# Properties

 Label 1850.2.a.u.1.2 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: $$\Q(\sqrt{13})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 3$$ x^2 - x - 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 74) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-1.30278$$ 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)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +0.302776 q^{3} +1.00000 q^{4} +0.302776 q^{6} -4.60555 q^{7} +1.00000 q^{8} -2.90833 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +0.302776 q^{3} +1.00000 q^{4} +0.302776 q^{6} -4.60555 q^{7} +1.00000 q^{8} -2.90833 q^{9} +1.30278 q^{11} +0.302776 q^{12} +2.30278 q^{13} -4.60555 q^{14} +1.00000 q^{16} +6.00000 q^{17} -2.90833 q^{18} +2.00000 q^{19} -1.39445 q^{21} +1.30278 q^{22} +6.90833 q^{23} +0.302776 q^{24} +2.30278 q^{26} -1.78890 q^{27} -4.60555 q^{28} +6.90833 q^{29} +3.30278 q^{31} +1.00000 q^{32} +0.394449 q^{33} +6.00000 q^{34} -2.90833 q^{36} -1.00000 q^{37} +2.00000 q^{38} +0.697224 q^{39} -0.908327 q^{41} -1.39445 q^{42} +6.60555 q^{43} +1.30278 q^{44} +6.90833 q^{46} +2.60555 q^{47} +0.302776 q^{48} +14.2111 q^{49} +1.81665 q^{51} +2.30278 q^{52} +6.00000 q^{53} -1.78890 q^{54} -4.60555 q^{56} +0.605551 q^{57} +6.90833 q^{58} +3.39445 q^{59} -10.5139 q^{61} +3.30278 q^{62} +13.3944 q^{63} +1.00000 q^{64} +0.394449 q^{66} -14.5139 q^{67} +6.00000 q^{68} +2.09167 q^{69} +6.00000 q^{71} -2.90833 q^{72} +8.69722 q^{73} -1.00000 q^{74} +2.00000 q^{76} -6.00000 q^{77} +0.697224 q^{78} -16.1194 q^{79} +8.18335 q^{81} -0.908327 q^{82} -17.2111 q^{83} -1.39445 q^{84} +6.60555 q^{86} +2.09167 q^{87} +1.30278 q^{88} +5.21110 q^{89} -10.6056 q^{91} +6.90833 q^{92} +1.00000 q^{93} +2.60555 q^{94} +0.302776 q^{96} -12.4222 q^{97} +14.2111 q^{98} -3.78890 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} - 3 q^{3} + 2 q^{4} - 3 q^{6} - 2 q^{7} + 2 q^{8} + 5 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 - 3 * q^3 + 2 * q^4 - 3 * q^6 - 2 * q^7 + 2 * q^8 + 5 * q^9 $$2 q + 2 q^{2} - 3 q^{3} + 2 q^{4} - 3 q^{6} - 2 q^{7} + 2 q^{8} + 5 q^{9} - q^{11} - 3 q^{12} + q^{13} - 2 q^{14} + 2 q^{16} + 12 q^{17} + 5 q^{18} + 4 q^{19} - 10 q^{21} - q^{22} + 3 q^{23} - 3 q^{24} + q^{26} - 18 q^{27} - 2 q^{28} + 3 q^{29} + 3 q^{31} + 2 q^{32} + 8 q^{33} + 12 q^{34} + 5 q^{36} - 2 q^{37} + 4 q^{38} + 5 q^{39} + 9 q^{41} - 10 q^{42} + 6 q^{43} - q^{44} + 3 q^{46} - 2 q^{47} - 3 q^{48} + 14 q^{49} - 18 q^{51} + q^{52} + 12 q^{53} - 18 q^{54} - 2 q^{56} - 6 q^{57} + 3 q^{58} + 14 q^{59} - 3 q^{61} + 3 q^{62} + 34 q^{63} + 2 q^{64} + 8 q^{66} - 11 q^{67} + 12 q^{68} + 15 q^{69} + 12 q^{71} + 5 q^{72} + 21 q^{73} - 2 q^{74} + 4 q^{76} - 12 q^{77} + 5 q^{78} - 7 q^{79} + 38 q^{81} + 9 q^{82} - 20 q^{83} - 10 q^{84} + 6 q^{86} + 15 q^{87} - q^{88} - 4 q^{89} - 14 q^{91} + 3 q^{92} + 2 q^{93} - 2 q^{94} - 3 q^{96} + 4 q^{97} + 14 q^{98} - 22 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 - 3 * q^3 + 2 * q^4 - 3 * q^6 - 2 * q^7 + 2 * q^8 + 5 * q^9 - q^11 - 3 * q^12 + q^13 - 2 * q^14 + 2 * q^16 + 12 * q^17 + 5 * q^18 + 4 * q^19 - 10 * q^21 - q^22 + 3 * q^23 - 3 * q^24 + q^26 - 18 * q^27 - 2 * q^28 + 3 * q^29 + 3 * q^31 + 2 * q^32 + 8 * q^33 + 12 * q^34 + 5 * q^36 - 2 * q^37 + 4 * q^38 + 5 * q^39 + 9 * q^41 - 10 * q^42 + 6 * q^43 - q^44 + 3 * q^46 - 2 * q^47 - 3 * q^48 + 14 * q^49 - 18 * q^51 + q^52 + 12 * q^53 - 18 * q^54 - 2 * q^56 - 6 * q^57 + 3 * q^58 + 14 * q^59 - 3 * q^61 + 3 * q^62 + 34 * q^63 + 2 * q^64 + 8 * q^66 - 11 * q^67 + 12 * q^68 + 15 * q^69 + 12 * q^71 + 5 * q^72 + 21 * q^73 - 2 * q^74 + 4 * q^76 - 12 * q^77 + 5 * q^78 - 7 * q^79 + 38 * q^81 + 9 * q^82 - 20 * q^83 - 10 * q^84 + 6 * q^86 + 15 * q^87 - q^88 - 4 * q^89 - 14 * q^91 + 3 * q^92 + 2 * q^93 - 2 * q^94 - 3 * q^96 + 4 * q^97 + 14 * q^98 - 22 * 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
$$3$$ 0.302776 0.174808 0.0874038 0.996173i $$-0.472143\pi$$
0.0874038 + 0.996173i $$0.472143\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ 0.302776 0.123608
$$7$$ −4.60555 −1.74073 −0.870367 0.492403i $$-0.836119\pi$$
−0.870367 + 0.492403i $$0.836119\pi$$
$$8$$ 1.00000 0.353553
$$9$$ −2.90833 −0.969442
$$10$$ 0 0
$$11$$ 1.30278 0.392802 0.196401 0.980524i $$-0.437075\pi$$
0.196401 + 0.980524i $$0.437075\pi$$
$$12$$ 0.302776 0.0874038
$$13$$ 2.30278 0.638675 0.319338 0.947641i $$-0.396540\pi$$
0.319338 + 0.947641i $$0.396540\pi$$
$$14$$ −4.60555 −1.23089
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 6.00000 1.45521 0.727607 0.685994i $$-0.240633\pi$$
0.727607 + 0.685994i $$0.240633\pi$$
$$18$$ −2.90833 −0.685499
$$19$$ 2.00000 0.458831 0.229416 0.973329i $$-0.426318\pi$$
0.229416 + 0.973329i $$0.426318\pi$$
$$20$$ 0 0
$$21$$ −1.39445 −0.304294
$$22$$ 1.30278 0.277753
$$23$$ 6.90833 1.44049 0.720243 0.693722i $$-0.244030\pi$$
0.720243 + 0.693722i $$0.244030\pi$$
$$24$$ 0.302776 0.0618038
$$25$$ 0 0
$$26$$ 2.30278 0.451611
$$27$$ −1.78890 −0.344273
$$28$$ −4.60555 −0.870367
$$29$$ 6.90833 1.28284 0.641422 0.767188i $$-0.278345\pi$$
