# Properties

 Label 2738.2.a.l.1.1 Level $2738$ Weight $2$ Character 2738.1 Self dual yes Analytic conductor $21.863$ 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] = [2738,2,Mod(1,2738)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2738, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2738.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2738 = 2 \cdot 37^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2738.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: $$21.8630400734$$ 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 1.1 Root $$-1.30278$$ of defining polynomial Character $$\chi$$ $$=$$ 2738.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -0.302776 q^{3} +1.00000 q^{4} -1.30278 q^{5} -0.302776 q^{6} +4.60555 q^{7} +1.00000 q^{8} -2.90833 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -0.302776 q^{3} +1.00000 q^{4} -1.30278 q^{5} -0.302776 q^{6} +4.60555 q^{7} +1.00000 q^{8} -2.90833 q^{9} -1.30278 q^{10} +1.30278 q^{11} -0.302776 q^{12} +2.30278 q^{13} +4.60555 q^{14} +0.394449 q^{15} +1.00000 q^{16} +6.00000 q^{17} -2.90833 q^{18} -2.00000 q^{19} -1.30278 q^{20} -1.39445 q^{21} +1.30278 q^{22} +6.90833 q^{23} -0.302776 q^{24} -3.30278 q^{25} +2.30278 q^{26} +1.78890 q^{27} +4.60555 q^{28} -6.90833 q^{29} +0.394449 q^{30} -3.30278 q^{31} +1.00000 q^{32} -0.394449 q^{33} +6.00000 q^{34} -6.00000 q^{35} -2.90833 q^{36} -2.00000 q^{38} -0.697224 q^{39} -1.30278 q^{40} -0.908327 q^{41} -1.39445 q^{42} +6.60555 q^{43} +1.30278 q^{44} +3.78890 q^{45} +6.90833 q^{46} -2.60555 q^{47} -0.302776 q^{48} +14.2111 q^{49} -3.30278 q^{50} -1.81665 q^{51} +2.30278 q^{52} -6.00000 q^{53} +1.78890 q^{54} -1.69722 q^{55} +4.60555 q^{56} +0.605551 q^{57} -6.90833 q^{58} -3.39445 q^{59} +0.394449 q^{60} +10.5139 q^{61} -3.30278 q^{62} -13.3944 q^{63} +1.00000 q^{64} -3.00000 q^{65} -0.394449 q^{66} +14.5139 q^{67} +6.00000 q^{68} -2.09167 q^{69} -6.00000 q^{70} +6.00000 q^{71} -2.90833 q^{72} -8.69722 q^{73} +1.00000 q^{75} -2.00000 q^{76} +6.00000 q^{77} -0.697224 q^{78} +16.1194 q^{79} -1.30278 q^{80} +8.18335 q^{81} -0.908327 q^{82} +17.2111 q^{83} -1.39445 q^{84} -7.81665 q^{85} +6.60555 q^{86} +2.09167 q^{87} +1.30278 q^{88} -5.21110 q^{89} +3.78890 q^{90} +10.6056 q^{91} +6.90833 q^{92} +1.00000 q^{93} -2.60555 q^{94} +2.60555 q^{95} -0.302776 q^{96} -12.4222 q^{97} +14.2111 q^{98} -3.78890 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 3 q^{3} + 2 q^{4} + q^{5} + 3 q^{6} + 2 q^{7} + 2 q^{8} + 5 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 + 3 * q^3 + 2 * q^4 + q^5 + 3 * q^6 + 2 * q^7 + 2 * q^8 + 5 * q^9 $$2 q + 2 q^{2} + 3 q^{3} + 2 q^{4} + q^{5} + 3 q^{6} + 2 q^{7} + 2 q^{8} + 5 q^{9} + q^{10} - q^{11} + 3 q^{12} + q^{13} + 2 q^{14} + 8 q^{15} + 2 q^{16} + 12 q^{17} + 5 q^{18} - 4 q^{19} + q^{20} - 10 q^{21} - q^{22} + 3 q^{23} + 3 q^{24} - 3 q^{25} + q^{26} + 18 q^{27} + 2 q^{28} - 3 q^{29} + 8 q^{30} - 3 q^{31} + 2 q^{32} - 8 q^{33} + 12 q^{34} - 12 q^{35} + 5 q^{36} - 4 q^{38} - 5 q^{39} + q^{40} + 9 q^{41} - 10 q^{42} + 6 q^{43} - q^{44} + 22 q^{45} + 3 q^{46} + 2 q^{47} + 3 q^{48} + 14 q^{49} - 3 q^{50} + 18 q^{51} + q^{52} - 12 q^{53} + 18 q^{54} - 7 q^{55} + 2 q^{56} - 6 q^{57} - 3 q^{58} - 14 q^{59} + 8 q^{60} + 3 q^{61} - 3 q^{62} - 34 q^{63} + 2 q^{64} - 6 q^{65} - 8 q^{66} + 11 q^{67} + 12 q^{68} - 15 q^{69} - 12 q^{70} + 12 q^{71} + 5 q^{72} - 21 q^{73} + 2 q^{75} - 4 q^{76} + 12 q^{77} - 5 q^{78} + 7 q^{79} + q^{80} + 38 q^{81} + 9 q^{82} + 20 q^{83} - 10 q^{84} + 6 q^{85} + 6 q^{86} + 15 q^{87} - q^{88} + 4 q^{89} + 22 q^{90} + 14 q^{91} + 3 q^{92} + 2 q^{93} + 2 q^{94} - 2 q^{95} + 3 q^{96} + 4 q^{97} + 14 q^{98} - 22 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 + 3 * q^3 + 2 * q^4 + q^5 + 3 * q^6 + 2 * q^7 + 2 * q^8 + 5 * q^9 + q^10 - q^11 + 3 * q^12 + q^13 + 2 * q^14 + 8 * q^15 + 2 * q^16 + 12 * q^17 + 5 * q^18 - 4 * q^19 + q^20 - 10 * q^21 - q^22 + 3 * q^23 + 3 * q^24 - 3 * q^25 + q^26 + 18 * q^27 + 2 * q^28 - 3 * q^29 + 8 * q^30 - 3 * q^31 + 2 * q^32 - 8 * q^33 + 12 * q^34 - 12 * q^35 + 5 * q^36 - 4 * q^38 - 5 * q^39 + q^40 + 9 * q^41 - 10 * q^42 + 6 * q^43 - q^44 + 22 * q^45 + 3 * q^46 + 2 * q^47 + 3 * q^48 + 14 * q^49 - 3 * q^50 + 18 * q^51 + q^52 - 12 * q^53 + 18 * q^54 - 7 * q^55 + 2 * q^56 - 6 * q^57 - 3 * q^58 - 14 * q^59 + 8 * q^60 + 3 * q^61 - 3 * q^62 - 34 * q^63 + 2 * q^64 - 6 * q^65 - 8 * q^66 + 11 * q^67 + 12 * q^68 - 15 * q^69 - 12 * q^70 + 12 * q^71 + 5 * q^72 - 21 * q^73 + 2 * q^75 - 4 * q^76 + 12 * q^77 - 5 * q^78 + 7 * q^79 + q^80 + 38 * q^81 + 9 * q^82 + 20 * q^83 - 10 * q^84 + 6 * q^85 + 6 * q^86 + 15 * q^87 - q^88 + 4 * q^89 + 22 * q^90 + 14 * q^91 + 3 * q^92 + 2 * q^93 + 2 * q^94 - 2 * q^95 + 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 $$-0.527857\pi$$
