# Properties

 Label 930.2.a.q.1.1 Level $930$ Weight $2$ Character 930.1 Self dual yes Analytic conductor $7.426$ 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] = [930,2,Mod(1,930)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(930, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("930.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$930 = 2 \cdot 3 \cdot 5 \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 930.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: $$7.42608738798$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{65})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 16$$ x^2 - x - 16 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$4.53113$$ of defining polynomial Character $$\chi$$ $$=$$ 930.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -4.53113 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -4.53113 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +6.53113 q^{11} +1.00000 q^{12} +6.00000 q^{13} -4.53113 q^{14} -1.00000 q^{15} +1.00000 q^{16} -4.00000 q^{17} +1.00000 q^{18} +4.53113 q^{19} -1.00000 q^{20} -4.53113 q^{21} +6.53113 q^{22} +6.53113 q^{23} +1.00000 q^{24} +1.00000 q^{25} +6.00000 q^{26} +1.00000 q^{27} -4.53113 q^{28} -1.00000 q^{30} +1.00000 q^{31} +1.00000 q^{32} +6.53113 q^{33} -4.00000 q^{34} +4.53113 q^{35} +1.00000 q^{36} -7.06226 q^{37} +4.53113 q^{38} +6.00000 q^{39} -1.00000 q^{40} +7.06226 q^{41} -4.53113 q^{42} -8.53113 q^{43} +6.53113 q^{44} -1.00000 q^{45} +6.53113 q^{46} -5.06226 q^{47} +1.00000 q^{48} +13.5311 q^{49} +1.00000 q^{50} -4.00000 q^{51} +6.00000 q^{52} -2.53113 q^{53} +1.00000 q^{54} -6.53113 q^{55} -4.53113 q^{56} +4.53113 q^{57} -9.06226 q^{59} -1.00000 q^{60} +5.06226 q^{61} +1.00000 q^{62} -4.53113 q^{63} +1.00000 q^{64} -6.00000 q^{65} +6.53113 q^{66} +5.06226 q^{67} -4.00000 q^{68} +6.53113 q^{69} +4.53113 q^{70} -12.5311 q^{71} +1.00000 q^{72} -8.53113 q^{73} -7.06226 q^{74} +1.00000 q^{75} +4.53113 q^{76} -29.5934 q^{77} +6.00000 q^{78} -8.53113 q^{79} -1.00000 q^{80} +1.00000 q^{81} +7.06226 q^{82} -8.00000 q^{83} -4.53113 q^{84} +4.00000 q^{85} -8.53113 q^{86} +6.53113 q^{88} -6.53113 q^{89} -1.00000 q^{90} -27.1868 q^{91} +6.53113 q^{92} +1.00000 q^{93} -5.06226 q^{94} -4.53113 q^{95} +1.00000 q^{96} +16.1245 q^{97} +13.5311 q^{98} +6.53113 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} - q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^5 + 2 * q^6 - q^7 + 2 * q^8 + 2 * q^9 $$2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} - q^{7} + 2 q^{8} + 2 q^{9} - 2 q^{10} + 5 q^{11} + 2 q^{12} + 12 q^{13} - q^{14} - 2 q^{15} + 2 q^{16} - 8 q^{17} + 2 q^{18} + q^{19} - 2 q^{20} - q^{21} + 5 q^{22} + 5 q^{23} + 2 q^{24} + 2 q^{25} + 12 q^{26} + 2 q^{27} - q^{28} - 2 q^{30} + 2 q^{31} + 2 q^{32} + 5 q^{33} - 8 q^{34} + q^{35} + 2 q^{36} + 2 q^{37} + q^{38} + 12 q^{39} - 2 q^{40} - 2 q^{41} - q^{42} - 9 q^{43} + 5 q^{44} - 2 q^{45} + 5 q^{46} + 6 q^{47} + 2 q^{48} + 19 q^{49} + 2 q^{50} - 8 q^{51} + 12 q^{52} + 3 q^{53} + 2 q^{54} - 5 q^{55} - q^{56} + q^{57} - 2 q^{59} - 2 q^{60} - 6 q^{61} + 2 q^{62} - q^{63} + 2 q^{64} - 12 q^{65} + 5 q^{66} - 6 q^{67} - 8 q^{68} + 5 q^{69} + q^{70} - 17 q^{71} + 2 q^{72} - 9 q^{73} + 2 q^{74} + 2 q^{75} + q^{76} - 35 q^{77} + 12 q^{78} - 9 q^{79} - 2 q^{80} + 2 q^{81} - 2 q^{82} - 16 q^{83} - q^{84} + 8 q^{85} - 9 q^{86} + 5 q^{88} - 5 q^{89} - 2 q^{90} - 6 q^{91} + 5 q^{92} + 2 q^{93} + 6 q^{94} - q^{95} + 2 q^{96} + 19 q^{98} + 5 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^5 + 2 * q^6 - q^7 + 2 * q^8 + 2 * q^9 - 2 * q^10 + 5 * q^11 + 2 * q^12 + 12 * q^13 - q^14 - 2 * q^15 + 2 * q^16 - 8 * q^17 + 2 * q^18 + q^19 - 2 * q^20 - q^21 + 5 * q^22 + 5 * q^23 + 2 * q^24 + 2 * q^25 + 12 * q^26 + 2 * q^27 - q^28 - 2 * q^30 + 2 * q^31 + 2 * q^32 + 5 * q^33 - 8 * q^34 + q^35 + 2 * q^36 + 2 * q^37 + q^38 + 12 * q^39 - 2 * q^40 - 2 * q^41 - q^42 - 9 * q^43 + 5 * q^44 - 2 * q^45 + 5 * q^46 + 6 * q^47 + 2 * q^48 + 19 * q^49 + 2 * q^50 - 8 * q^51 + 12 * q^52 + 3 * q^53 + 2 * q^54 - 5 * q^55 - q^56 + q^57 - 2 * q^59 - 2 * q^60 - 6 * q^61 + 2 * q^62 - q^63 + 2 * q^64 - 12 * q^65 + 5 * q^66 - 6 * q^67 - 8 * q^68 + 5 * q^69 + q^70 - 17 * q^71 + 2 * q^72 - 9 * q^73 + 2 * q^74 + 2 * q^75 + q^76 - 35 * q^77 + 12 * q^78 - 9 * q^79 - 2 * q^80 + 2 * q^81 - 2 * q^82 - 16 * q^83 - q^84 + 8 * q^85 - 9 * q^86 + 5 * q^88 - 5 * q^89 - 2 * q^90 - 6 * q^91 + 5 * q^92 + 2 * q^93 + 6 * q^94 - q^95 + 2 * q^96 + 19 * q^98 + 5 * 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
$$3$$ 1.00000 0.577350
$$4$$ 1.00000 0.500000
$$5$$ −1.00000 −0.447214
$$6$$ 1.00000 0.408248
$$7$$ −4.53113 −1.71261 −0.856303 0.516474i $$-0.827244\pi$$
−0.856303 + 0.516474i $$0.827244\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 1.00000 0.333333
$$10$$ −1.00000 −0.316228
$$11$$ 6.53113 1.96921 0.984605 0.174796i $$-0.0559265\pi$$
0.984605 + 0.174796i $$0.0559265\pi$$
$$12$$ 1.00000 0.288675
$$13$$ 6.00000 1.66410 0.832050 0.554700i $$-0.187167\pi$$
0.832050 + 0.554700i $$0.187167\pi$$
$$14$$ −4.53113 −1.21100
$$15$$ −1.00000 −0.258199
$$16$$ 1.00000 0.250000
$$17$$ −4.00000 −0.970143 −0.485071 0.874475i $$-0.661206\pi$$
−0.485071 + 0.874475i $$0.661206\pi$$
$$18$$ 1.00000 0.235702
$$19$$ 4.53113 1.03951 0.519756 0.854315i $$-0.326023\pi$$
