Properties

 Label 1850.2.a.s.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)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -2.44949 q^{3} +1.00000 q^{4} +2.44949 q^{6} +0.449490 q^{7} -1.00000 q^{8} +3.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} -2.44949 q^{3} +1.00000 q^{4} +2.44949 q^{6} +0.449490 q^{7} -1.00000 q^{8} +3.00000 q^{9} -4.89898 q^{11} -2.44949 q^{12} -4.00000 q^{13} -0.449490 q^{14} +1.00000 q^{16} -4.89898 q^{17} -3.00000 q^{18} +8.44949 q^{19} -1.10102 q^{21} +4.89898 q^{22} -0.898979 q^{23} +2.44949 q^{24} +4.00000 q^{26} +0.449490 q^{28} -6.44949 q^{31} -1.00000 q^{32} +12.0000 q^{33} +4.89898 q^{34} +3.00000 q^{36} -1.00000 q^{37} -8.44949 q^{38} +9.79796 q^{39} +2.00000 q^{41} +1.10102 q^{42} -4.00000 q^{43} -4.89898 q^{44} +0.898979 q^{46} +0.449490 q^{47} -2.44949 q^{48} -6.79796 q^{49} +12.0000 q^{51} -4.00000 q^{52} -7.79796 q^{53} -0.449490 q^{56} -20.6969 q^{57} -8.44949 q^{59} +12.0000 q^{61} +6.44949 q^{62} +1.34847 q^{63} +1.00000 q^{64} -12.0000 q^{66} +10.4495 q^{67} -4.89898 q^{68} +2.20204 q^{69} -4.89898 q^{71} -3.00000 q^{72} -4.00000 q^{73} +1.00000 q^{74} +8.44949 q^{76} -2.20204 q^{77} -9.79796 q^{78} +1.55051 q^{79} -9.00000 q^{81} -2.00000 q^{82} +14.4495 q^{83} -1.10102 q^{84} +4.00000 q^{86} +4.89898 q^{88} -3.79796 q^{89} -1.79796 q^{91} -0.898979 q^{92} +15.7980 q^{93} -0.449490 q^{94} +2.44949 q^{96} +2.00000 q^{97} +6.79796 q^{98} -14.6969 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{4} - 4 q^{7} - 2 q^{8} + 6 q^{9}+O(q^{10})$$ 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
$$7$$ 0.449490 0.169891 0.0849456 0.996386i $$-0.472928\pi$$
0.0849456 + 0.996386i $$0.472928\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 3.00000 1.00000
$$10$$ 0 0
$$11$$ −4.89898 −1.47710 −0.738549 0.674200i $$-0.764489\pi$$
−0.738549 + 0.674200i $$0.764489\pi$$
$$12$$ −2.44949 −0.707107
$$13$$ −4.00000 −1.10940 −0.554700 0.832050i $$-0.687167\pi$$
−0.554700 + 0.832050i $$0.687167\pi$$
$$14$$ −0.449490 −0.120131
$$15$$ 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
$$19$$ 8.44949 1.93845 0.969223 0.246185i $$-0.0791770\pi$$
0.969223 + 0.246185i $$0.0791770\pi$$
$$20$$ 0 0
$$21$$ −1.10102 −0.240262
$$22$$ 4.89898 1.04447
$$23$$ −0.898979 −0.187450 −0.0937251 0.995598i $$-0.529877\pi$$
−0.0937251 + 0.995598i $$0.529877\pi$$
$$24$$ 2.44949 0.500000
$$25$$ 0 0
$$26$$ 4.00000 0.784465
$$27$$ 0 0
$$28$$ 0.449490 0.0849456
$$29$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$30$$ 0 0
$$31$$ −6.44949 −1.15836 −0.579181 0.815199i $$-0.696628\pi$$
−0.579181 + 0.815199i $$0.696628\pi$$
$$32$$ −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
$$38$$ −8.44949 −1.37069
$$39$$ 9.79796 1.56893
$$40$$ 0 0
$$41$$ 2.00000 0.312348 0.156174 0.987730i $$-0.450084\pi$$
0.156174 + 0.987730i $$0.450084\pi$$
$$42$$ 1.10102 0.169891
$$43$$ −4.00000 −0.609994 −0.304997 0.952353i $$-0.598656\pi$$
−0.304997 + 0.952353i $$0.598656\pi$$
$$44$$ −4.89898 −0.738549
$$45$$ 0 0
$$46$$ 0.898979 0.132547
$$47$$ 0.449490 0.0655648 0.0327824 0.999463i $$-0.489563\pi$$
0.0327824 + 0.999463i $$0.489563\pi$$
$$48$$ −2.44949 −0.353553
$$49$$ −6.79796 −0.971137
$$50$$ 0 0
$$51$$ 12.0000 1.68034
$$52$$ −4.00000 −0.554700
$$53$$ −7.79796 −1.07113 −0.535566 0.844493i $$-0.679902\pi$$
−0.535566 + 0.844493i $$0.679902\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −0.449490 −0.0600656
$$57$$ −20.6969 −2.74138
$$58$$ 0 0
$$59$$ −8.44949 −1.10003 −0.550015 0.835155i $$-0.685378\pi$$
−0.550015 + 0.835155i $$0.685378\pi$$
$$60$$ 0 0
$$61$$ 12.0000 1.53644 0.768221 0.640184i $$-0.221142\pi$$
0.768221 + 0.640184i $$0.221142\pi$$
$$62$$ 6.44949 0.819086
$$63$$ 1.34847 0.169891
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ −12.0000 −1.47710
$$67$$ 10.4495 1.27661 0.638304 0.769784i $$-0.279636\pi$$
0.638304 + 0.769784i $$0.279636\pi$$
$$68$$ −4.89898 −0.594089
$$69$$ 2.20204 0.265095
$$70$$ 0 0
$$71$$ −4.89898 −0.581402 −0.290701 0.956814i $$-0.593888\pi$$
−0.290701 + 0.956814i $$0.593888\pi$$
$$72$$ −3.00000 −0.353553
$$73$$ −4.00000 −0.468165 −0.234082 0.972217i $$-0.575209\pi$$
−0.234082 + 0.972217i $$0.575209\pi$$
$$74$$ 1.00000 0.116248
$$75$$ 0 0
$$76$$ 8.44949 0.969223
$$77$$ −2.20204 −0.250946
$$78$$ −9.79796 −1.10940
$$79$$ 1.55051 0.174446 0.0872230 0.996189i $$-0.472201\pi$$
0.0872230 + 0.996189i $$0.472201\pi$$
$$80$$ 0 0
$$81$$ −9.00000 −1.00000
$$82$$ −2.00000 −0.220863
$$83$$ 14.4495 1.58604 0.793019 0.609197i $$-0.208508\pi$$