−0.0874038 + 0.996173i $$0.527857\pi$$
$$4$$ 1.00000 0.500000
$$5$$ −1.30278 −0.582619 −0.291309 0.956629i $$-0.594091\pi$$
−0.291309 + 0.956629i $$0.594091\pi$$
$$6$$ −0.302776 −0.123608
$$7$$ 4.60555 1.74073 0.870367 0.492403i $$-0.163881\pi$$
0.870367 + 0.492403i $$0.163881\pi$$
$$8$$ 1.00000 0.353553
$$9$$ −2.90833 −0.969442
$$10$$ −1.30278 −0.411974
$$11$$ 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
$$15$$ 0.394449 0.101846
$$16$$ 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 $$-0.573682\pi$$
−0.229416 + 0.973329i $$0.573682\pi$$
$$20$$ −1.30278 −0.291309
$$21$$ −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
$$25$$ −3.30278 −0.660555
$$26$$ 2.30278 0.451611
$$27$$ 1.78890 0.344273
$$28$$ 4.60555 0.870367
$$29$$ −6.90833 −1.28284 −0.641422 0.767188i $$-0.721655\pi$$
−0.641422 + 0.767188i $$0.721655\pi$$
$$30$$ 0.394449 0.0720162
$$31$$ −3.30278 −0.593196 −0.296598 0.955002i $$-0.595852\pi$$
−0.296598 + 0.955002i $$0.595852\pi$$
$$32$$ 1.00000 0.176777
$$33$$ −0.394449 −0.0686647
$$34$$ 6.00000 1.02899
$$35$$ −6.00000 −1.01419
$$36$$ −2.90833 −0.484721
$$37$$ 0 0
$$38$$ −2.00000 −0.324443
$$39$$ −0.697224 −0.111645
$$40$$ −1.30278 −0.205987
$$41$$ −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
$$45$$ 3.78890 0.564815
$$46$$ 6.90833 1.01858
$$47$$ −2.60555 −0.380059 −0.190029 0.981778i $$-0.560858\pi$$
−0.190029 + 0.981778i $$0.560858\pi$$
$$48$$ −0.302776 −0.0437019
$$49$$ 14.2111 2.03016
$$50$$ −3.30278 −0.467083
$$51$$ −1.81665 −0.254382
$$52$$ 2.30278 0.319338
$$53$$ −6.00000 −0.824163 −0.412082 0.911147i $$-0.635198\pi$$
−0.412082 + 0.911147i $$0.635198\pi$$
$$54$$ 1.78890 0.243438
$$55$$ −1.69722 −0.228854
$$56$$ 4.60555 0.615443
$$57$$ 0.605551 0.0802072
$$58$$ −6.90833 −0.907108
$$59$$ −3.39445 −0.441920 −0.220960 0.975283i $$-0.570919\pi$$
−0.220960 + 0.975283i $$0.570919\pi$$
$$60$$ 0.394449 0.0509231
$$61$$ 10.5139 1.34616 0.673082 0.739568i $$-0.264970\pi$$
0.673082 + 0.739568i $$0.264970\pi$$
$$62$$ −3.30278 −0.419453
$$63$$ −13.3944 −1.68754
$$64$$ 1.00000 0.125000
$$65$$ −3.00000 −0.372104
$$66$$ −0.394449 −0.0485533
$$67$$ 14.5139 1.77315 0.886576 0.462583i $$-0.153077\pi$$
0.886576 + 0.462583i $$0.153077\pi$$
$$68$$ 6.00000 0.727607
$$69$$ −2.09167 −0.251808
$$70$$ −6.00000 −0.717137
$$71$$ 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 $$-0.669972\pi$$
−0.508967 + 0.860786i $$0.669972\pi$$
$$74$$ 0 0
$$75$$ 1.00000 0.115470
$$76$$ −2.00000 −0.229416
$$77$$ 6.00000 0.683763
$$78$$ −0.697224 −0.0789451
$$79$$ 16.1194 1.81358 0.906789 0.421585i $$-0.138526\pi$$
0.906789 + 0.421585i $$0.138526\pi$$
$$80$$ −1.30278 −0.145655
$$81$$ 8.18335 0.909261
$$82$$ −0.908327 −0.100308
$$83$$ 17.2111 1.88916 0.944582 0.328276i $$-0.106467\pi$$
0.944582 + 0.328276i $$0.106467\pi$$
$$84$$ −1.39445 −0.152147
$$85$$ −7.81665 −0.847835
$$86$$ 6.60555 0.712295
$$87$$ 2.09167 0.224251
$$88$$ 1.30278 0.138876
$$89$$ −5.21110 −0.552376 −0.276188 0.961104i $$-0.589071\pi$$
−0.276188 + 0.961104i $$0.589071\pi$$
$$90$$ 3.78890 0.399385
$$91$$ 10.6056 1.11176
$$92$$ 6.90833 0.720243
$$93$$ 1.00000 0.103695
$$94$$ −2.60555 −0.268742
$$95$$ 2.60555 0.267324
$$96$$ −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
$$100$$ −3.30278 −0.330278
$$101$$ 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
$$105$$ 1.81665 0.177287
$$106$$ −6.00000 −0.582772
$$107$$ 4.30278 0.415965 0.207983 0.978133i $$-0.433310\pi$$
0.207983 + 0.978133i $$0.433310\pi$$
$$108$$ 1.78890 0.172137
$$109$$ −2.00000 −0.191565 −0.0957826 0.995402i $$-0.530535\pi$$
−0.0957826 + 0.995402i $$0.530535\pi$$
$$110$$ −1.69722 −0.161824
$$111$$ 0 0
$$112$$ 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
$$115$$ −9.00000 −0.839254
$$116$$ −6.90833 −0.641422
$$117$$ −6.69722 −0.619159
$$118$$ −3.39445 −0.312484
$$119$$ 27.6333 2.53314
$$120$$ 0.394449 0.0360081