0.641422 + 0.767188i $$0.278345\pi$$
$$30$$ 0 0
$$31$$ 3.30278 0.593196 0.296598 0.955002i $$-0.404148\pi$$
0.296598 + 0.955002i $$0.404148\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 0.394449 0.0686647
$$34$$ 6.00000 1.02899
$$35$$ 0 0
$$36$$ −2.90833 −0.484721
$$37$$ −1.00000 −0.164399
$$38$$ 2.00000 0.324443
$$39$$ 0.697224 0.111645
$$40$$ 0 0
$$41$$ −0.908327 −0.141857 −0.0709284 0.997481i $$-0.522596\pi$$
−0.0709284 + 0.997481i $$0.522596\pi$$
$$42$$ −1.39445 −0.215168
$$43$$ 6.60555 1.00734 0.503669 0.863897i $$-0.331983\pi$$
0.503669 + 0.863897i $$0.331983\pi$$
$$44$$ 1.30278 0.196401
$$45$$ 0 0
$$46$$ 6.90833 1.01858
$$47$$ 2.60555 0.380059 0.190029 0.981778i $$-0.439142\pi$$
0.190029 + 0.981778i $$0.439142\pi$$
$$48$$ 0.302776 0.0437019
$$49$$ 14.2111 2.03016
$$50$$ 0 0
$$51$$ 1.81665 0.254382
$$52$$ 2.30278 0.319338
$$53$$ 6.00000 0.824163 0.412082 0.911147i $$-0.364802\pi$$
0.412082 + 0.911147i $$0.364802\pi$$
$$54$$ −1.78890 −0.243438
$$55$$ 0 0
$$56$$ −4.60555 −0.615443
$$57$$ 0.605551 0.0802072
$$58$$ 6.90833 0.907108
$$59$$ 3.39445 0.441920 0.220960 0.975283i $$-0.429081\pi$$
0.220960 + 0.975283i $$0.429081\pi$$
$$60$$ 0 0
$$61$$ −10.5139 −1.34616 −0.673082 0.739568i $$-0.735030\pi$$
−0.673082 + 0.739568i $$0.735030\pi$$
$$62$$ 3.30278 0.419453
$$63$$ 13.3944 1.68754
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 0.394449 0.0485533
$$67$$ −14.5139 −1.77315 −0.886576 0.462583i $$-0.846923\pi$$
−0.886576 + 0.462583i $$0.846923\pi$$
$$68$$ 6.00000 0.727607
$$69$$ 2.09167 0.251808
$$70$$ 0 0
$$71$$ 6.00000 0.712069 0.356034 0.934473i $$-0.384129\pi$$
0.356034 + 0.934473i $$0.384129\pi$$
$$72$$ −2.90833 −0.342750
$$73$$ 8.69722 1.01793 0.508967 0.860786i $$-0.330028\pi$$
0.508967 + 0.860786i $$0.330028\pi$$
$$74$$ −1.00000 −0.116248
$$75$$ 0 0
$$76$$ 2.00000 0.229416
$$77$$ −6.00000 −0.683763
$$78$$ 0.697224 0.0789451
$$79$$ −16.1194 −1.81358 −0.906789 0.421585i $$-0.861474\pi$$
−0.906789 + 0.421585i $$0.861474\pi$$
$$80$$ 0 0
$$81$$ 8.18335 0.909261
$$82$$ −0.908327 −0.100308
$$83$$ −17.2111 −1.88916 −0.944582 0.328276i $$-0.893533\pi$$
−0.944582 + 0.328276i $$0.893533\pi$$
$$84$$ −1.39445 −0.152147
$$85$$ 0 0
$$86$$ 6.60555 0.712295
$$87$$ 2.09167 0.224251
$$88$$ 1.30278 0.138876
$$89$$ 5.21110 0.552376 0.276188 0.961104i $$-0.410929\pi$$
0.276188 + 0.961104i $$0.410929\pi$$
$$90$$ 0 0
$$91$$ −10.6056 −1.11176
$$92$$ 6.90833 0.720243
$$93$$ 1.00000 0.103695
$$94$$ 2.60555 0.268742
$$95$$ 0 0
$$96$$ 0.302776 0.0309019
$$97$$ −12.4222 −1.26128 −0.630642 0.776074i $$-0.717208\pi$$
−0.630642 + 0.776074i $$0.717208\pi$$
$$98$$ 14.2111 1.43554
$$99$$ −3.78890 −0.380799
$$100$$ 0 0
$$101$$ 16.4222 1.63407 0.817035 0.576588i $$-0.195616\pi$$
0.817035 + 0.576588i $$0.195616\pi$$
$$102$$ 1.81665 0.179876
$$103$$ −3.30278 −0.325432 −0.162716 0.986673i $$-0.552025\pi$$
−0.162716 + 0.986673i $$0.552025\pi$$
$$104$$ 2.30278 0.225806
$$105$$ 0 0
$$106$$ 6.00000 0.582772
$$107$$ −4.30278 −0.415965 −0.207983 0.978133i $$-0.566690\pi$$
−0.207983 + 0.978133i $$0.566690\pi$$
$$108$$ −1.78890 −0.172137
$$109$$ 2.00000 0.191565 0.0957826 0.995402i $$-0.469465\pi$$
0.0957826 + 0.995402i $$0.469465\pi$$
$$110$$ 0 0
$$111$$ −0.302776 −0.0287382
$$112$$ −4.60555 −0.435184
$$113$$ −11.2111 −1.05465 −0.527326 0.849663i $$-0.676805\pi$$
−0.527326 + 0.849663i $$0.676805\pi$$
$$114$$ 0.605551 0.0567151
$$115$$ 0 0
$$116$$ 6.90833 0.641422
$$117$$ −6.69722 −0.619159
$$118$$ 3.39445 0.312484
$$119$$ −27.6333 −2.53314
$$120$$ 0 0
$$121$$ −9.30278 −0.845707
$$122$$ −10.5139 −0.951882
$$123$$ −0.275019 −0.0247977
$$124$$ 3.30278 0.296598
$$125$$ 0 0
$$126$$ 13.3944 1.19327
$$127$$ 4.78890 0.424946 0.212473 0.977167i $$-0.431848\pi$$
0.212473 + 0.977167i $$0.431848\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 2.00000 0.176090
$$130$$ 0 0
$$131$$ 3.39445 0.296574 0.148287 0.988944i $$-0.452624\pi$$
0.148287 + 0.988944i $$0.452624\pi$$
$$132$$ 0.394449 0.0343324
$$133$$ −9.21110 −0.798704
$$134$$ −14.5139 −1.25381
$$135$$ 0 0
$$136$$ 6.00000 0.514496
$$137$$ 9.90833 0.846525 0.423263 0.906007i $$-0.360885\pi$$
0.423263 + 0.906007i $$0.360885\pi$$
$$138$$ 2.09167 0.178055
$$139$$ 8.90833 0.755594 0.377797 0.925888i $$-0.376682\pi$$
0.377797 + 0.925888i $$0.376682\pi$$
$$140$$ 0 0
$$141$$ 0.788897 0.0664372
$$142$$ 6.00000 0.503509
$$143$$ 3.00000 0.250873
$$144$$ −2.90833 −0.242361
$$145$$ 0 0
$$146$$ 8.69722 0.719787
$$147$$ 4.30278 0.354887
$$148$$ −1.00000 −0.0821995
$$149$$ −1.81665 −0.148826 −0.0744130 0.997228i $$-0.523708\pi$$
−0.0744130 + 0.997228i $$0.523708\pi$$
$$150$$ 0 0
$$151$$ −13.3944 −1.09002 −0.545012 0.838428i $$-0.683475\pi$$
−0.545012 + 0.838428i $$0.683475\pi$$
$$152$$ 2.00000 0.162221
$$153$$ −17.4500 −1.41075
$$154$$ −6.00000 −0.483494
$$155$$ 0 0
$$156$$ 0.697224 0.0558226
$$157$$ −7.21110 −0.575509 −0.287754 0.957704i $$-0.592909\pi$$
−0.287754 + 0.957704i $$0.592909\pi$$
$$158$$ −16.1194 −1.28239
$$159$$ 1.81665 0.144070
$$160$$ 0 0
$$161$$ −31.8167 −2.50750
