# Properties

 Label 825.2.a.k.1.2 Level $825$ Weight $2$ Character 825.1 Self dual yes Analytic conductor $6.588$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,2,Mod(1,825)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(825, 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("825.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$825 = 3 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 825.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: $$6.58765816676$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.148.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 3x + 1$$ x^3 - x^2 - 3*x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 165) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-1.48119$$ of defining polynomial Character $$\chi$$ $$=$$ 825.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+0.193937 q^{2} -1.00000 q^{3} -1.96239 q^{4} -0.193937 q^{6} -3.35026 q^{7} -0.768452 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+0.193937 q^{2} -1.00000 q^{3} -1.96239 q^{4} -0.193937 q^{6} -3.35026 q^{7} -0.768452 q^{8} +1.00000 q^{9} +1.00000 q^{11} +1.96239 q^{12} -2.96239 q^{13} -0.649738 q^{14} +3.77575 q^{16} +4.57452 q^{17} +0.193937 q^{18} -4.31265 q^{19} +3.35026 q^{21} +0.193937 q^{22} +6.70052 q^{23} +0.768452 q^{24} -0.574515 q^{26} -1.00000 q^{27} +6.57452 q^{28} -3.61213 q^{29} +9.92478 q^{31} +2.26916 q^{32} -1.00000 q^{33} +0.887166 q^{34} -1.96239 q^{36} +2.00000 q^{37} -0.836381 q^{38} +2.96239 q^{39} -4.38787 q^{41} +0.649738 q^{42} +9.27504 q^{43} -1.96239 q^{44} +1.29948 q^{46} +9.92478 q^{47} -3.77575 q^{48} +4.22425 q^{49} -4.57452 q^{51} +5.81336 q^{52} -4.70052 q^{53} -0.193937 q^{54} +2.57452 q^{56} +4.31265 q^{57} -0.700523 q^{58} +10.7005 q^{59} -8.70052 q^{61} +1.92478 q^{62} -3.35026 q^{63} -7.11142 q^{64} -0.193937 q^{66} -5.92478 q^{67} -8.97698 q^{68} -6.70052 q^{69} +9.92478 q^{71} -0.768452 q^{72} +7.73813 q^{73} +0.387873 q^{74} +8.46310 q^{76} -3.35026 q^{77} +0.574515 q^{78} +11.5369 q^{79} +1.00000 q^{81} -0.850969 q^{82} -10.8872 q^{83} -6.57452 q^{84} +1.79877 q^{86} +3.61213 q^{87} -0.768452 q^{88} -2.77575 q^{89} +9.92478 q^{91} -13.1490 q^{92} -9.92478 q^{93} +1.92478 q^{94} -2.26916 q^{96} -0.0752228 q^{97} +0.819237 q^{98} +1.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + q^{2} - 3 q^{3} + 5 q^{4} - q^{6} + 9 q^{8} + 3 q^{9}+O(q^{10})$$ 3 * q + q^2 - 3 * q^3 + 5 * q^4 - q^6 + 9 * q^8 + 3 * q^9 $$3 q + q^{2} - 3 q^{3} + 5 q^{4} - q^{6} + 9 q^{8} + 3 q^{9} + 3 q^{11} - 5 q^{12} + 2 q^{13} - 12 q^{14} + 13 q^{16} + 2 q^{17} + q^{18} + 8 q^{19} + q^{22} - 9 q^{24} + 10 q^{26} - 3 q^{27} + 8 q^{28} - 10 q^{29} + 8 q^{31} + 29 q^{32} - 3 q^{33} - 30 q^{34} + 5 q^{36} + 6 q^{37} - 2 q^{39} - 14 q^{41} + 12 q^{42} - 4 q^{43} + 5 q^{44} + 24 q^{46} + 8 q^{47} - 13 q^{48} + 11 q^{49} - 2 q^{51} + 30 q^{52} + 6 q^{53} - q^{54} - 4 q^{56} - 8 q^{57} + 18 q^{58} + 12 q^{59} - 6 q^{61} - 16 q^{62} + 13 q^{64} - q^{66} + 4 q^{67} - 42 q^{68} + 8 q^{71} + 9 q^{72} + 14 q^{73} + 2 q^{74} + 48 q^{76} - 10 q^{78} + 12 q^{79} + 3 q^{81} - 26 q^{82} - 8 q^{84} - 8 q^{86} + 10 q^{87} + 9 q^{88} - 10 q^{89} + 8 q^{91} - 16 q^{92} - 8 q^{93} - 16 q^{94} - 29 q^{96} - 22 q^{97} - 39 q^{98} + 3 q^{99}+O(q^{100})$$ 3 * q + q^2 - 3 * q^3 + 5 * q^4 - q^6 + 9 * q^8 + 3 * q^9 + 3 * q^11 - 5 * q^12 + 2 * q^13 - 12 * q^14 + 13 * q^16 + 2 * q^17 + q^18 + 8 * q^19 + q^22 - 9 * q^24 + 10 * q^26 - 3 * q^27 + 8 * q^28 - 10 * q^29 + 8 * q^31 + 29 * q^32 - 3 * q^33 - 30 * q^34 + 5 * q^36 + 6 * q^37 - 2 * q^39 - 14 * q^41 + 12 * q^42 - 4 * q^43 + 5 * q^44 + 24 * q^46 + 8 * q^47 - 13 * q^48 + 11 * q^49 - 2 * q^51 + 30 * q^52 + 6 * q^53 - q^54 - 4 * q^56 - 8 * q^57 + 18 * q^58 + 12 * q^59 - 6 * q^61 - 16 * q^62 + 13 * q^64 - q^66 + 4 * q^67 - 42 * q^68 + 8 * q^71 + 9 * q^72 + 14 * q^73 + 2 * q^74 + 48 * q^76 - 10 * q^78 + 12 * q^79 + 3 * q^81 - 26 * q^82 - 8 * q^84 - 8 * q^86 + 10 * q^87 + 9 * q^88 - 10 * q^89 + 8 * q^91 - 16 * q^92 - 8 * q^93 - 16 * q^94 - 29 * q^96 - 22 * q^97 - 39 * q^98 + 3 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0.193937 0.137134 0.0685669 0.997647i $$-0.478157\pi$$
0.0685669 + 0.997647i $$0.478157\pi$$
$$3$$ −1.00000 −0.577350
$$4$$ −1.96239 −0.981194
$$5$$ 0 0
$$6$$ −0.193937 −0.0791743
$$7$$ −3.35026 −1.26628 −0.633140 0.774037i $$-0.718234\pi$$
−0.633140 + 0.774037i $$0.718234\pi$$
$$8$$ −0.768452 −0.271689
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 1.00000 0.301511
$$12$$ 1.96239 0.566493
$$13$$ −2.96239 −0.821619 −0.410809 0.911721i $$-0.634754\pi$$
−0.410809 + 0.911721i $$0.634754\pi$$
$$14$$ −0.649738 −0.173650
$$15$$ 0 0
$$16$$ 3.77575 0.943937
$$17$$ 4.57452 1.10948 0.554741 0.832023i $$-0.312817\pi$$
0.554741 + 0.832023i $$0.312817\pi$$
$$18$$ 0.193937 0.0457113
$$19$$ −4.31265 −0.989390 −0.494695 0.869067i $$-0.664720\pi$$
−0.494695 + 0.869067i $$0.664720\pi$$
$$20$$ 0 0
$$21$$ 3.35026 0.731087
$$22$$ 0.193937 0.0413474
$$23$$ 6.70052 1.39716 0.698578 0.715534i $$-0.253817\pi$$