0.519756 + 0.854315i $$0.326023\pi$$
$$20$$ −1.00000 −0.223607
$$21$$ −4.53113 −0.988773
$$22$$ 6.53113 1.39244
$$23$$ 6.53113 1.36183 0.680917 0.732360i $$-0.261581\pi$$
0.680917 + 0.732360i $$0.261581\pi$$
$$24$$ 1.00000 0.204124
$$25$$ 1.00000 0.200000
$$26$$ 6.00000 1.17670
$$27$$ 1.00000 0.192450
$$28$$ −4.53113 −0.856303
$$29$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$30$$ −1.00000 −0.182574
$$31$$ 1.00000 0.179605
$$32$$ 1.00000 0.176777
$$33$$ 6.53113 1.13692
$$34$$ −4.00000 −0.685994
$$35$$ 4.53113 0.765901
$$36$$ 1.00000 0.166667
$$37$$ −7.06226 −1.16103 −0.580514 0.814250i $$-0.697148\pi$$
−0.580514 + 0.814250i $$0.697148\pi$$
$$38$$ 4.53113 0.735046
$$39$$ 6.00000 0.960769
$$40$$ −1.00000 −0.158114
$$41$$ 7.06226 1.10294 0.551470 0.834195i $$-0.314067\pi$$
0.551470 + 0.834195i $$0.314067\pi$$
$$42$$ −4.53113 −0.699168
$$43$$ −8.53113 −1.30098 −0.650492 0.759513i $$-0.725437\pi$$
−0.650492 + 0.759513i $$0.725437\pi$$
$$44$$ 6.53113 0.984605
$$45$$ −1.00000 −0.149071
$$46$$ 6.53113 0.962962
$$47$$ −5.06226 −0.738406 −0.369203 0.929349i $$-0.620369\pi$$
−0.369203 + 0.929349i $$0.620369\pi$$
$$48$$ 1.00000 0.144338
$$49$$ 13.5311 1.93302
$$50$$ 1.00000 0.141421
$$51$$ −4.00000 −0.560112
$$52$$ 6.00000 0.832050
$$53$$ −2.53113 −0.347677 −0.173839 0.984774i $$-0.555617\pi$$
−0.173839 + 0.984774i $$0.555617\pi$$
$$54$$ 1.00000 0.136083
$$55$$ −6.53113 −0.880657
$$56$$ −4.53113 −0.605498
$$57$$ 4.53113 0.600163
$$58$$ 0 0
$$59$$ −9.06226 −1.17981 −0.589903 0.807474i $$-0.700834\pi$$
−0.589903 + 0.807474i $$0.700834\pi$$
$$60$$ −1.00000 −0.129099
$$61$$ 5.06226 0.648156 0.324078 0.946030i $$-0.394946\pi$$
0.324078 + 0.946030i $$0.394946\pi$$
$$62$$ 1.00000 0.127000
$$63$$ −4.53113 −0.570869
$$64$$ 1.00000 0.125000
$$65$$ −6.00000 −0.744208
$$66$$ 6.53113 0.803926
$$67$$ 5.06226 0.618453 0.309227 0.950988i $$-0.399930\pi$$
0.309227 + 0.950988i $$0.399930\pi$$
$$68$$ −4.00000 −0.485071
$$69$$ 6.53113 0.786256
$$70$$ 4.53113 0.541573
$$71$$ −12.5311 −1.48717 −0.743586 0.668641i $$-0.766876\pi$$
−0.743586 + 0.668641i $$0.766876\pi$$
$$72$$ 1.00000 0.117851
$$73$$ −8.53113 −0.998493 −0.499247 0.866460i $$-0.666390\pi$$
−0.499247 + 0.866460i $$0.666390\pi$$
$$74$$ −7.06226 −0.820971
$$75$$ 1.00000 0.115470
$$76$$ 4.53113 0.519756
$$77$$ −29.5934 −3.37248
$$78$$ 6.00000 0.679366
$$79$$ −8.53113 −0.959827 −0.479913 0.877316i $$-0.659332\pi$$
−0.479913 + 0.877316i $$0.659332\pi$$
$$80$$ −1.00000 −0.111803
$$81$$ 1.00000 0.111111
$$82$$ 7.06226 0.779896
$$83$$ −8.00000 −0.878114 −0.439057 0.898459i $$-0.644687\pi$$
−0.439057 + 0.898459i $$0.644687\pi$$
$$84$$ −4.53113 −0.494387
$$85$$ 4.00000 0.433861
$$86$$ −8.53113 −0.919935
$$87$$ 0 0
$$88$$ 6.53113 0.696221
$$89$$ −6.53113 −0.692298 −0.346149 0.938180i $$-0.612511\pi$$
−0.346149 + 0.938180i $$0.612511\pi$$
$$90$$ −1.00000 −0.105409
$$91$$ −27.1868 −2.84995
$$92$$ 6.53113 0.680917
$$93$$ 1.00000 0.103695
$$94$$ −5.06226 −0.522132
$$95$$ −4.53113 −0.464884
$$96$$ 1.00000 0.102062
$$97$$ 16.1245 1.63720 0.818598 0.574367i $$-0.194752\pi$$
0.818598 + 0.574367i $$0.194752\pi$$
$$98$$ 13.5311 1.36685
$$99$$ 6.53113 0.656403
$$100$$ 1.00000 0.100000
$$101$$ −9.46887 −0.942188 −0.471094 0.882083i $$-0.656141\pi$$
−0.471094 + 0.882083i $$0.656141\pi$$
$$102$$ −4.00000 −0.396059
$$103$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$104$$ 6.00000 0.588348
$$105$$ 4.53113 0.442193
$$106$$ −2.53113 −0.245845
$$107$$ 9.59339 0.927428 0.463714 0.885985i $$-0.346517\pi$$
0.463714 + 0.885985i $$0.346517\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ 15.0623 1.44270 0.721351 0.692569i $$-0.243521\pi$$
0.721351 + 0.692569i $$0.243521\pi$$
$$110$$ −6.53113 −0.622719
$$111$$ −7.06226 −0.670320
$$112$$ −4.53113 −0.428151
$$113$$ −7.59339 −0.714326 −0.357163 0.934042i $$-0.616256\pi$$
−0.357163 + 0.934042i $$0.616256\pi$$
$$114$$ 4.53113 0.424379
$$115$$ −6.53113 −0.609031
$$116$$ 0 0
$$117$$ 6.00000 0.554700
$$118$$ −9.06226 −0.834248
$$119$$ 18.1245 1.66147
$$120$$ −1.00000 −0.0912871
$$121$$ 31.6556 2.87779
$$122$$ 5.06226 0.458315
$$123$$ 7.06226 0.636782
$$124$$ 1.00000 0.0898027
$$125$$ −1.00000 −0.0894427
$$126$$ −4.53113 −0.403665
$$127$$ 7.06226 0.626674 0.313337 0.949642i $$-0.398553\pi$$
0.313337 + 0.949642i $$0.398553\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ −8.53113 −0.751124
$$130$$ −6.00000 −0.526235
$$131$$ −9.06226 −0.791773 −0.395887 0.918299i $$-0.629563\pi$$
−0.395887 + 0.918299i $$0.629563\pi$$
$$132$$ 6.53113 0.568462
$$133$$ −20.5311 −1.78027
$$134$$ 5.06226 0.437312
$$135$$ −1.00000 −0.0860663
$$136$$ −4.00000 −0.342997
$$137$$ 17.0623 1.45773 0.728864 0.684659i $$-0.240049\pi$$
0.728864 + 0.684659i $$0.240049\pi$$
$$138$$ 6.53113 0.555967
$$139$$ 6.00000 0.508913 0.254457 0.967084i $$-0.418103\pi$$
0.254457 + 0.967084i $$0.418103\pi$$
$$140$$ 4.53113 0.382950
$$141$$ −5.06226 −0.426319
$$142$$ −12.5311 −1.05159
$$143$$ 39.1868 3.27696
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ −8.53113 −0.706041
$$147$$ 13.5311 1.11603
$$148$$ −7.06226 −0.580514
$$149$$ −18.5311 −1.51813 −0.759065 0.651015i $$-0.774343\pi$$
−0.759065 + 0.651015i $$0.774343\pi$$
$$150$$ 1.00000 0.0816497
$$151$$ 14.1245 1.14944 0.574718 0.818351i $$-0.305112\pi$$
0.574718 + 0.818351i $$0.305112\pi$$
$$152$$ 4.53113 0.367523
$$153$$ −4.00000 −0.323381