$$162$$ 8.18335 0.642944
$$163$$ 20.4222 1.59959 0.799795 0.600273i $$-0.204941\pi$$
0.799795 + 0.600273i $$0.204941\pi$$
$$164$$ −0.908327 −0.0709284
$$165$$ 0 0
$$166$$ −17.2111 −1.33584
$$167$$ −12.5139 −0.968353 −0.484176 0.874970i $$-0.660881\pi$$
−0.484176 + 0.874970i $$0.660881\pi$$
$$168$$ −1.39445 −0.107584
$$169$$ −7.69722 −0.592094
$$170$$ 0 0
$$171$$ −5.81665 −0.444811
$$172$$ 6.60555 0.503669
$$173$$ 23.2111 1.76471 0.882354 0.470587i $$-0.155958\pi$$
0.882354 + 0.470587i $$0.155958\pi$$
$$174$$ 2.09167 0.158569
$$175$$ 0 0
$$176$$ 1.30278 0.0982004
$$177$$ 1.02776 0.0772509
$$178$$ 5.21110 0.390589
$$179$$ 7.81665 0.584244 0.292122 0.956381i $$-0.405639\pi$$
0.292122 + 0.956381i $$0.405639\pi$$
$$180$$ 0 0
$$181$$ 20.0000 1.48659 0.743294 0.668965i $$-0.233262\pi$$
0.743294 + 0.668965i $$0.233262\pi$$
$$182$$ −10.6056 −0.786136
$$183$$ −3.18335 −0.235320
$$184$$ 6.90833 0.509289
$$185$$ 0 0
$$186$$ 1.00000 0.0733236
$$187$$ 7.81665 0.571610
$$188$$ 2.60555 0.190029
$$189$$ 8.23886 0.599289
$$190$$ 0 0
$$191$$ 12.5139 0.905472 0.452736 0.891644i $$-0.350448\pi$$
0.452736 + 0.891644i $$0.350448\pi$$
$$192$$ 0.302776 0.0218509
$$193$$ 4.00000 0.287926 0.143963 0.989583i $$-0.454015\pi$$
0.143963 + 0.989583i $$0.454015\pi$$
$$194$$ −12.4222 −0.891862
$$195$$ 0 0
$$196$$ 14.2111 1.01508
$$197$$ 6.00000 0.427482 0.213741 0.976890i $$-0.431435\pi$$
0.213741 + 0.976890i $$0.431435\pi$$
$$198$$ −3.78890 −0.269265
$$199$$ −2.42221 −0.171706 −0.0858528 0.996308i $$-0.527361\pi$$
−0.0858528 + 0.996308i $$0.527361\pi$$
$$200$$ 0 0
$$201$$ −4.39445 −0.309961
$$202$$ 16.4222 1.15546
$$203$$ −31.8167 −2.23309
$$204$$ 1.81665 0.127191
$$205$$ 0 0
$$206$$ −3.30278 −0.230115
$$207$$ −20.0917 −1.39647
$$208$$ 2.30278 0.159669
$$209$$ 2.60555 0.180230
$$210$$ 0 0
$$211$$ 6.69722 0.461056 0.230528 0.973066i $$-0.425955\pi$$
0.230528 + 0.973066i $$0.425955\pi$$
$$212$$ 6.00000 0.412082
$$213$$ 1.81665 0.124475
$$214$$ −4.30278 −0.294132
$$215$$ 0 0
$$216$$ −1.78890 −0.121719
$$217$$ −15.2111 −1.03260
$$218$$ 2.00000 0.135457
$$219$$ 2.63331 0.177942
$$220$$ 0 0
$$221$$ 13.8167 0.929409
$$222$$ −0.302776 −0.0203210
$$223$$ −15.8167 −1.05916 −0.529581 0.848260i $$-0.677651\pi$$
−0.529581 + 0.848260i $$0.677651\pi$$
$$224$$ −4.60555 −0.307721
$$225$$ 0 0
$$226$$ −11.2111 −0.745751
$$227$$ 7.81665 0.518810 0.259405 0.965769i $$-0.416474\pi$$
0.259405 + 0.965769i $$0.416474\pi$$
$$228$$ 0.605551 0.0401036
$$229$$ 17.3944 1.14946 0.574729 0.818344i $$-0.305108\pi$$
0.574729 + 0.818344i $$0.305108\pi$$
$$230$$ 0 0
$$231$$ −1.81665 −0.119527
$$232$$ 6.90833 0.453554
$$233$$ 9.51388 0.623275 0.311637 0.950201i $$-0.399123\pi$$
0.311637 + 0.950201i $$0.399123\pi$$
$$234$$ −6.69722 −0.437811
$$235$$ 0 0
$$236$$ 3.39445 0.220960
$$237$$ −4.88057 −0.317027
$$238$$ −27.6333 −1.79120
$$239$$ 0.513878 0.0332400 0.0166200 0.999862i $$-0.494709\pi$$
0.0166200 + 0.999862i $$0.494709\pi$$
$$240$$ 0 0
$$241$$ 8.00000 0.515325 0.257663 0.966235i $$-0.417048\pi$$
0.257663 + 0.966235i $$0.417048\pi$$
$$242$$ −9.30278 −0.598005
$$243$$ 7.84441 0.503219
$$244$$ −10.5139 −0.673082
$$245$$ 0 0
$$246$$ −0.275019 −0.0175346
$$247$$ 4.60555 0.293044
$$248$$ 3.30278 0.209726
$$249$$ −5.21110 −0.330240
$$250$$ 0 0
$$251$$ −6.78890 −0.428511 −0.214256 0.976778i $$-0.568733\pi$$
−0.214256 + 0.976778i $$0.568733\pi$$
$$252$$ 13.3944 0.843771
$$253$$ 9.00000 0.565825
$$254$$ 4.78890 0.300482
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 11.2111 0.699329 0.349665 0.936875i $$-0.386296\pi$$
0.349665 + 0.936875i $$0.386296\pi$$
$$258$$ 2.00000 0.124515
$$259$$ 4.60555 0.286175
$$260$$ 0 0
$$261$$ −20.0917 −1.24364
$$262$$ 3.39445 0.209710
$$263$$ 7.81665 0.481996 0.240998 0.970526i $$-0.422525\pi$$
0.240998 + 0.970526i $$0.422525\pi$$
$$264$$ 0.394449 0.0242766
$$265$$ 0 0
$$266$$ −9.21110 −0.564769
$$267$$ 1.57779 0.0965595
$$268$$ −14.5139 −0.886576
$$269$$ −6.78890 −0.413926 −0.206963 0.978349i $$-0.566358\pi$$
−0.206963 + 0.978349i $$0.566358\pi$$
$$270$$ 0 0
$$271$$ 6.42221 0.390121 0.195061 0.980791i $$-0.437510\pi$$
0.195061 + 0.980791i $$0.437510\pi$$
$$272$$ 6.00000 0.363803
$$273$$ −3.21110 −0.194345
$$274$$ 9.90833 0.598584
$$275$$ 0 0
$$276$$ 2.09167 0.125904
$$277$$ 25.1194 1.50928 0.754640 0.656139i $$-0.227811\pi$$
0.754640 + 0.656139i $$0.227811\pi$$
$$278$$ 8.90833 0.534286
$$279$$ −9.60555 −0.575069
$$280$$ 0 0
$$281$$ −12.0000 −0.715860 −0.357930 0.933748i $$-0.616517\pi$$
−0.357930 + 0.933748i $$0.616517\pi$$
$$282$$ 0.788897 0.0469782
$$283$$ −17.3944 −1.03399 −0.516996 0.855988i $$-0.672950\pi$$
−0.516996 + 0.855988i $$0.672950\pi$$
$$284$$ 6.00000 0.356034
$$285$$ 0 0
$$286$$ 3.00000 0.177394
$$287$$ 4.18335 0.246935
$$288$$ −2.90833 −0.171375
$$289$$ 19.0000 1.11765
$$290$$ 0 0
$$291$$ −3.76114 −0.220482
$$292$$ 8.69722 0.508967
$$293$$ 25.0278 1.46214 0.731069 0.682304i $$-0.239022\pi$$
0.731069 + 0.682304i $$0.239022\pi$$
$$294$$ 4.30278 0.250943
$$295$$ 0 0
$$296$$ −1.00000 −0.0581238
$$297$$ −2.33053 −0.135231