0.698578 + 0.715534i $$0.253817\pi$$
$$24$$ 0.768452 0.156860
$$25$$ 0 0
$$26$$ −0.574515 −0.112672
$$27$$ −1.00000 −0.192450
$$28$$ 6.57452 1.24247
$$29$$ −3.61213 −0.670755 −0.335378 0.942084i $$-0.608864\pi$$
−0.335378 + 0.942084i $$0.608864\pi$$
$$30$$ 0 0
$$31$$ 9.92478 1.78254 0.891271 0.453470i $$-0.149814\pi$$
0.891271 + 0.453470i $$0.149814\pi$$
$$32$$ 2.26916 0.401134
$$33$$ −1.00000 −0.174078
$$34$$ 0.887166 0.152148
$$35$$ 0 0
$$36$$ −1.96239 −0.327065
$$37$$ 2.00000 0.328798 0.164399 0.986394i $$-0.447432\pi$$
0.164399 + 0.986394i $$0.447432\pi$$
$$38$$ −0.836381 −0.135679
$$39$$ 2.96239 0.474362
$$40$$ 0 0
$$41$$ −4.38787 −0.685271 −0.342635 0.939468i $$-0.611320\pi$$
−0.342635 + 0.939468i $$0.611320\pi$$
$$42$$ 0.649738 0.100257
$$43$$ 9.27504 1.41443 0.707215 0.706998i $$-0.249951\pi$$
0.707215 + 0.706998i $$0.249951\pi$$
$$44$$ −1.96239 −0.295841
$$45$$ 0 0
$$46$$ 1.29948 0.191597
$$47$$ 9.92478 1.44768 0.723839 0.689969i $$-0.242376\pi$$
0.723839 + 0.689969i $$0.242376\pi$$
$$48$$ −3.77575 −0.544982
$$49$$ 4.22425 0.603465
$$50$$ 0 0
$$51$$ −4.57452 −0.640560
$$52$$ 5.81336 0.806168
$$53$$ −4.70052 −0.645667 −0.322833 0.946456i $$-0.604635\pi$$
−0.322833 + 0.946456i $$0.604635\pi$$
$$54$$ −0.193937 −0.0263914
$$55$$ 0 0
$$56$$ 2.57452 0.344034
$$57$$ 4.31265 0.571224
$$58$$ −0.700523 −0.0919832
$$59$$ 10.7005 1.39309 0.696545 0.717513i $$-0.254720\pi$$
0.696545 + 0.717513i $$0.254720\pi$$
$$60$$ 0 0
$$61$$ −8.70052 −1.11399 −0.556994 0.830517i $$-0.688045\pi$$
−0.556994 + 0.830517i $$0.688045\pi$$
$$62$$ 1.92478 0.244447
$$63$$ −3.35026 −0.422093
$$64$$ −7.11142 −0.888927
$$65$$ 0 0
$$66$$ −0.193937 −0.0238719
$$67$$ −5.92478 −0.723827 −0.361913 0.932212i $$-0.617876\pi$$
−0.361913 + 0.932212i $$0.617876\pi$$
$$68$$ −8.97698 −1.08862
$$69$$ −6.70052 −0.806648
$$70$$ 0 0
$$71$$ 9.92478 1.17785 0.588927 0.808186i $$-0.299550\pi$$
0.588927 + 0.808186i $$0.299550\pi$$
$$72$$ −0.768452 −0.0905629
$$73$$ 7.73813 0.905680 0.452840 0.891592i $$-0.350411\pi$$
0.452840 + 0.891592i $$0.350411\pi$$
$$74$$ 0.387873 0.0450893
$$75$$ 0 0
$$76$$ 8.46310 0.970784
$$77$$ −3.35026 −0.381798
$$78$$ 0.574515 0.0650511
$$79$$ 11.5369 1.29800 0.649002 0.760787i $$-0.275187\pi$$
0.649002 + 0.760787i $$0.275187\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ −0.850969 −0.0939738
$$83$$ −10.8872 −1.19502 −0.597511 0.801861i $$-0.703844\pi$$
−0.597511 + 0.801861i $$0.703844\pi$$
$$84$$ −6.57452 −0.717338
$$85$$ 0 0
$$86$$ 1.79877 0.193966
$$87$$ 3.61213 0.387261
$$88$$ −0.768452 −0.0819173
$$89$$ −2.77575 −0.294229 −0.147114 0.989120i $$-0.546999\pi$$
−0.147114 + 0.989120i $$0.546999\pi$$
$$90$$ 0 0
$$91$$ 9.92478 1.04040
$$92$$ −13.1490 −1.37088
$$93$$ −9.92478 −1.02915
$$94$$ 1.92478 0.198526
$$95$$ 0 0
$$96$$ −2.26916 −0.231595
$$97$$ −0.0752228 −0.00763772 −0.00381886 0.999993i $$-0.501216\pi$$
−0.00381886 + 0.999993i $$0.501216\pi$$
$$98$$ 0.819237 0.0827555
$$99$$ 1.00000 0.100504
$$100$$ 0 0
$$101$$ −15.0884 −1.50135 −0.750676 0.660671i $$-0.770272\pi$$
−0.750676 + 0.660671i $$0.770272\pi$$
$$102$$ −0.887166 −0.0878425
$$103$$ 3.22425 0.317695 0.158848 0.987303i $$-0.449222\pi$$
0.158848 + 0.987303i $$0.449222\pi$$
$$104$$ 2.27645 0.223225
$$105$$ 0 0
$$106$$ −0.911603 −0.0885427
$$107$$ 0.962389 0.0930376 0.0465188 0.998917i $$-0.485187\pi$$
0.0465188 + 0.998917i $$0.485187\pi$$
$$108$$ 1.96239 0.188831
$$109$$ 11.4010 1.09202 0.546011 0.837778i $$-0.316146\pi$$
0.546011 + 0.837778i $$0.316146\pi$$
$$110$$ 0 0
$$111$$ −2.00000 −0.189832
$$112$$ −12.6497 −1.19529
$$113$$ 6.00000 0.564433 0.282216 0.959351i $$-0.408930\pi$$
0.282216 + 0.959351i $$0.408930\pi$$
$$114$$ 0.836381 0.0783342
$$115$$ 0 0
$$116$$ 7.08840 0.658141
$$117$$ −2.96239 −0.273873
$$118$$ 2.07522 0.191040
$$119$$ −15.3258 −1.40492
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ −1.68735 −0.152765
$$123$$ 4.38787 0.395641
$$124$$ −19.4763 −1.74902
$$125$$ 0 0
$$126$$ −0.649738 −0.0578833
$$127$$ 14.5745 1.29328 0.646640 0.762796i $$-0.276174\pi$$
0.646640 + 0.762796i $$0.276174\pi$$
$$128$$ −5.91748 −0.523037
$$129$$ −9.27504 −0.816622
$$130$$ 0 0
$$131$$ −5.92478 −0.517650 −0.258825 0.965924i $$-0.583335\pi$$
−0.258825 + 0.965924i $$0.583335\pi$$
$$132$$ 1.96239 0.170804
$$133$$ 14.4485 1.25284
$$134$$ −1.14903 −0.0992612
$$135$$ 0 0
$$136$$ −3.51530 −0.301434
$$137$$ −13.8496 −1.18325 −0.591624 0.806214i $$-0.701513\pi$$
−0.591624 + 0.806214i $$0.701513\pi$$
$$138$$ −1.29948 −0.110619
$$139$$ 13.6121 1.15457 0.577283 0.816544i $$-0.304113\pi$$
0.577283 + 0.816544i $$0.304113\pi$$
$$140$$ 0 0
$$141$$ −9.92478 −0.835817
$$142$$ 1.92478 0.161524
$$143$$ −2.96239 −0.247727
$$144$$ 3.77575 0.314646
$$145$$ 0 0
$$146$$ 1.50071 0.124199
$$147$$ −4.22425 −0.348411
$$148$$ −3.92478 −0.322615
$$149$$ 1.53690 0.125908 0.0629540 0.998016i $$-0.479948\pi$$
0.0629540 + 0.998016i $$0.479948\pi$$
$$150$$ 0 0
$$151$$ −6.76116 −0.550215 −0.275108 0.961413i $$-0.588713\pi$$
−0.275108 + 0.961413i $$0.588713\pi$$