$$121$$ −9.30278 −0.845707
$$122$$ 10.5139 0.951882
$$123$$ 0.275019 0.0247977
$$124$$ −3.30278 −0.296598
$$125$$ 10.8167 0.967471
$$126$$ −13.3944 −1.19327
$$127$$ −4.78890 −0.424946 −0.212473 0.977167i $$-0.568152\pi$$
−0.212473 + 0.977167i $$0.568152\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ −2.00000 −0.176090
$$130$$ −3.00000 −0.263117
$$131$$ −3.39445 −0.296574 −0.148287 0.988944i $$-0.547376\pi$$
−0.148287 + 0.988944i $$0.547376\pi$$
$$132$$ −0.394449 −0.0343324
$$133$$ −9.21110 −0.798704
$$134$$ 14.5139 1.25381
$$135$$ −2.33053 −0.200580
$$136$$ 6.00000 0.514496
$$137$$ −9.90833 −0.846525 −0.423263 0.906007i $$-0.639115\pi$$
−0.423263 + 0.906007i $$0.639115\pi$$
$$138$$ −2.09167 −0.178055
$$139$$ 8.90833 0.755594 0.377797 0.925888i $$-0.376682\pi$$
0.377797 + 0.925888i $$0.376682\pi$$
$$140$$ −6.00000 −0.507093
$$141$$ 0.788897 0.0664372
$$142$$ 6.00000 0.503509
$$143$$ 3.00000 0.250873
$$144$$ −2.90833 −0.242361
$$145$$ 9.00000 0.747409
$$146$$ −8.69722 −0.719787
$$147$$ −4.30278 −0.354887
$$148$$ 0 0
$$149$$ −1.81665 −0.148826 −0.0744130 0.997228i $$-0.523708\pi$$
−0.0744130 + 0.997228i $$0.523708\pi$$
$$150$$ 1.00000 0.0816497
$$151$$ −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
$$155$$ 4.30278 0.345607
$$156$$ −0.697224 −0.0558226
$$157$$ 7.21110 0.575509 0.287754 0.957704i $$-0.407091\pi$$
0.287754 + 0.957704i $$0.407091\pi$$
$$158$$ 16.1194 1.28239
$$159$$ 1.81665 0.144070
$$160$$ −1.30278 −0.102993
$$161$$ 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
$$165$$ 0.513878 0.0400054
$$166$$ 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
$$170$$ −7.81665 −0.599510
$$171$$ 5.81665 0.444811
$$172$$ 6.60555 0.503669
$$173$$ −23.2111 −1.76471 −0.882354 0.470587i $$-0.844042\pi$$
−0.882354 + 0.470587i $$0.844042\pi$$
$$174$$ 2.09167 0.158569
$$175$$ −15.2111 −1.14985
$$176$$ 1.30278 0.0982004
$$177$$ 1.02776 0.0772509
$$178$$ −5.21110 −0.390589
$$179$$ −7.81665 −0.584244 −0.292122 0.956381i $$-0.594361\pi$$
−0.292122 + 0.956381i $$0.594361\pi$$
$$180$$ 3.78890 0.282408
$$181$$ 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
$$190$$ 2.60555 0.189027
$$191$$ −12.5139 −0.905472 −0.452736 0.891644i $$-0.649552\pi$$
−0.452736 + 0.891644i $$0.649552\pi$$
$$192$$ −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
$$195$$ 0.908327 0.0650466
$$196$$ 14.2111 1.01508
$$197$$ −6.00000 −0.427482 −0.213741 0.976890i $$-0.568565\pi$$
−0.213741 + 0.976890i $$0.568565\pi$$
$$198$$ −3.78890 −0.269265
$$199$$ 2.42221 0.171706 0.0858528 0.996308i $$-0.472639\pi$$
0.0858528 + 0.996308i $$0.472639\pi$$
$$200$$ −3.30278 −0.233542
$$201$$ −4.39445 −0.309961
$$202$$ 16.4222 1.15546
$$203$$ −31.8167 −2.23309
$$204$$ −1.81665 −0.127191
$$205$$ 1.18335 0.0826485
$$206$$ −3.30278 −0.230115
$$207$$ −20.0917 −1.39647
$$208$$ 2.30278 0.159669
$$209$$ −2.60555 −0.180230
$$210$$ 1.81665 0.125361
$$211$$ 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
$$215$$ −8.60555 −0.586894
$$216$$ 1.78890 0.121719
$$217$$ −15.2111 −1.03260
$$218$$ −2.00000 −0.135457
$$219$$ 2.63331 0.177942
$$220$$ −1.69722 −0.114427
$$221$$ 13.8167 0.929409
$$222$$ 0 0
$$223$$ 15.8167 1.05916 0.529581 0.848260i $$-0.322349\pi$$
0.529581 + 0.848260i $$0.322349\pi$$
$$224$$ 4.60555 0.307721
$$225$$ 9.60555 0.640370
$$226$$ −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 $$0.305108\pi$$
$$230$$ −9.00000 −0.593442
$$231$$ −1.81665 −0.119527
$$232$$ −6.90833 −0.453554
$$233$$ −9.51388 −0.623275 −0.311637 0.950201i $$-0.600877\pi$$
−0.311637 + 0.950201i $$0.600877\pi$$
$$234$$ −6.69722 −0.437811
$$235$$ 3.39445 0.221429
$$236$$ −3.39445 −0.220960
$$237$$ −4.88057 −0.317027
$$238$$ 27.6333 1.79120
$$239$$ −0.513878 −0.0332400 −0.0166200 0.999862i $$-0.505291\pi$$
−0.0166200 + 0.999862i $$0.505291\pi$$
$$240$$ 0.394449 0.0254616
$$241$$ −8.00000 −0.515325 −0.257663 0.966235i $$-0.582952\pi$$
−0.257663 + 0.966235i $$0.582952\pi$$
$$242$$ −9.30278 −0.598005
$$243$$ −7.84441 −0.503219
$$244$$ 10.5139 0.673082
$$245$$ −18.5139 −1.18281
$$246$$ 0.275019 0.0175346
$$247$$ −4.60555 −0.293044
$$248$$ −3.30278 −0.209726
$$249$$ −5.21110 −0.330240
$$250$$ 10.8167 0.684105
$$251$$ 6.78890 0.428511 0.214256 0.976778i $$-0.431267\pi$$
0.214256 + 0.976778i $$0.431267\pi$$
$$252$$ −13.3944 −0.843771
$$253$$ 9.00000 0.565825
