 0
$$571$$ −25.3939 −1.06270 −0.531350 0.847152i $$-0.678315\pi$$
−0.531350 + 0.847152i $$0.678315\pi$$
$$572$$ 19.5959 0.819346
$$573$$ −27.7980 −1.16128
$$574$$ 8.89898 0.371436
$$575$$ 0 0
$$576$$ 3.00000 0.125000
$$577$$ −30.6969 −1.27793 −0.638965 0.769236i $$-0.720637\pi$$
−0.638965 + 0.769236i $$0.720637\pi$$
$$578$$ 7.00000 0.291162
$$579$$ −34.2929 −1.42516
$$580$$ 0 0
$$581$$ −42.4949 −1.76299
$$582$$ 4.89898 0.203069
$$583$$ −57.7980 −2.39375
$$584$$ 4.00000 0.165521
$$585$$ 0 0
$$586$$ 25.5959 1.05736
$$587$$ −12.0000 −0.495293 −0.247647 0.968850i $$-0.579657\pi$$
−0.247647 + 0.968850i $$0.579657\pi$$
$$588$$ −31.3485 −1.29279
$$589$$ −5.50510 −0.226834
$$590$$ 0 0
$$591$$ 4.89898 0.201517
$$592$$ 1.00000 0.0410997
$$593$$ 24.0000 0.985562 0.492781 0.870153i $$-0.335980\pi$$
0.492781 + 0.870153i $$0.335980\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −18.6969 −0.765856
$$597$$ −15.7980 −0.646567
$$598$$ −35.5959 −1.45563
$$599$$ 11.5959 0.473796 0.236898 0.971534i $$-0.423869\pi$$
0.236898 + 0.971534i $$0.423869\pi$$
$$600$$ 0 0
$$601$$ −8.00000 −0.326327 −0.163163 0.986599i $$-0.552170\pi$$
−0.163163 + 0.986599i $$0.552170\pi$$
$$602$$ 17.7980 0.725391
$$603$$ −16.6515 −0.678103
$$604$$ −8.00000 −0.325515
$$605$$ 0 0
$$606$$ −26.2020 −1.06439
$$607$$ 26.6969 1.08360 0.541798 0.840509i $$-0.317744\pi$$
0.541798 + 0.840509i $$0.317744\pi$$
$$608$$ 3.55051 0.143992
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 17.7980 0.720028
$$612$$ −14.6969 −0.594089
$$613$$ 39.7980 1.60742 0.803712 0.595018i $$-0.202855\pi$$
0.803712 + 0.595018i $$0.202855\pi$$
$$614$$ 13.1464 0.530547
$$615$$ 0 0
$$616$$ 21.7980 0.878265
$$617$$ 19.5959 0.788902 0.394451 0.918917i $$-0.370935\pi$$
0.394451 + 0.918917i $$0.370935\pi$$
$$618$$ −24.0000 −0.965422
$$619$$ −12.8990 −0.518454 −0.259227 0.965816i $$-0.583468\pi$$
−0.259227 + 0.965816i $$0.583468\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ −7.34847 −0.294647
$$623$$ 70.2929 2.81622
$$624$$ −9.79796 −0.392232
$$625$$ 0 0
$$626$$ −17.5959 −0.703274
$$627$$ −42.6061 −1.70152
$$628$$ −7.79796 −0.311172
$$629$$ −4.89898 −0.195335
$$630$$ 0 0
$$631$$ −20.2474 −0.806038 −0.403019 0.915192i $$-0.632039\pi$$
−0.403019 + 0.915192i $$0.632039\pi$$
$$632$$ 6.44949 0.256547
$$633$$ 16.4041 0.652004
$$634$$ 18.0000 0.714871
$$635$$ 0 0
$$636$$ 28.8990 1.14592
$$637$$ 51.1918 2.02829
$$638$$ 0 0
$$639$$ 14.6969 0.581402
$$640$$ 0 0
$$641$$ 17.7980 0.702977 0.351489 0.936192i $$-0.385676\pi$$