$$152$$ 3.31406 0.268806
$$153$$ 4.57452 0.369828
$$154$$ −0.649738 −0.0523574
$$155$$ 0 0
$$156$$ −5.81336 −0.465441
$$157$$ 5.47627 0.437054 0.218527 0.975831i $$-0.429875\pi$$
0.218527 + 0.975831i $$0.429875\pi$$
$$158$$ 2.23743 0.178000
$$159$$ 4.70052 0.372776
$$160$$ 0 0
$$161$$ −22.4485 −1.76919
$$162$$ 0.193937 0.0152371
$$163$$ −12.6253 −0.988890 −0.494445 0.869209i $$-0.664629\pi$$
−0.494445 + 0.869209i $$0.664629\pi$$
$$164$$ 8.61071 0.672384
$$165$$ 0 0
$$166$$ −2.11142 −0.163878
$$167$$ −18.3634 −1.42101 −0.710503 0.703695i $$-0.751532\pi$$
−0.710503 + 0.703695i $$0.751532\pi$$
$$168$$ −2.57452 −0.198628
$$169$$ −4.22425 −0.324943
$$170$$ 0 0
$$171$$ −4.31265 −0.329797
$$172$$ −18.2012 −1.38783
$$173$$ 8.57452 0.651908 0.325954 0.945386i $$-0.394314\pi$$
0.325954 + 0.945386i $$0.394314\pi$$
$$174$$ 0.700523 0.0531065
$$175$$ 0 0
$$176$$ 3.77575 0.284608
$$177$$ −10.7005 −0.804301
$$178$$ −0.538319 −0.0403487
$$179$$ 14.1768 1.05962 0.529812 0.848115i $$-0.322263\pi$$
0.529812 + 0.848115i $$0.322263\pi$$
$$180$$ 0 0
$$181$$ −5.22425 −0.388316 −0.194158 0.980970i $$-0.562197\pi$$
−0.194158 + 0.980970i $$0.562197\pi$$
$$182$$ 1.92478 0.142674
$$183$$ 8.70052 0.643161
$$184$$ −5.14903 −0.379592
$$185$$ 0 0
$$186$$ −1.92478 −0.141132
$$187$$ 4.57452 0.334522
$$188$$ −19.4763 −1.42045
$$189$$ 3.35026 0.243696
$$190$$ 0 0
$$191$$ −16.6253 −1.20296 −0.601482 0.798886i $$-0.705423\pi$$
−0.601482 + 0.798886i $$0.705423\pi$$
$$192$$ 7.11142 0.513222
$$193$$ 16.3634 1.17787 0.588933 0.808182i $$-0.299548\pi$$
0.588933 + 0.808182i $$0.299548\pi$$
$$194$$ −0.0145884 −0.00104739
$$195$$ 0 0
$$196$$ −8.28963 −0.592116
$$197$$ 20.4241 1.45515 0.727577 0.686026i $$-0.240646\pi$$
0.727577 + 0.686026i $$0.240646\pi$$
$$198$$ 0.193937 0.0137825
$$199$$ −8.62530 −0.611431 −0.305716 0.952123i $$-0.598896\pi$$
−0.305716 + 0.952123i $$0.598896\pi$$
$$200$$ 0 0
$$201$$ 5.92478 0.417902
$$202$$ −2.92619 −0.205886
$$203$$ 12.1016 0.849364
$$204$$ 8.97698 0.628514
$$205$$ 0 0
$$206$$ 0.625301 0.0435668
$$207$$ 6.70052 0.465719
$$208$$ −11.1852 −0.775556
$$209$$ −4.31265 −0.298312
$$210$$ 0 0
$$211$$ 9.08840 0.625671 0.312836 0.949807i $$-0.398721\pi$$
0.312836 + 0.949807i $$0.398721\pi$$
$$212$$ 9.22425 0.633524
$$213$$ −9.92478 −0.680035
$$214$$ 0.186642 0.0127586
$$215$$ 0 0
$$216$$ 0.768452 0.0522865
$$217$$ −33.2506 −2.25720
$$218$$ 2.21108 0.149753
$$219$$ −7.73813 −0.522895
$$220$$ 0 0
$$221$$ −13.5515 −0.911572
$$222$$ −0.387873 −0.0260323
$$223$$ 6.70052 0.448700 0.224350 0.974509i $$-0.427974\pi$$
0.224350 + 0.974509i $$0.427974\pi$$
$$224$$ −7.60228 −0.507949
$$225$$ 0 0
$$226$$ 1.16362 0.0774028
$$227$$ −16.9624 −1.12583 −0.562917 0.826514i $$-0.690321\pi$$
−0.562917 + 0.826514i $$0.690321\pi$$
$$228$$ −8.46310 −0.560482
$$229$$ 25.8496 1.70819 0.854093 0.520120i $$-0.174113\pi$$
0.854093 + 0.520120i $$0.174113\pi$$
$$230$$ 0 0
$$231$$ 3.35026 0.220431
$$232$$ 2.77575 0.182237
$$233$$ 19.2750 1.26275 0.631375 0.775478i $$-0.282491\pi$$
0.631375 + 0.775478i $$0.282491\pi$$
$$234$$ −0.574515 −0.0375573
$$235$$ 0 0
$$236$$ −20.9986 −1.36689
$$237$$ −11.5369 −0.749402
$$238$$ −2.97224 −0.192662
$$239$$ 26.5501 1.71738 0.858691 0.512494i $$-0.171278\pi$$
0.858691 + 0.512494i $$0.171278\pi$$
$$240$$ 0 0
$$241$$ 28.5501 1.83907 0.919536 0.393006i $$-0.128565\pi$$
0.919536 + 0.393006i $$0.128565\pi$$
$$242$$ 0.193937 0.0124667
$$243$$ −1.00000 −0.0641500
$$244$$ 17.0738 1.09304
$$245$$ 0 0
$$246$$ 0.850969 0.0542558
$$247$$ 12.7757 0.812901
$$248$$ −7.62672 −0.484297
$$249$$ 10.8872 0.689946
$$250$$ 0 0
$$251$$ 29.9248 1.88884 0.944418 0.328748i $$-0.106627\pi$$
0.944418 + 0.328748i $$0.106627\pi$$
$$252$$ 6.57452 0.414156
$$253$$ 6.70052 0.421258
$$254$$ 2.82653 0.177352
$$255$$ 0 0
$$256$$ 13.0752 0.817201
$$257$$ −8.70052 −0.542724 −0.271362 0.962477i $$-0.587474\pi$$
−0.271362 + 0.962477i $$0.587474\pi$$
$$258$$ −1.79877 −0.111986
$$259$$ −6.70052 −0.416350
$$260$$ 0 0
$$261$$ −3.61213 −0.223585
$$262$$ −1.14903 −0.0709874
$$263$$ −12.2882 −0.757724 −0.378862 0.925453i $$-0.623684\pi$$
−0.378862 + 0.925453i $$0.623684\pi$$
$$264$$ 0.768452 0.0472950
$$265$$ 0 0
$$266$$ 2.80209 0.171807
$$267$$ 2.77575 0.169873
$$268$$ 11.6267 0.710215
$$269$$ −5.84955 −0.356654 −0.178327 0.983971i $$-0.557068\pi$$
−0.178327 + 0.983971i $$0.557068\pi$$
$$270$$ 0 0
$$271$$ −5.08840 −0.309098 −0.154549 0.987985i $$-0.549392\pi$$
−0.154549 + 0.987985i $$0.549392\pi$$
$$272$$ 17.2722 1.04728
$$273$$ −9.92478 −0.600675
$$274$$ −2.68594 −0.162263
$$275$$ 0 0
$$276$$ 13.1490 0.791479
$$277$$ −1.41090 −0.0847725 −0.0423863 0.999101i $$-0.513496\pi$$
−0.0423863 + 0.999101i $$0.513496\pi$$
$$278$$ 2.63989 0.158330
$$279$$ 9.92478 0.594181
$$280$$ 0 0
$$281$$ −4.38787 −0.261759 −0.130879 0.991398i $$-0.541780\pi$$
−0.130879 + 0.991398i $$0.541780\pi$$
$$282$$ −1.92478 −0.114619
$$283$$ −26.5745 −1.57969 −0.789845 0.613306i $$-0.789839\pi$$
−0.789845 + 0.613306i $$0.789839\pi$$
$$284$$ −19.4763 −1.15570
$$285$$ 0 0