$$254$$ −4.78890 −0.300482
$$255$$ 2.36669 0.148208
$$256$$ 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
$$259$$ 0 0
$$260$$ −3.00000 −0.186052
$$261$$ 20.0917 1.24364
$$262$$ −3.39445 −0.209710
$$263$$ −7.81665 −0.481996 −0.240998 0.970526i $$-0.577475\pi$$
−0.240998 + 0.970526i $$0.577475\pi$$
$$264$$ −0.394449 −0.0242766
$$265$$ 7.81665 0.480173
$$266$$ −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 $$0.566358\pi$$
$$270$$ −2.33053 −0.141832
$$271$$ 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
$$275$$ −4.30278 −0.259467
$$276$$ −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
$$280$$ −6.00000 −0.358569
$$281$$ 12.0000 0.715860 0.357930 0.933748i $$-0.383483\pi$$
0.357930 + 0.933748i $$0.383483\pi$$
$$282$$ 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
$$285$$ −0.788897 −0.0467303
$$286$$ 3.00000 0.177394
$$287$$ −4.18335 −0.246935
$$288$$ −2.90833 −0.171375
$$289$$ 19.0000 1.11765
$$290$$ 9.00000 0.528498
$$291$$ 3.76114 0.220482
$$292$$ −8.69722 −0.508967
$$293$$ −25.0278 −1.46214 −0.731069 0.682304i $$-0.760978\pi$$
−0.731069 + 0.682304i $$0.760978\pi$$
$$294$$ −4.30278 −0.250943
$$295$$ 4.42221 0.257471
$$296$$ 0 0
$$297$$ 2.33053 0.135231
$$298$$ −1.81665 −0.105236
$$299$$ 15.9083 0.920002
$$300$$ 1.00000 0.0577350
$$301$$ 30.4222 1.75351
$$302$$ −13.3944 −0.770764
$$303$$ −4.97224 −0.285648
$$304$$ −2.00000 −0.114708
$$305$$ −13.6972 −0.784301
$$306$$ −17.4500 −0.997548
$$307$$ 7.09167 0.404743 0.202372 0.979309i $$-0.435135\pi$$
0.202372 + 0.979309i $$0.435135\pi$$
$$308$$ 6.00000 0.341882
$$309$$ 1.00000 0.0568880
$$310$$ 4.30278 0.244381
$$311$$ −5.09167 −0.288722 −0.144361 0.989525i $$-0.546113\pi$$
−0.144361 + 0.989525i $$0.546113\pi$$
$$312$$ −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
$$315$$ 17.4500 0.983194
$$316$$ 16.1194 0.906789
$$317$$ −5.21110 −0.292685 −0.146342 0.989234i $$-0.546750\pi$$
−0.146342 + 0.989234i $$0.546750\pi$$
$$318$$ 1.81665 0.101873
$$319$$ −9.00000 −0.503903
$$320$$ −1.30278 −0.0728274
$$321$$ −1.30278 −0.0727138
$$322$$ 31.8167 1.77307
$$323$$ −12.0000 −0.667698
$$324$$ 8.18335 0.454630
$$325$$ −7.60555 −0.421880
$$326$$ 20.4222 1.13108
$$327$$ 0.605551 0.0334871
$$328$$ −0.908327 −0.0501540
$$329$$ −12.0000 −0.661581
$$330$$ 0.513878 0.0282881
$$331$$ −1.21110 −0.0665682 −0.0332841 0.999446i $$-0.510597\pi$$
−0.0332841 + 0.999446i $$0.510597\pi$$
$$332$$ 17.2111 0.944582
$$333$$ 0 0
$$334$$ −12.5139 −0.684729
$$335$$ −18.9083 −1.03307
$$336$$ −1.39445 −0.0760734
$$337$$ −19.1194 −1.04150 −0.520751 0.853709i $$-0.674348\pi$$
−0.520751 + 0.853709i $$0.674348\pi$$
$$338$$ −7.69722 −0.418674
$$339$$ 3.39445 0.184361
$$340$$ −7.81665 −0.423918
$$341$$ −4.30278 −0.233008
$$342$$ 5.81665 0.314529
$$343$$ 33.2111 1.79323
$$344$$ 6.60555 0.356147
$$345$$ 2.72498 0.146708
$$346$$ −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 $$0.702931\pi$$
$$350$$ −15.2111 −0.813068
$$351$$ 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
$$355$$ −7.81665 −0.414865
$$356$$ −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 $$0.595603\pi$$
$$360$$ 3.78890 0.199692
$$361$$ −15.0000 −0.789474
$$362$$ 20.0000 1.05118
$$363$$ 2.81665 0.147836
$$364$$ 10.6056 0.555882
$$365$$ 11.3305 0.593067
$$366$$ −3.18335 −0.166396
$$367$$ −17.8167 −0.930022 −0.465011 0.885305i $$-0.653950\pi$$
−0.465011 + 0.885305i $$0.653950\pi$$
$$368$$ 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 $$-0.468497\pi$$
0.0988094 + 0.995106i $$0.468497\pi$$
$$374$$ 7.81665 0.404190
$$375$$ −3.27502 −0.169121
$$376$$ −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 $$0.628818\pi$$
$$380$$ 2.60555 0.133662
$$381$$ 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
$$385$$ −7.81665 −0.398374
$$386$$ 4.00000 0.203595
$$387$$ −19.2111 −0.976555
$$388$$ −12.4222 −0.630642
$$389$$ 11.8806 0.602369 0.301184 0.953566i $$-0.402618\pi$$
0.301184 + 0.953566i $$0.402618\pi$$
$$390$$ 0.908327 0.0459949
$$391$$ 41.4500 2.09621
$$392$$ 14.2111 0.717769
$$393$$ 1.02776 0.0518435
$$394$$ −6.00000 −0.302276
$$395$$ −21.0000 −1.05662
$$396$$ −3.78890 −0.190399
$$397$$ 27.8167 1.39608 0.698039 0.716060i $$-0.254056\pi$$
0.698039 + 0.716060i $$0.254056\pi$$
$$398$$ 2.42221 0.121414
$$399$$ 2.78890 0.139620
$$400$$ −3.30278 −0.165139