0.351489 + 0.936192i $$0.385676\pi$$
$$642$$ 13.5959 0.536588
$$643$$ −23.1010 −0.911015 −0.455508 0.890232i $$-0.650542\pi$$
−0.455508 + 0.890232i $$0.650542\pi$$
$$644$$ −39.5959 −1.56030
$$645$$ 0 0
$$646$$ −17.3939 −0.684353
$$647$$ −32.0000 −1.25805 −0.629025 0.777385i $$-0.716546\pi$$
−0.629025 + 0.777385i $$0.716546\pi$$
$$648$$ −9.00000 −0.353553
$$649$$ −17.3939 −0.682769
$$650$$ 0 0
$$651$$ 16.8990 0.662323
$$652$$ −21.7980 −0.853674
$$653$$ 34.0000 1.33052 0.665261 0.746611i $$-0.268320\pi$$
0.665261 + 0.746611i $$0.268320\pi$$
$$654$$ 14.2020 0.555344
$$655$$ 0 0
$$656$$ 2.00000 0.0780869
$$657$$ 12.0000 0.468165
$$658$$ 19.7980 0.771805
$$659$$ 31.5959 1.23080 0.615401 0.788214i $$-0.288994\pi$$
0.615401 + 0.788214i $$0.288994\pi$$
$$660$$ 0 0
$$661$$ −31.1918 −1.21322 −0.606611 0.794999i $$-0.707471\pi$$
−0.606611 + 0.794999i $$0.707471\pi$$
$$662$$ 14.2474 0.553743
$$663$$ 48.0000 1.86417
$$664$$ −9.55051 −0.370632
$$665$$ 0 0
$$666$$ 3.00000 0.116248
$$667$$ 0 0
$$668$$ 8.00000 0.309529
$$669$$ −1.10102 −0.0425679
$$670$$ 0 0
$$671$$ 58.7878 2.26948
$$672$$ −10.8990 −0.420437
$$673$$ 24.0000 0.925132 0.462566 0.886585i $$-0.346929\pi$$
0.462566 + 0.886585i $$0.346929\pi$$
$$674$$ −3.59592 −0.138510
$$675$$ 0 0
$$676$$ 3.00000 0.115385
$$677$$ 18.0000 0.691796 0.345898 0.938272i $$-0.387574\pi$$
0.345898 + 0.938272i $$0.387574\pi$$
$$678$$ 7.59592 0.291719
$$679$$ −8.89898 −0.341511
$$680$$ 0 0
$$681$$ 60.9898 2.33713
$$682$$ −7.59592 −0.290863
$$683$$ 15.5959 0.596761 0.298381 0.954447i $$-0.403554\pi$$
0.298381 + 0.954447i $$0.403554\pi$$
$$684$$ 10.6515 0.407271
$$685$$ 0 0
$$686$$ 25.7980 0.984971
$$687$$ 24.4949 0.934539
$$688$$ 4.00000 0.152499
$$689$$ −47.1918 −1.79787
$$690$$ 0 0
$$691$$ 37.7980 1.43790 0.718951 0.695061i $$-0.244623\pi$$
0.718951 + 0.695061i $$0.244623\pi$$
$$692$$ 19.7980 0.752605
$$693$$ 65.3939 2.48411
$$694$$ 2.20204 0.0835883
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −9.79796 −0.371124
$$698$$ 7.10102 0.268778
$$699$$ −24.0000 −0.907763
$$700$$ 0 0
$$701$$ −13.7980 −0.521142 −0.260571 0.965455i $$-0.583911\pi$$
−0.260571 + 0.965455i $$0.583911\pi$$
$$702$$ 0 0
$$703$$ 3.55051 0.133910
$$704$$ 4.89898 0.184637
$$705$$ 0 0
$$706$$ −6.00000 −0.225813
$$707$$ 47.5959 1.79003
$$708$$ 8.69694 0.326851
$$709$$ 5.79796 0.217747 0.108873 0.994056i $$-0.465276\pi$$
0.108873 + 0.994056i $$0.465276\pi$$
$$710$$ 0 0
$$711$$ 19.3485 0.725624
$$712$$ 15.7980 0.592054
$$713$$ 13.7980 0.516738
$$714$$ 53.3939 1.99822