$$286$$ −0.574515 −0.0339718
$$287$$ 14.7005 0.867744
$$288$$ 2.26916 0.133711
$$289$$ 3.92619 0.230952
$$290$$ 0 0
$$291$$ 0.0752228 0.00440964
$$292$$ −15.1852 −0.888648
$$293$$ 3.42548 0.200119 0.100059 0.994981i $$-0.468097\pi$$
0.100059 + 0.994981i $$0.468097\pi$$
$$294$$ −0.819237 −0.0477789
$$295$$ 0 0
$$296$$ −1.53690 −0.0893307
$$297$$ −1.00000 −0.0580259
$$298$$ 0.298062 0.0172663
$$299$$ −19.8496 −1.14793
$$300$$ 0 0
$$301$$ −31.0738 −1.79106
$$302$$ −1.31124 −0.0754531
$$303$$ 15.0884 0.866806
$$304$$ −16.2835 −0.933921
$$305$$ 0 0
$$306$$ 0.887166 0.0507159
$$307$$ 16.6497 0.950251 0.475125 0.879918i $$-0.342403\pi$$
0.475125 + 0.879918i $$0.342403\pi$$
$$308$$ 6.57452 0.374618
$$309$$ −3.22425 −0.183421
$$310$$ 0 0
$$311$$ 32.9986 1.87118 0.935589 0.353091i $$-0.114869\pi$$
0.935589 + 0.353091i $$0.114869\pi$$
$$312$$ −2.27645 −0.128879
$$313$$ −15.4010 −0.870519 −0.435259 0.900305i $$-0.643343\pi$$
−0.435259 + 0.900305i $$0.643343\pi$$
$$314$$ 1.06205 0.0599349
$$315$$ 0 0
$$316$$ −22.6399 −1.27359
$$317$$ −2.15045 −0.120781 −0.0603905 0.998175i $$-0.519235\pi$$
−0.0603905 + 0.998175i $$0.519235\pi$$
$$318$$ 0.911603 0.0511202
$$319$$ −3.61213 −0.202240
$$320$$ 0 0
$$321$$ −0.962389 −0.0537153
$$322$$ −4.35359 −0.242616
$$323$$ −19.7283 −1.09771
$$324$$ −1.96239 −0.109022
$$325$$ 0 0
$$326$$ −2.44851 −0.135610
$$327$$ −11.4010 −0.630479
$$328$$ 3.37187 0.186180
$$329$$ −33.2506 −1.83316
$$330$$ 0 0
$$331$$ −14.5501 −0.799745 −0.399872 0.916571i $$-0.630946\pi$$
−0.399872 + 0.916571i $$0.630946\pi$$
$$332$$ 21.3649 1.17255
$$333$$ 2.00000 0.109599
$$334$$ −3.56134 −0.194868
$$335$$ 0 0
$$336$$ 12.6497 0.690100
$$337$$ −16.2619 −0.885840 −0.442920 0.896561i $$-0.646057\pi$$
−0.442920 + 0.896561i $$0.646057\pi$$
$$338$$ −0.819237 −0.0445606
$$339$$ −6.00000 −0.325875
$$340$$ 0 0
$$341$$ 9.92478 0.537457
$$342$$ −0.836381 −0.0452263
$$343$$ 9.29948 0.502125
$$344$$ −7.12742 −0.384285
$$345$$ 0 0
$$346$$ 1.66291 0.0893987
$$347$$ 0.962389 0.0516637 0.0258319 0.999666i $$-0.491777\pi$$
0.0258319 + 0.999666i $$0.491777\pi$$
$$348$$ −7.08840 −0.379978
$$349$$ 20.7005 1.10807 0.554037 0.832492i $$-0.313087\pi$$
0.554037 + 0.832492i $$0.313087\pi$$
$$350$$ 0 0
$$351$$ 2.96239 0.158121
$$352$$ 2.26916 0.120947
$$353$$ −20.5501 −1.09377 −0.546885 0.837208i $$-0.684187\pi$$
−0.546885 + 0.837208i $$0.684187\pi$$
$$354$$ −2.07522 −0.110297
$$355$$ 0 0
$$356$$ 5.44709 0.288695
$$357$$ 15.3258 0.811129
$$358$$ 2.74940 0.145310
$$359$$ 17.9248 0.946034 0.473017 0.881053i $$-0.343165\pi$$
0.473017 + 0.881053i $$0.343165\pi$$
$$360$$ 0 0
$$361$$ −0.401047 −0.0211077
$$362$$ −1.01317 −0.0532512
$$363$$ −1.00000 −0.0524864
$$364$$ −19.4763 −1.02083
$$365$$ 0 0
$$366$$ 1.68735 0.0881992
$$367$$ 29.6531 1.54788 0.773939 0.633261i $$-0.218284\pi$$
0.773939 + 0.633261i $$0.218284\pi$$
$$368$$ 25.2995 1.31883
$$369$$ −4.38787 −0.228424
$$370$$ 0 0
$$371$$ 15.7480 0.817595
$$372$$ 19.4763 1.00980
$$373$$ 9.13918 0.473209 0.236604 0.971606i $$-0.423965\pi$$
0.236604 + 0.971606i $$0.423965\pi$$
$$374$$ 0.887166 0.0458743
$$375$$ 0 0
$$376$$ −7.62672 −0.393318
$$377$$ 10.7005 0.551105
$$378$$ 0.649738 0.0334189
$$379$$ −20.0000 −1.02733 −0.513665 0.857991i $$-0.671713\pi$$
−0.513665 + 0.857991i $$0.671713\pi$$
$$380$$ 0 0
$$381$$ −14.5745 −0.746675
$$382$$ −3.22425 −0.164967
$$383$$ 34.9234 1.78450 0.892250 0.451541i $$-0.149126\pi$$
0.892250 + 0.451541i $$0.149126\pi$$
$$384$$ 5.91748 0.301975
$$385$$ 0 0
$$386$$ 3.17347 0.161525
$$387$$ 9.27504 0.471477
$$388$$ 0.147616 0.00749408
$$389$$ 2.77575 0.140736 0.0703680 0.997521i $$-0.477583\pi$$
0.0703680 + 0.997521i $$0.477583\pi$$
$$390$$ 0 0
$$391$$ 30.6516 1.55012
$$392$$ −3.24614 −0.163955
$$393$$ 5.92478 0.298865
$$394$$ 3.96097 0.199551
$$395$$ 0 0
$$396$$ −1.96239 −0.0986137
$$397$$ 19.9248 0.999996 0.499998 0.866027i $$-0.333334\pi$$
0.499998 + 0.866027i $$0.333334\pi$$
$$398$$ −1.67276 −0.0838479
$$399$$ −14.4485 −0.723330
$$400$$ 0 0
$$401$$ 2.00000 0.0998752 0.0499376 0.998752i $$-0.484098\pi$$
0.0499376 + 0.998752i $$0.484098\pi$$
$$402$$ 1.14903 0.0573085
$$403$$ −29.4010 −1.46457
$$404$$ 29.6093 1.47312
$$405$$ 0 0
$$406$$ 2.34694 0.116477
$$407$$ 2.00000 0.0991363
$$408$$ 3.51530 0.174033
$$409$$ −13.0738 −0.646458 −0.323229 0.946321i $$-0.604768\pi$$
−0.323229 + 0.946321i $$0.604768\pi$$
$$410$$ 0 0
$$411$$ 13.8496 0.683148
$$412$$ −6.32724 −0.311721
$$413$$ −35.8496 −1.76404
$$414$$ 1.29948 0.0638658
$$415$$ 0 0
$$416$$ −6.72213 −0.329580
$$417$$ −13.6121 −0.666589
$$418$$ −0.836381 −0.0409087
$$419$$ 7.22425 0.352928 0.176464 0.984307i $$-0.443534\pi$$
0.176464 + 0.984307i $$0.443534\pi$$
$$420$$ 0 0
$$421$$ 30.6253 1.49259 0.746293 0.665618i $$-0.231832\pi$$
0.746293 + 0.665618i $$0.231832\pi$$
$$422$$ 1.76257 0.0858007
$$423$$ 9.92478 0.482559
$$424$$ 3.61213 0.175420
$$425$$ 0 0
$$426$$ −1.92478 −0.0932558
$$427$$ 29.1490 1.41062
$$428$$ −1.88858 −0.0912880
$$429$$ 2.96239 0.143025
$$430$$ 0 0