$$401$$ −13.8167 −0.689971 −0.344985 0.938608i $$-0.612116\pi$$
−0.344985 + 0.938608i $$0.612116\pi$$
$$402$$ −4.39445 −0.219175
$$403$$ −7.60555 −0.378859
$$404$$ 16.4222 0.817035
$$405$$ −10.6611 −0.529753
$$406$$ −31.8167 −1.57903
$$407$$ 0 0
$$408$$ −1.81665 −0.0899378
$$409$$ 5.02776 0.248607 0.124303 0.992244i $$-0.460330\pi$$
0.124303 + 0.992244i $$0.460330\pi$$
$$410$$ 1.18335 0.0584413
$$411$$ 3.00000 0.147979
$$412$$ −3.30278 −0.162716
$$413$$ −15.6333 −0.769265
$$414$$ −20.0917 −0.987452
$$415$$ −22.4222 −1.10066
$$416$$ 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 $$0.710545\pi$$
$$420$$ 1.81665 0.0886436
$$421$$ −28.7250 −1.39997 −0.699985 0.714158i $$-0.746810\pi$$
−0.699985 + 0.714158i $$0.746810\pi$$
$$422$$ 6.69722 0.326016
$$423$$ 7.57779 0.368445
$$424$$ −6.00000 −0.291386
$$425$$ −19.8167 −0.961249
$$426$$ −1.81665 −0.0880172
$$427$$ 48.4222 2.34331
$$428$$ 4.30278 0.207983
$$429$$ −0.908327 −0.0438544
$$430$$ −8.60555 −0.414997
$$431$$ 5.21110 0.251010 0.125505 0.992093i $$-0.459945\pi$$
0.125505 + 0.992093i $$0.459945\pi$$
$$432$$ 1.78890 0.0860684
$$433$$ −11.9361 −0.573612 −0.286806 0.957989i $$-0.592593\pi$$
−0.286806 + 0.957989i $$0.592593\pi$$
$$434$$ −15.2111 −0.730156
$$435$$ −2.72498 −0.130653
$$436$$ −2.00000 −0.0957826
$$437$$ −13.8167 −0.660940
$$438$$ 2.63331 0.125824
$$439$$ 9.33053 0.445322 0.222661 0.974896i $$-0.428526\pi$$
0.222661 + 0.974896i $$0.428526\pi$$
$$440$$ −1.69722 −0.0809120
$$441$$ −41.3305 −1.96812
$$442$$ 13.8167 0.657191
$$443$$ 0.275019 0.0130666 0.00653328 0.999979i $$-0.497920\pi$$
0.00653328 + 0.999979i $$0.497920\pi$$
$$444$$ 0 0
$$445$$ 6.78890 0.321825
$$446$$ 15.8167 0.748940
$$447$$ 0.550039 0.0260159
$$448$$ 4.60555 0.217592
$$449$$ 0.788897 0.0372304 0.0186152 0.999827i $$-0.494074\pi$$
0.0186152 + 0.999827i $$0.494074\pi$$
$$450$$ 9.60555 0.452810
$$451$$ −1.18335 −0.0557216
$$452$$ −11.2111 −0.527326
$$453$$ 4.05551 0.190545
$$454$$ 7.81665 0.366854
$$455$$ −13.8167 −0.647735
$$456$$ 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
$$460$$ −9.00000 −0.419627
$$461$$ 16.4222 0.764858 0.382429 0.923985i $$-0.375088\pi$$
0.382429 + 0.923985i $$0.375088\pi$$
$$462$$ −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
$$465$$ −1.30278 −0.0604148
$$466$$ −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
$$470$$ 3.39445 0.156574
$$471$$ −2.18335 −0.100603
$$472$$ −3.39445 −0.156242
$$473$$ 8.60555 0.395684
$$474$$ −4.88057 −0.224172
$$475$$ 6.60555 0.303083
$$476$$ 27.6333 1.26657
$$477$$ 17.4500 0.798979
$$478$$ −0.513878 −0.0235042
$$479$$ −12.1194 −0.553751 −0.276875 0.960906i $$-0.589299\pi$$
−0.276875 + 0.960906i $$0.589299\pi$$
$$480$$ 0.394449 0.0180040
$$481$$ 0 0
$$482$$ −8.00000 −0.364390
$$483$$ −9.63331 −0.438331
$$484$$ −9.30278 −0.422853
$$485$$ 16.1833 0.734848
$$486$$ −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
$$490$$ −18.5139 −0.836372
$$491$$ −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
$$495$$ 4.93608 0.221860
$$496$$ −3.30278 −0.148299
$$497$$ 27.6333 1.23952
$$498$$ −5.21110 −0.233515
$$499$$ −8.23886 −0.368822 −0.184411 0.982849i $$-0.559038\pi$$
−0.184411 + 0.982849i $$0.559038\pi$$
$$500$$ 10.8167 0.483735
$$501$$ 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
$$505$$ −21.3944 −0.952040
$$506$$ 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 $$0.693891\pi$$
$$510$$ 2.36669 0.104799
$$511$$ −40.0555 −1.77195
$$512$$ 1.00000 0.0441942
$$513$$ −3.57779 −0.157964
$$514$$ 11.2111 0.494501
$$515$$ 4.30278 0.189603
$$516$$ −2.00000 −0.0880451
$$517$$ −3.39445 −0.149288
$$518$$ 0 0
$$519$$ 7.02776 0.308484
$$520$$ −3.00000 −0.131559
$$521$$ −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
$$525$$ 4.60555 0.201003
$$526$$ −7.81665 −0.340822
$$527$$ −19.8167 −0.863227
$$528$$ −0.394449 −0.0171662
$$529$$ 24.7250 1.07500
$$530$$ 7.81665 0.339534
$$531$$ 9.87217 0.428416
$$532$$ −9.21110 −0.399352
$$533$$ −2.09167 −0.0906004
$$534$$ 1.57779 0.0682779
$$535$$ −5.60555 −0.242349
$$536$$ 14.5139 0.626904
$$537$$ 2.36669 0.102130
$$538$$ −6.78890 −0.292690
$$539$$ 18.5139 0.797449
$$540$$ −2.33053 −0.100290
$$541$$ 20.9361 0.900113 0.450056 0.893000i $$-0.351404\pi$$
0.450056 + 0.893000i $$0.351404\pi$$
$$542$$ 6.42221 0.275857
$$543$$ −6.05551 −0.259867
$$544$$ 6.00000 0.257248
$$545$$ 2.60555 0.111610