0.793019 + 0.609197i $$0.208508\pi$$
$$84$$ −1.10102 −0.120131
$$85$$ 0 0
$$86$$ 4.00000 0.431331
$$87$$ 0 0
$$88$$ 4.89898 0.522233
$$89$$ −3.79796 −0.402583 −0.201291 0.979531i $$-0.564514\pi$$
−0.201291 + 0.979531i $$0.564514\pi$$
$$90$$ 0 0
$$91$$ −1.79796 −0.188477
$$92$$ −0.898979 −0.0937251
$$93$$ 15.7980 1.63817
$$94$$ −0.449490 −0.0463613
$$95$$ 0 0
$$96$$ 2.44949 0.250000
$$97$$ 2.00000 0.203069 0.101535 0.994832i $$-0.467625\pi$$
0.101535 + 0.994832i $$0.467625\pi$$
$$98$$ 6.79796 0.686698
$$99$$ −14.6969 −1.47710
$$100$$ 0 0
$$101$$ −18.6969 −1.86041 −0.930207 0.367034i $$-0.880373\pi$$
−0.930207 + 0.367034i $$0.880373\pi$$
$$102$$ −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
$$106$$ 7.79796 0.757405
$$107$$ 10.4495 1.01019 0.505095 0.863064i $$-0.331457\pi$$
0.505095 + 0.863064i $$0.331457\pi$$
$$108$$ 0 0
$$109$$ 13.7980 1.32160 0.660802 0.750560i $$-0.270216\pi$$
0.660802 + 0.750560i $$0.270216\pi$$
$$110$$ 0 0
$$111$$ 2.44949 0.232495
$$112$$ 0.449490 0.0424728
$$113$$ 12.8990 1.21343 0.606717 0.794918i $$-0.292486\pi$$
0.606717 + 0.794918i $$0.292486\pi$$
$$114$$ 20.6969 1.93845
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −12.0000 −1.10940
$$118$$ 8.44949 0.777839
$$119$$ −2.20204 −0.201861
$$120$$ 0 0
$$121$$ 13.0000 1.18182
$$122$$ −12.0000 −1.08643
$$123$$ −4.89898 −0.441726
$$124$$ −6.44949 −0.579181
$$125$$ 0 0
$$126$$ −1.34847 −0.120131
$$127$$ 17.3485 1.53943 0.769714 0.638389i $$-0.220399\pi$$
0.769714 + 0.638389i $$0.220399\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 9.79796 0.862662
$$130$$ 0 0
$$131$$ 14.2474 1.24481 0.622403 0.782697i $$-0.286157\pi$$
0.622403 + 0.782697i $$0.286157\pi$$
$$132$$ 12.0000 1.04447
$$133$$ 3.79796 0.329325
$$134$$ −10.4495 −0.902698
$$135$$ 0 0
$$136$$ 4.89898 0.420084
$$137$$ 19.5959 1.67419 0.837096 0.547056i $$-0.184251\pi$$
0.837096 + 0.547056i $$0.184251\pi$$
$$138$$ −2.20204 −0.187450
$$139$$ 13.7980 1.17033 0.585164 0.810915i $$-0.301030\pi$$
0.585164 + 0.810915i $$0.301030\pi$$
$$140$$ 0 0
$$141$$ −1.10102 −0.0927227
$$142$$ 4.89898 0.411113
$$143$$ 19.5959 1.63869
$$144$$ 3.00000 0.250000
$$145$$ 0 0
$$146$$ 4.00000 0.331042
$$147$$ 16.6515 1.37340
$$148$$ −1.00000 −0.0821995
$$149$$ 10.6969 0.876327 0.438164 0.898895i $$-0.355629\pi$$
0.438164 + 0.898895i $$0.355629\pi$$
$$150$$ 0 0
$$151$$ −8.00000 −0.651031 −0.325515 0.945537i $$-0.605538\pi$$
−0.325515 + 0.945537i $$0.605538\pi$$
$$152$$ −8.44949 −0.685344
$$153$$ −14.6969 −1.18818
$$154$$ 2.20204 0.177446
$$155$$ 0 0
$$156$$ 9.79796 0.784465
$$157$$ −11.7980 −0.941580 −0.470790 0.882245i $$-0.656031\pi$$
−0.470790 + 0.882245i $$0.656031\pi$$
$$158$$ −1.55051 −0.123352
$$159$$ 19.1010 1.51481
$$160$$ 0 0
$$161$$ −0.404082 −0.0318461
$$162$$ 9.00000 0.707107
$$163$$ 2.20204 0.172477 0.0862386 0.996275i $$-0.472515\pi$$
0.0862386 + 0.996275i $$0.472515\pi$$
$$164$$ 2.00000 0.156174
$$165$$ 0 0
$$166$$ −14.4495 −1.12150
$$167$$ −8.00000 −0.619059 −0.309529 0.950890i $$-0.600171\pi$$
−0.309529 + 0.950890i $$0.600171\pi$$
$$168$$ 1.10102 0.0849456
$$169$$ 3.00000 0.230769
$$170$$ 0 0
$$171$$ 25.3485 1.93845
$$172$$ −4.00000 −0.304997
$$173$$ −0.202041 −0.0153609 −0.00768045 0.999971i $$-0.502445\pi$$
−0.00768045 + 0.999971i $$0.502445\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −4.89898 −0.369274
$$177$$ 20.6969 1.55568
$$178$$ 3.79796 0.284669
$$179$$ 5.34847 0.399763 0.199882 0.979820i $$-0.435944\pi$$
0.199882 + 0.979820i $$0.435944\pi$$
$$180$$ 0 0
$$181$$ −18.6969 −1.38973 −0.694866 0.719139i $$-0.744536\pi$$
−0.694866 + 0.719139i $$0.744536\pi$$
$$182$$ 1.79796 0.133274
$$183$$ −29.3939 −2.17286
$$184$$ 0.898979 0.0662736
$$185$$ 0 0
$$186$$ −15.7980 −1.15836
$$187$$ 24.0000 1.75505
$$188$$ 0.449490 0.0327824
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −3.34847 −0.242287 −0.121143 0.992635i $$-0.538656\pi$$
−0.121143 + 0.992635i $$0.538656\pi$$
$$192$$ −2.44949 −0.176777
$$193$$ −14.0000 −1.00774 −0.503871 0.863779i $$-0.668091\pi$$
−0.503871 + 0.863779i $$0.668091\pi$$
$$194$$ −2.00000 −0.143592
$$195$$ 0 0
$$196$$ −6.79796 −0.485568
$$197$$ 2.00000 0.142494 0.0712470 0.997459i $$-0.477302\pi$$
0.0712470 + 0.997459i $$0.477302\pi$$
$$198$$ 14.6969 1.04447
$$199$$ 1.55051 0.109913 0.0549564 0.998489i $$-0.482498\pi$$
0.0549564 + 0.998489i $$0.482498\pi$$
$$200$$ 0 0
$$201$$ −25.5959 −1.80540
$$202$$ 18.6969 1.31551
$$203$$ 0 0
$$204$$ 12.0000 0.840168
$$205$$ 0 0
$$206$$ −9.79796 −0.682656
$$207$$ −2.69694 −0.187450
$$208$$ −4.00000 −0.277350