$$431$$ −33.8759 −1.63174 −0.815872 0.578232i $$-0.803743\pi$$
−0.815872 + 0.578232i $$0.803743\pi$$
$$432$$ −3.77575 −0.181661
$$433$$ 9.47627 0.455400 0.227700 0.973731i $$-0.426879\pi$$
0.227700 + 0.973731i $$0.426879\pi$$
$$434$$ −6.44851 −0.309538
$$435$$ 0 0
$$436$$ −22.3733 −1.07149
$$437$$ −28.8970 −1.38233
$$438$$ −1.50071 −0.0717066
$$439$$ −29.4617 −1.40613 −0.703065 0.711126i $$-0.748186\pi$$
−0.703065 + 0.711126i $$0.748186\pi$$
$$440$$ 0 0
$$441$$ 4.22425 0.201155
$$442$$ −2.62813 −0.125007
$$443$$ 19.0738 0.906224 0.453112 0.891454i $$-0.350314\pi$$
0.453112 + 0.891454i $$0.350314\pi$$
$$444$$ 3.92478 0.186262
$$445$$ 0 0
$$446$$ 1.29948 0.0615320
$$447$$ −1.53690 −0.0726931
$$448$$ 23.8251 1.12563
$$449$$ 35.8759 1.69309 0.846544 0.532318i $$-0.178679\pi$$
0.846544 + 0.532318i $$0.178679\pi$$
$$450$$ 0 0
$$451$$ −4.38787 −0.206617
$$452$$ −11.7743 −0.553818
$$453$$ 6.76116 0.317667
$$454$$ −3.28963 −0.154390
$$455$$ 0 0
$$456$$ −3.31406 −0.155195
$$457$$ −5.28963 −0.247438 −0.123719 0.992317i $$-0.539482\pi$$
−0.123719 + 0.992317i $$0.539482\pi$$
$$458$$ 5.01317 0.234250
$$459$$ −4.57452 −0.213520
$$460$$ 0 0
$$461$$ 36.3390 1.69248 0.846238 0.532805i $$-0.178862\pi$$
0.846238 + 0.532805i $$0.178862\pi$$
$$462$$ 0.649738 0.0302286
$$463$$ −10.5501 −0.490304 −0.245152 0.969485i $$-0.578838\pi$$
−0.245152 + 0.969485i $$0.578838\pi$$
$$464$$ −13.6385 −0.633150
$$465$$ 0 0
$$466$$ 3.73813 0.173166
$$467$$ −18.7005 −0.865357 −0.432679 0.901548i $$-0.642431\pi$$
−0.432679 + 0.901548i $$0.642431\pi$$
$$468$$ 5.81336 0.268723
$$469$$ 19.8496 0.916567
$$470$$ 0 0
$$471$$ −5.47627 −0.252333
$$472$$ −8.22284 −0.378487
$$473$$ 9.27504 0.426467
$$474$$ −2.23743 −0.102768
$$475$$ 0 0
$$476$$ 30.0752 1.37850
$$477$$ −4.70052 −0.215222
$$478$$ 5.14903 0.235511
$$479$$ −9.29948 −0.424904 −0.212452 0.977172i $$-0.568145\pi$$
−0.212452 + 0.977172i $$0.568145\pi$$
$$480$$ 0 0
$$481$$ −5.92478 −0.270147
$$482$$ 5.53690 0.252199
$$483$$ 22.4485 1.02144
$$484$$ −1.96239 −0.0891995
$$485$$ 0 0
$$486$$ −0.193937 −0.00879714
$$487$$ 35.4763 1.60758 0.803792 0.594911i $$-0.202813\pi$$
0.803792 + 0.594911i $$0.202813\pi$$
$$488$$ 6.68594 0.302658
$$489$$ 12.6253 0.570936
$$490$$ 0 0
$$491$$ 24.7757 1.11811 0.559057 0.829129i $$-0.311163\pi$$
0.559057 + 0.829129i $$0.311163\pi$$
$$492$$ −8.61071 −0.388201
$$493$$ −16.5237 −0.744191
$$494$$ 2.47768 0.111476
$$495$$ 0 0
$$496$$ 37.4734 1.68261
$$497$$ −33.2506 −1.49149
$$498$$ 2.11142 0.0946150
$$499$$ 14.1768 0.634640 0.317320 0.948318i $$-0.397217\pi$$
0.317320 + 0.948318i $$0.397217\pi$$
$$500$$ 0 0
$$501$$ 18.3634 0.820418
$$502$$ 5.80351 0.259023
$$503$$ 8.43866 0.376261 0.188131 0.982144i $$-0.439757\pi$$
0.188131 + 0.982144i $$0.439757\pi$$
$$504$$ 2.57452 0.114678
$$505$$ 0 0
$$506$$ 1.29948 0.0577688
$$507$$ 4.22425 0.187606
$$508$$ −28.6009 −1.26896
$$509$$ 1.10299 0.0488890 0.0244445 0.999701i $$-0.492218\pi$$
0.0244445 + 0.999701i $$0.492218\pi$$
$$510$$ 0 0
$$511$$ −25.9248 −1.14684
$$512$$ 14.3707 0.635103
$$513$$ 4.31265 0.190408
$$514$$ −1.68735 −0.0744258
$$515$$ 0 0
$$516$$ 18.2012 0.801265
$$517$$ 9.92478 0.436491
$$518$$ −1.29948 −0.0570957
$$519$$ −8.57452 −0.376379
$$520$$ 0 0
$$521$$ −12.4485 −0.545379 −0.272690 0.962102i $$-0.587913\pi$$
−0.272690 + 0.962102i $$0.587913\pi$$
$$522$$ −0.700523 −0.0306611
$$523$$ −30.0508 −1.31403 −0.657015 0.753878i $$-0.728181\pi$$
−0.657015 + 0.753878i $$0.728181\pi$$
$$524$$ 11.6267 0.507915
$$525$$ 0 0
$$526$$ −2.38313 −0.103910
$$527$$ 45.4010 1.97770
$$528$$ −3.77575 −0.164318
$$529$$ 21.8970 0.952044
$$530$$ 0 0
$$531$$ 10.7005 0.464363
$$532$$ −28.3536 −1.22928
$$533$$ 12.9986 0.563031
$$534$$ 0.538319 0.0232953
$$535$$ 0 0
$$536$$ 4.55291 0.196656
$$537$$ −14.1768 −0.611774
$$538$$ −1.13444 −0.0489093
$$539$$ 4.22425 0.181951
$$540$$ 0 0
$$541$$ −18.0000 −0.773880 −0.386940 0.922105i $$-0.626468\pi$$
−0.386940 + 0.922105i $$0.626468\pi$$
$$542$$ −0.986826 −0.0423878
$$543$$ 5.22425 0.224194
$$544$$ 10.3803 0.445052
$$545$$ 0 0
$$546$$ −1.92478 −0.0823729
$$547$$ 14.3028 0.611544 0.305772 0.952105i $$-0.401086\pi$$
0.305772 + 0.952105i $$0.401086\pi$$
$$548$$ 27.1782 1.16100
$$549$$ −8.70052 −0.371329
$$550$$ 0 0
$$551$$ 15.5778 0.663638
$$552$$ 5.14903 0.219157
$$553$$ −38.6516 −1.64364
$$554$$ −0.273624 −0.0116252
$$555$$ 0 0
$$556$$ −26.7123 −1.13285
$$557$$ 11.7988 0.499930 0.249965 0.968255i $$-0.419581\pi$$
0.249965 + 0.968255i $$0.419581\pi$$
$$558$$ 1.92478 0.0814823
$$559$$ −27.4763 −1.16212
$$560$$ 0 0
$$561$$ −4.57452 −0.193136
$$562$$ −0.850969 −0.0358960
$$563$$ −30.4847 −1.28478 −0.642389 0.766379i $$-0.722056\pi$$
−0.642389 + 0.766379i $$0.722056\pi$$
$$564$$ 19.4763 0.820099
$$565$$ 0 0
$$566$$ −5.15377 −0.216629
$$567$$ −3.35026 −0.140698
$$568$$ −7.62672 −0.320010
$$569$$ −27.0884 −1.13560 −0.567802 0.823165i $$-0.692206\pi$$
−0.567802 + 0.823165i $$0.692206\pi$$
$$570$$ 0 0
$$571$$ 7.28489 0.304863 0.152432 0.988314i $$-0.451290\pi$$
0.152432 + 0.988314i $$0.451290\pi$$
$$572$$ 5.81336 0.243069