$$546$$ −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
$$550$$ −4.30278 −0.183471
$$551$$ 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
$$560$$ −6.00000 −0.253546
$$561$$ −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
$$565$$ 14.6056 0.614460
$$566$$ −17.3944 −0.731143
$$567$$ 37.6888 1.58278
$$568$$ 6.00000 0.251754
$$569$$ 10.4222 0.436922 0.218461 0.975846i $$-0.429896\pi$$
0.218461 + 0.975846i $$0.429896\pi$$
$$570$$ −0.788897 −0.0330433
$$571$$ −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
$$575$$ −22.8167 −0.951520
$$576$$ −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
$$580$$ 9.00000 0.373705
$$581$$ 79.2666 3.28853
$$582$$ 3.76114 0.155904
$$583$$ −7.81665 −0.323733
$$584$$ −8.69722 −0.359894
$$585$$ 8.72498 0.360734
$$586$$ −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
$$590$$ 4.42221 0.182059
$$591$$ 1.81665 0.0747272
$$592$$ 0 0
$$593$$ 36.5139 1.49945 0.749723 0.661752i $$-0.230187\pi$$
0.749723 + 0.661752i $$0.230187\pi$$
$$594$$ 2.33053 0.0956229
$$595$$ −36.0000 −1.47586
$$596$$ −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 $$0.755557\pi$$
$$600$$ 1.00000 0.0408248
$$601$$ −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
$$605$$ 12.1194 0.492725
$$606$$ −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
$$610$$ −13.6972 −0.554584
$$611$$ −6.00000 −0.242734
$$612$$ −17.4500 −0.705373
$$613$$ −8.18335 −0.330522 −0.165261 0.986250i $$-0.552847\pi$$
−0.165261 + 0.986250i $$0.552847\pi$$
$$614$$ 7.09167 0.286197
$$615$$ −0.358288 −0.0144476
$$616$$ 6.00000 0.241747
$$617$$ 47.5694 1.91507 0.957536 0.288314i $$-0.0930948\pi$$
0.957536 + 0.288314i $$0.0930948\pi$$
$$618$$ 1.00000 0.0402259
$$619$$ −2.69722 −0.108411 −0.0542053 0.998530i $$-0.517263\pi$$
−0.0542053 + 0.998530i $$0.517263\pi$$
$$620$$ 4.30278 0.172804
$$621$$ 12.3583 0.495921
$$622$$ −5.09167 −0.204157
$$623$$ −24.0000 −0.961540
$$624$$ −0.697224 −0.0279113
$$625$$ 2.42221 0.0968882
$$626$$ −27.0278 −1.08025
$$627$$ 0.788897 0.0315055
$$628$$ 7.21110 0.287754
$$629$$ 0 0
$$630$$ 17.4500 0.695223
$$631$$ −18.3028 −0.728622 −0.364311 0.931277i $$-0.618695\pi$$
−0.364311 + 0.931277i $$0.618695\pi$$
$$632$$ 16.1194 0.641196
$$633$$ −2.02776 −0.0805961
$$634$$ −5.21110 −0.206959
$$635$$ 6.23886 0.247582
$$636$$ 1.81665 0.0720350
$$637$$ 32.7250 1.29661
$$638$$ −9.00000 −0.356313
$$639$$ −17.4500 −0.690310
$$640$$ −1.30278 −0.0514967
$$641$$ −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
$$645$$ 2.60555 0.102593
$$646$$ −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
$$650$$ −7.60555 −0.298314
$$651$$ 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
$$655$$ 4.42221 0.172790
$$656$$ −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 $$0.806233\pi$$
$$660$$ 0.513878 0.0200027
$$661$$ 12.4861 0.485654 0.242827 0.970070i $$-0.421925\pi$$
0.242827 + 0.970070i $$0.421925\pi$$
$$662$$ −1.21110 −0.0470708
$$663$$ −4.18335 −0.162468
$$664$$ 17.2111 0.667920
$$665$$ 12.0000 0.465340
$$666$$ 0 0
$$667$$ −47.7250 −1.84792
$$668$$ −12.5139 −0.484176
$$669$$ −4.78890 −0.185149
$$670$$ −18.9083 −0.730492
$$671$$ 13.6972 0.528775
$$672$$ −1.39445 −0.0537920
$$673$$ 24.3028 0.936803 0.468402 0.883516i $$-0.344830\pi$$
0.468402 + 0.883516i $$0.344830\pi$$
$$674$$ −19.1194 −0.736453
$$675$$ −5.90833 −0.227412
$$676$$ −7.69722 −0.296047
$$677$$ −36.2389 −1.39277 −0.696386 0.717667i $$-0.745210\pi$$
−0.696386 + 0.717667i $$0.745210\pi$$
$$678$$ 3.39445 0.130363
$$679$$ −57.2111 −2.19556
$$680$$ −7.81665 −0.299755
$$681$$ −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
$$685$$ 12.9083 0.493202
$$686$$ 33.2111 1.26801
$$687$$ −5.26662 −0.200934
$$688$$ 6.60555 0.251834
$$689$$ −13.8167 −0.526373
$$690$$ 2.72498 0.103738
$$691$$ 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
$$695$$ −11.6056 −0.440224
$$696$$ 2.09167 0.0792847
$$697$$ −5.44996 −0.206432
$$698$$ −22.2389 −0.841753
$$699$$ 2.88057 0.108953
$$700$$ −15.2111 −0.574926
$$701$$ −14.8806 −0.562031 −0.281016 0.959703i $$-0.590671\pi$$
−0.281016 + 0.959703i $$0.590671\pi$$
$$702$$ 4.11943 0.155478
$$703$$ 0 0
$$704$$ 1.30278 0.0491002
$$705$$ −1.02776 −0.0387075
$$706$$ −31.8167 −1.19744
$$707$$ 75.6333 2.84448
$$708$$ 1.02776 0.0386254
$$709$$ 1.66947 0.0626982 0.0313491 0.999508i $$-0.490020\pi$$