$$209$$ −41.3939 −2.86327
$$210$$ 0 0
$$211$$ 22.6969 1.56252 0.781261 0.624205i $$-0.214577\pi$$
0.781261 + 0.624205i $$0.214577\pi$$
$$212$$ −7.79796 −0.535566
$$213$$ 12.0000 0.822226
$$214$$ −10.4495 −0.714312
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −2.89898 −0.196796
$$218$$ −13.7980 −0.934516
$$219$$ 9.79796 0.662085
$$220$$ 0 0
$$221$$ 19.5959 1.31816
$$222$$ −2.44949 −0.164399
$$223$$ 4.44949 0.297960 0.148980 0.988840i $$-0.452401\pi$$
0.148980 + 0.988840i $$0.452401\pi$$
$$224$$ −0.449490 −0.0300328
$$225$$ 0 0
$$226$$ −12.8990 −0.858027
$$227$$ 15.1010 1.00229 0.501145 0.865363i $$-0.332912\pi$$
0.501145 + 0.865363i $$0.332912\pi$$
$$228$$ −20.6969 −1.37069
$$229$$ −10.0000 −0.660819 −0.330409 0.943838i $$-0.607187\pi$$
−0.330409 + 0.943838i $$0.607187\pi$$
$$230$$ 0 0
$$231$$ 5.39388 0.354891
$$232$$ 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
$$236$$ −8.44949 −0.550015
$$237$$ −3.79796 −0.246704
$$238$$ 2.20204 0.142737
$$239$$ 26.0454 1.68474 0.842369 0.538902i $$-0.181161\pi$$
0.842369 + 0.538902i $$0.181161\pi$$
$$240$$ 0 0
$$241$$ −11.7980 −0.759973 −0.379987 0.924992i $$-0.624071\pi$$
−0.379987 + 0.924992i $$0.624071\pi$$
$$242$$ −13.0000 −0.835672
$$243$$ 22.0454 1.41421
$$244$$ 12.0000 0.768221
$$245$$ 0 0
$$246$$ 4.89898 0.312348
$$247$$ −33.7980 −2.15051
$$248$$ 6.44949 0.409543
$$249$$ −35.3939 −2.24300
$$250$$ 0 0
$$251$$ 6.65153 0.419841 0.209920 0.977718i $$-0.432679\pi$$
0.209920 + 0.977718i $$0.432679\pi$$
$$252$$ 1.34847 0.0849456
$$253$$ 4.40408 0.276882
$$254$$ −17.3485 −1.08854
$$255$$ 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
$$259$$ −0.449490 −0.0279299
$$260$$ 0 0
$$261$$ 0 0
$$262$$ −14.2474 −0.880210
$$263$$ −26.2474 −1.61849 −0.809244 0.587473i $$-0.800123\pi$$
−0.809244 + 0.587473i $$0.800123\pi$$
$$264$$ −12.0000 −0.738549
$$265$$ 0 0
$$266$$ −3.79796 −0.232868
$$267$$ 9.30306 0.569338
$$268$$ 10.4495 0.638304
$$269$$ 10.6969 0.652204 0.326102 0.945335i $$-0.394265\pi$$
0.326102 + 0.945335i $$0.394265\pi$$
$$270$$ 0 0
$$271$$ 16.4949 1.00199 0.500997 0.865449i $$-0.332967\pi$$
0.500997 + 0.865449i $$0.332967\pi$$
$$272$$ −4.89898 −0.297044
$$273$$ 4.40408 0.266547
$$274$$ −19.5959 −1.18383
$$275$$ 0 0
$$276$$ 2.20204 0.132547
$$277$$ 19.5959 1.17740 0.588702 0.808350i $$-0.299639\pi$$
0.588702 + 0.808350i $$0.299639\pi$$
$$278$$ −13.7980 −0.827547
$$279$$ −19.3485 −1.15836
$$280$$ 0 0
$$281$$ 8.20204 0.489293 0.244646 0.969612i $$-0.421328\pi$$
0.244646 + 0.969612i $$0.421328\pi$$
$$282$$ 1.10102 0.0655648
$$283$$ 23.5959 1.40263 0.701316 0.712851i $$-0.252596\pi$$
0.701316 + 0.712851i $$0.252596\pi$$
$$284$$ −4.89898 −0.290701
$$285$$ 0 0
$$286$$ −19.5959 −1.15873
$$287$$ 0.898979 0.0530651
$$288$$ −3.00000 −0.176777
$$289$$ 7.00000 0.411765
$$290$$ 0 0
$$291$$ −4.89898 −0.287183
$$292$$ −4.00000 −0.234082
$$293$$ 13.5959 0.794282 0.397141 0.917758i $$-0.370002\pi$$
0.397141 + 0.917758i $$0.370002\pi$$
$$294$$ −16.6515 −0.971137
$$295$$ 0 0
$$296$$ 1.00000 0.0581238
$$297$$ 0 0
$$298$$ −10.6969 −0.619657
$$299$$ 3.59592 0.207957
$$300$$ 0 0
$$301$$ −1.79796 −0.103633
$$302$$ 8.00000 0.460348
$$303$$ 45.7980 2.63102
$$304$$ 8.44949 0.484611
$$305$$ 0 0
$$306$$ 14.6969 0.840168
$$307$$ 21.1464 1.20689 0.603445 0.797404i $$-0.293794\pi$$
0.603445 + 0.797404i $$0.293794\pi$$
$$308$$ −2.20204 −0.125473
$$309$$ −24.0000 −1.36531
$$310$$ 0 0
$$311$$ 7.34847 0.416693 0.208347 0.978055i $$-0.433192\pi$$
0.208347 + 0.978055i $$0.433192\pi$$
$$312$$ −9.79796 −0.554700
$$313$$ −21.5959 −1.22067 −0.610337 0.792142i $$-0.708966\pi$$
−0.610337 + 0.792142i $$0.708966\pi$$
$$314$$ 11.7980 0.665797
$$315$$ 0 0
$$316$$ 1.55051 0.0872230
$$317$$ −18.0000 −1.01098 −0.505490 0.862832i $$-0.668688\pi$$
−0.505490 + 0.862832i $$0.668688\pi$$
$$318$$ −19.1010 −1.07113
$$319$$ 0 0
$$320$$ 0 0
$$321$$ −25.5959 −1.42862
$$322$$ 0.404082 0.0225186
$$323$$ −41.3939 −2.30322
$$324$$ −9.00000 −0.500000
$$325$$ 0 0
$$326$$ −2.20204 −0.121960
$$327$$ −33.7980 −1.86903
$$328$$ −2.00000 −0.110432
$$329$$ 0.202041 0.0111389
$$330$$ 0 0
$$331$$ −10.2474 −0.563251 −0.281625 0.959524i $$-0.590874\pi$$
−0.281625 + 0.959524i $$0.590874\pi$$
$$332$$ 14.4495 0.793019
$$333$$ −3.00000 −0.164399
$$334$$ 8.00000 0.437741
$$335$$ 0 0
$$336$$ −1.10102 −0.0600656
$$337$$ −35.5959 −1.93903 −0.969517 0.245026i $$-0.921204\pi$$
−0.969517 + 0.245026i $$0.921204\pi$$
$$338$$ −3.00000 −0.163178
$$339$$ −31.5959 −1.71605
$$340$$ 0 0
$$341$$ 31.5959 1.71101
$$342$$ −25.3485 −1.37069