$$298$$ −1.81665 −0.105236
$$299$$ 15.9083 0.920002
$$300$$ 0 0
$$301$$ −30.4222 −1.75351
$$302$$ −13.3944 −0.770764
$$303$$ 4.97224 0.285648
$$304$$ 2.00000 0.114708
$$305$$ 0 0
$$306$$ −17.4500 −0.997548
$$307$$ −7.09167 −0.404743 −0.202372 0.979309i $$-0.564865\pi$$
−0.202372 + 0.979309i $$0.564865\pi$$
$$308$$ −6.00000 −0.341882
$$309$$ −1.00000 −0.0568880
$$310$$ 0 0
$$311$$ 5.09167 0.288722 0.144361 0.989525i $$-0.453887\pi$$
0.144361 + 0.989525i $$0.453887\pi$$
$$312$$ 0.697224 0.0394726
$$313$$ −27.0278 −1.52770 −0.763850 0.645394i $$-0.776693\pi$$
−0.763850 + 0.645394i $$0.776693\pi$$
$$314$$ −7.21110 −0.406946
$$315$$ 0 0
$$316$$ −16.1194 −0.906789
$$317$$ 5.21110 0.292685 0.146342 0.989234i $$-0.453250\pi$$
0.146342 + 0.989234i $$0.453250\pi$$
$$318$$ 1.81665 0.101873
$$319$$ 9.00000 0.503903
$$320$$ 0 0
$$321$$ −1.30278 −0.0727138
$$322$$ −31.8167 −1.77307
$$323$$ 12.0000 0.667698
$$324$$ 8.18335 0.454630
$$325$$ 0 0
$$326$$ 20.4222 1.13108
$$327$$ 0.605551 0.0334871
$$328$$ −0.908327 −0.0501540
$$329$$ −12.0000 −0.661581
$$330$$ 0 0
$$331$$ 1.21110 0.0665682 0.0332841 0.999446i $$-0.489403\pi$$
0.0332841 + 0.999446i $$0.489403\pi$$
$$332$$ −17.2111 −0.944582
$$333$$ 2.90833 0.159375
$$334$$ −12.5139 −0.684729
$$335$$ 0 0
$$336$$ −1.39445 −0.0760734
$$337$$ 19.1194 1.04150 0.520751 0.853709i $$-0.325652\pi$$
0.520751 + 0.853709i $$0.325652\pi$$
$$338$$ −7.69722 −0.418674
$$339$$ −3.39445 −0.184361
$$340$$ 0 0
$$341$$ 4.30278 0.233008
$$342$$ −5.81665 −0.314529
$$343$$ −33.2111 −1.79323
$$344$$ 6.60555 0.356147
$$345$$ 0 0
$$346$$ 23.2111 1.24784
$$347$$ −31.8167 −1.70801 −0.854004 0.520267i $$-0.825832\pi$$
−0.854004 + 0.520267i $$0.825832\pi$$
$$348$$ 2.09167 0.112125
$$349$$ −22.2389 −1.19042 −0.595209 0.803571i $$-0.702931\pi$$
−0.595209 + 0.803571i $$0.702931\pi$$
$$350$$ 0 0
$$351$$ −4.11943 −0.219879
$$352$$ 1.30278 0.0694382
$$353$$ −31.8167 −1.69343 −0.846715 0.532047i $$-0.821423\pi$$
−0.846715 + 0.532047i $$0.821423\pi$$
$$354$$ 1.02776 0.0546246
$$355$$ 0 0
$$356$$ 5.21110 0.276188
$$357$$ −8.36669 −0.442812
$$358$$ 7.81665 0.413123
$$359$$ −11.2111 −0.591699 −0.295850 0.955235i $$-0.595603\pi$$
−0.295850 + 0.955235i $$0.595603\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ 20.0000 1.05118
$$363$$ −2.81665 −0.147836
$$364$$ −10.6056 −0.555882
$$365$$ 0 0
$$366$$ −3.18335 −0.166396
$$367$$ 17.8167 0.930022 0.465011 0.885305i $$-0.346050\pi$$
0.465011 + 0.885305i $$0.346050\pi$$
$$368$$ 6.90833 0.360121
$$369$$ 2.64171 0.137522
$$370$$ 0 0
$$371$$ −27.6333 −1.43465
$$372$$ 1.00000 0.0518476
$$373$$ −3.81665 −0.197619 −0.0988094 0.995106i $$-0.531503\pi$$
−0.0988094 + 0.995106i $$0.531503\pi$$
$$374$$ 7.81665 0.404190
$$375$$ 0 0
$$376$$ 2.60555 0.134371
$$377$$ 15.9083 0.819321
$$378$$ 8.23886 0.423761
$$379$$ −15.3305 −0.787477 −0.393738 0.919223i $$-0.628818\pi$$
−0.393738 + 0.919223i $$0.628818\pi$$
$$380$$ 0 0
$$381$$ 1.44996 0.0742838
$$382$$ 12.5139 0.640266
$$383$$ −20.8444 −1.06510 −0.532550 0.846399i $$-0.678766\pi$$
−0.532550 + 0.846399i $$0.678766\pi$$
$$384$$ 0.302776 0.0154510
$$385$$ 0 0
$$386$$ 4.00000 0.203595
$$387$$ −19.2111 −0.976555
$$388$$ −12.4222 −0.630642
$$389$$ −11.8806 −0.602369 −0.301184 0.953566i $$-0.597382\pi$$
−0.301184 + 0.953566i $$0.597382\pi$$
$$390$$ 0 0
$$391$$ 41.4500 2.09621
$$392$$ 14.2111 0.717769
$$393$$ 1.02776 0.0518435
$$394$$ 6.00000 0.302276
$$395$$ 0 0
$$396$$ −3.78890 −0.190399
$$397$$ −27.8167 −1.39608 −0.698039 0.716060i $$-0.745944\pi$$
−0.698039 + 0.716060i $$0.745944\pi$$
$$398$$ −2.42221 −0.121414
$$399$$ −2.78890 −0.139620
$$400$$ 0 0
$$401$$ 13.8167 0.689971 0.344985 0.938608i $$-0.387884\pi$$
0.344985 + 0.938608i $$0.387884\pi$$
$$402$$ −4.39445 −0.219175
$$403$$ 7.60555 0.378859
$$404$$ 16.4222 0.817035
$$405$$ 0 0
$$406$$ −31.8167 −1.57903
$$407$$ −1.30278 −0.0645762
$$408$$ 1.81665 0.0899378
$$409$$ −5.02776 −0.248607 −0.124303 0.992244i $$-0.539670\pi$$
−0.124303 + 0.992244i $$0.539670\pi$$
$$410$$ 0 0
$$411$$ 3.00000 0.147979
$$412$$ −3.30278 −0.162716
$$413$$ −15.6333 −0.769265
$$414$$ −20.0917 −0.987452
$$415$$ 0 0
$$416$$ 2.30278 0.112903
$$417$$ 2.69722 0.132084
$$418$$ 2.60555 0.127442
$$419$$ −25.1472 −1.22852 −0.614260 0.789104i $$-0.710545\pi$$
−0.614260 + 0.789104i $$0.710545\pi$$
$$420$$ 0 0
$$421$$ 28.7250 1.39997 0.699985 0.714158i $$-0.253190\pi$$
0.699985 + 0.714158i $$0.253190\pi$$
$$422$$ 6.69722 0.326016
$$423$$ −7.57779 −0.368445
$$424$$ 6.00000 0.291386
$$425$$ 0 0
$$426$$ 1.81665 0.0880172
$$427$$ 48.4222 2.34331
$$428$$ −4.30278 −0.207983
$$429$$ 0.908327 0.0438544
$$430$$ 0 0
$$431$$ −5.21110 −0.251010 −0.125505 0.992093i $$-0.540055\pi$$
−0.125505 + 0.992093i $$0.540055\pi$$
$$432$$ −1.78890 −0.0860684
$$433$$ 11.9361 0.573612 0.286806 0.957989i $$-0.407407\pi$$
0.286806 + 0.957989i $$0.407407\pi$$
$$434$$ −15.2111 −0.730156
$$435$$ 0 0
$$436$$ 2.00000 0.0957826
$$437$$ 13.8167 0.660940
$$438$$ 2.63331 0.125824
$$439$$ −9.33053 −0.445322 −0.222661 0.974896i $$-0.571474\pi$$