$$573$$ 16.6253 0.694532
$$574$$ 2.85097 0.118997
$$575$$ 0 0
$$576$$ −7.11142 −0.296309
$$577$$ 31.6239 1.31652 0.658260 0.752791i $$-0.271293\pi$$
0.658260 + 0.752791i $$0.271293\pi$$
$$578$$ 0.761432 0.0316714
$$579$$ −16.3634 −0.680041
$$580$$ 0 0
$$581$$ 36.4749 1.51323
$$582$$ 0.0145884 0.000604711 0
$$583$$ −4.70052 −0.194676
$$584$$ −5.94639 −0.246063
$$585$$ 0 0
$$586$$ 0.664327 0.0274431
$$587$$ −33.1490 −1.36821 −0.684103 0.729385i $$-0.739806\pi$$
−0.684103 + 0.729385i $$0.739806\pi$$
$$588$$ 8.28963 0.341858
$$589$$ −42.8021 −1.76363
$$590$$ 0 0
$$591$$ −20.4241 −0.840134
$$592$$ 7.55149 0.310364
$$593$$ −34.4993 −1.41672 −0.708358 0.705853i $$-0.750564\pi$$
−0.708358 + 0.705853i $$0.750564\pi$$
$$594$$ −0.193937 −0.00795731
$$595$$ 0 0
$$596$$ −3.01600 −0.123540
$$597$$ 8.62530 0.353010
$$598$$ −3.84955 −0.157420
$$599$$ −14.4485 −0.590350 −0.295175 0.955443i $$-0.595378\pi$$
−0.295175 + 0.955443i $$0.595378\pi$$
$$600$$ 0 0
$$601$$ −15.9248 −0.649585 −0.324793 0.945785i $$-0.605295\pi$$
−0.324793 + 0.945785i $$0.605295\pi$$
$$602$$ −6.02635 −0.245616
$$603$$ −5.92478 −0.241276
$$604$$ 13.2680 0.539868
$$605$$ 0 0
$$606$$ 2.92619 0.118868
$$607$$ 14.5745 0.591561 0.295781 0.955256i $$-0.404420\pi$$
0.295781 + 0.955256i $$0.404420\pi$$
$$608$$ −9.78609 −0.396878
$$609$$ −12.1016 −0.490380
$$610$$ 0 0
$$611$$ −29.4010 −1.18944
$$612$$ −8.97698 −0.362873
$$613$$ −16.4123 −0.662887 −0.331443 0.943475i $$-0.607536\pi$$
−0.331443 + 0.943475i $$0.607536\pi$$
$$614$$ 3.22899 0.130312
$$615$$ 0 0
$$616$$ 2.57452 0.103730
$$617$$ 17.8496 0.718596 0.359298 0.933223i $$-0.383016\pi$$
0.359298 + 0.933223i $$0.383016\pi$$
$$618$$ −0.625301 −0.0251533
$$619$$ −0.402462 −0.0161763 −0.00808815 0.999967i $$-0.502575\pi$$
−0.00808815 + 0.999967i $$0.502575\pi$$
$$620$$ 0 0
$$621$$ −6.70052 −0.268883
$$622$$ 6.39963 0.256602
$$623$$ 9.29948 0.372576
$$624$$ 11.1852 0.447767
$$625$$ 0 0
$$626$$ −2.98683 −0.119378
$$627$$ 4.31265 0.172231
$$628$$ −10.7466 −0.428835
$$629$$ 9.14903 0.364796
$$630$$ 0 0
$$631$$ −38.0263 −1.51380 −0.756902 0.653528i $$-0.773288\pi$$
−0.756902 + 0.653528i $$0.773288\pi$$
$$632$$ −8.86556 −0.352653
$$633$$ −9.08840 −0.361231
$$634$$ −0.417050 −0.0165632
$$635$$ 0 0
$$636$$ −9.22425 −0.365765
$$637$$ −12.5139 −0.495818
$$638$$ −0.700523 −0.0277340
$$639$$ 9.92478 0.392618
$$640$$ 0 0
$$641$$ −28.0263 −1.10697 −0.553487 0.832858i $$-0.686703\pi$$
−0.553487 + 0.832858i $$0.686703\pi$$
$$642$$ −0.186642 −0.00736619
$$643$$ 4.62530 0.182404 0.0912020 0.995832i $$-0.470929\pi$$
0.0912020 + 0.995832i $$0.470929\pi$$
$$644$$ 44.0527 1.73592
$$645$$ 0 0
$$646$$ −3.82604 −0.150533
$$647$$ −23.5778 −0.926941 −0.463470 0.886112i $$-0.653396\pi$$
−0.463470 + 0.886112i $$0.653396\pi$$
$$648$$ −0.768452 −0.0301876
$$649$$ 10.7005 0.420032
$$650$$ 0 0
$$651$$ 33.2506 1.30319
$$652$$ 24.7757 0.970293
$$653$$ 2.25202 0.0881282 0.0440641 0.999029i $$-0.485969\pi$$
0.0440641 + 0.999029i $$0.485969\pi$$
$$654$$ −2.21108 −0.0864601
$$655$$ 0 0
$$656$$ −16.5675 −0.646852
$$657$$ 7.73813 0.301893
$$658$$ −6.44851 −0.251389
$$659$$ −41.4010 −1.61276 −0.806378 0.591401i $$-0.798575\pi$$
−0.806378 + 0.591401i $$0.798575\pi$$
$$660$$ 0 0
$$661$$ 3.40105 0.132285 0.0661427 0.997810i $$-0.478931\pi$$
0.0661427 + 0.997810i $$0.478931\pi$$
$$662$$ −2.82179 −0.109672
$$663$$ 13.5515 0.526296
$$664$$ 8.36626 0.324674
$$665$$ 0 0
$$666$$ 0.387873 0.0150298
$$667$$ −24.2031 −0.937149
$$668$$ 36.0362 1.39428
$$669$$ −6.70052 −0.259057
$$670$$ 0 0
$$671$$ −8.70052 −0.335880
$$672$$ 7.60228 0.293264
$$673$$ −0.887166 −0.0341977 −0.0170989 0.999854i $$-0.505443\pi$$
−0.0170989 + 0.999854i $$0.505443\pi$$
$$674$$ −3.15377 −0.121479
$$675$$ 0 0
$$676$$ 8.28963 0.318832
$$677$$ −18.9018 −0.726453 −0.363227 0.931701i $$-0.618325\pi$$
−0.363227 + 0.931701i $$0.618325\pi$$
$$678$$ −1.16362 −0.0446885
$$679$$ 0.252016 0.00967149
$$680$$ 0 0
$$681$$ 16.9624 0.650000
$$682$$ 1.92478 0.0737035
$$683$$ 20.8773 0.798848 0.399424 0.916766i $$-0.369210\pi$$
0.399424 + 0.916766i $$0.369210\pi$$
$$684$$ 8.46310 0.323595
$$685$$ 0 0
$$686$$ 1.80351 0.0688583
$$687$$ −25.8496 −0.986222
$$688$$ 35.0202 1.33513
$$689$$ 13.9248 0.530492
$$690$$ 0 0
$$691$$ −2.44851 −0.0931456 −0.0465728 0.998915i $$-0.514830\pi$$
−0.0465728 + 0.998915i $$0.514830\pi$$
$$692$$ −16.8265 −0.639649
$$693$$ −3.35026 −0.127266
$$694$$ 0.186642 0.00708485
$$695$$ 0 0
$$696$$ −2.77575 −0.105214
$$697$$ −20.0724 −0.760296
$$698$$ 4.01459 0.151954
$$699$$ −19.2750 −0.729049
$$700$$ 0 0
$$701$$ −2.98683 −0.112811 −0.0564054 0.998408i $$-0.517964\pi$$
−0.0564054 + 0.998408i $$0.517964\pi$$
$$702$$ 0.574515 0.0216837
$$703$$ −8.62530 −0.325309
$$704$$ −7.11142 −0.268022
$$705$$ 0 0
$$706$$ −3.98541 −0.149993
$$707$$ 50.5501 1.90113
$$708$$ 20.9986 0.789175
$$709$$ 24.1768 0.907979 0.453989 0.891007i $$-0.350000\pi$$
0.453989 + 0.891007i $$0.350000\pi$$
$$710$$ 0 0
$$711$$ 11.5369 0.432668
$$712$$ 2.13303 0.0799386
$$713$$ 66.5012 2.49049