$$343$$ −6.20204 −0.334879
$$344$$ 4.00000 0.215666
$$345$$ 0 0
$$346$$ 0.202041 0.0108618
$$347$$ −21.7980 −1.17018 −0.585088 0.810970i $$-0.698940\pi$$
−0.585088 + 0.810970i $$0.698940\pi$$
$$348$$ 0 0
$$349$$ 16.8990 0.904582 0.452291 0.891871i $$-0.350607\pi$$
0.452291 + 0.891871i $$0.350607\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 4.89898 0.261116
$$353$$ 6.00000 0.319348 0.159674 0.987170i $$-0.448956\pi$$
0.159674 + 0.987170i $$0.448956\pi$$
$$354$$ −20.6969 −1.10003
$$355$$ 0 0
$$356$$ −3.79796 −0.201291
$$357$$ 5.39388 0.285474
$$358$$ −5.34847 −0.282675
$$359$$ −3.10102 −0.163666 −0.0818328 0.996646i $$-0.526077\pi$$
−0.0818328 + 0.996646i $$0.526077\pi$$
$$360$$ 0 0
$$361$$ 52.3939 2.75757
$$362$$ 18.6969 0.982689
$$363$$ −31.8434 −1.67134
$$364$$ −1.79796 −0.0942387
$$365$$ 0 0
$$366$$ 29.3939 1.53644
$$367$$ 17.3485 0.905583 0.452791 0.891617i $$-0.350428\pi$$
0.452791 + 0.891617i $$0.350428\pi$$
$$368$$ −0.898979 −0.0468625
$$369$$ 6.00000 0.312348
$$370$$ 0 0
$$371$$ −3.50510 −0.181976
$$372$$ 15.7980 0.819086
$$373$$ −7.79796 −0.403763 −0.201882 0.979410i $$-0.564706\pi$$
−0.201882 + 0.979410i $$0.564706\pi$$
$$374$$ −24.0000 −1.24101
$$375$$ 0 0
$$376$$ −0.449490 −0.0231807
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 7.59592 0.390176 0.195088 0.980786i $$-0.437501\pi$$
0.195088 + 0.980786i $$0.437501\pi$$
$$380$$ 0 0
$$381$$ −42.4949 −2.17708
$$382$$ 3.34847 0.171323
$$383$$ 5.30306 0.270974 0.135487 0.990779i $$-0.456740\pi$$
0.135487 + 0.990779i $$0.456740\pi$$
$$384$$ 2.44949 0.125000
$$385$$ 0 0
$$386$$ 14.0000 0.712581
$$387$$ −12.0000 −0.609994
$$388$$ 2.00000 0.101535
$$389$$ −6.20204 −0.314456 −0.157228 0.987562i $$-0.550256\pi$$
−0.157228 + 0.987562i $$0.550256\pi$$
$$390$$ 0 0
$$391$$ 4.40408 0.222724
$$392$$ 6.79796 0.343349
$$393$$ −34.8990 −1.76042
$$394$$ −2.00000 −0.100759
$$395$$ 0 0
$$396$$ −14.6969 −0.738549
$$397$$ 35.7980 1.79665 0.898324 0.439334i $$-0.144785\pi$$
0.898324 + 0.439334i $$0.144785\pi$$
$$398$$ −1.55051 −0.0777201
$$399$$ −9.30306 −0.465736
$$400$$ 0 0
$$401$$ −4.20204 −0.209840 −0.104920 0.994481i $$-0.533459\pi$$
−0.104920 + 0.994481i $$0.533459\pi$$
$$402$$ 25.5959 1.27661
$$403$$ 25.7980 1.28509
$$404$$ −18.6969 −0.930207
$$405$$ 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
$$413$$ −3.79796 −0.186885
$$414$$ 2.69694 0.132547
$$415$$ 0 0
$$416$$ 4.00000 0.196116
$$417$$ −33.7980 −1.65509
$$418$$ 41.3939 2.02464
$$419$$ −13.7980 −0.674074 −0.337037 0.941491i $$-0.609425\pi$$
−0.337037 + 0.941491i $$0.609425\pi$$
$$420$$ 0 0
$$421$$ 19.5959 0.955047 0.477523 0.878619i $$-0.341535\pi$$
0.477523 + 0.878619i $$0.341535\pi$$
$$422$$ −22.6969 −1.10487
$$423$$ 1.34847 0.0655648
$$424$$ 7.79796 0.378702
$$425$$ 0 0
$$426$$ −12.0000 −0.581402
$$427$$ 5.39388 0.261028
$$428$$ 10.4495 0.505095
$$429$$ −48.0000 −2.31746
$$430$$ 0 0
$$431$$ −40.2474 −1.93865 −0.969326 0.245780i $$-0.920956\pi$$
−0.969326 + 0.245780i $$0.920956\pi$$
$$432$$ 0 0
$$433$$ 37.3939 1.79704 0.898518 0.438938i $$-0.144645\pi$$
0.898518 + 0.438938i $$0.144645\pi$$
$$434$$ 2.89898 0.139155
$$435$$ 0 0
$$436$$ 13.7980 0.660802
$$437$$ −7.59592 −0.363362
$$438$$ −9.79796 −0.468165
$$439$$ 35.3485 1.68709 0.843545 0.537058i $$-0.180464\pi$$
0.843545 + 0.537058i $$0.180464\pi$$
$$440$$ 0 0
$$441$$ −20.3939 −0.971137
$$442$$ −19.5959 −0.932083
$$443$$ −39.3485 −1.86950 −0.934751 0.355303i $$-0.884378\pi$$
−0.934751 + 0.355303i $$0.884378\pi$$
$$444$$ 2.44949 0.116248
$$445$$ 0 0
$$446$$ −4.44949 −0.210689
$$447$$ −26.2020 −1.23931
$$448$$ 0.449490 0.0212364
$$449$$ 3.79796 0.179237 0.0896184 0.995976i $$-0.471435\pi$$
0.0896184 + 0.995976i $$0.471435\pi$$
$$450$$ 0 0
$$451$$ −9.79796 −0.461368
$$452$$ 12.8990 0.606717
$$453$$ 19.5959 0.920697
$$454$$ −15.1010 −0.708726
$$455$$ 0 0
$$456$$ 20.6969 0.969223
$$457$$ 2.00000 0.0935561 0.0467780 0.998905i $$-0.485105\pi$$
0.0467780 + 0.998905i $$0.485105\pi$$
$$458$$ 10.0000 0.467269
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −14.2020 −0.661455 −0.330727 0.943726i $$-0.607294\pi$$
−0.330727 + 0.943726i $$0.607294\pi$$
$$462$$ −5.39388 −0.250946
$$463$$ −28.4949 −1.32427 −0.662135 0.749384i $$-0.730350\pi$$
−0.662135 + 0.749384i $$0.730350\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ −9.79796 −0.453882
$$467$$ 12.0000 0.555294 0.277647 0.960683i $$-0.410445\pi$$
0.277647 + 0.960683i $$0.410445\pi$$
$$468$$ −12.0000 −0.554700
$$469$$ 4.69694 0.216884
$$470$$ 0 0
$$471$$ 28.8990 1.33159