$$154$$ −29.5934 −2.38470
$$155$$ −1.00000 −0.0803219
$$156$$ 6.00000 0.480384
$$157$$ −22.5311 −1.79818 −0.899090 0.437764i $$-0.855771\pi$$
−0.899090 + 0.437764i $$0.855771\pi$$
$$158$$ −8.53113 −0.678700
$$159$$ −2.53113 −0.200732
$$160$$ −1.00000 −0.0790569
$$161$$ −29.5934 −2.33229
$$162$$ 1.00000 0.0785674
$$163$$ 5.06226 0.396507 0.198253 0.980151i $$-0.436473\pi$$
0.198253 + 0.980151i $$0.436473\pi$$
$$164$$ 7.06226 0.551470
$$165$$ −6.53113 −0.508448
$$166$$ −8.00000 −0.620920
$$167$$ −23.5934 −1.82571 −0.912856 0.408283i $$-0.866128\pi$$
−0.912856 + 0.408283i $$0.866128\pi$$
$$168$$ −4.53113 −0.349584
$$169$$ 23.0000 1.76923
$$170$$ 4.00000 0.306786
$$171$$ 4.53113 0.346504
$$172$$ −8.53113 −0.650492
$$173$$ 14.0000 1.06440 0.532200 0.846619i $$-0.321365\pi$$
0.532200 + 0.846619i $$0.321365\pi$$
$$174$$ 0 0
$$175$$ −4.53113 −0.342521
$$176$$ 6.53113 0.492302
$$177$$ −9.06226 −0.681161
$$178$$ −6.53113 −0.489529
$$179$$ −3.06226 −0.228884 −0.114442 0.993430i $$-0.536508\pi$$
−0.114442 + 0.993430i $$0.536508\pi$$
$$180$$ −1.00000 −0.0745356
$$181$$ 2.40661 0.178882 0.0894411 0.995992i $$-0.471492\pi$$
0.0894411 + 0.995992i $$0.471492\pi$$
$$182$$ −27.1868 −2.01522
$$183$$ 5.06226 0.374213
$$184$$ 6.53113 0.481481
$$185$$ 7.06226 0.519228
$$186$$ 1.00000 0.0733236
$$187$$ −26.1245 −1.91041
$$188$$ −5.06226 −0.369203
$$189$$ −4.53113 −0.329591
$$190$$ −4.53113 −0.328723
$$191$$ −8.00000 −0.578860 −0.289430 0.957199i $$-0.593466\pi$$
−0.289430 + 0.957199i $$0.593466\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ 14.0000 1.00774 0.503871 0.863779i $$-0.331909\pi$$
0.503871 + 0.863779i $$0.331909\pi$$
$$194$$ 16.1245 1.15767
$$195$$ −6.00000 −0.429669
$$196$$ 13.5311 0.966509
$$197$$ 6.00000 0.427482 0.213741 0.976890i $$-0.431435\pi$$
0.213741 + 0.976890i $$0.431435\pi$$
$$198$$ 6.53113 0.464147
$$199$$ −8.53113 −0.604756 −0.302378 0.953188i $$-0.597780\pi$$
−0.302378 + 0.953188i $$0.597780\pi$$
$$200$$ 1.00000 0.0707107
$$201$$ 5.06226 0.357064
$$202$$ −9.46887 −0.666227
$$203$$ 0 0
$$204$$ −4.00000 −0.280056
$$205$$ −7.06226 −0.493249
$$206$$ 0 0
$$207$$ 6.53113 0.453945
$$208$$ 6.00000 0.416025
$$209$$ 29.5934 2.04702
$$210$$ 4.53113 0.312678
$$211$$ −1.59339 −0.109693 −0.0548466 0.998495i $$-0.517467\pi$$
−0.0548466 + 0.998495i $$0.517467\pi$$
$$212$$ −2.53113 −0.173839
$$213$$ −12.5311 −0.858619
$$214$$ 9.59339 0.655790
$$215$$ 8.53113 0.581818
$$216$$ 1.00000 0.0680414
$$217$$ −4.53113 −0.307593
$$218$$ 15.0623 1.02014
$$219$$ −8.53113 −0.576480
$$220$$ −6.53113 −0.440329
$$221$$ −24.0000 −1.61441
$$222$$ −7.06226 −0.473988
$$223$$ 2.00000 0.133930 0.0669650 0.997755i $$-0.478668\pi$$
0.0669650 + 0.997755i $$0.478668\pi$$
$$224$$ −4.53113 −0.302749
$$225$$ 1.00000 0.0666667
$$226$$ −7.59339 −0.505105
$$227$$ −24.5311 −1.62819 −0.814094 0.580733i $$-0.802766\pi$$
−0.814094 + 0.580733i $$0.802766\pi$$
$$228$$ 4.53113 0.300081
$$229$$ −5.59339 −0.369621 −0.184811 0.982774i $$-0.559167\pi$$
−0.184811 + 0.982774i $$0.559167\pi$$
$$230$$ −6.53113 −0.430650
$$231$$ −29.5934 −1.94710
$$232$$ 0 0
$$233$$ 1.46887 0.0962289 0.0481145 0.998842i $$-0.484679\pi$$
0.0481145 + 0.998842i $$0.484679\pi$$
$$234$$ 6.00000 0.392232
$$235$$ 5.06226 0.330225
$$236$$ −9.06226 −0.589903
$$237$$ −8.53113 −0.554156
$$238$$ 18.1245 1.17484
$$239$$ −8.00000 −0.517477 −0.258738 0.965947i $$-0.583307\pi$$
−0.258738 + 0.965947i $$0.583307\pi$$
$$240$$ −1.00000 −0.0645497
$$241$$ 2.00000 0.128831 0.0644157 0.997923i $$-0.479482\pi$$
0.0644157 + 0.997923i $$0.479482\pi$$
$$242$$ 31.6556 2.03490
$$243$$ 1.00000 0.0641500
$$244$$ 5.06226 0.324078
$$245$$ −13.5311 −0.864472
$$246$$ 7.06226 0.450273
$$247$$ 27.1868 1.72985
$$248$$ 1.00000 0.0635001
$$249$$ −8.00000 −0.506979
$$250$$ −1.00000 −0.0632456
$$251$$ 3.06226 0.193288 0.0966440 0.995319i $$-0.469189\pi$$
0.0966440 + 0.995319i $$0.469189\pi$$
$$252$$ −4.53113 −0.285434
$$253$$ 42.6556 2.68174
$$254$$ 7.06226 0.443125
$$255$$ 4.00000 0.250490
$$256$$ 1.00000 0.0625000
$$257$$ 12.6556 0.789437 0.394719 0.918802i $$-0.370842\pi$$
0.394719 + 0.918802i $$0.370842\pi$$
$$258$$ −8.53113 −0.531125
$$259$$ 32.0000 1.98838
$$260$$ −6.00000 −0.372104
$$261$$ 0 0
$$262$$ −9.06226 −0.559868
$$263$$ −23.0623 −1.42208 −0.711040 0.703152i $$-0.751775\pi$$
−0.711040 + 0.703152i $$0.751775\pi$$
$$264$$ 6.53113 0.401963
$$265$$ 2.53113 0.155486
$$266$$ −20.5311 −1.25884
$$267$$ −6.53113 −0.399699
$$268$$ 5.06226 0.309227
$$269$$ −19.1868 −1.16984 −0.584919 0.811092i $$-0.698874\pi$$
−0.584919 + 0.811092i $$0.698874\pi$$
$$270$$ −1.00000 −0.0608581
$$271$$ −11.4689 −0.696684 −0.348342 0.937367i $$-0.613255\pi$$
−0.348342 + 0.937367i $$0.613255\pi$$
$$272$$ −4.00000 −0.242536
$$273$$ −27.1868 −1.64542
$$274$$ 17.0623 1.03077
$$275$$ 6.53113 0.393842
$$276$$ 6.53113 0.393128
$$277$$ −15.0623 −0.905003 −0.452502 0.891764i $$-0.649468\pi$$
−0.452502 + 0.891764i $$0.649468\pi$$
$$278$$ 6.00000 0.359856
$$279$$ 1.00000 0.0598684
$$280$$ 4.53113 0.270787
$$281$$ 15.0623 0.898539 0.449269 0.893396i $$-0.351684\pi$$
0.449269 + 0.893396i $$0.351684\pi$$
$$282$$ −5.06226 −0.301453
$$283$$ −12.0000 −0.713326 −0.356663 0.934233i $$-0.616086\pi$$
−0.356663 + 0.934233i $$0.616086\pi$$
$$284$$ −12.5311 −0.743586
$$285$$ −4.53113 −0.268401
$$286$$ 39.1868 2.31716