$$715$$ 0 0
$$716$$ −9.34847 −0.349369
$$717$$ 44.2020 1.65076
$$718$$ −12.8990 −0.481386
$$719$$ 51.5959 1.92420 0.962102 0.272692i $$-0.0879138\pi$$
0.962102 + 0.272692i $$0.0879138\pi$$
$$720$$ 0 0
$$721$$ 43.5959 1.62360
$$722$$ −6.39388 −0.237955
$$723$$ −19.1010 −0.710375
$$724$$ 10.6969 0.397549
$$725$$ 0 0
$$726$$ −31.8434 −1.18182
$$727$$ 0.898979 0.0333413 0.0166707 0.999861i $$-0.494693\pi$$
0.0166707 + 0.999861i $$0.494693\pi$$
$$728$$ 17.7980 0.659636
$$729$$ −27.0000 −1.00000
$$730$$ 0 0
$$731$$ −19.5959 −0.724781
$$732$$ −29.3939 −1.08643
$$733$$ −17.5959 −0.649920 −0.324960 0.945728i $$-0.605351\pi$$
−0.324960 + 0.945728i $$0.605351\pi$$
$$734$$ −2.65153 −0.0978698
$$735$$ 0 0
$$736$$ −8.89898 −0.328021
$$737$$ −27.1918 −1.00162
$$738$$ 6.00000 0.220863
$$739$$ −45.7980 −1.68471 −0.842353 0.538927i $$-0.818830\pi$$
−0.842353 + 0.538927i $$0.818830\pi$$
$$740$$ 0 0
$$741$$ −34.7878 −1.27796
$$742$$ −52.4949 −1.92715
$$743$$ −13.7526 −0.504532 −0.252266 0.967658i $$-0.581176\pi$$
−0.252266 + 0.967658i $$0.581176\pi$$
$$744$$ 3.79796 0.139240
$$745$$ 0 0
$$746$$ −11.7980 −0.431954
$$747$$ −28.6515 −1.04830
$$748$$ −24.0000 −0.877527
$$749$$ −24.6969 −0.902406
$$750$$ 0 0
$$751$$ 30.6969 1.12015 0.560074 0.828443i $$-0.310773\pi$$
0.560074 + 0.828443i $$0.310773\pi$$
$$752$$ 4.44949 0.162256
$$753$$ −52.2929 −1.90566
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −33.5959 −1.22106 −0.610532 0.791991i $$-0.709044\pi$$
−0.610532 + 0.791991i $$0.709044\pi$$
$$758$$ −31.5959 −1.14762
$$759$$ 106.788 3.87615
$$760$$ 0 0
$$761$$ 25.1918 0.913203 0.456602 0.889671i $$-0.349066\pi$$
0.456602 + 0.889671i $$0.349066\pi$$
$$762$$ 6.49490 0.235285
$$763$$ −25.7980 −0.933949
$$764$$ 11.3485 0.410573
$$765$$ 0 0
$$766$$ −34.6969 −1.25365
$$767$$ −14.2020 −0.512806
$$768$$ −2.44949 −0.0883883
$$769$$ −35.7980 −1.29091 −0.645454 0.763799i $$-0.723332\pi$$
−0.645454 + 0.763799i $$0.723332\pi$$
$$770$$ 0 0
$$771$$ 12.0000 0.432169
$$772$$ 14.0000 0.503871
$$773$$ −6.00000 −0.215805 −0.107903 0.994161i $$-0.534413\pi$$
−0.107903 + 0.994161i $$0.534413\pi$$
$$774$$ 12.0000 0.431331
$$775$$ 0 0
$$776$$ −2.00000 −0.0717958
$$777$$ −10.8990 −0.390999
$$778$$ −25.7980 −0.924902
$$779$$ 7.10102 0.254420
$$780$$ 0 0
$$781$$ 24.0000 0.858788
$$782$$ 43.5959 1.55899
$$783$$ 0 0
$$784$$ 12.7980 0.457070
$$785$$ 0 0
$$786$$ 25.1010 0.895324
$$787$$ −28.7423 −1.02455 −0.512277 0.858820i $$-0.671198\pi$$
−0.512277 + 0.858820i $$0.671198\pi$$
$$788$$ −2.00000 −0.0712470