$$714$$ 2.97224 0.111233
$$715$$ 0 0
$$716$$ −27.8204 −1.03970
$$717$$ −26.5501 −0.991531
$$718$$ 3.47627 0.129733
$$719$$ −30.0263 −1.11979 −0.559897 0.828562i $$-0.689159\pi$$
−0.559897 + 0.828562i $$0.689159\pi$$
$$720$$ 0 0
$$721$$ −10.8021 −0.402291
$$722$$ −0.0777777 −0.00289459
$$723$$ −28.5501 −1.06179
$$724$$ 10.2520 0.381013
$$725$$ 0 0
$$726$$ −0.193937 −0.00719766
$$727$$ −14.9525 −0.554559 −0.277279 0.960789i $$-0.589433\pi$$
−0.277279 + 0.960789i $$0.589433\pi$$
$$728$$ −7.62672 −0.282665
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 42.4288 1.56929
$$732$$ −17.0738 −0.631066
$$733$$ 19.1128 0.705949 0.352974 0.935633i $$-0.385170\pi$$
0.352974 + 0.935633i $$0.385170\pi$$
$$734$$ 5.75081 0.212266
$$735$$ 0 0
$$736$$ 15.2046 0.560447
$$737$$ −5.92478 −0.218242
$$738$$ −0.850969 −0.0313246
$$739$$ −3.31406 −0.121910 −0.0609549 0.998141i $$-0.519415\pi$$
−0.0609549 + 0.998141i $$0.519415\pi$$
$$740$$ 0 0
$$741$$ −12.7757 −0.469329
$$742$$ 3.05411 0.112120
$$743$$ 34.9887 1.28361 0.641806 0.766867i $$-0.278185\pi$$
0.641806 + 0.766867i $$0.278185\pi$$
$$744$$ 7.62672 0.279609
$$745$$ 0 0
$$746$$ 1.77242 0.0648930
$$747$$ −10.8872 −0.398341
$$748$$ −8.97698 −0.328231
$$749$$ −3.22425 −0.117812
$$750$$ 0 0
$$751$$ −26.9234 −0.982447 −0.491224 0.871033i $$-0.663450\pi$$
−0.491224 + 0.871033i $$0.663450\pi$$
$$752$$ 37.4734 1.36652
$$753$$ −29.9248 −1.09052
$$754$$ 2.07522 0.0755752
$$755$$ 0 0
$$756$$ −6.57452 −0.239113
$$757$$ −15.9248 −0.578796 −0.289398 0.957209i $$-0.593455\pi$$
−0.289398 + 0.957209i $$0.593455\pi$$
$$758$$ −3.87873 −0.140882
$$759$$ −6.70052 −0.243214
$$760$$ 0 0
$$761$$ −30.9380 −1.12150 −0.560750 0.827985i $$-0.689487\pi$$
−0.560750 + 0.827985i $$0.689487\pi$$
$$762$$ −2.82653 −0.102394
$$763$$ −38.1965 −1.38281
$$764$$ 32.6253 1.18034
$$765$$ 0 0
$$766$$ 6.77292 0.244715
$$767$$ −31.6991 −1.14459
$$768$$ −13.0752 −0.471811
$$769$$ 9.32582 0.336298 0.168149 0.985762i $$-0.446221\pi$$
0.168149 + 0.985762i $$0.446221\pi$$
$$770$$ 0 0
$$771$$ 8.70052 0.313342
$$772$$ −32.1114 −1.15572
$$773$$ −44.7005 −1.60777 −0.803883 0.594787i $$-0.797236\pi$$
−0.803883 + 0.594787i $$0.797236\pi$$
$$774$$ 1.79877 0.0646554
$$775$$ 0 0
$$776$$ 0.0578051 0.00207508
$$777$$ 6.70052 0.240380
$$778$$ 0.538319 0.0192997
$$779$$ 18.9234 0.678000
$$780$$ 0 0
$$781$$ 9.92478 0.355136
$$782$$ 5.94448 0.212574
$$783$$ 3.61213 0.129087
$$784$$ 15.9497 0.569633
$$785$$ 0 0
$$786$$ 1.14903 0.0409846
$$787$$ −21.6775 −0.772719 −0.386360 0.922348i $$-0.626268\pi$$
−0.386360 + 0.922348i $$0.626268\pi$$
$$788$$ −40.0800 −1.42779
$$789$$ 12.2882 0.437472
$$790$$ 0 0
$$791$$ −20.1016 −0.714730
$$792$$ −0.768452 −0.0273058
$$793$$ 25.7743 0.915273
$$794$$ 3.86414 0.137133
$$795$$ 0 0
$$796$$ 16.9262 0.599933
$$797$$ −22.7466 −0.805725 −0.402862 0.915261i $$-0.631985\pi$$
−0.402862 + 0.915261i $$0.631985\pi$$
$$798$$ −2.80209 −0.0991930
$$799$$ 45.4010 1.60617
$$800$$ 0 0
$$801$$ −2.77575 −0.0980762
$$802$$ 0.387873 0.0136963
$$803$$ 7.73813 0.273073
$$804$$ −11.6267 −0.410043
$$805$$ 0 0
$$806$$ −5.70194 −0.200842
$$807$$ 5.84955 0.205914
$$808$$ 11.5947 0.407900
$$809$$ −23.6121 −0.830158 −0.415079 0.909785i $$-0.636246\pi$$
−0.415079 + 0.909785i $$0.636246\pi$$
$$810$$ 0 0
$$811$$ −26.0870 −0.916038 −0.458019 0.888942i $$-0.651441\pi$$
−0.458019 + 0.888942i $$0.651441\pi$$
$$812$$ −23.7480 −0.833391
$$813$$ 5.08840 0.178458
$$814$$ 0.387873 0.0135949
$$815$$ 0 0
$$816$$ −17.2722 −0.604648
$$817$$ −40.0000 −1.39942
$$818$$ −2.53549 −0.0886513
$$819$$ 9.92478 0.346800
$$820$$ 0 0
$$821$$ −54.4142 −1.89907 −0.949535 0.313662i $$-0.898444\pi$$
−0.949535 + 0.313662i $$0.898444\pi$$
$$822$$ 2.68594 0.0936827
$$823$$ −0.121269 −0.00422716 −0.00211358 0.999998i $$-0.500673\pi$$
−0.00211358 + 0.999998i $$0.500673\pi$$
$$824$$ −2.47768 −0.0863142
$$825$$ 0 0
$$826$$ −6.95254 −0.241910
$$827$$ 18.2130 0.633328 0.316664 0.948538i $$-0.397437\pi$$
0.316664 + 0.948538i $$0.397437\pi$$
$$828$$ −13.1490 −0.456960
$$829$$ 13.0738 0.454072 0.227036 0.973886i $$-0.427096\pi$$
0.227036 + 0.973886i $$0.427096\pi$$
$$830$$ 0 0
$$831$$ 1.41090 0.0489434
$$832$$ 21.0668 0.730359
$$833$$ 19.3239 0.669534
$$834$$ −2.63989 −0.0914119
$$835$$ 0 0
$$836$$ 8.46310 0.292702
$$837$$ −9.92478 −0.343050
$$838$$ 1.40105 0.0483984
$$839$$ −26.5501 −0.916610 −0.458305 0.888795i $$-0.651543\pi$$
−0.458305 + 0.888795i $$0.651543\pi$$
$$840$$ 0 0
$$841$$ −15.9525 −0.550088
$$842$$ 5.93937 0.204684
$$843$$ 4.38787 0.151126
$$844$$ −17.8350 −0.613905
$$845$$ 0 0
$$846$$ 1.92478 0.0661752
$$847$$ −3.35026 −0.115116
$$848$$ −17.7480 −0.609468
$$849$$ 26.5745 0.912035
$$850$$ 0 0
$$851$$ 13.4010 0.459382
$$852$$ 19.4763 0.667246
$$853$$ −40.6155 −1.39065 −0.695323 0.718697i $$-0.744739\pi$$
−0.695323 + 0.718697i $$0.744739\pi$$
$$854$$ 5.65306 0.193444
$$855$$ 0 0
$$856$$ −0.739549 −0.0252773
$$857$$ −20.1721 −0.689064 −0.344532 0.938775i $$-0.611962\pi$$
−0.344532 + 0.938775i $$0.611962\pi$$
$$858$$ 0.574515 0.0196136