−0.222661 + 0.974896i $$0.571474\pi$$
$$440$$ 0 0
$$441$$ −41.3305 −1.96812
$$442$$ 13.8167 0.657191
$$443$$ −0.275019 −0.0130666 −0.00653328 0.999979i $$-0.502080\pi$$
−0.00653328 + 0.999979i $$0.502080\pi$$
$$444$$ −0.302776 −0.0143691
$$445$$ 0 0
$$446$$ −15.8167 −0.748940
$$447$$ −0.550039 −0.0260159
$$448$$ −4.60555 −0.217592
$$449$$ −0.788897 −0.0372304 −0.0186152 0.999827i $$-0.505926\pi$$
−0.0186152 + 0.999827i $$0.505926\pi$$
$$450$$ 0 0
$$451$$ −1.18335 −0.0557216
$$452$$ −11.2111 −0.527326
$$453$$ −4.05551 −0.190545
$$454$$ 7.81665 0.366854
$$455$$ 0 0
$$456$$ 0.605551 0.0283575
$$457$$ −4.60555 −0.215439 −0.107719 0.994181i $$-0.534355\pi$$
−0.107719 + 0.994181i $$0.534355\pi$$
$$458$$ 17.3944 0.812789
$$459$$ −10.7334 −0.500991
$$460$$ 0 0
$$461$$ −16.4222 −0.764858 −0.382429 0.923985i $$-0.624912\pi$$
−0.382429 + 0.923985i $$0.624912\pi$$
$$462$$ −1.81665 −0.0845184
$$463$$ −30.3028 −1.40829 −0.704145 0.710056i $$-0.748669\pi$$
−0.704145 + 0.710056i $$0.748669\pi$$
$$464$$ 6.90833 0.320711
$$465$$ 0 0
$$466$$ 9.51388 0.440722
$$467$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$468$$ −6.69722 −0.309579
$$469$$ 66.8444 3.08659
$$470$$ 0 0
$$471$$ −2.18335 −0.100603
$$472$$ 3.39445 0.156242
$$473$$ 8.60555 0.395684
$$474$$ −4.88057 −0.224172
$$475$$ 0 0
$$476$$ −27.6333 −1.26657
$$477$$ −17.4500 −0.798979
$$478$$ 0.513878 0.0235042
$$479$$ 12.1194 0.553751 0.276875 0.960906i $$-0.410701\pi$$
0.276875 + 0.960906i $$0.410701\pi$$
$$480$$ 0 0
$$481$$ −2.30278 −0.104998
$$482$$ 8.00000 0.364390
$$483$$ −9.63331 −0.438331
$$484$$ −9.30278 −0.422853
$$485$$ 0 0
$$486$$ 7.84441 0.355830
$$487$$ 22.7889 1.03266 0.516332 0.856389i $$-0.327297\pi$$
0.516332 + 0.856389i $$0.327297\pi$$
$$488$$ −10.5139 −0.475941
$$489$$ 6.18335 0.279621
$$490$$ 0 0
$$491$$ −14.7250 −0.664529 −0.332265 0.943186i $$-0.607813\pi$$
−0.332265 + 0.943186i $$0.607813\pi$$
$$492$$ −0.275019 −0.0123988
$$493$$ 41.4500 1.86681
$$494$$ 4.60555 0.207214
$$495$$ 0 0
$$496$$ 3.30278 0.148299
$$497$$ −27.6333 −1.23952
$$498$$ −5.21110 −0.233515
$$499$$ 8.23886 0.368822 0.184411 0.982849i $$-0.440962\pi$$
0.184411 + 0.982849i $$0.440962\pi$$
$$500$$ 0 0
$$501$$ −3.78890 −0.169275
$$502$$ −6.78890 −0.303003
$$503$$ 24.5139 1.09302 0.546510 0.837453i $$-0.315956\pi$$
0.546510 + 0.837453i $$0.315956\pi$$
$$504$$ 13.3944 0.596636
$$505$$ 0 0
$$506$$ 9.00000 0.400099
$$507$$ −2.33053 −0.103503
$$508$$ 4.78890 0.212473
$$509$$ −25.8167 −1.14430 −0.572152 0.820148i $$-0.693891\pi$$
−0.572152 + 0.820148i $$0.693891\pi$$
$$510$$ 0 0
$$511$$ −40.0555 −1.77195
$$512$$ 1.00000 0.0441942
$$513$$ −3.57779 −0.157964
$$514$$ 11.2111 0.494501
$$515$$ 0 0
$$516$$ 2.00000 0.0880451
$$517$$ 3.39445 0.149288
$$518$$ 4.60555 0.202356
$$519$$ 7.02776 0.308484
$$520$$ 0 0
$$521$$ −9.63331 −0.422043 −0.211021 0.977481i $$-0.567679\pi$$
−0.211021 + 0.977481i $$0.567679\pi$$
$$522$$ −20.0917 −0.879389
$$523$$ −32.2389 −1.40971 −0.704853 0.709353i $$-0.748987\pi$$
−0.704853 + 0.709353i $$0.748987\pi$$
$$524$$ 3.39445 0.148287
$$525$$ 0 0
$$526$$ 7.81665 0.340822
$$527$$ 19.8167 0.863227
$$528$$ 0.394449 0.0171662
$$529$$ 24.7250 1.07500
$$530$$ 0 0
$$531$$ −9.87217 −0.428416
$$532$$ −9.21110 −0.399352
$$533$$ −2.09167 −0.0906004
$$534$$ 1.57779 0.0682779
$$535$$ 0 0
$$536$$ −14.5139 −0.626904
$$537$$ 2.36669 0.102130
$$538$$ −6.78890 −0.292690
$$539$$ 18.5139 0.797449
$$540$$ 0 0
$$541$$ −20.9361 −0.900113 −0.450056 0.893000i $$-0.648596\pi$$
−0.450056 + 0.893000i $$0.648596\pi$$
$$542$$ 6.42221 0.275857
$$543$$ 6.05551 0.259867
$$544$$ 6.00000 0.257248
$$545$$ 0 0
$$546$$ −3.21110 −0.137423
$$547$$ 13.3944 0.572705 0.286353 0.958124i $$-0.407557\pi$$
0.286353 + 0.958124i $$0.407557\pi$$
$$548$$ 9.90833 0.423263
$$549$$ 30.5778 1.30503
$$550$$ 0 0
$$551$$ 13.8167 0.588609
$$552$$ 2.09167 0.0890275
$$553$$ 74.2389 3.15696
$$554$$ 25.1194 1.06722
$$555$$ 0 0
$$556$$ 8.90833 0.377797
$$557$$ 6.51388 0.276002 0.138001 0.990432i $$-0.455932\pi$$
0.138001 + 0.990432i $$0.455932\pi$$
$$558$$ −9.60555 −0.406635
$$559$$ 15.2111 0.643361
$$560$$ 0 0
$$561$$ 2.36669 0.0999218
$$562$$ −12.0000 −0.506189
$$563$$ −44.0555 −1.85672 −0.928359 0.371684i $$-0.878780\pi$$
−0.928359 + 0.371684i $$0.878780\pi$$
$$564$$ 0.788897 0.0332186
$$565$$ 0 0
$$566$$ −17.3944 −0.731143
$$567$$ −37.6888 −1.58278
$$568$$ 6.00000 0.251754
$$569$$ −10.4222 −0.436922 −0.218461 0.975846i $$-0.570104\pi$$
−0.218461 + 0.975846i $$0.570104\pi$$
$$570$$ 0 0
$$571$$ −20.3028 −0.849645 −0.424822 0.905277i $$-0.639663\pi$$
−0.424822 + 0.905277i $$0.639663\pi$$
$$572$$ 3.00000 0.125436
$$573$$ 3.78890 0.158283
$$574$$ 4.18335 0.174609
$$575$$ 0 0
$$576$$ −2.90833 −0.121180
$$577$$ 28.2389 1.17560 0.587800 0.809007i $$-0.299994\pi$$
0.587800 + 0.809007i $$0.299994\pi$$
$$578$$ 19.0000 0.790296
$$579$$ 1.21110 0.0503317
$$580$$ 0 0
$$581$$ 79.2666 3.28853
$$582$$ −3.76114 −0.155904
$$583$$ 7.81665 0.323733
$$584$$ 8.69722 0.359894
$$585$$ 0 0
$$586$$ 25.0278 1.03389