$$789$$ −4.29286 −0.152830
$$790$$ 0 0
$$791$$ −13.7980 −0.490599
$$792$$ 14.6969 0.522233
$$793$$ 48.0000 1.70453
$$794$$ −16.2020 −0.574989
$$795$$ 0 0
$$796$$ 6.44949 0.228596
$$797$$ −43.5959 −1.54425 −0.772123 0.635473i $$-0.780805\pi$$
−0.772123 + 0.635473i $$0.780805\pi$$
$$798$$ −38.6969 −1.36986
$$799$$ −21.7980 −0.771156
$$800$$ 0 0
$$801$$ 47.3939 1.67458
$$802$$ −23.7980 −0.840335
$$803$$ 19.5959 0.691525
$$804$$ 13.5959 0.479491
$$805$$ 0 0
$$806$$ −6.20204 −0.218458
$$807$$ 45.7980 1.61216
$$808$$ 10.6969 0.376317
$$809$$ −33.1918 −1.16696 −0.583481 0.812127i $$-0.698310\pi$$
−0.583481 + 0.812127i $$0.698310\pi$$
$$810$$ 0 0
$$811$$ 26.2020 0.920078 0.460039 0.887899i $$-0.347835\pi$$
0.460039 + 0.887899i $$0.347835\pi$$
$$812$$ 0 0
$$813$$ 79.5959 2.79155
$$814$$ 4.89898 0.171709
$$815$$ 0 0
$$816$$ 12.0000 0.420084
$$817$$ 14.2020 0.496867
$$818$$ −10.0000 −0.349642
$$819$$ 53.3939 1.86573
$$820$$ 0 0
$$821$$ −6.40408 −0.223504 −0.111752 0.993736i $$-0.535646\pi$$
−0.111752 + 0.993736i $$0.535646\pi$$
$$822$$ −48.0000 −1.67419
$$823$$ 33.3485 1.16245 0.581227 0.813741i $$-0.302573\pi$$
0.581227 + 0.813741i $$0.302573\pi$$
$$824$$ 9.79796 0.341328
$$825$$ 0 0
$$826$$ −15.7980 −0.549681
$$827$$ −36.4949 −1.26905 −0.634526 0.772902i $$-0.718805\pi$$
−0.634526 + 0.772902i $$0.718805\pi$$
$$828$$ −26.6969 −0.927783
$$829$$ 8.40408 0.291886 0.145943 0.989293i $$-0.453378\pi$$
0.145943 + 0.989293i $$0.453378\pi$$
$$830$$ 0 0
$$831$$ −48.0000 −1.66510
$$832$$ 4.00000 0.138675
$$833$$ −62.6969 −2.17232
$$834$$ 14.2020 0.491776
$$835$$ 0 0
$$836$$ 17.3939 0.601580
$$837$$ 0 0
$$838$$ 5.79796 0.200287
$$839$$ 14.2020 0.490309 0.245154 0.969484i $$-0.421161\pi$$
0.245154 + 0.969484i $$0.421161\pi$$
$$840$$ 0 0
$$841$$ −29.0000 −1.00000
$$842$$ −19.5959 −0.675320
$$843$$ −68.0908 −2.34517
$$844$$ −6.69694 −0.230518
$$845$$ 0 0
$$846$$ 13.3485 0.458930
$$847$$ 57.8434 1.98752
$$848$$ −11.7980 −0.405144
$$849$$ −38.2020 −1.31109
$$850$$ 0 0
$$851$$ −8.89898 −0.305053
$$852$$ −12.0000 −0.411113
$$853$$ −37.5959 −1.28726 −0.643630 0.765337i $$-0.722572\pi$$
−0.643630 + 0.765337i $$0.722572\pi$$
$$854$$ 53.3939 1.82710
$$855$$ 0 0
$$856$$ −5.55051 −0.189713
$$857$$ 9.59592 0.327790 0.163895 0.986478i $$-0.447594\pi$$
0.163895 + 0.986478i $$0.447594\pi$$
$$858$$ −48.0000 −1.63869
$$859$$ −55.1464 −1.88157 −0.940786 0.339001i $$-0.889911\pi$$
−0.940786 + 0.339001i $$0.889911\pi$$
$$860$$ 0 0
$$861$$ −21.7980 −0.742872
$$862$$ −15.7526 −0.536534