$$287$$ −32.0000 −1.88890
$$288$$ 1.00000 0.0589256
$$289$$ −1.00000 −0.0588235
$$290$$ 0 0
$$291$$ 16.1245 0.945236
$$292$$ −8.53113 −0.499247
$$293$$ −14.0000 −0.817889 −0.408944 0.912559i $$-0.634103\pi$$
−0.408944 + 0.912559i $$0.634103\pi$$
$$294$$ 13.5311 0.789151
$$295$$ 9.06226 0.527625
$$296$$ −7.06226 −0.410485
$$297$$ 6.53113 0.378975
$$298$$ −18.5311 −1.07348
$$299$$ 39.1868 2.26623
$$300$$ 1.00000 0.0577350
$$301$$ 38.6556 2.22807
$$302$$ 14.1245 0.812775
$$303$$ −9.46887 −0.543972
$$304$$ 4.53113 0.259878
$$305$$ −5.06226 −0.289864
$$306$$ −4.00000 −0.228665
$$307$$ −30.1245 −1.71930 −0.859648 0.510886i $$-0.829317\pi$$
−0.859648 + 0.510886i $$0.829317\pi$$
$$308$$ −29.5934 −1.68624
$$309$$ 0 0
$$310$$ −1.00000 −0.0567962
$$311$$ −8.00000 −0.453638 −0.226819 0.973937i $$-0.572833\pi$$
−0.226819 + 0.973937i $$0.572833\pi$$
$$312$$ 6.00000 0.339683
$$313$$ 10.9377 0.618238 0.309119 0.951023i $$-0.399966\pi$$
0.309119 + 0.951023i $$0.399966\pi$$
$$314$$ −22.5311 −1.27151
$$315$$ 4.53113 0.255300
$$316$$ −8.53113 −0.479913
$$317$$ 12.9377 0.726656 0.363328 0.931661i $$-0.381640\pi$$
0.363328 + 0.931661i $$0.381640\pi$$
$$318$$ −2.53113 −0.141939
$$319$$ 0 0
$$320$$ −1.00000 −0.0559017
$$321$$ 9.59339 0.535451
$$322$$ −29.5934 −1.64917
$$323$$ −18.1245 −1.00848
$$324$$ 1.00000 0.0555556
$$325$$ 6.00000 0.332820
$$326$$ 5.06226 0.280373
$$327$$ 15.0623 0.832945
$$328$$ 7.06226 0.389948
$$329$$ 22.9377 1.26460
$$330$$ −6.53113 −0.359527
$$331$$ 12.9377 0.711123 0.355561 0.934653i $$-0.384290\pi$$
0.355561 + 0.934653i $$0.384290\pi$$
$$332$$ −8.00000 −0.439057
$$333$$ −7.06226 −0.387009
$$334$$ −23.5934 −1.29097
$$335$$ −5.06226 −0.276581
$$336$$ −4.53113 −0.247193
$$337$$ −19.1868 −1.04517 −0.522585 0.852587i $$-0.675032\pi$$
−0.522585 + 0.852587i $$0.675032\pi$$
$$338$$ 23.0000 1.25104
$$339$$ −7.59339 −0.412416
$$340$$ 4.00000 0.216930
$$341$$ 6.53113 0.353680
$$342$$ 4.53113 0.245015
$$343$$ −29.5934 −1.59789
$$344$$ −8.53113 −0.459968
$$345$$ −6.53113 −0.351624
$$346$$ 14.0000 0.752645
$$347$$ 10.1245 0.543512 0.271756 0.962366i $$-0.412396\pi$$
0.271756 + 0.962366i $$0.412396\pi$$
$$348$$ 0 0
$$349$$ 8.93774 0.478426 0.239213 0.970967i $$-0.423111\pi$$
0.239213 + 0.970967i $$0.423111\pi$$
$$350$$ −4.53113 −0.242199
$$351$$ 6.00000 0.320256
$$352$$ 6.53113 0.348110
$$353$$ −6.93774 −0.369259 −0.184629 0.982808i $$-0.559108\pi$$
−0.184629 + 0.982808i $$0.559108\pi$$
$$354$$ −9.06226 −0.481654
$$355$$ 12.5311 0.665083
$$356$$ −6.53113 −0.346149
$$357$$ 18.1245 0.959251
$$358$$ −3.06226 −0.161845
$$359$$ 3.46887 0.183080 0.0915400 0.995801i $$-0.470821\pi$$
0.0915400 + 0.995801i $$0.470821\pi$$
$$360$$ −1.00000 −0.0527046
$$361$$ 1.53113 0.0805857
$$362$$ 2.40661 0.126489
$$363$$ 31.6556 1.66149
$$364$$ −27.1868 −1.42497
$$365$$ 8.53113 0.446540
$$366$$ 5.06226 0.264608
$$367$$ −10.0000 −0.521996 −0.260998 0.965339i $$-0.584052\pi$$
−0.260998 + 0.965339i $$0.584052\pi$$
$$368$$ 6.53113 0.340459
$$369$$ 7.06226 0.367646
$$370$$ 7.06226 0.367149
$$371$$ 11.4689 0.595434
$$372$$ 1.00000 0.0518476
$$373$$ −18.5311 −0.959505 −0.479753 0.877404i $$-0.659274\pi$$
−0.479753 + 0.877404i $$0.659274\pi$$
$$374$$ −26.1245 −1.35087
$$375$$ −1.00000 −0.0516398
$$376$$ −5.06226 −0.261066
$$377$$ 0 0
$$378$$ −4.53113 −0.233056
$$379$$ 21.5934 1.10918 0.554589 0.832124i $$-0.312876\pi$$
0.554589 + 0.832124i $$0.312876\pi$$
$$380$$ −4.53113 −0.232442
$$381$$ 7.06226 0.361810
$$382$$ −8.00000 −0.409316
$$383$$ −3.06226 −0.156474 −0.0782370 0.996935i $$-0.524929\pi$$
−0.0782370 + 0.996935i $$0.524929\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 29.5934 1.50822
$$386$$ 14.0000 0.712581
$$387$$ −8.53113 −0.433662
$$388$$ 16.1245 0.818598
$$389$$ 10.9377 0.554566 0.277283 0.960788i $$-0.410566\pi$$
0.277283 + 0.960788i $$0.410566\pi$$
$$390$$ −6.00000 −0.303822
$$391$$ −26.1245 −1.32117
$$392$$ 13.5311 0.683425
$$393$$ −9.06226 −0.457130
$$394$$ 6.00000 0.302276
$$395$$ 8.53113 0.429248
$$396$$ 6.53113 0.328202
$$397$$ 37.7179 1.89301 0.946504 0.322693i $$-0.104588\pi$$
0.946504 + 0.322693i $$0.104588\pi$$
$$398$$ −8.53113 −0.427627
$$399$$ −20.5311 −1.02784
$$400$$ 1.00000 0.0500000
$$401$$ 21.7179 1.08454 0.542270 0.840204i $$-0.317565\pi$$
0.542270 + 0.840204i $$0.317565\pi$$
$$402$$ 5.06226 0.252482
$$403$$ 6.00000 0.298881
$$404$$ −9.46887 −0.471094
$$405$$ −1.00000 −0.0496904
$$406$$ 0 0
$$407$$ −46.1245 −2.28631
$$408$$ −4.00000 −0.198030
$$409$$ 7.06226 0.349206 0.174603 0.984639i $$-0.444136\pi$$
0.174603 + 0.984639i $$0.444136\pi$$
$$410$$ −7.06226 −0.348780
$$411$$ 17.0623 0.841619
$$412$$ 0 0
$$413$$ 41.0623 2.02054
$$414$$ 6.53113 0.320987
$$415$$ 8.00000 0.392705
$$416$$ 6.00000 0.294174
$$417$$ 6.00000 0.293821
$$418$$ 29.5934 1.44746
$$419$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$420$$ 4.53113 0.221096
$$421$$ −26.2490 −1.27930 −0.639650 0.768667i $$-0.720921\pi$$
−0.639650 + 0.768667i $$0.720921\pi$$
$$422$$ −1.59339 −0.0775648
$$423$$ −5.06226 −0.246135
$$424$$ −2.53113 −0.122922
$$425$$ −4.00000 −0.194029
$$426$$ −12.5311 −0.607135
$$427$$ −22.9377 −1.11004
$$428$$ 9.59339 0.463714
$$429$$ 39.1868 1.89196
$$430$$ 8.53113 0.411408
$$431$$ −24.0000 −1.15604 −0.578020 0.816023i $$-0.696174\pi$$
−0.578020 + 0.816023i $$0.696174\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ −25.5934 −1.22994 −0.614970 0.788551i $$-0.710832\pi$$