$$472$$ 8.44949 0.388919
$$473$$ 19.5959 0.901021
$$474$$ 3.79796 0.174446
$$475$$ 0 0
$$476$$ −2.20204 −0.100930
$$477$$ −23.3939 −1.07113
$$478$$ −26.0454 −1.19129
$$479$$ −29.1464 −1.33173 −0.665867 0.746070i $$-0.731938\pi$$
−0.665867 + 0.746070i $$0.731938\pi$$
$$480$$ 0 0
$$481$$ 4.00000 0.182384
$$482$$ 11.7980 0.537382
$$483$$ 0.989795 0.0450372
$$484$$ 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
$$489$$ −5.39388 −0.243920
$$490$$ 0 0
$$491$$ −38.6969 −1.74637 −0.873184 0.487390i $$-0.837949\pi$$
−0.873184 + 0.487390i $$0.837949\pi$$
$$492$$ −4.89898 −0.220863
$$493$$ 0 0
$$494$$ 33.7980 1.52064
$$495$$ 0 0
$$496$$ −6.44949 −0.289591
$$497$$ −2.20204 −0.0987750
$$498$$ 35.3939 1.58604
$$499$$ −25.3485 −1.13475 −0.567377 0.823458i $$-0.692042\pi$$
−0.567377 + 0.823458i $$0.692042\pi$$
$$500$$ 0 0
$$501$$ 19.5959 0.875481
$$502$$ −6.65153 −0.296872
$$503$$ −17.7980 −0.793572 −0.396786 0.917911i $$-0.629874\pi$$
−0.396786 + 0.917911i $$0.629874\pi$$
$$504$$ −1.34847 −0.0600656
$$505$$ 0 0
$$506$$ −4.40408 −0.195785
$$507$$ −7.34847 −0.326357
$$508$$ 17.3485 0.769714
$$509$$ 30.0000 1.32973 0.664863 0.746965i $$-0.268490\pi$$
0.664863 + 0.746965i $$0.268490\pi$$
$$510$$ 0 0
$$511$$ −1.79796 −0.0795370
$$512$$ −1.00000 −0.0441942
$$513$$ 0 0
$$514$$ 4.89898 0.216085
$$515$$ 0 0
$$516$$ 9.79796 0.431331
$$517$$ −2.20204 −0.0968457
$$518$$ 0.449490 0.0197494
$$519$$ 0.494897 0.0217236
$$520$$ 0 0
$$521$$ −1.79796 −0.0787700 −0.0393850 0.999224i $$-0.512540\pi$$
−0.0393850 + 0.999224i $$0.512540\pi$$
$$522$$ 0 0
$$523$$ −0.898979 −0.0393096 −0.0196548 0.999807i $$-0.506257\pi$$
−0.0196548 + 0.999807i $$0.506257\pi$$
$$524$$ 14.2474 0.622403
$$525$$ 0 0
$$526$$ 26.2474 1.14444
$$527$$ 31.5959 1.37634
$$528$$ 12.0000 0.522233
$$529$$ −22.1918 −0.964862
$$530$$ 0 0
$$531$$ −25.3485 −1.10003
$$532$$ 3.79796 0.164662
$$533$$ −8.00000 −0.346518
$$534$$ −9.30306 −0.402583
$$535$$ 0 0
$$536$$ −10.4495 −0.451349
$$537$$ −13.1010 −0.565351
$$538$$ −10.6969 −0.461178
$$539$$ 33.3031 1.43446
$$540$$ 0 0
$$541$$ 39.5959 1.70236 0.851181 0.524873i $$-0.175887\pi$$
0.851181 + 0.524873i $$0.175887\pi$$
$$542$$ −16.4949 −0.708517
$$543$$ 45.7980 1.96538
$$544$$ 4.89898 0.210042
$$545$$ 0 0
$$546$$ −4.40408 −0.188477
$$547$$ −18.6969 −0.799423 −0.399712 0.916641i $$-0.630890\pi$$
−0.399712 + 0.916641i $$0.630890\pi$$
$$548$$ 19.5959 0.837096
$$549$$ 36.0000 1.53644
$$550$$ 0 0
$$551$$ 0 0
$$552$$ −2.20204 −0.0937251
$$553$$ 0.696938 0.0296368
$$554$$ −19.5959 −0.832551
$$555$$ 0 0
$$556$$ 13.7980 0.585164
$$557$$ 12.0000 0.508456 0.254228 0.967144i $$-0.418179\pi$$
0.254228 + 0.967144i $$0.418179\pi$$
$$558$$ 19.3485 0.819086
$$559$$ 16.0000 0.676728
$$560$$ 0 0
$$561$$ −58.7878 −2.48202
$$562$$ −8.20204 −0.345982
$$563$$ 5.30306 0.223497 0.111749 0.993736i $$-0.464355\pi$$
0.111749 + 0.993736i $$0.464355\pi$$
$$564$$ −1.10102 −0.0463613
$$565$$ 0 0
$$566$$ −23.5959 −0.991810
$$567$$ −4.04541 −0.169891
$$568$$ 4.89898 0.205557
$$569$$ −17.5959 −0.737659 −0.368830 0.929497i $$-0.620241\pi$$
−0.368830 + 0.929497i $$0.620241\pi$$
$$570$$ 0 0
$$571$$ 33.3939 1.39749 0.698745 0.715371i $$-0.253742\pi$$
0.698745 + 0.715371i $$0.253742\pi$$
$$572$$ 19.5959 0.819346
$$573$$ 8.20204 0.342645
$$574$$ −0.898979 −0.0375227
$$575$$ 0 0
$$576$$ 3.00000 0.125000
$$577$$ 1.30306 0.0542472 0.0271236 0.999632i $$-0.491365\pi$$
0.0271236 + 0.999632i $$0.491365\pi$$
$$578$$ −7.00000 −0.291162
$$579$$ 34.2929 1.42516
$$580$$ 0 0
$$581$$ 6.49490 0.269454
$$582$$ 4.89898 0.203069
$$583$$ 38.2020 1.58217
$$584$$ 4.00000 0.165521
$$585$$ 0 0
$$586$$ −13.5959 −0.561642
$$587$$ 12.0000 0.495293 0.247647 0.968850i $$-0.420343\pi$$
0.247647 + 0.968850i $$0.420343\pi$$
$$588$$ 16.6515 0.686698
$$589$$ −54.4949 −2.24542
$$590$$ 0 0
$$591$$ −4.89898 −0.201517
$$592$$ −1.00000 −0.0410997
$$593$$ −24.0000 −0.985562 −0.492781 0.870153i $$-0.664020\pi$$
−0.492781 + 0.870153i $$0.664020\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 10.6969 0.438164
$$597$$ −3.79796 −0.155440
$$598$$ −3.59592 −0.147048
$$599$$ −27.5959 −1.12754 −0.563769 0.825932i $$-0.690649\pi$$
−0.563769 + 0.825932i $$0.690649\pi$$
$$600$$ 0 0
$$601$$ −8.00000 −0.326327 −0.163163 0.986599i $$-0.552170\pi$$
−0.163163 + 0.986599i $$0.552170\pi$$
$$602$$ 1.79796 0.0732793
$$603$$ 31.3485 1.27661
$$604$$ −8.00000 −0.325515
$$605$$ 0 0
$$606$$ −45.7980 −1.86041
$$607$$ 2.69694 0.109465 0.0547327 0.998501i $$-0.482569\pi$$
0.0547327 + 0.998501i $$0.482569\pi$$
$$608$$ −8.44949 −0.342672