$$863$$ −42.7423 −1.45497 −0.727483 0.686126i $$-0.759310\pi$$
−0.727483 + 0.686126i $$0.759310\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 21.3939 0.726994
$$867$$ −17.1464 −0.582323
$$868$$ −6.89898 −0.234167
$$869$$ 31.5959 1.07182
$$870$$ 0 0
$$871$$ −22.2020 −0.752287
$$872$$ −5.79796 −0.196344
$$873$$ −6.00000 −0.203069
$$874$$ −31.5959 −1.06875
$$875$$ 0 0
$$876$$ −9.79796 −0.331042
$$877$$ 15.3939 0.519814 0.259907 0.965634i $$-0.416308\pi$$
0.259907 + 0.965634i $$0.416308\pi$$
$$878$$ 20.6515 0.696955
$$879$$ −62.6969 −2.11472
$$880$$ 0 0
$$881$$ −5.39388 −0.181724 −0.0908622 0.995863i $$-0.528962\pi$$
−0.0908622 + 0.995863i $$0.528962\pi$$
$$882$$ 38.3939 1.29279
$$883$$ 42.6969 1.43687 0.718433 0.695596i $$-0.244860\pi$$
0.718433 + 0.695596i $$0.244860\pi$$
$$884$$ −19.5959 −0.659082
$$885$$ 0 0
$$886$$ 24.6515 0.828184
$$887$$ −20.0454 −0.673059 −0.336529 0.941673i $$-0.609253\pi$$
−0.336529 + 0.941673i $$0.609253\pi$$
$$888$$ −2.44949 −0.0821995
$$889$$ −11.7980 −0.395691
$$890$$ 0 0
$$891$$ −44.0908 −1.47710
$$892$$ 0.449490 0.0150500
$$893$$ 15.7980 0.528659
$$894$$ 45.7980 1.53171
$$895$$ 0 0
$$896$$ 4.44949 0.148647
$$897$$ 87.1918 2.91125
$$898$$ −15.7980 −0.527185
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 57.7980 1.92553
$$902$$ 9.79796 0.326236
$$903$$ −43.5959 −1.45078
$$904$$ −3.10102 −0.103138
$$905$$ 0 0
$$906$$ 19.5959 0.651031
$$907$$ 0.898979 0.0298501 0.0149251 0.999889i $$-0.495249\pi$$
0.0149251 + 0.999889i $$0.495249\pi$$
$$908$$ −24.8990 −0.826302
$$909$$ 32.0908 1.06439
$$910$$ 0 0
$$911$$ −31.8434 −1.05502 −0.527509 0.849550i $$-0.676874\pi$$
−0.527509 + 0.849550i $$0.676874\pi$$
$$912$$ −8.69694 −0.287984
$$913$$ −46.7878 −1.54845
$$914$$ −2.00000 −0.0661541
$$915$$ 0 0
$$916$$ −10.0000 −0.330409
$$917$$ −45.5959 −1.50571
$$918$$ 0 0
$$919$$ 25.1464 0.829504 0.414752 0.909934i $$-0.363868\pi$$
0.414752 + 0.909934i $$0.363868\pi$$
$$920$$ 0 0
$$921$$ −32.2020 −1.06109
$$922$$ −33.7980 −1.11308
$$923$$ 19.5959 0.645007
$$924$$ −53.3939 −1.75653
$$925$$ 0 0
$$926$$ −20.4949 −0.673504
$$927$$ 29.3939 0.965422
$$928$$ 0 0
$$929$$ −1.59592 −0.0523604 −0.0261802 0.999657i $$-0.508334\pi$$
−0.0261802 + 0.999657i $$0.508334\pi$$
$$930$$ 0 0
$$931$$ 45.4393 1.48921
$$932$$ 9.79796 0.320943
$$933$$ 18.0000 0.589294
$$934$$ −12.0000 −0.392652
$$935$$ 0 0
$$936$$ 12.0000 0.392232
$$937$$ 8.00000 0.261349 0.130674 0.991425i $$-0.458286\pi$$
0.130674 + 0.991425i $$0.458286\pi$$
$$938$$ −24.6969 −0.806384