−0.614970 + 0.788551i $$0.710832\pi$$
$$434$$ −4.53113 −0.217501
$$435$$ 0 0
$$436$$ 15.0623 0.721351
$$437$$ 29.5934 1.41564
$$438$$ −8.53113 −0.407633
$$439$$ −25.0623 −1.19616 −0.598078 0.801438i $$-0.704069\pi$$
−0.598078 + 0.801438i $$0.704069\pi$$
$$440$$ −6.53113 −0.311359
$$441$$ 13.5311 0.644339
$$442$$ −24.0000 −1.14156
$$443$$ 14.4066 0.684479 0.342239 0.939613i $$-0.388815\pi$$
0.342239 + 0.939613i $$0.388815\pi$$
$$444$$ −7.06226 −0.335160
$$445$$ 6.53113 0.309605
$$446$$ 2.00000 0.0947027
$$447$$ −18.5311 −0.876492
$$448$$ −4.53113 −0.214076
$$449$$ −4.12452 −0.194648 −0.0973240 0.995253i $$-0.531028\pi$$
−0.0973240 + 0.995253i $$0.531028\pi$$
$$450$$ 1.00000 0.0471405
$$451$$ 46.1245 2.17192
$$452$$ −7.59339 −0.357163
$$453$$ 14.1245 0.663628
$$454$$ −24.5311 −1.15130
$$455$$ 27.1868 1.27454
$$456$$ 4.53113 0.212190
$$457$$ 14.9377 0.698758 0.349379 0.936981i $$-0.386393\pi$$
0.349379 + 0.936981i $$0.386393\pi$$
$$458$$ −5.59339 −0.261362
$$459$$ −4.00000 −0.186704
$$460$$ −6.53113 −0.304515
$$461$$ 24.0000 1.11779 0.558896 0.829238i $$-0.311225\pi$$
0.558896 + 0.829238i $$0.311225\pi$$
$$462$$ −29.5934 −1.37681
$$463$$ 13.1868 0.612841 0.306421 0.951896i $$-0.400869\pi$$
0.306421 + 0.951896i $$0.400869\pi$$
$$464$$ 0 0
$$465$$ −1.00000 −0.0463739
$$466$$ 1.46887 0.0680441
$$467$$ −22.1245 −1.02380 −0.511900 0.859045i $$-0.671058\pi$$
−0.511900 + 0.859045i $$0.671058\pi$$
$$468$$ 6.00000 0.277350
$$469$$ −22.9377 −1.05917
$$470$$ 5.06226 0.233505
$$471$$ −22.5311 −1.03818
$$472$$ −9.06226 −0.417124
$$473$$ −55.7179 −2.56191
$$474$$ −8.53113 −0.391848
$$475$$ 4.53113 0.207902
$$476$$ 18.1245 0.830736
$$477$$ −2.53113 −0.115892
$$478$$ −8.00000 −0.365911
$$479$$ 1.34436 0.0614252 0.0307126 0.999528i $$-0.490222\pi$$
0.0307126 + 0.999528i $$0.490222\pi$$
$$480$$ −1.00000 −0.0456435
$$481$$ −42.3735 −1.93207
$$482$$ 2.00000 0.0910975
$$483$$ −29.5934 −1.34655
$$484$$ 31.6556 1.43889
$$485$$ −16.1245 −0.732177
$$486$$ 1.00000 0.0453609
$$487$$ 23.0623 1.04505 0.522525 0.852624i $$-0.324990\pi$$
0.522525 + 0.852624i $$0.324990\pi$$
$$488$$ 5.06226 0.229158
$$489$$ 5.06226 0.228923
$$490$$ −13.5311 −0.611274
$$491$$ −7.59339 −0.342685 −0.171342 0.985212i $$-0.554810\pi$$
−0.171342 + 0.985212i $$0.554810\pi$$
$$492$$ 7.06226 0.318391
$$493$$ 0 0
$$494$$ 27.1868 1.22319
$$495$$ −6.53113 −0.293552
$$496$$ 1.00000 0.0449013
$$497$$ 56.7802 2.54694
$$498$$ −8.00000 −0.358489
$$499$$ −7.06226 −0.316150 −0.158075 0.987427i $$-0.550529\pi$$
−0.158075 + 0.987427i $$0.550529\pi$$
$$500$$ −1.00000 −0.0447214
$$501$$ −23.5934 −1.05407
$$502$$ 3.06226 0.136675
$$503$$ 16.0000 0.713405 0.356702 0.934218i $$-0.383901\pi$$
0.356702 + 0.934218i $$0.383901\pi$$
$$504$$ −4.53113 −0.201833
$$505$$ 9.46887 0.421359
$$506$$ 42.6556 1.89627
$$507$$ 23.0000 1.02147
$$508$$ 7.06226 0.313337
$$509$$ 38.1245 1.68984 0.844920 0.534893i $$-0.179648\pi$$
0.844920 + 0.534893i $$0.179648\pi$$
$$510$$ 4.00000 0.177123
$$511$$ 38.6556 1.71003
$$512$$ 1.00000 0.0441942
$$513$$ 4.53113 0.200054
$$514$$ 12.6556 0.558217
$$515$$ 0 0
$$516$$ −8.53113 −0.375562
$$517$$ −33.0623 −1.45408
$$518$$ 32.0000 1.40600
$$519$$ 14.0000 0.614532
$$520$$ −6.00000 −0.263117
$$521$$ −12.1245 −0.531185 −0.265592 0.964085i $$-0.585568\pi$$
−0.265592 + 0.964085i $$0.585568\pi$$
$$522$$ 0 0
$$523$$ 9.59339 0.419490 0.209745 0.977756i $$-0.432737\pi$$
0.209745 + 0.977756i $$0.432737\pi$$
$$524$$ −9.06226 −0.395887
$$525$$ −4.53113 −0.197755
$$526$$ −23.0623 −1.00556
$$527$$ −4.00000 −0.174243
$$528$$ 6.53113 0.284231
$$529$$ 19.6556 0.854593
$$530$$ 2.53113 0.109945
$$531$$ −9.06226 −0.393268
$$532$$ −20.5311 −0.890137
$$533$$ 42.3735 1.83540
$$534$$ −6.53113 −0.282630
$$535$$ −9.59339 −0.414758
$$536$$ 5.06226 0.218656
$$537$$ −3.06226 −0.132146
$$538$$ −19.1868 −0.827201
$$539$$ 88.3735 3.80652
$$540$$ −1.00000 −0.0430331
$$541$$ −4.12452 −0.177327 −0.0886634 0.996062i $$-0.528260\pi$$
−0.0886634 + 0.996062i $$0.528260\pi$$
$$542$$ −11.4689 −0.492630
$$543$$ 2.40661 0.103278
$$544$$ −4.00000 −0.171499
$$545$$ −15.0623 −0.645196
$$546$$ −27.1868 −1.16349
$$547$$ −7.18677 −0.307284 −0.153642 0.988127i $$-0.549100\pi$$
−0.153642 + 0.988127i $$0.549100\pi$$
$$548$$ 17.0623 0.728864
$$549$$ 5.06226 0.216052
$$550$$ 6.53113 0.278488
$$551$$ 0 0
$$552$$ 6.53113 0.277983
$$553$$ 38.6556 1.64381
$$554$$ −15.0623 −0.639934
$$555$$ 7.06226 0.299776
$$556$$ 6.00000 0.254457
$$557$$ −26.7802 −1.13471 −0.567356 0.823473i $$-0.692034\pi$$
−0.567356 + 0.823473i $$0.692034\pi$$
$$558$$ 1.00000 0.0423334
$$559$$ −51.1868 −2.16497
$$560$$ 4.53113 0.191475
$$561$$ −26.1245 −1.10298
$$562$$ 15.0623 0.635363
$$563$$ −4.00000 −0.168580 −0.0842900 0.996441i $$-0.526862\pi$$
−0.0842900 + 0.996441i $$0.526862\pi$$
$$564$$ −5.06226 −0.213160
$$565$$ 7.59339 0.319456
$$566$$ −12.0000 −0.504398
$$567$$ −4.53113 −0.190290
$$568$$ −12.5311 −0.525794
$$569$$ 9.46887 0.396956 0.198478 0.980105i $$-0.436400\pi$$
0.198478 + 0.980105i $$0.436400\pi$$
$$570$$ −4.53113 −0.189788
$$571$$ 36.1245 1.51176 0.755882 0.654708i $$-0.227208\pi$$
0.755882 + 0.654708i $$0.227208\pi$$
$$572$$ 39.1868 1.63848
$$573$$ −8.00000 −0.334205
$$574$$ −32.0000 −1.33565
$$575$$ 6.53113 0.272367
$$576$$ 1.00000 0.0416667