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −1.79796 −0.0727376
$$612$$ −14.6969 −0.594089
$$613$$ −20.2020 −0.815953 −0.407976 0.912992i $$-0.633765\pi$$
−0.407976 + 0.912992i $$0.633765\pi$$
$$614$$ −21.1464 −0.853400
$$615$$ 0 0
$$616$$ 2.20204 0.0887228
$$617$$ 19.5959 0.788902 0.394451 0.918917i $$-0.370935\pi$$
0.394451 + 0.918917i $$0.370935\pi$$
$$618$$ 24.0000 0.965422
$$619$$ −3.10102 −0.124641 −0.0623203 0.998056i $$-0.519850\pi$$
−0.0623203 + 0.998056i $$0.519850\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ −7.34847 −0.294647
$$623$$ −1.70714 −0.0683953
$$624$$ 9.79796 0.392232
$$625$$ 0 0
$$626$$ 21.5959 0.863146
$$627$$ 101.394 4.04928
$$628$$ −11.7980 −0.470790
$$629$$ 4.89898 0.195335
$$630$$ 0 0
$$631$$ 4.24745 0.169088 0.0845441 0.996420i $$-0.473057\pi$$
0.0845441 + 0.996420i $$0.473057\pi$$
$$632$$ −1.55051 −0.0616760
$$633$$ −55.5959 −2.20974
$$634$$ 18.0000 0.714871
$$635$$ 0 0
$$636$$ 19.1010 0.757405
$$637$$ 27.1918 1.07738
$$638$$ 0 0
$$639$$ −14.6969 −0.581402
$$640$$ 0 0
$$641$$ −1.79796 −0.0710151 −0.0355076 0.999369i $$-0.511305\pi$$
−0.0355076 + 0.999369i $$0.511305\pi$$
$$642$$ 25.5959 1.01019
$$643$$ 32.8990 1.29741 0.648705 0.761040i $$-0.275311\pi$$
0.648705 + 0.761040i $$0.275311\pi$$
$$644$$ −0.404082 −0.0159231
$$645$$ 0 0
$$646$$ 41.3939 1.62862
$$647$$ 32.0000 1.25805 0.629025 0.777385i $$-0.283454\pi$$
0.629025 + 0.777385i $$0.283454\pi$$
$$648$$ 9.00000 0.353553
$$649$$ 41.3939 1.62485
$$650$$ 0 0
$$651$$ 7.10102 0.278311
$$652$$ 2.20204 0.0862386
$$653$$ −34.0000 −1.33052 −0.665261 0.746611i $$-0.731680\pi$$
−0.665261 + 0.746611i $$0.731680\pi$$
$$654$$ 33.7980 1.32160
$$655$$ 0 0
$$656$$ 2.00000 0.0780869
$$657$$ −12.0000 −0.468165
$$658$$ −0.202041 −0.00787638
$$659$$ −7.59592 −0.295895 −0.147947 0.988995i $$-0.547267\pi$$
−0.147947 + 0.988995i $$0.547267\pi$$
$$660$$ 0 0
$$661$$ 47.1918 1.83555 0.917775 0.397101i $$-0.129984\pi$$
0.917775 + 0.397101i $$0.129984\pi$$
$$662$$ 10.2474 0.398278
$$663$$ −48.0000 −1.86417
$$664$$ −14.4495 −0.560749
$$665$$ 0 0
$$666$$ 3.00000 0.116248
$$667$$ 0 0
$$668$$ −8.00000 −0.309529
$$669$$ −10.8990 −0.421379
$$670$$ 0 0
$$671$$ −58.7878 −2.26948
$$672$$ 1.10102 0.0424728
$$673$$ −24.0000 −0.925132 −0.462566 0.886585i $$-0.653071\pi$$
−0.462566 + 0.886585i $$0.653071\pi$$
$$674$$ 35.5959 1.37110
$$675$$ 0 0
$$676$$ 3.00000 0.115385
$$677$$ −18.0000 −0.691796 −0.345898 0.938272i $$-0.612426\pi$$
−0.345898 + 0.938272i $$0.612426\pi$$
$$678$$ 31.5959 1.21343
$$679$$ 0.898979 0.0344997
$$680$$ 0 0
$$681$$ −36.9898 −1.41745
$$682$$ −31.5959 −1.20987
$$683$$ 23.5959 0.902873 0.451436 0.892303i $$-0.350912\pi$$
0.451436 + 0.892303i $$0.350912\pi$$
$$684$$ 25.3485 0.969223
$$685$$ 0 0
$$686$$ 6.20204 0.236795
$$687$$ 24.4949 0.934539
$$688$$ −4.00000 −0.152499
$$689$$ 31.1918 1.18831
$$690$$ 0 0
$$691$$ 18.2020 0.692438 0.346219 0.938154i $$-0.387465\pi$$
0.346219 + 0.938154i $$0.387465\pi$$
$$692$$ −0.202041 −0.00768045
$$693$$ −6.60612 −0.250946
$$694$$ 21.7980 0.827439
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −9.79796 −0.371124
$$698$$ −16.8990 −0.639636
$$699$$ −24.0000 −0.907763
$$700$$ 0 0
$$701$$ 5.79796 0.218986 0.109493 0.993988i $$-0.465077\pi$$
0.109493 + 0.993988i $$0.465077\pi$$
$$702$$ 0 0
$$703$$ −8.44949 −0.318679
$$704$$ −4.89898 −0.184637
$$705$$ 0 0
$$706$$ −6.00000 −0.225813
$$707$$ −8.40408 −0.316068
$$708$$ 20.6969 0.777839
$$709$$ −13.7980 −0.518193 −0.259097 0.965851i $$-0.583425\pi$$
−0.259097 + 0.965851i $$0.583425\pi$$
$$710$$ 0 0
$$711$$ 4.65153 0.174446
$$712$$ 3.79796 0.142335
$$713$$ 5.79796 0.217135
$$714$$ −5.39388 −0.201861
$$715$$ 0 0
$$716$$ 5.34847 0.199882
$$717$$ −63.7980 −2.38258
$$718$$ 3.10102 0.115729
$$719$$ 12.4041 0.462594 0.231297 0.972883i $$-0.425703\pi$$
0.231297 + 0.972883i $$0.425703\pi$$
$$720$$ 0 0
$$721$$ 4.40408 0.164017
$$722$$ −52.3939 −1.94990
$$723$$ 28.8990 1.07476
$$724$$ −18.6969 −0.694866
$$725$$ 0 0
$$726$$ 31.8434 1.18182
$$727$$ 8.89898 0.330045 0.165022 0.986290i $$-0.447230\pi$$
0.165022 + 0.986290i $$0.447230\pi$$
$$728$$ 1.79796 0.0666368
$$729$$ −27.0000 −1.00000
$$730$$ 0 0
$$731$$ 19.5959 0.724781
$$732$$ −29.3939 −1.08643
$$733$$ −21.5959 −0.797663 −0.398832 0.917024i $$-0.630584\pi$$
−0.398832 + 0.917024i $$0.630584\pi$$
$$734$$ −17.3485 −0.640344
$$735$$ 0 0
$$736$$ 0.898979 0.0331368
$$737$$ −51.1918 −1.88568
$$738$$ −6.00000 −0.220863
$$739$$ −26.2020 −0.963858 −0.481929 0.876210i $$-0.660064\pi$$