$$587$$ 2.36669 0.0976838 0.0488419 0.998807i $$-0.484447\pi$$
0.0488419 + 0.998807i $$0.484447\pi$$
$$588$$ 4.30278 0.177443
$$589$$ 6.60555 0.272177
$$590$$ 0 0
$$591$$ 1.81665 0.0747272
$$592$$ −1.00000 −0.0410997
$$593$$ −36.5139 −1.49945 −0.749723 0.661752i $$-0.769813\pi$$
−0.749723 + 0.661752i $$0.769813\pi$$
$$594$$ −2.33053 −0.0956229
$$595$$ 0 0
$$596$$ −1.81665 −0.0744130
$$597$$ −0.733385 −0.0300154
$$598$$ 15.9083 0.650540
$$599$$ −35.2111 −1.43869 −0.719343 0.694655i $$-0.755557\pi$$
−0.719343 + 0.694655i $$0.755557\pi$$
$$600$$ 0 0
$$601$$ −20.6972 −0.844257 −0.422129 0.906536i $$-0.638717\pi$$
−0.422129 + 0.906536i $$0.638717\pi$$
$$602$$ −30.4222 −1.23992
$$603$$ 42.2111 1.71897
$$604$$ −13.3944 −0.545012
$$605$$ 0 0
$$606$$ 4.97224 0.201984
$$607$$ 31.5139 1.27911 0.639554 0.768746i $$-0.279119\pi$$
0.639554 + 0.768746i $$0.279119\pi$$
$$608$$ 2.00000 0.0811107
$$609$$ −9.63331 −0.390361
$$610$$ 0 0
$$611$$ 6.00000 0.242734
$$612$$ −17.4500 −0.705373
$$613$$ 8.18335 0.330522 0.165261 0.986250i $$-0.447153\pi$$
0.165261 + 0.986250i $$0.447153\pi$$
$$614$$ −7.09167 −0.286197
$$615$$ 0 0
$$616$$ −6.00000 −0.241747
$$617$$ −47.5694 −1.91507 −0.957536 0.288314i $$-0.906905\pi$$
−0.957536 + 0.288314i $$0.906905\pi$$
$$618$$ −1.00000 −0.0402259
$$619$$ −2.69722 −0.108411 −0.0542053 0.998530i $$-0.517263\pi$$
−0.0542053 + 0.998530i $$0.517263\pi$$
$$620$$ 0 0
$$621$$ −12.3583 −0.495921
$$622$$ 5.09167 0.204157
$$623$$ −24.0000 −0.961540
$$624$$ 0.697224 0.0279113
$$625$$ 0 0
$$626$$ −27.0278 −1.08025
$$627$$ 0.788897 0.0315055
$$628$$ −7.21110 −0.287754
$$629$$ −6.00000 −0.239236
$$630$$ 0 0
$$631$$ 18.3028 0.728622 0.364311 0.931277i $$-0.381305\pi$$
0.364311 + 0.931277i $$0.381305\pi$$
$$632$$ −16.1194 −0.641196
$$633$$ 2.02776 0.0805961
$$634$$ 5.21110 0.206959
$$635$$ 0 0
$$636$$ 1.81665 0.0720350
$$637$$ 32.7250 1.29661
$$638$$ 9.00000 0.356313
$$639$$ −17.4500 −0.690310
$$640$$ 0 0
$$641$$ −2.48612 −0.0981959 −0.0490980 0.998794i $$-0.515635\pi$$
−0.0490980 + 0.998794i $$0.515635\pi$$
$$642$$ −1.30278 −0.0514165
$$643$$ 29.8167 1.17585 0.587927 0.808914i $$-0.299944\pi$$
0.587927 + 0.808914i $$0.299944\pi$$
$$644$$ −31.8167 −1.25375
$$645$$ 0 0
$$646$$ 12.0000 0.472134
$$647$$ 25.9361 1.01965 0.509826 0.860277i $$-0.329710\pi$$
0.509826 + 0.860277i $$0.329710\pi$$
$$648$$ 8.18335 0.321472
$$649$$ 4.42221 0.173587
$$650$$ 0 0
$$651$$ −4.60555 −0.180506
$$652$$ 20.4222 0.799795
$$653$$ −6.90833 −0.270344 −0.135172 0.990822i $$-0.543159\pi$$
−0.135172 + 0.990822i $$0.543159\pi$$
$$654$$ 0.605551 0.0236789
$$655$$ 0 0
$$656$$ −0.908327 −0.0354642
$$657$$ −25.2944 −0.986827
$$658$$ −12.0000 −0.467809
$$659$$ −42.1194 −1.64074 −0.820370 0.571833i $$-0.806233\pi$$
−0.820370 + 0.571833i $$0.806233\pi$$
$$660$$ 0 0
$$661$$ −12.4861 −0.485654 −0.242827 0.970070i $$-0.578075\pi$$
−0.242827 + 0.970070i $$0.578075\pi$$
$$662$$ 1.21110 0.0470708
$$663$$ 4.18335 0.162468
$$664$$ −17.2111 −0.667920
$$665$$ 0 0
$$666$$ 2.90833 0.112695
$$667$$ 47.7250 1.84792
$$668$$ −12.5139 −0.484176
$$669$$ −4.78890 −0.185149
$$670$$ 0 0
$$671$$ −13.6972 −0.528775
$$672$$ −1.39445 −0.0537920
$$673$$ −24.3028 −0.936803 −0.468402 0.883516i $$-0.655170\pi$$
−0.468402 + 0.883516i $$0.655170\pi$$
$$674$$ 19.1194 0.736453
$$675$$ 0 0
$$676$$ −7.69722 −0.296047
$$677$$ 36.2389 1.39277 0.696386 0.717667i $$-0.254790\pi$$
0.696386 + 0.717667i $$0.254790\pi$$
$$678$$ −3.39445 −0.130363
$$679$$ 57.2111 2.19556
$$680$$ 0 0
$$681$$ 2.36669 0.0906918
$$682$$ 4.30278 0.164762
$$683$$ −12.0000 −0.459167 −0.229584 0.973289i $$-0.573736\pi$$
−0.229584 + 0.973289i $$0.573736\pi$$
$$684$$ −5.81665 −0.222405
$$685$$ 0 0
$$686$$ −33.2111 −1.26801
$$687$$ 5.26662 0.200934
$$688$$ 6.60555 0.251834
$$689$$ 13.8167 0.526373
$$690$$ 0 0
$$691$$ 8.00000 0.304334 0.152167 0.988355i $$-0.451375\pi$$
0.152167 + 0.988355i $$0.451375\pi$$
$$692$$ 23.2111 0.882354
$$693$$ 17.4500 0.662869
$$694$$ −31.8167 −1.20774
$$695$$ 0 0
$$696$$ 2.09167 0.0792847
$$697$$ −5.44996 −0.206432
$$698$$ −22.2389 −0.841753
$$699$$ 2.88057 0.108953
$$700$$ 0 0
$$701$$ 14.8806 0.562031 0.281016 0.959703i $$-0.409329\pi$$
0.281016 + 0.959703i $$0.409329\pi$$
$$702$$ −4.11943 −0.155478
$$703$$ −2.00000 −0.0754314
$$704$$ 1.30278 0.0491002
$$705$$ 0 0
$$706$$ −31.8167 −1.19744
$$707$$ −75.6333 −2.84448
$$708$$ 1.02776 0.0386254
$$709$$ −1.66947 −0.0626982 −0.0313491 0.999508i $$-0.509980\pi$$
−0.0313491 + 0.999508i $$0.509980\pi$$
$$710$$ 0 0
$$711$$ 46.8806 1.75816
$$712$$ 5.21110 0.195294
$$713$$ 22.8167 0.854490
$$714$$ −8.36669 −0.313116
$$715$$ 0 0
$$716$$ 7.81665 0.292122
$$717$$ 0.155590 0.00581061
$$718$$ −11.2111 −0.418395
$$719$$ −8.36669 −0.312025 −0.156012 0.987755i $$-0.549864\pi$$
−0.156012 + 0.987755i $$0.549864\pi$$
$$720$$ 0 0
$$721$$ 15.2111 0.566491
$$722$$ −15.0000 −0.558242
$$723$$ 2.42221 0.0900828
$$724$$ 20.0000 0.743294
$$725$$ 0 0
$$726$$ −2.81665 −0.104536
$$727$$ −29.9083 −1.10924 −0.554619 0.832104i $$-0.687136\pi$$
−0.554619 + 0.832104i $$0.687136\pi$$