$$577$$ 37.1868 1.54811 0.774053 0.633121i $$-0.218226\pi$$
0.774053 + 0.633121i $$0.218226\pi$$
$$578$$ −1.00000 −0.0415945
$$579$$ 14.0000 0.581820
$$580$$ 0 0
$$581$$ 36.2490 1.50386
$$582$$ 16.1245 0.668383
$$583$$ −16.5311 −0.684649
$$584$$ −8.53113 −0.353021
$$585$$ −6.00000 −0.248069
$$586$$ −14.0000 −0.578335
$$587$$ −44.2490 −1.82635 −0.913176 0.407564i $$-0.866378\pi$$
−0.913176 + 0.407564i $$0.866378\pi$$
$$588$$ 13.5311 0.558014
$$589$$ 4.53113 0.186702
$$590$$ 9.06226 0.373087
$$591$$ 6.00000 0.246807
$$592$$ −7.06226 −0.290257
$$593$$ −14.0000 −0.574911 −0.287456 0.957794i $$-0.592809\pi$$
−0.287456 + 0.957794i $$0.592809\pi$$
$$594$$ 6.53113 0.267975
$$595$$ −18.1245 −0.743033
$$596$$ −18.5311 −0.759065
$$597$$ −8.53113 −0.349156
$$598$$ 39.1868 1.60247
$$599$$ −13.5934 −0.555411 −0.277705 0.960666i $$-0.589574\pi$$
−0.277705 + 0.960666i $$0.589574\pi$$
$$600$$ 1.00000 0.0408248
$$601$$ 30.0000 1.22373 0.611863 0.790964i $$-0.290420\pi$$
0.611863 + 0.790964i $$0.290420\pi$$
$$602$$ 38.6556 1.57549
$$603$$ 5.06226 0.206151
$$604$$ 14.1245 0.574718
$$605$$ −31.6556 −1.28698
$$606$$ −9.46887 −0.384647
$$607$$ −14.4066 −0.584746 −0.292373 0.956304i $$-0.594445\pi$$
−0.292373 + 0.956304i $$0.594445\pi$$
$$608$$ 4.53113 0.183762
$$609$$ 0 0
$$610$$ −5.06226 −0.204965
$$611$$ −30.3735 −1.22878
$$612$$ −4.00000 −0.161690
$$613$$ −30.0000 −1.21169 −0.605844 0.795583i $$-0.707165\pi$$
−0.605844 + 0.795583i $$0.707165\pi$$
$$614$$ −30.1245 −1.21573
$$615$$ −7.06226 −0.284778
$$616$$ −29.5934 −1.19235
$$617$$ −3.59339 −0.144664 −0.0723321 0.997381i $$-0.523044\pi$$
−0.0723321 + 0.997381i $$0.523044\pi$$
$$618$$ 0 0
$$619$$ −34.0000 −1.36658 −0.683288 0.730149i $$-0.739451\pi$$
−0.683288 + 0.730149i $$0.739451\pi$$
$$620$$ −1.00000 −0.0401610
$$621$$ 6.53113 0.262085
$$622$$ −8.00000 −0.320771
$$623$$ 29.5934 1.18563
$$624$$ 6.00000 0.240192
$$625$$ 1.00000 0.0400000
$$626$$ 10.9377 0.437160
$$627$$ 29.5934 1.18185
$$628$$ −22.5311 −0.899090
$$629$$ 28.2490 1.12636
$$630$$ 4.53113 0.180524
$$631$$ 30.6556 1.22038 0.610191 0.792254i $$-0.291093\pi$$
0.610191 + 0.792254i $$0.291093\pi$$
$$632$$ −8.53113 −0.339350
$$633$$ −1.59339 −0.0633314
$$634$$ 12.9377 0.513823
$$635$$ −7.06226 −0.280257
$$636$$ −2.53113 −0.100366
$$637$$ 81.1868 3.21674
$$638$$ 0 0
$$639$$ −12.5311 −0.495724
$$640$$ −1.00000 −0.0395285
$$641$$ −22.0000 −0.868948 −0.434474 0.900684i $$-0.643066\pi$$
−0.434474 + 0.900684i $$0.643066\pi$$
$$642$$ 9.59339 0.378621
$$643$$ −17.5934 −0.693815 −0.346908 0.937899i $$-0.612768\pi$$
−0.346908 + 0.937899i $$0.612768\pi$$
$$644$$ −29.5934 −1.16614
$$645$$ 8.53113 0.335913
$$646$$ −18.1245 −0.713100
$$647$$ −6.53113 −0.256765 −0.128383 0.991725i $$-0.540979\pi$$
−0.128383 + 0.991725i $$0.540979\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ −59.1868 −2.32328
$$650$$ 6.00000 0.235339
$$651$$ −4.53113 −0.177589
$$652$$ 5.06226 0.198253
$$653$$ 38.2490 1.49680 0.748400 0.663248i $$-0.230822\pi$$
0.748400 + 0.663248i $$0.230822\pi$$
$$654$$ 15.0623 0.588981
$$655$$ 9.06226 0.354092
$$656$$ 7.06226 0.275735
$$657$$ −8.53113 −0.332831
$$658$$ 22.9377 0.894206
$$659$$ 16.0000 0.623272 0.311636 0.950202i $$-0.399123\pi$$
0.311636 + 0.950202i $$0.399123\pi$$
$$660$$ −6.53113 −0.254224
$$661$$ −10.0000 −0.388955 −0.194477 0.980907i $$-0.562301\pi$$
−0.194477 + 0.980907i $$0.562301\pi$$
$$662$$ 12.9377 0.502840
$$663$$ −24.0000 −0.932083
$$664$$ −8.00000 −0.310460
$$665$$ 20.5311 0.796163
$$666$$ −7.06226 −0.273657
$$667$$ 0 0
$$668$$ −23.5934 −0.912856
$$669$$ 2.00000 0.0773245
$$670$$ −5.06226 −0.195572
$$671$$ 33.0623 1.27635
$$672$$ −4.53113 −0.174792
$$673$$ −9.06226 −0.349324 −0.174662 0.984628i $$-0.555883\pi$$
−0.174662 + 0.984628i $$0.555883\pi$$
$$674$$ −19.1868 −0.739047
$$675$$ 1.00000 0.0384900
$$676$$ 23.0000 0.884615
$$677$$ 27.5934 1.06050 0.530250 0.847841i $$-0.322098\pi$$
0.530250 + 0.847841i $$0.322098\pi$$
$$678$$ −7.59339 −0.291622
$$679$$ −73.0623 −2.80387
$$680$$ 4.00000 0.153393
$$681$$ −24.5311 −0.940035
$$682$$ 6.53113 0.250090
$$683$$ 7.46887 0.285788 0.142894 0.989738i $$-0.454359\pi$$
0.142894 + 0.989738i $$0.454359\pi$$
$$684$$ 4.53113 0.173252
$$685$$ −17.0623 −0.651915
$$686$$ −29.5934 −1.12988
$$687$$ −5.59339 −0.213401
$$688$$ −8.53113 −0.325246
$$689$$ −15.1868 −0.578570
$$690$$ −6.53113 −0.248636
$$691$$ 0.531129 0.0202051 0.0101025 0.999949i $$-0.496784\pi$$
0.0101025 + 0.999949i $$0.496784\pi$$
$$692$$ 14.0000 0.532200
$$693$$ −29.5934 −1.12416
$$694$$ 10.1245 0.384321
$$695$$ −6.00000 −0.227593
$$696$$ 0 0
$$697$$ −28.2490 −1.07001
$$698$$ 8.93774 0.338299
$$699$$ 1.46887 0.0555578
$$700$$ −4.53113 −0.171261
$$701$$ −13.4689 −0.508712 −0.254356 0.967111i $$-0.581864\pi$$
−0.254356 + 0.967111i $$0.581864\pi$$
$$702$$ 6.00000 0.226455
$$703$$ −32.0000 −1.20690
$$704$$ 6.53113 0.246151
$$705$$ 5.06226 0.190656
$$706$$ −6.93774 −0.261105
$$707$$ 42.9047 1.61360
$$708$$ −9.06226 −0.340581
$$709$$ 49.5934 1.86252 0.931259 0.364357i $$-0.118711\pi$$
0.931259 + 0.364357i $$0.118711\pi$$
$$710$$ 12.5311 0.470285
$$711$$ −8.53113 −0.319942
$$712$$ −6.53113 −0.244764
$$713$$ 6.53113 0.244593
$$714$$ 18.1245 0.678293
$$715$$ −39.1868 −1.46550
$$716$$ −3.06226 −0.114442
$$717$$ −8.00000 −0.298765
$$718$$ 3.46887 0.129457
$$719$$ 15.1868 0.566371 0.283186 0.959065i $$-0.408609\pi$$