−0.481929 + 0.876210i $$0.660064\pi$$
$$740$$ 0 0
$$741$$ 82.7878 3.04128
$$742$$ 3.50510 0.128676
$$743$$ 38.2474 1.40316 0.701581 0.712589i $$-0.252478\pi$$
0.701581 + 0.712589i $$0.252478\pi$$
$$744$$ −15.7980 −0.579181
$$745$$ 0 0
$$746$$ 7.79796 0.285504
$$747$$ 43.3485 1.58604
$$748$$ 24.0000 0.877527
$$749$$ 4.69694 0.171622
$$750$$ 0 0
$$751$$ 1.30306 0.0475494 0.0237747 0.999717i $$-0.492432\pi$$
0.0237747 + 0.999717i $$0.492432\pi$$
$$752$$ 0.449490 0.0163912
$$753$$ −16.2929 −0.593745
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −5.59592 −0.203387 −0.101694 0.994816i $$-0.532426\pi$$
−0.101694 + 0.994816i $$0.532426\pi$$
$$758$$ −7.59592 −0.275896
$$759$$ −10.7878 −0.391571
$$760$$ 0 0
$$761$$ −53.1918 −1.92820 −0.964101 0.265535i $$-0.914451\pi$$
−0.964101 + 0.265535i $$0.914451\pi$$
$$762$$ 42.4949 1.53943
$$763$$ 6.20204 0.224529
$$764$$ −3.34847 −0.121143
$$765$$ 0 0
$$766$$ −5.30306 −0.191607
$$767$$ 33.7980 1.22037
$$768$$ −2.44949 −0.0883883
$$769$$ −16.2020 −0.584261 −0.292130 0.956379i $$-0.594364\pi$$
−0.292130 + 0.956379i $$0.594364\pi$$
$$770$$ 0 0
$$771$$ 12.0000 0.432169
$$772$$ −14.0000 −0.503871
$$773$$ 6.00000 0.215805 0.107903 0.994161i $$-0.465587\pi$$
0.107903 + 0.994161i $$0.465587\pi$$
$$774$$ 12.0000 0.431331
$$775$$ 0 0
$$776$$ −2.00000 −0.0717958
$$777$$ 1.10102 0.0394989
$$778$$ 6.20204 0.222354
$$779$$ 16.8990 0.605469
$$780$$ 0 0
$$781$$ 24.0000 0.858788
$$782$$ −4.40408 −0.157490
$$783$$ 0 0
$$784$$ −6.79796 −0.242784
$$785$$ 0 0
$$786$$ 34.8990 1.24481
$$787$$ −44.7423 −1.59489 −0.797446 0.603390i $$-0.793816\pi$$
−0.797446 + 0.603390i $$0.793816\pi$$
$$788$$ 2.00000 0.0712470
$$789$$ 64.2929 2.28889
$$790$$ 0 0
$$791$$ 5.79796 0.206152
$$792$$ 14.6969 0.522233
$$793$$ −48.0000 −1.70453
$$794$$ −35.7980 −1.27042
$$795$$ 0 0
$$796$$ 1.55051 0.0549564
$$797$$ 4.40408 0.156001 0.0780003 0.996953i $$-0.475146\pi$$
0.0780003 + 0.996953i $$0.475146\pi$$
$$798$$ 9.30306 0.329325
$$799$$ −2.20204 −0.0779026
$$800$$ 0 0
$$801$$ −11.3939 −0.402583
$$802$$ 4.20204 0.148379
$$803$$ 19.5959 0.691525
$$804$$ −25.5959 −0.902698
$$805$$ 0 0
$$806$$ −25.7980 −0.908694
$$807$$ −26.2020 −0.922356
$$808$$ 18.6969 0.657756
$$809$$ 45.1918 1.58886 0.794430 0.607356i $$-0.207770\pi$$
0.794430 + 0.607356i $$0.207770\pi$$
$$810$$ 0 0
$$811$$ 45.7980 1.60818 0.804092 0.594505i $$-0.202652\pi$$
0.804092 + 0.594505i $$0.202652\pi$$
$$812$$ 0 0
$$813$$ −40.4041 −1.41703
$$814$$ −4.89898 −0.171709
$$815$$ 0 0
$$816$$ 12.0000 0.420084
$$817$$ −33.7980 −1.18244
$$818$$ 10.0000 0.349642
$$819$$ −5.39388 −0.188477
$$820$$ 0 0
$$821$$ −45.5959 −1.59131 −0.795654 0.605751i $$-0.792873\pi$$
−0.795654 + 0.605751i $$0.792873\pi$$
$$822$$ 48.0000 1.67419
$$823$$ −18.6515 −0.650151 −0.325076 0.945688i $$-0.605390\pi$$
−0.325076 + 0.945688i $$0.605390\pi$$
$$824$$ −9.79796 −0.341328
$$825$$ 0 0
$$826$$ 3.79796 0.132148
$$827$$ −12.4949 −0.434490 −0.217245 0.976117i $$-0.569707\pi$$
−0.217245 + 0.976117i $$0.569707\pi$$
$$828$$ −2.69694 −0.0937251
$$829$$ 47.5959 1.65307 0.826537 0.562882i $$-0.190307\pi$$
0.826537 + 0.562882i $$0.190307\pi$$
$$830$$ 0 0
$$831$$ −48.0000 −1.66510
$$832$$ −4.00000 −0.138675
$$833$$ 33.3031 1.15388
$$834$$ 33.7980 1.17033
$$835$$ 0 0
$$836$$ −41.3939 −1.43164
$$837$$ 0 0
$$838$$ 13.7980 0.476643
$$839$$ 33.7980 1.16684 0.583418 0.812172i $$-0.301715\pi$$
0.583418 + 0.812172i $$0.301715\pi$$
$$840$$ 0 0
$$841$$ −29.0000 −1.00000
$$842$$ −19.5959 −0.675320
$$843$$ −20.0908 −0.691964
$$844$$ 22.6969 0.781261
$$845$$ 0 0
$$846$$ −1.34847 −0.0463613
$$847$$ 5.84337 0.200780
$$848$$ −7.79796 −0.267783
$$849$$ −57.7980 −1.98362
$$850$$ 0 0
$$851$$ 0.898979 0.0308166
$$852$$ 12.0000 0.411113
$$853$$ −1.59592 −0.0546432 −0.0273216 0.999627i $$-0.508698\pi$$
−0.0273216 + 0.999627i $$0.508698\pi$$
$$854$$ −5.39388 −0.184575
$$855$$ 0 0
$$856$$ −10.4495 −0.357156
$$857$$ 29.5959 1.01098 0.505489 0.862833i $$-0.331312\pi$$
0.505489 + 0.862833i $$0.331312\pi$$
$$858$$ 48.0000 1.63869
$$859$$ −20.8536 −0.711515 −0.355757 0.934578i $$-0.615777\pi$$
−0.355757 + 0.934578i $$0.615777\pi$$
$$860$$ 0 0
$$861$$ −2.20204 −0.0750454
$$862$$ 40.2474 1.37083
$$863$$ −30.7423 −1.04648 −0.523241 0.852185i $$-0.675277\pi$$
−0.523241 + 0.852185i $$0.675277\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −37.3939 −1.27070
$$867$$ −17.1464 −0.582323
$$868$$ −2.89898 −0.0983978
$$869$$ −7.59592 −0.257674
$$870$$ 0 0
$$871$$ −41.7980 −1.41627
$$872$$ −13.7980 −0.467258
$$873$$ 6.00000 0.203069
$$874$$ 7.59592 0.256936