0.0313491 + 0.999508i $$0.490020\pi$$
$$710$$ −7.81665 −0.293354
$$711$$ −46.8806 −1.75816
$$712$$ −5.21110 −0.195294
$$713$$ −22.8167 −0.854490
$$714$$ −8.36669 −0.313116
$$715$$ −3.90833 −0.146163
$$716$$ −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 $$0.549864\pi$$
$$720$$ 3.78890 0.141204
$$721$$ −15.2111 −0.566491
$$722$$ −15.0000 −0.558242
$$723$$ 2.42221 0.0900828
$$724$$ 20.0000 0.743294
$$725$$ 22.8167 0.847389
$$726$$ 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
$$730$$ 11.3305 0.419362
$$731$$ 39.6333 1.46589
$$732$$ −3.18335 −0.117660
$$733$$ 29.6333 1.09453 0.547266 0.836959i $$-0.315669\pi$$
0.547266 + 0.836959i $$0.315669\pi$$
$$734$$ −17.8167 −0.657625
$$735$$ 5.60555 0.206764
$$736$$ 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 $$-0.274657\pi$$
0.650266 + 0.759706i $$0.274657\pi$$
$$744$$ 1.00000 0.0366618
$$745$$ 2.36669 0.0867089
$$746$$ 3.81665 0.139738
$$747$$ −50.0555 −1.83144
$$748$$ 7.81665 0.285805
$$749$$ 19.8167 0.724085
$$750$$ −3.27502 −0.119587
$$751$$ 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
$$755$$ 17.4500 0.635069
$$756$$ 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
$$760$$ 2.60555 0.0945133
$$761$$ 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
$$765$$ 22.7334 0.821927
$$766$$ −20.8444 −0.753139
$$767$$ −7.81665 −0.282243
$$768$$ −0.302776 −0.0109255
$$769$$ 22.0000 0.793340 0.396670 0.917961i $$-0.370166\pi$$
0.396670 + 0.917961i $$0.370166\pi$$
$$770$$ −7.81665 −0.281693
$$771$$ −3.39445 −0.122248
$$772$$ 4.00000 0.143963
$$773$$ −50.0555 −1.80037 −0.900186 0.435506i $$-0.856569\pi$$
−0.900186 + 0.435506i $$0.856569\pi$$
$$774$$ −19.2111 −0.690529
$$775$$ 10.9083 0.391839
$$776$$ −12.4222 −0.445931
$$777$$ 0 0
$$778$$ 11.8806 0.425939
$$779$$ 1.81665 0.0650884
$$780$$ 0.908327 0.0325233
$$781$$ 7.81665 0.279702
$$782$$ 41.4500 1.48225
$$783$$ −12.3583 −0.441649
$$784$$ 14.2111 0.507539
$$785$$ −9.39445 −0.335302
$$786$$ 1.02776 0.0366589
$$787$$ 25.2111 0.898679 0.449339 0.893361i $$-0.351659\pi$$
0.449339 + 0.893361i $$0.351659\pi$$
$$788$$ −6.00000 −0.213741
$$789$$ 2.36669 0.0842565
$$790$$ −21.0000 −0.747146
$$791$$ −51.6333 −1.83587
$$792$$ −3.78890 −0.134633
$$793$$ 24.2111 0.859761
$$794$$ 27.8167 0.987176
$$795$$ −2.36669 −0.0839379
$$796$$ 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
$$800$$ −3.30278 −0.116771
$$801$$ 15.1556 0.535496
$$802$$ −13.8167 −0.487883
$$803$$ −11.3305 −0.399846
$$804$$ −4.39445 −0.154980
$$805$$ −41.4500 −1.46092
$$806$$ −7.60555 −0.267894
$$807$$ 2.05551 0.0723575
$$808$$ 16.4222 0.577731
$$809$$ −29.4500 −1.03541 −0.517703 0.855561i $$-0.673213\pi$$
−0.517703 + 0.855561i $$0.673213\pi$$
$$810$$ −10.6611 −0.374592
$$811$$ 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
$$814$$ 0 0
$$815$$ −26.6056 −0.931952
$$816$$ −1.81665 −0.0635956
$$817$$ −13.2111 −0.462198
$$818$$ 5.02776 0.175791
$$819$$ −30.8444 −1.07779
$$820$$ 1.18335 0.0413242
$$821$$ 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 $$-0.571867\pi$$
−0.223864 + 0.974620i $$0.571867\pi$$
$$824$$ −3.30278 −0.115058
$$825$$ 1.30278 0.0453568
$$826$$ −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 $$-0.526148\pi$$
−0.0820527 + 0.996628i $$0.526148\pi$$
$$830$$ −22.4222 −0.778286
$$831$$ −7.60555 −0.263834
$$832$$ 2.30278 0.0798344
$$833$$ 85.2666 2.95431
$$834$$ −2.69722 −0.0933972
$$835$$ 16.3028 0.564181
$$836$$ −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 $$0.821183\pi$$
$$840$$ 1.81665 0.0626805
$$841$$ 18.7250 0.645689
$$842$$ −28.7250 −0.989928
$$843$$ −3.63331 −0.125138
$$844$$ 6.69722 0.230528
$$845$$ 10.0278 0.344965
$$846$$ 7.57779 0.260530
$$847$$ −42.8444 −1.47215
$$848$$ −6.00000 −0.206041
$$849$$ 5.26662 0.180750
$$850$$ −19.8167 −0.679706
$$851$$ 0 0
$$852$$ −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
$$855$$ −7.57779 −0.259155
$$856$$ 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 $$-0.365395\pi$$
0.410382 + 0.911914i $$0.365395\pi$$
$$860$$ −8.60555 −0.293447
$$861$$ 1.26662 0.0431661
$$862$$ 5.21110 0.177491
$$863$$ 12.0000 0.408485 0.204242 0.978920i $$-0.434527\pi$$
0.204242 + 0.978920i $$0.434527\pi$$
$$864$$ 1.78890 0.0608595
$$865$$ 30.2389 1.02815
$$866$$ −11.9361 −0.405605
$$867$$ −5.75274 −0.195373
$$868$$ −15.2111 −0.516298
$$869$$ 21.0000 0.712376