0.283186 + 0.959065i $$0.408609\pi$$
$$720$$ −1.00000 −0.0372678
$$721$$ 0 0
$$722$$ 1.53113 0.0569827
$$723$$ 2.00000 0.0743808
$$724$$ 2.40661 0.0894411
$$725$$ 0 0
$$726$$ 31.6556 1.17485
$$727$$ 37.8424 1.40350 0.701749 0.712424i $$-0.252403\pi$$
0.701749 + 0.712424i $$0.252403\pi$$
$$728$$ −27.1868 −1.00761
$$729$$ 1.00000 0.0370370
$$730$$ 8.53113 0.315751
$$731$$ 34.1245 1.26214
$$732$$ 5.06226 0.187106
$$733$$ −28.1245 −1.03880 −0.519401 0.854530i $$-0.673845\pi$$
−0.519401 + 0.854530i $$0.673845\pi$$
$$734$$ −10.0000 −0.369107
$$735$$ −13.5311 −0.499103
$$736$$ 6.53113 0.240741
$$737$$ 33.0623 1.21786
$$738$$ 7.06226 0.259965
$$739$$ −21.1868 −0.779368 −0.389684 0.920949i $$-0.627416\pi$$
−0.389684 + 0.920949i $$0.627416\pi$$
$$740$$ 7.06226 0.259614
$$741$$ 27.1868 0.998731
$$742$$ 11.4689 0.421036
$$743$$ −28.6556 −1.05127 −0.525637 0.850709i $$-0.676173\pi$$
−0.525637 + 0.850709i $$0.676173\pi$$
$$744$$ 1.00000 0.0366618
$$745$$ 18.5311 0.678928
$$746$$ −18.5311 −0.678473
$$747$$ −8.00000 −0.292705
$$748$$ −26.1245 −0.955207
$$749$$ −43.4689 −1.58832
$$750$$ −1.00000 −0.0365148
$$751$$ 30.9377 1.12893 0.564467 0.825456i $$-0.309082\pi$$
0.564467 + 0.825456i $$0.309082\pi$$
$$752$$ −5.06226 −0.184602
$$753$$ 3.06226 0.111595
$$754$$ 0 0
$$755$$ −14.1245 −0.514044
$$756$$ −4.53113 −0.164796
$$757$$ 26.0000 0.944986 0.472493 0.881334i $$-0.343354\pi$$
0.472493 + 0.881334i $$0.343354\pi$$
$$758$$ 21.5934 0.784307
$$759$$ 42.6556 1.54830
$$760$$ −4.53113 −0.164361
$$761$$ 11.5934 0.420260 0.210130 0.977673i $$-0.432611\pi$$
0.210130 + 0.977673i $$0.432611\pi$$
$$762$$ 7.06226 0.255839
$$763$$ −68.2490 −2.47078
$$764$$ −8.00000 −0.289430
$$765$$ 4.00000 0.144620
$$766$$ −3.06226 −0.110644
$$767$$ −54.3735 −1.96331
$$768$$ 1.00000 0.0360844
$$769$$ −4.40661 −0.158907 −0.0794533 0.996839i $$-0.525317\pi$$
−0.0794533 + 0.996839i $$0.525317\pi$$
$$770$$ 29.5934 1.06647
$$771$$ 12.6556 0.455782
$$772$$ 14.0000 0.503871
$$773$$ 22.5311 0.810388 0.405194 0.914231i $$-0.367204\pi$$
0.405194 + 0.914231i $$0.367204\pi$$
$$774$$ −8.53113 −0.306645
$$775$$ 1.00000 0.0359211
$$776$$ 16.1245 0.578836
$$777$$ 32.0000 1.14799
$$778$$ 10.9377 0.392137
$$779$$ 32.0000 1.14652
$$780$$ −6.00000 −0.214834
$$781$$ −81.8424 −2.92855
$$782$$ −26.1245 −0.934211
$$783$$ 0 0
$$784$$ 13.5311 0.483255
$$785$$ 22.5311 0.804170
$$786$$ −9.06226 −0.323240
$$787$$ −36.5311 −1.30219 −0.651097 0.758994i $$-0.725691\pi$$
−0.651097 + 0.758994i $$0.725691\pi$$
$$788$$ 6.00000 0.213741
$$789$$ −23.0623 −0.821038
$$790$$ 8.53113 0.303524
$$791$$ 34.4066 1.22336
$$792$$ 6.53113 0.232074
$$793$$ 30.3735 1.07860
$$794$$ 37.7179 1.33856
$$795$$ 2.53113 0.0897699
$$796$$ −8.53113 −0.302378
$$797$$ 12.1245 0.429472 0.214736 0.976672i $$-0.431111\pi$$
0.214736 + 0.976672i $$0.431111\pi$$
$$798$$ −20.5311 −0.726794
$$799$$ 20.2490 0.716359
$$800$$ 1.00000 0.0353553
$$801$$ −6.53113 −0.230766
$$802$$ 21.7179 0.766886
$$803$$ −55.7179 −1.96624
$$804$$ 5.06226 0.178532
$$805$$ 29.5934 1.04303
$$806$$ 6.00000 0.211341
$$807$$ −19.1868 −0.675406
$$808$$ −9.46887 −0.333114
$$809$$ 15.5934 0.548234 0.274117 0.961696i $$-0.411614\pi$$
0.274117 + 0.961696i $$0.411614\pi$$
$$810$$ −1.00000 −0.0351364
$$811$$ 32.7802 1.15107 0.575534 0.817778i $$-0.304794\pi$$
0.575534 + 0.817778i $$0.304794\pi$$
$$812$$ 0 0
$$813$$ −11.4689 −0.402231
$$814$$ −46.1245 −1.61666
$$815$$ −5.06226 −0.177323
$$816$$ −4.00000 −0.140028
$$817$$ −38.6556 −1.35239
$$818$$ 7.06226 0.246926
$$819$$ −27.1868 −0.949983
$$820$$ −7.06226 −0.246625
$$821$$ 42.1245 1.47016 0.735078 0.677983i $$-0.237146\pi$$
0.735078 + 0.677983i $$0.237146\pi$$
$$822$$ 17.0623 0.595115
$$823$$ 40.1245 1.39865 0.699326 0.714803i $$-0.253483\pi$$
0.699326 + 0.714803i $$0.253483\pi$$
$$824$$ 0 0
$$825$$ 6.53113 0.227385
$$826$$ 41.0623 1.42874
$$827$$ 8.00000 0.278187 0.139094 0.990279i $$-0.455581\pi$$
0.139094 + 0.990279i $$0.455581\pi$$
$$828$$ 6.53113 0.226972
$$829$$ 30.6556 1.06471 0.532357 0.846520i $$-0.321306\pi$$
0.532357 + 0.846520i $$0.321306\pi$$
$$830$$ 8.00000 0.277684
$$831$$ −15.0623 −0.522504
$$832$$ 6.00000 0.208013
$$833$$ −54.1245 −1.87530
$$834$$ 6.00000 0.207763
$$835$$ 23.5934 0.816483
$$836$$ 29.5934 1.02351
$$837$$ 1.00000 0.0345651
$$838$$ 0 0
$$839$$ 33.8424 1.16837 0.584185 0.811621i $$-0.301414\pi$$
0.584185 + 0.811621i $$0.301414\pi$$
$$840$$ 4.53113 0.156339
$$841$$ −29.0000 −1.00000
$$842$$ −26.2490 −0.904601
$$843$$ 15.0623 0.518772
$$844$$ −1.59339 −0.0548466
$$845$$ −23.0000 −0.791224
$$846$$ −5.06226 −0.174044
$$847$$ −143.436 −4.92851
$$848$$ −2.53113 −0.0869193
$$849$$ −12.0000 −0.411839
$$850$$ −4.00000 −0.137199
$$851$$ −46.1245 −1.58113
$$852$$ −12.5311 −0.429309
$$853$$ 57.7179 1.97622 0.988112 0.153738i $$-0.0491311\pi$$
0.988112 + 0.153738i $$0.0491311\pi$$
$$854$$ −22.9377 −0.784913
$$855$$ −4.53113 −0.154961
$$856$$ 9.59339 0.327895
$$857$$ 14.0000 0.478231 0.239115 0.970991i $$-0.423143\pi$$
0.239115 + 0.970991i $$0.423143\pi$$
$$858$$ 39.1868 1.33781
$$859$$ −23.0623 −0.786874 −0.393437 0.919352i $$-0.628714\pi$$
−0.393437 + 0.919352i $$0.628714\pi$$
$$860$$ 8.53113 0.290909
$$861$$ −32.0000 −1.09056
$$862$$ −24.0000 −0.817443
$$863$$ 33.4689 1.13929 0.569647 0.821890i $$-0.307080\pi$$