$$728$$ −10.6056 −0.393068
$$729$$ −22.1749 −0.821294
$$730$$ 0 0
$$731$$ 39.6333 1.46589
$$732$$ −3.18335 −0.117660
$$733$$ −29.6333 −1.09453 −0.547266 0.836959i $$-0.684331\pi$$
−0.547266 + 0.836959i $$0.684331\pi$$
$$734$$ 17.8167 0.657625
$$735$$ 0 0
$$736$$ 6.90833 0.254644
$$737$$ −18.9083 −0.696497
$$738$$ 2.64171 0.0972427
$$739$$ −42.3305 −1.55715 −0.778577 0.627549i $$-0.784058\pi$$
−0.778577 + 0.627549i $$0.784058\pi$$
$$740$$ 0 0
$$741$$ 1.39445 0.0512264
$$742$$ −27.6333 −1.01445
$$743$$ −35.4500 −1.30053 −0.650266 0.759706i $$-0.725343\pi$$
−0.650266 + 0.759706i $$0.725343\pi$$
$$744$$ 1.00000 0.0366618
$$745$$ 0 0
$$746$$ −3.81665 −0.139738
$$747$$ 50.0555 1.83144
$$748$$ 7.81665 0.285805
$$749$$ 19.8167 0.724085
$$750$$ 0 0
$$751$$ 14.0000 0.510867 0.255434 0.966827i $$-0.417782\pi$$
0.255434 + 0.966827i $$0.417782\pi$$
$$752$$ 2.60555 0.0950147
$$753$$ −2.05551 −0.0749070
$$754$$ 15.9083 0.579347
$$755$$ 0 0
$$756$$ 8.23886 0.299644
$$757$$ −9.30278 −0.338115 −0.169058 0.985606i $$-0.554072\pi$$
−0.169058 + 0.985606i $$0.554072\pi$$
$$758$$ −15.3305 −0.556830
$$759$$ 2.72498 0.0989105
$$760$$ 0 0
$$761$$ 42.1194 1.52683 0.763414 0.645909i $$-0.223522\pi$$
0.763414 + 0.645909i $$0.223522\pi$$
$$762$$ 1.44996 0.0525266
$$763$$ −9.21110 −0.333464
$$764$$ 12.5139 0.452736
$$765$$ 0 0
$$766$$ −20.8444 −0.753139
$$767$$ 7.81665 0.282243
$$768$$ 0.302776 0.0109255
$$769$$ −22.0000 −0.793340 −0.396670 0.917961i $$-0.629834\pi$$
−0.396670 + 0.917961i $$0.629834\pi$$
$$770$$ 0 0
$$771$$ 3.39445 0.122248
$$772$$ 4.00000 0.143963
$$773$$ 50.0555 1.80037 0.900186 0.435506i $$-0.143431\pi$$
0.900186 + 0.435506i $$0.143431\pi$$
$$774$$ −19.2111 −0.690529
$$775$$ 0 0
$$776$$ −12.4222 −0.445931
$$777$$ 1.39445 0.0500256
$$778$$ −11.8806 −0.425939
$$779$$ −1.81665 −0.0650884
$$780$$ 0 0
$$781$$ 7.81665 0.279702
$$782$$ 41.4500 1.48225
$$783$$ −12.3583 −0.441649
$$784$$ 14.2111 0.507539
$$785$$ 0 0
$$786$$ 1.02776 0.0366589
$$787$$ −25.2111 −0.898679 −0.449339 0.893361i $$-0.648341\pi$$
−0.449339 + 0.893361i $$0.648341\pi$$
$$788$$ 6.00000 0.213741
$$789$$ 2.36669 0.0842565
$$790$$ 0 0
$$791$$ 51.6333 1.83587
$$792$$ −3.78890 −0.134633
$$793$$ −24.2111 −0.859761
$$794$$ −27.8167 −0.987176
$$795$$ 0 0
$$796$$ −2.42221 −0.0858528
$$797$$ 17.3305 0.613879 0.306939 0.951729i $$-0.400695\pi$$
0.306939 + 0.951729i $$0.400695\pi$$
$$798$$ −2.78890 −0.0987259
$$799$$ 15.6333 0.553067
$$800$$ 0 0
$$801$$ −15.1556 −0.535496
$$802$$ 13.8167 0.487883
$$803$$ 11.3305 0.399846
$$804$$ −4.39445 −0.154980
$$805$$ 0 0
$$806$$ 7.60555 0.267894
$$807$$ −2.05551 −0.0723575
$$808$$ 16.4222 0.577731
$$809$$ 29.4500 1.03541 0.517703 0.855561i $$-0.326787\pi$$
0.517703 + 0.855561i $$0.326787\pi$$
$$810$$ 0 0
$$811$$ 54.1472 1.90136 0.950682 0.310166i $$-0.100385\pi$$
0.950682 + 0.310166i $$0.100385\pi$$
$$812$$ −31.8167 −1.11655
$$813$$ 1.94449 0.0681961
$$814$$ −1.30278 −0.0456623
$$815$$ 0 0
$$816$$ 1.81665 0.0635956
$$817$$ 13.2111 0.462198
$$818$$ −5.02776 −0.175791
$$819$$ 30.8444 1.07779
$$820$$ 0 0
$$821$$ 11.2111 0.391270 0.195635 0.980677i $$-0.437323\pi$$
0.195635 + 0.980677i $$0.437323\pi$$
$$822$$ 3.00000 0.104637
$$823$$ 12.8444 0.447728 0.223864 0.974620i $$-0.428133\pi$$
0.223864 + 0.974620i $$0.428133\pi$$
$$824$$ −3.30278 −0.115058
$$825$$ 0 0
$$826$$ −15.6333 −0.543952
$$827$$ −27.3944 −0.952598 −0.476299 0.879283i $$-0.658022\pi$$
−0.476299 + 0.879283i $$0.658022\pi$$
$$828$$ −20.0917 −0.698234
$$829$$ 4.72498 0.164105 0.0820527 0.996628i $$-0.473852\pi$$
0.0820527 + 0.996628i $$0.473852\pi$$
$$830$$ 0 0
$$831$$ 7.60555 0.263834
$$832$$ 2.30278 0.0798344
$$833$$ 85.2666 2.95431
$$834$$ 2.69722 0.0933972
$$835$$ 0 0
$$836$$ 2.60555 0.0901149
$$837$$ −5.90833 −0.204222
$$838$$ −25.1472 −0.868695
$$839$$ −49.0278 −1.69263 −0.846313 0.532686i $$-0.821183\pi$$
−0.846313 + 0.532686i $$0.821183\pi$$
$$840$$ 0 0
$$841$$ 18.7250 0.645689
$$842$$ 28.7250 0.989928
$$843$$ −3.63331 −0.125138
$$844$$ 6.69722 0.230528
$$845$$ 0 0
$$846$$ −7.57779 −0.260530
$$847$$ 42.8444 1.47215
$$848$$ 6.00000 0.206041
$$849$$ −5.26662 −0.180750
$$850$$ 0 0
$$851$$ −6.90833 −0.236814
$$852$$ 1.81665 0.0622375
$$853$$ 11.5416 0.395178 0.197589 0.980285i $$-0.436689\pi$$
0.197589 + 0.980285i $$0.436689\pi$$
$$854$$ 48.4222 1.65697
$$855$$ 0 0
$$856$$ −4.30278 −0.147066
$$857$$ 14.8444 0.507075 0.253538 0.967326i $$-0.418406\pi$$
0.253538 + 0.967326i $$0.418406\pi$$
$$858$$ 0.908327 0.0310098
$$859$$ −24.0555 −0.820764 −0.410382 0.911914i $$-0.634605\pi$$
−0.410382 + 0.911914i $$0.634605\pi$$
$$860$$ 0 0
$$861$$ 1.26662 0.0431661
$$862$$ −5.21110 −0.177491
$$863$$ −12.0000 −0.408485 −0.204242 0.978920i $$-0.565473\pi$$
−0.204242 + 0.978920i $$0.565473\pi$$
$$864$$ −1.78890 −0.0608595
$$865$$ 0 0
$$866$$ 11.9361 0.405605
$$867$$ 5.75274 0.195373
$$868$$ −15.2111 −0.516298
$$869$$ −21.0000 −0.712376
$$870$$ 0 0
$$871$$ −33.4222 −1.13247
$$872$$ 2.00000 0.0677285
$$873$$ 36.1278 1.22274
$$874$$ 13.8167 0.467355
$$875$$ 0 0
$$876$$ 2.63331 0.0889712