$$875$$ 0 0
$$876$$ 9.79796 0.331042
$$877$$ 43.3939 1.46531 0.732654 0.680602i $$-0.238282\pi$$
0.732654 + 0.680602i $$0.238282\pi$$
$$878$$ −35.3485 −1.19295
$$879$$ −33.3031 −1.12328
$$880$$ 0 0
$$881$$ 53.3939 1.79889 0.899443 0.437039i $$-0.143973\pi$$
0.899443 + 0.437039i $$0.143973\pi$$
$$882$$ 20.3939 0.686698
$$883$$ −13.3031 −0.447684 −0.223842 0.974625i $$-0.571860\pi$$
−0.223842 + 0.974625i $$0.571860\pi$$
$$884$$ 19.5959 0.659082
$$885$$ 0 0
$$886$$ 39.3485 1.32194
$$887$$ −24.0454 −0.807366 −0.403683 0.914899i $$-0.632270\pi$$
−0.403683 + 0.914899i $$0.632270\pi$$
$$888$$ −2.44949 −0.0821995
$$889$$ 7.79796 0.261535
$$890$$ 0 0
$$891$$ 44.0908 1.47710
$$892$$ 4.44949 0.148980
$$893$$ 3.79796 0.127094
$$894$$ 26.2020 0.876327
$$895$$ 0 0
$$896$$ −0.449490 −0.0150164
$$897$$ −8.80816 −0.294096
$$898$$ −3.79796 −0.126740
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 38.2020 1.27269
$$902$$ 9.79796 0.326236
$$903$$ 4.40408 0.146559
$$904$$ −12.8990 −0.429014
$$905$$ 0 0
$$906$$ −19.5959 −0.651031
$$907$$ 8.89898 0.295486 0.147743 0.989026i $$-0.452799\pi$$
0.147743 + 0.989026i $$0.452799\pi$$
$$908$$ 15.1010 0.501145
$$909$$ −56.0908 −1.86041
$$910$$ 0 0
$$911$$ 31.8434 1.05502 0.527509 0.849550i $$-0.323126\pi$$
0.527509 + 0.849550i $$0.323126\pi$$
$$912$$ −20.6969 −0.685344
$$913$$ −70.7878 −2.34273
$$914$$ −2.00000 −0.0661541
$$915$$ 0 0
$$916$$ −10.0000 −0.330409
$$917$$ 6.40408 0.211481
$$918$$ 0 0
$$919$$ −9.14643 −0.301713 −0.150856 0.988556i $$-0.548203\pi$$
−0.150856 + 0.988556i $$0.548203\pi$$
$$920$$ 0 0
$$921$$ −51.7980 −1.70680
$$922$$ 14.2020 0.467719
$$923$$ 19.5959 0.645007
$$924$$ 5.39388 0.177446
$$925$$ 0 0
$$926$$ 28.4949 0.936400
$$927$$ 29.3939 0.965422
$$928$$ 0 0
$$929$$ 37.5959 1.23348 0.616741 0.787166i $$-0.288453\pi$$
0.616741 + 0.787166i $$0.288453\pi$$
$$930$$ 0 0
$$931$$ −57.4393 −1.88250
$$932$$ 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 $$-0.541714\pi$$
−0.130674 + 0.991425i $$0.541714\pi$$
$$938$$ −4.69694 −0.153360
$$939$$ 52.8990 1.72629
$$940$$ 0 0
$$941$$ 30.2929 0.987519 0.493759 0.869599i $$-0.335622\pi$$
0.493759 + 0.869599i $$0.335622\pi$$
$$942$$ −28.8990 −0.941580
$$943$$ −1.79796 −0.0585496
$$944$$ −8.44949 −0.275007
$$945$$ 0 0
$$946$$ −19.5959 −0.637118
$$947$$ 33.3939 1.08516 0.542578 0.840006i $$-0.317448\pi$$
0.542578 + 0.840006i $$0.317448\pi$$
$$948$$ −3.79796 −0.123352
$$949$$ 16.0000 0.519382
$$950$$ 0 0
$$951$$ 44.0908 1.42974
$$952$$ 2.20204 0.0713686
$$953$$ 2.20204 0.0713311 0.0356656 0.999364i $$-0.488645\pi$$
0.0356656 + 0.999364i $$0.488645\pi$$
$$954$$ 23.3939 0.757405
$$955$$ 0 0
$$956$$ 26.0454 0.842369
$$957$$ 0 0
$$958$$ 29.1464 0.941678
$$959$$ 8.80816 0.284430
$$960$$ 0 0
$$961$$ 10.5959 0.341804
$$962$$ −4.00000 −0.128965
$$963$$ 31.3485 1.01019
$$964$$ −11.7980 −0.379987
$$965$$ 0 0
$$966$$ −0.989795 −0.0318461
$$967$$ −52.4949 −1.68812 −0.844061 0.536247i $$-0.819842\pi$$
−0.844061 + 0.536247i $$0.819842\pi$$
$$968$$ −13.0000 −0.417836
$$969$$ 101.394 3.25724
$$970$$ 0 0
$$971$$ −44.8990 −1.44088 −0.720438 0.693519i $$-0.756059\pi$$
−0.720438 + 0.693519i $$0.756059\pi$$
$$972$$ 22.0454 0.707107
$$973$$ 6.20204 0.198828
$$974$$ 29.3939 0.941841
$$975$$ 0 0
$$976$$ 12.0000 0.384111
$$977$$ 49.5959 1.58671 0.793357 0.608757i $$-0.208331\pi$$
0.793357 + 0.608757i $$0.208331\pi$$
$$978$$ 5.39388 0.172477
$$979$$ 18.6061 0.594654
$$980$$ 0 0
$$981$$ 41.3939 1.32160
$$982$$ 38.6969 1.23487
$$983$$ 48.9444 1.56108 0.780542 0.625104i $$-0.214943\pi$$
0.780542 + 0.625104i $$0.214943\pi$$
$$984$$ 4.89898 0.156174
$$985$$ 0 0
$$986$$ 0 0
$$987$$ −0.494897 −0.0157528
$$988$$ −33.7980 −1.07526
$$989$$ 3.59592 0.114344
$$990$$ 0 0
$$991$$ 31.8434 1.01154 0.505769 0.862669i $$-0.331209\pi$$
0.505769 + 0.862669i $$0.331209\pi$$
$$992$$ 6.44949 0.204772
$$993$$ 25.1010 0.796557
$$994$$ 2.20204 0.0698445
$$995$$ 0 0
$$996$$ −35.3939 −1.12150
$$997$$ 22.0000 0.696747 0.348373 0.937356i $$-0.386734\pi$$
0.348373 + 0.937356i $$0.386734\pi$$
$$998$$ 25.3485 0.802392
$$999$$ 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.s.1.1 2
5.2 odd 4 370.2.b.c.149.2 4
5.3 odd 4 370.2.b.c.149.3 yes 4
5.4 even 2 1850.2.a.v.1.2 2
15.2 even 4 3330.2.d.m.1999.4 4
15.8 even 4 3330.2.d.m.1999.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.b.c.149.2 4 5.2 odd 4
370.2.b.c.149.3 yes 4 5.3 odd 4
1850.2.a.s.1.1 2 1.1 even 1 trivial
1850.2.a.v.1.2 2 5.4 even 2
3330.2.d.m.1999.1 4 15.8 even 4
3330.2.d.m.1999.4 4 15.2 even 4