0.569647 + 0.821890i $$0.307080\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ −14.0000 −0.476014
$$866$$ −25.5934 −0.869699
$$867$$ −1.00000 −0.0339618
$$868$$ −4.53113 −0.153797
$$869$$ −55.7179 −1.89010
$$870$$ 0 0
$$871$$ 30.3735 1.02917
$$872$$ 15.0623 0.510072
$$873$$ 16.1245 0.545732
$$874$$ 29.5934 1.00101
$$875$$ 4.53113 0.153180
$$876$$ −8.53113 −0.288240
$$877$$ 11.8755 0.401007 0.200503 0.979693i $$-0.435742\pi$$
0.200503 + 0.979693i $$0.435742\pi$$
$$878$$ −25.0623 −0.845810
$$879$$ −14.0000 −0.472208
$$880$$ −6.53113 −0.220164
$$881$$ 3.87548 0.130568 0.0652842 0.997867i $$-0.479205\pi$$
0.0652842 + 0.997867i $$0.479205\pi$$
$$882$$ 13.5311 0.455617
$$883$$ −13.3444 −0.449073 −0.224537 0.974466i $$-0.572087\pi$$
−0.224537 + 0.974466i $$0.572087\pi$$
$$884$$ −24.0000 −0.807207
$$885$$ 9.06226 0.304624
$$886$$ 14.4066 0.484000
$$887$$ 23.1868 0.778536 0.389268 0.921125i $$-0.372728\pi$$
0.389268 + 0.921125i $$0.372728\pi$$
$$888$$ −7.06226 −0.236994
$$889$$ −32.0000 −1.07325
$$890$$ 6.53113 0.218924
$$891$$ 6.53113 0.218801
$$892$$ 2.00000 0.0669650
$$893$$ −22.9377 −0.767582
$$894$$ −18.5311 −0.619774
$$895$$ 3.06226 0.102360
$$896$$ −4.53113 −0.151374
$$897$$ 39.1868 1.30841
$$898$$ −4.12452 −0.137637
$$899$$ 0 0
$$900$$ 1.00000 0.0333333
$$901$$ 10.1245 0.337297
$$902$$ 46.1245 1.53578
$$903$$ 38.6556 1.28638
$$904$$ −7.59339 −0.252552
$$905$$ −2.40661 −0.0799985
$$906$$ 14.1245 0.469256
$$907$$ 32.2490 1.07081 0.535406 0.844595i $$-0.320159\pi$$
0.535406 + 0.844595i $$0.320159\pi$$
$$908$$ −24.5311 −0.814094
$$909$$ −9.46887 −0.314063
$$910$$ 27.1868 0.901233
$$911$$ −16.0000 −0.530104 −0.265052 0.964234i $$-0.585389\pi$$
−0.265052 + 0.964234i $$0.585389\pi$$
$$912$$ 4.53113 0.150041
$$913$$ −52.2490 −1.72919
$$914$$ 14.9377 0.494097
$$915$$ −5.06226 −0.167353
$$916$$ −5.59339 −0.184811
$$917$$ 41.0623 1.35600
$$918$$ −4.00000 −0.132020
$$919$$ −25.0623 −0.826728 −0.413364 0.910566i $$-0.635646\pi$$
−0.413364 + 0.910566i $$0.635646\pi$$
$$920$$ −6.53113 −0.215325
$$921$$ −30.1245 −0.992637
$$922$$ 24.0000 0.790398
$$923$$ −75.1868 −2.47480
$$924$$ −29.5934 −0.973551
$$925$$ −7.06226 −0.232206
$$926$$ 13.1868 0.433344
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −51.8424 −1.70089 −0.850447 0.526060i $$-0.823669\pi$$
−0.850447 + 0.526060i $$0.823669\pi$$
$$930$$ −1.00000 −0.0327913
$$931$$ 61.3113 2.00940
$$932$$ 1.46887 0.0481145
$$933$$ −8.00000 −0.261908
$$934$$ −22.1245 −0.723936
$$935$$ 26.1245 0.854363
$$936$$ 6.00000 0.196116
$$937$$ −8.93774 −0.291983 −0.145992 0.989286i $$-0.546637\pi$$
−0.145992 + 0.989286i $$0.546637\pi$$
$$938$$ −22.9377 −0.748944
$$939$$ 10.9377 0.356940
$$940$$ 5.06226 0.165113
$$941$$ −54.1245 −1.76441 −0.882204 0.470867i $$-0.843941\pi$$
−0.882204 + 0.470867i $$0.843941\pi$$
$$942$$ −22.5311 −0.734104
$$943$$ 46.1245 1.50202
$$944$$ −9.06226 −0.294951
$$945$$ 4.53113 0.147398
$$946$$ −55.7179 −1.81155
$$947$$ 18.9377 0.615394 0.307697 0.951484i $$-0.400442\pi$$
0.307697 + 0.951484i $$0.400442\pi$$
$$948$$ −8.53113 −0.277078
$$949$$ −51.1868 −1.66159
$$950$$ 4.53113 0.147009
$$951$$ 12.9377 0.419535
$$952$$ 18.1245 0.587419
$$953$$ 21.8755 0.708616 0.354308 0.935129i $$-0.384716\pi$$
0.354308 + 0.935129i $$0.384716\pi$$
$$954$$ −2.53113 −0.0819483
$$955$$ 8.00000 0.258874
$$956$$ −8.00000 −0.258738
$$957$$ 0 0
$$958$$ 1.34436 0.0434342
$$959$$ −77.3113 −2.49651
$$960$$ −1.00000 −0.0322749
$$961$$ 1.00000 0.0322581
$$962$$ −42.3735 −1.36618
$$963$$ 9.59339 0.309143
$$964$$ 2.00000 0.0644157
$$965$$ −14.0000 −0.450676
$$966$$ −29.5934 −0.952152
$$967$$ −0.937742 −0.0301558 −0.0150779 0.999886i $$-0.504800\pi$$
−0.0150779 + 0.999886i $$0.504800\pi$$
$$968$$ 31.6556 1.01745
$$969$$ −18.1245 −0.582243
$$970$$ −16.1245 −0.517727
$$971$$ 15.1868 0.487367 0.243683 0.969855i $$-0.421644\pi$$
0.243683 + 0.969855i $$0.421644\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ −27.1868 −0.871568
$$974$$ 23.0623 0.738962
$$975$$ 6.00000 0.192154
$$976$$ 5.06226 0.162039
$$977$$ −38.0000 −1.21573 −0.607864 0.794041i $$-0.707973\pi$$
−0.607864 + 0.794041i $$0.707973\pi$$
$$978$$ 5.06226 0.161873
$$979$$ −42.6556 −1.36328
$$980$$ −13.5311 −0.432236
$$981$$ 15.0623 0.480901
$$982$$ −7.59339 −0.242315
$$983$$ 0.937742 0.0299093 0.0149547 0.999888i $$-0.495240\pi$$
0.0149547 + 0.999888i $$0.495240\pi$$
$$984$$ 7.06226 0.225137
$$985$$ −6.00000 −0.191176
$$986$$ 0 0
$$987$$ 22.9377 0.730116
$$988$$ 27.1868 0.864926
$$989$$ −55.7179 −1.77173
$$990$$ −6.53113 −0.207573
$$991$$ −32.5311 −1.03339 −0.516693 0.856171i $$-0.672837\pi$$
−0.516693 + 0.856171i $$0.672837\pi$$
$$992$$ 1.00000 0.0317500
$$993$$ 12.9377 0.410567
$$994$$ 56.7802 1.80096
$$995$$ 8.53113 0.270455
$$996$$ −8.00000 −0.253490
$$997$$ −2.24903 −0.0712275 −0.0356138 0.999366i $$-0.511339\pi$$
−0.0356138 + 0.999366i $$0.511339\pi$$
$$998$$ −7.06226 −0.223552
$$999$$ −7.06226 −0.223440
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 930.2.a.q.1.1 2
3.2 odd 2 2790.2.a.bf.1.1 2
4.3 odd 2 7440.2.a.bd.1.2 2
5.2 odd 4 4650.2.d.bg.3349.3 4
5.3 odd 4 4650.2.d.bg.3349.2 4
5.4 even 2 4650.2.a.bz.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.a.q.1.1 2 1.1 even 1 trivial
2790.2.a.bf.1.1 2 3.2 odd 2
4650.2.a.bz.1.2 2 5.4 even 2
4650.2.d.bg.3349.2 4 5.3 odd 4
4650.2.d.bg.3349.3 4 5.2 odd 4
7440.2.a.bd.1.2 2 4.3 odd 2