$$877$$ −7.21110 −0.243502 −0.121751 0.992561i $$-0.538851\pi$$
−0.121751 + 0.992561i $$0.538851\pi$$
$$878$$ −9.33053 −0.314890
$$879$$ 7.57779 0.255593
$$880$$ 0 0
$$881$$ 25.5416 0.860520 0.430260 0.902705i $$-0.358422\pi$$
0.430260 + 0.902705i $$0.358422\pi$$
$$882$$ −41.3305 −1.39167
$$883$$ 2.42221 0.0815137 0.0407568 0.999169i $$-0.487023\pi$$
0.0407568 + 0.999169i $$0.487023\pi$$
$$884$$ 13.8167 0.464704
$$885$$ 0 0
$$886$$ −0.275019 −0.00923945
$$887$$ −28.4222 −0.954324 −0.477162 0.878815i $$-0.658335\pi$$
−0.477162 + 0.878815i $$0.658335\pi$$
$$888$$ −0.302776 −0.0101605
$$889$$ −22.0555 −0.739718
$$890$$ 0 0
$$891$$ 10.6611 0.357159
$$892$$ −15.8167 −0.529581
$$893$$ 5.21110 0.174383
$$894$$ −0.550039 −0.0183960
$$895$$ 0 0
$$896$$ −4.60555 −0.153861
$$897$$ 4.81665 0.160823
$$898$$ −0.788897 −0.0263258
$$899$$ 22.8167 0.760978
$$900$$ 0 0
$$901$$ 36.0000 1.19933
$$902$$ −1.18335 −0.0394011
$$903$$ −9.21110 −0.306526
$$904$$ −11.2111 −0.372876
$$905$$ 0 0
$$906$$ −4.05551 −0.134735
$$907$$ −26.0000 −0.863316 −0.431658 0.902037i $$-0.642071\pi$$
−0.431658 + 0.902037i $$0.642071\pi$$
$$908$$ 7.81665 0.259405
$$909$$ −47.7611 −1.58414
$$910$$ 0 0
$$911$$ −46.4222 −1.53804 −0.769018 0.639227i $$-0.779254\pi$$
−0.769018 + 0.639227i $$0.779254\pi$$
$$912$$ 0.605551 0.0200518
$$913$$ −22.4222 −0.742067
$$914$$ −4.60555 −0.152338
$$915$$ 0 0
$$916$$ 17.3944 0.574729
$$917$$ −15.6333 −0.516257
$$918$$ −10.7334 −0.354254
$$919$$ −38.4222 −1.26743 −0.633716 0.773566i $$-0.718471\pi$$
−0.633716 + 0.773566i $$0.718471\pi$$
$$920$$ 0 0
$$921$$ −2.14719 −0.0707522
$$922$$ −16.4222 −0.540837
$$923$$ 13.8167 0.454781
$$924$$ −1.81665 −0.0597635
$$925$$ 0 0
$$926$$ −30.3028 −0.995811
$$927$$ 9.60555 0.315488
$$928$$ 6.90833 0.226777
$$929$$ −36.5139 −1.19798 −0.598991 0.800756i $$-0.704431\pi$$
−0.598991 + 0.800756i $$0.704431\pi$$
$$930$$ 0 0
$$931$$ 28.4222 0.931500
$$932$$ 9.51388 0.311637
$$933$$ 1.54163 0.0504708
$$934$$ 0 0
$$935$$ 0 0
$$936$$ −6.69722 −0.218906
$$937$$ 28.9083 0.944394 0.472197 0.881493i $$-0.343461\pi$$
0.472197 + 0.881493i $$0.343461\pi$$
$$938$$ 66.8444 2.18255
$$939$$ −8.18335 −0.267053
$$940$$ 0 0
$$941$$ −7.81665 −0.254816 −0.127408 0.991850i $$-0.540666\pi$$
−0.127408 + 0.991850i $$0.540666\pi$$
$$942$$ −2.18335 −0.0711373
$$943$$ −6.27502 −0.204343
$$944$$ 3.39445 0.110480
$$945$$ 0 0
$$946$$ 8.60555 0.279791
$$947$$ −39.6333 −1.28791 −0.643955 0.765064i $$-0.722707\pi$$
−0.643955 + 0.765064i $$0.722707\pi$$
$$948$$ −4.88057 −0.158514
$$949$$ 20.0278 0.650128
$$950$$ 0 0
$$951$$ 1.57779 0.0511635
$$952$$ −27.6333 −0.895601
$$953$$ 18.7527 0.607461 0.303730 0.952758i $$-0.401768\pi$$
0.303730 + 0.952758i $$0.401768\pi$$
$$954$$ −17.4500 −0.564963
$$955$$ 0 0
$$956$$ 0.513878 0.0166200
$$957$$ 2.72498 0.0880861
$$958$$ 12.1194 0.391561
$$959$$ −45.6333 −1.47358
$$960$$ 0 0
$$961$$ −20.0917 −0.648118
$$962$$ −2.30278 −0.0742445
$$963$$ 12.5139 0.403254
$$964$$ 8.00000 0.257663
$$965$$ 0 0
$$966$$ −9.63331 −0.309947
$$967$$ −25.7250 −0.827260 −0.413630 0.910445i $$-0.635739\pi$$
−0.413630 + 0.910445i $$0.635739\pi$$
$$968$$ −9.30278 −0.299003
$$969$$ 3.63331 0.116719
$$970$$ 0 0
$$971$$ 31.5416 1.01222 0.506110 0.862469i $$-0.331083\pi$$
0.506110 + 0.862469i $$0.331083\pi$$
$$972$$ 7.84441 0.251610
$$973$$ −41.0278 −1.31529
$$974$$ 22.7889 0.730203
$$975$$ 0 0
$$976$$ −10.5139 −0.336541
$$977$$ −18.0000 −0.575871 −0.287936 0.957650i $$-0.592969\pi$$
−0.287936 + 0.957650i $$0.592969\pi$$
$$978$$ 6.18335 0.197722
$$979$$ 6.78890 0.216974
$$980$$ 0 0
$$981$$ −5.81665 −0.185711
$$982$$ −14.7250 −0.469893
$$983$$ 12.0000 0.382741 0.191370 0.981518i $$-0.438707\pi$$
0.191370 + 0.981518i $$0.438707\pi$$
$$984$$ −0.275019 −0.00876729
$$985$$ 0 0
$$986$$ 41.4500 1.32004
$$987$$ −3.63331 −0.115649
$$988$$ 4.60555 0.146522
$$989$$ 45.6333 1.45105
$$990$$ 0 0
$$991$$ 54.3028 1.72498 0.862492 0.506070i $$-0.168902\pi$$
0.862492 + 0.506070i $$0.168902\pi$$
$$992$$ 3.30278 0.104863
$$993$$ 0.366692 0.0116366
$$994$$ −27.6333 −0.876475
$$995$$ 0 0
$$996$$ −5.21110 −0.165120
$$997$$ 23.5778 0.746716 0.373358 0.927687i $$-0.378206\pi$$
0.373358 + 0.927687i $$0.378206\pi$$
$$998$$ 8.23886 0.260797
$$999$$ 1.78890 0.0565982
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.u.1.2 2
5.2 odd 4 1850.2.b.i.149.3 4
5.3 odd 4 1850.2.b.i.149.2 4
5.4 even 2 74.2.a.a.1.1 2
15.14 odd 2 666.2.a.j.1.1 2
20.19 odd 2 592.2.a.f.1.2 2
35.34 odd 2 3626.2.a.a.1.2 2
40.19 odd 2 2368.2.a.ba.1.1 2
40.29 even 2 2368.2.a.s.1.2 2
55.54 odd 2 8954.2.a.p.1.1 2
60.59 even 2 5328.2.a.bf.1.1 2
185.184 even 2 2738.2.a.l.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.a.a.1.1 2 5.4 even 2
592.2.a.f.1.2 2 20.19 odd 2
666.2.a.j.1.1 2 15.14 odd 2
1850.2.a.u.1.2 2 1.1 even 1 trivial
1850.2.b.i.149.2 4 5.3 odd 4
1850.2.b.i.149.3 4 5.2 odd 4
2368.2.a.s.1.2 2 40.29 even 2
2368.2.a.ba.1.1 2 40.19 odd 2
2738.2.a.l.1.1 2 185.184 even 2
3626.2.a.a.1.2 2 35.34 odd 2
5328.2.a.bf.1.1 2 60.59 even 2
8954.2.a.p.1.1 2 55.54 odd 2