$$870$$ −2.72498 −0.0923855
$$871$$ 33.4222 1.13247
$$872$$ −2.00000 −0.0677285
$$873$$ 36.1278 1.22274
$$874$$ −13.8167 −0.467355
$$875$$ 49.8167 1.68411
$$876$$ 2.63331 0.0889712
$$877$$ 7.21110 0.243502 0.121751 0.992561i $$-0.461149\pi$$
0.121751 + 0.992561i $$0.461149\pi$$
$$878$$ 9.33053 0.314890
$$879$$ 7.57779 0.255593
$$880$$ −1.69722 −0.0572134
$$881$$ 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
$$885$$ −1.33894 −0.0450078
$$886$$ 0.275019 0.00923945
$$887$$ 28.4222 0.954324 0.477162 0.878815i $$-0.341665\pi$$
0.477162 + 0.878815i $$0.341665\pi$$
$$888$$ 0 0
$$889$$ −22.0555 −0.739718
$$890$$ 6.78890 0.227564
$$891$$ 10.6611 0.357159
$$892$$ 15.8167 0.529581
$$893$$ 5.21110 0.174383
$$894$$ 0.550039 0.0183960
$$895$$ 10.1833 0.340392
$$896$$ 4.60555 0.153861
$$897$$ −4.81665 −0.160823
$$898$$ 0.788897 0.0263258
$$899$$ 22.8167 0.760978
$$900$$ 9.60555 0.320185
$$901$$ −36.0000 −1.19933
$$902$$ −1.18335 −0.0394011
$$903$$ −9.21110 −0.306526
$$904$$ −11.2111 −0.372876
$$905$$ −26.0555 −0.866115
$$906$$ 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
$$910$$ −13.8167 −0.458018
$$911$$ 46.4222 1.53804 0.769018 0.639227i $$-0.220746\pi$$
0.769018 + 0.639227i $$0.220746\pi$$
$$912$$ 0.605551 0.0200518
$$913$$ 22.4222 0.742067
$$914$$ −4.60555 −0.152338
$$915$$ 4.14719 0.137102
$$916$$ 17.3944 0.574729
$$917$$ −15.6333 −0.516257
$$918$$ 10.7334 0.354254
$$919$$ 38.4222 1.26743 0.633716 0.773566i $$-0.281529\pi$$
0.633716 + 0.773566i $$0.281529\pi$$
$$920$$ −9.00000 −0.296721
$$921$$ −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 $$0.704431\pi$$
$$930$$ −1.30278 −0.0427197
$$931$$ −28.4222 −0.931500
$$932$$ −9.51388 −0.311637
$$933$$ 1.54163 0.0504708
$$934$$ 0 0
$$935$$ −10.1833 −0.333031
$$936$$ −6.69722 −0.218906
$$937$$ −28.9083 −0.944394 −0.472197 0.881493i $$-0.656539\pi$$
−0.472197 + 0.881493i $$0.656539\pi$$
$$938$$ 66.8444 2.18255
$$939$$ 8.18335 0.267053
$$940$$ 3.39445 0.110715
$$941$$ −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
$$945$$ −10.7334 −0.349157
$$946$$ 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
$$950$$ 6.60555 0.214312
$$951$$ 1.57779 0.0511635
$$952$$ 27.6333 0.895601
$$953$$ −18.7527 −0.607461 −0.303730 0.952758i $$-0.598232\pi$$
−0.303730 + 0.952758i $$0.598232\pi$$
$$954$$ 17.4500 0.564963
$$955$$ 16.3028 0.527545
$$956$$ −0.513878 −0.0166200
$$957$$ 2.72498 0.0880861
$$958$$ −12.1194 −0.391561
$$959$$ −45.6333 −1.47358
$$960$$ 0.394449 0.0127308
$$961$$ −20.0917 −0.648118
$$962$$ 0 0
$$963$$ −12.5139 −0.403254
$$964$$ −8.00000 −0.257663
$$965$$ −5.21110 −0.167751
$$966$$ −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
$$970$$ 16.1833 0.519616
$$971$$ 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
$$975$$ 2.30278 0.0737478
$$976$$ 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
$$980$$ −18.5139 −0.591404
$$981$$ 5.81665 0.185711
$$982$$ −14.7250 −0.469893
$$983$$ −12.0000 −0.382741 −0.191370 0.981518i $$-0.561293\pi$$
−0.191370 + 0.981518i $$0.561293\pi$$
$$984$$ 0.275019 0.00876729
$$985$$ 7.81665 0.249059
$$986$$ −41.4500 −1.32004
$$987$$ 3.63331 0.115649
$$988$$ −4.60555 −0.146522
$$989$$ 45.6333 1.45105
$$990$$ 4.93608 0.156879
$$991$$ −54.3028 −1.72498 −0.862492 0.506070i $$-0.831098\pi$$
−0.862492 + 0.506070i $$0.831098\pi$$
$$992$$ −3.30278 −0.104863
$$993$$ 0.366692 0.0116366
$$994$$ 27.6333 0.876475
$$995$$ −3.15559 −0.100039
$$996$$ −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
$$999$$ 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 2738.2.a.l.1.1 2
37.36 even 2 74.2.a.a.1.1 2
111.110 odd 2 666.2.a.j.1.1 2
148.147 odd 2 592.2.a.f.1.2 2
185.73 odd 4 1850.2.b.i.149.3 4
185.147 odd 4 1850.2.b.i.149.2 4
185.184 even 2 1850.2.a.u.1.2 2
259.258 odd 2 3626.2.a.a.1.2 2
296.147 odd 2 2368.2.a.ba.1.1 2
296.221 even 2 2368.2.a.s.1.2 2
407.406 odd 2 8954.2.a.p.1.1 2
444.443 even 2 5328.2.a.bf.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.a.a.1.1 2 37.36 even 2
592.2.a.f.1.2 2 148.147 odd 2
666.2.a.j.1.1 2 111.110 odd 2
1850.2.a.u.1.2 2 185.184 even 2
1850.2.b.i.149.2 4 185.147 odd 4
1850.2.b.i.149.3 4 185.73 odd 4
2368.2.a.s.1.2 2 296.221 even 2
2368.2.a.ba.1.1 2 296.147 odd 2
2738.2.a.l.1.1 2 1.1 even 1 trivial
3626.2.a.a.1.2 2 259.258 odd 2
5328.2.a.bf.1.1 2 444.443 even 2
8954.2.a.p.1.1 2 407.406 odd 2