$$939$$ 43.1010 1.40655
$$940$$ 0 0
$$941$$ −38.2929 −1.24831 −0.624156 0.781300i $$-0.714557\pi$$
−0.624156 + 0.781300i $$0.714557\pi$$
$$942$$ 19.1010 0.622345
$$943$$ −17.7980 −0.579581
$$944$$ −3.55051 −0.115559
$$945$$ 0 0
$$946$$ 19.5959 0.637118
$$947$$ 25.3939 0.825190 0.412595 0.910915i $$-0.364622\pi$$
0.412595 + 0.910915i $$0.364622\pi$$
$$948$$ −15.7980 −0.513094
$$949$$ 16.0000 0.519382
$$950$$ 0 0
$$951$$ −44.0908 −1.42974
$$952$$ −21.7980 −0.706476
$$953$$ −21.7980 −0.706105 −0.353053 0.935603i $$-0.614856\pi$$
−0.353053 + 0.935603i $$0.614856\pi$$
$$954$$ −35.3939 −1.14592
$$955$$ 0 0
$$956$$ −18.0454 −0.583630
$$957$$ 0 0
$$958$$ 5.14643 0.166274
$$959$$ 87.1918 2.81557
$$960$$ 0 0
$$961$$ −28.5959 −0.922449
$$962$$ 4.00000 0.128965
$$963$$ −16.6515 −0.536588
$$964$$ 7.79796 0.251155
$$965$$ 0 0
$$966$$ 96.9898 3.12060
$$967$$ 3.50510 0.112716 0.0563582 0.998411i $$-0.482051\pi$$
0.0563582 + 0.998411i $$0.482051\pi$$
$$968$$ 13.0000 0.417836
$$969$$ 42.6061 1.36871
$$970$$ 0 0
$$971$$ −35.1010 −1.12645 −0.563223 0.826305i $$-0.690439\pi$$
−0.563223 + 0.826305i $$0.690439\pi$$
$$972$$ 22.0454 0.707107
$$973$$ −25.7980 −0.827045
$$974$$ −29.3939 −0.941841
$$975$$ 0 0
$$976$$ 12.0000 0.384111
$$977$$ −10.4041 −0.332856 −0.166428 0.986054i $$-0.553223\pi$$
−0.166428 + 0.986054i $$0.553223\pi$$
$$978$$ 53.3939 1.70735
$$979$$ 77.3939 2.47352
$$980$$ 0 0
$$981$$ −17.3939 −0.555344
$$982$$ −9.30306 −0.296873
$$983$$ 4.94439 0.157701 0.0788507 0.996886i $$-0.474875\pi$$
0.0788507 + 0.996886i $$0.474875\pi$$
$$984$$ −4.89898 −0.156174
$$985$$ 0 0
$$986$$ 0 0
$$987$$ −48.4949 −1.54361
$$988$$ 14.2020 0.451827
$$989$$ −35.5959 −1.13188
$$990$$ 0 0
$$991$$ −31.8434 −1.01154 −0.505769 0.862669i $$-0.668791\pi$$
−0.505769 + 0.862669i $$0.668791\pi$$
$$992$$ −1.55051 −0.0492287
$$993$$ −34.8990 −1.10749
$$994$$ 21.7980 0.691389
$$995$$ 0 0
$$996$$ 23.3939 0.741263
$$997$$ −22.0000 −0.696747 −0.348373 0.937356i $$-0.613266\pi$$
−0.348373 + 0.937356i $$0.613266\pi$$
$$998$$ −10.6515 −0.337168
$$999$$ 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 1850.2.a.v.1.1 2
5.2 odd 4 370.2.b.c.149.4 yes 4
5.3 odd 4 370.2.b.c.149.1 4
5.4 even 2 1850.2.a.s.1.2 2
15.2 even 4 3330.2.d.m.1999.2 4
15.8 even 4 3330.2.d.m.1999.3 4

By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.b.c.149.1 4 5.3 odd 4
370.2.b.c.149.4 yes 4 5.2 odd 4
1850.2.a.s.1.2 2 5.4 even 2
1850.2.a.v.1.1 2 1.1 even 1 trivial
3330.2.d.m.1999.2 4 15.2 even 4
3330.2.d.m.1999.3 4 15.8 even 4