$$859$$ 21.8035 0.743926 0.371963 0.928248i $$-0.378685\pi$$
0.371963 + 0.928248i $$0.378685\pi$$
$$860$$ 0 0
$$861$$ −14.7005 −0.500993
$$862$$ −6.56978 −0.223767
$$863$$ −35.4274 −1.20596 −0.602981 0.797755i $$-0.706021\pi$$
−0.602981 + 0.797755i $$0.706021\pi$$
$$864$$ −2.26916 −0.0771984
$$865$$ 0 0
$$866$$ 1.83780 0.0624508
$$867$$ −3.92619 −0.133340
$$868$$ 65.2506 2.21475
$$869$$ 11.5369 0.391363
$$870$$ 0 0
$$871$$ 17.5515 0.594710
$$872$$ −8.76116 −0.296690
$$873$$ −0.0752228 −0.00254591
$$874$$ −5.60419 −0.189564
$$875$$ 0 0
$$876$$ 15.1852 0.513061
$$877$$ 14.0362 0.473969 0.236984 0.971513i $$-0.423841\pi$$
0.236984 + 0.971513i $$0.423841\pi$$
$$878$$ −5.71370 −0.192828
$$879$$ −3.42548 −0.115539
$$880$$ 0 0
$$881$$ −21.0738 −0.709995 −0.354997 0.934867i $$-0.615518\pi$$
−0.354997 + 0.934867i $$0.615518\pi$$
$$882$$ 0.819237 0.0275852
$$883$$ 42.1476 1.41838 0.709190 0.705017i $$-0.249061\pi$$
0.709190 + 0.705017i $$0.249061\pi$$
$$884$$ 26.5933 0.894429
$$885$$ 0 0
$$886$$ 3.69911 0.124274
$$887$$ −6.93604 −0.232889 −0.116445 0.993197i $$-0.537150\pi$$
−0.116445 + 0.993197i $$0.537150\pi$$
$$888$$ 1.53690 0.0515751
$$889$$ −48.8284 −1.63765
$$890$$ 0 0
$$891$$ 1.00000 0.0335013
$$892$$ −13.1490 −0.440262
$$893$$ −42.8021 −1.43232
$$894$$ −0.298062 −0.00996868
$$895$$ 0 0
$$896$$ 19.8251 0.662311
$$897$$ 19.8496 0.662757
$$898$$ 6.95765 0.232180
$$899$$ −35.8496 −1.19565
$$900$$ 0 0
$$901$$ −21.5026 −0.716356
$$902$$ −0.850969 −0.0283342
$$903$$ 31.0738 1.03407
$$904$$ −4.61071 −0.153350
$$905$$ 0 0
$$906$$ 1.31124 0.0435629
$$907$$ −53.2017 −1.76653 −0.883267 0.468870i $$-0.844661\pi$$
−0.883267 + 0.468870i $$0.844661\pi$$
$$908$$ 33.2868 1.10466
$$909$$ −15.0884 −0.500451
$$910$$ 0 0
$$911$$ 36.4749 1.20847 0.604233 0.796808i $$-0.293480\pi$$
0.604233 + 0.796808i $$0.293480\pi$$
$$912$$ 16.2835 0.539200
$$913$$ −10.8872 −0.360313
$$914$$ −1.02585 −0.0339322
$$915$$ 0 0
$$916$$ −50.7269 −1.67606
$$917$$ 19.8496 0.655490
$$918$$ −0.887166 −0.0292808
$$919$$ 9.73340 0.321075 0.160538 0.987030i $$-0.448677\pi$$
0.160538 + 0.987030i $$0.448677\pi$$
$$920$$ 0 0
$$921$$ −16.6497 −0.548628
$$922$$ 7.04746 0.232096
$$923$$ −29.4010 −0.967747
$$924$$ −6.57452 −0.216286
$$925$$ 0 0
$$926$$ −2.04605 −0.0672372
$$927$$ 3.22425 0.105898
$$928$$ −8.19649 −0.269063
$$929$$ −24.1768 −0.793215 −0.396607 0.917988i $$-0.629813\pi$$
−0.396607 + 0.917988i $$0.629813\pi$$
$$930$$ 0 0
$$931$$ −18.2177 −0.597062
$$932$$ −37.8251 −1.23900
$$933$$ −32.9986 −1.08033
$$934$$ −3.62672 −0.118670
$$935$$ 0 0
$$936$$ 2.27645 0.0744082
$$937$$ 7.48612 0.244561 0.122280 0.992496i $$-0.460979\pi$$
0.122280 + 0.992496i $$0.460979\pi$$
$$938$$ 3.84955 0.125692
$$939$$ 15.4010 0.502594
$$940$$ 0 0
$$941$$ −21.2360 −0.692274 −0.346137 0.938184i $$-0.612507\pi$$
−0.346137 + 0.938184i $$0.612507\pi$$
$$942$$ −1.06205 −0.0346034
$$943$$ −29.4010 −0.957430
$$944$$ 40.4025 1.31499
$$945$$ 0 0
$$946$$ 1.79877 0.0584830
$$947$$ 15.4763 0.502911 0.251456 0.967869i $$-0.419091\pi$$
0.251456 + 0.967869i $$0.419091\pi$$
$$948$$ 22.6399 0.735309
$$949$$ −22.9234 −0.744124
$$950$$ 0 0
$$951$$ 2.15045 0.0697330
$$952$$ 11.7772 0.381700
$$953$$ 32.0508 1.03823 0.519113 0.854705i $$-0.326262\pi$$
0.519113 + 0.854705i $$0.326262\pi$$
$$954$$ −0.911603 −0.0295142
$$955$$ 0 0
$$956$$ −52.1016 −1.68509
$$957$$ 3.61213 0.116763
$$958$$ −1.80351 −0.0582687
$$959$$ 46.3996 1.49832
$$960$$ 0 0
$$961$$ 67.5012 2.17746
$$962$$ −1.14903 −0.0370462
$$963$$ 0.962389 0.0310125
$$964$$ −56.0263 −1.80449
$$965$$ 0 0
$$966$$ 4.35359 0.140074
$$967$$ 17.3766 0.558794 0.279397 0.960176i $$-0.409865\pi$$
0.279397 + 0.960176i $$0.409865\pi$$
$$968$$ −0.768452 −0.0246990
$$969$$ 19.7283 0.633764
$$970$$ 0 0
$$971$$ 36.2031 1.16181 0.580907 0.813970i $$-0.302698\pi$$
0.580907 + 0.813970i $$0.302698\pi$$
$$972$$ 1.96239 0.0629436
$$973$$ −45.6042 −1.46200
$$974$$ 6.88015 0.220454
$$975$$ 0 0
$$976$$ −32.8510 −1.05153
$$977$$ 28.1476 0.900522 0.450261 0.892897i $$-0.351331\pi$$
0.450261 + 0.892897i $$0.351331\pi$$
$$978$$ 2.44851 0.0782946
$$979$$ −2.77575 −0.0887132
$$980$$ 0 0
$$981$$ 11.4010 0.364007
$$982$$ 4.80492 0.153331
$$983$$ −7.07381 −0.225619 −0.112810 0.993617i $$-0.535985\pi$$
−0.112810 + 0.993617i $$0.535985\pi$$
$$984$$ −3.37187 −0.107491
$$985$$ 0 0
$$986$$ −3.20456 −0.102054
$$987$$ 33.2506 1.05838
$$988$$ −25.0710 −0.797614
$$989$$ 62.1476 1.97618
$$990$$ 0 0
$$991$$ 44.4260 1.41124 0.705619 0.708592i $$-0.250669\pi$$
0.705619 + 0.708592i $$0.250669\pi$$
$$992$$ 22.5209 0.715039
$$993$$ 14.5501 0.461733
$$994$$ −6.44851 −0.204534
$$995$$ 0 0
$$996$$ −21.3649 −0.676971
$$997$$ 28.4847 0.902120 0.451060 0.892494i $$-0.351046\pi$$
0.451060 + 0.892494i $$0.351046\pi$$
$$998$$ 2.74940 0.0870307
$$999$$ −2.00000 −0.0632772
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 825.2.a.k.1.2 3
3.2 odd 2 2475.2.a.bb.1.2 3
5.2 odd 4 825.2.c.g.199.4 6
5.3 odd 4 825.2.c.g.199.3 6
5.4 even 2 165.2.a.c.1.2 3
11.10 odd 2 9075.2.a.cf.1.2 3
15.2 even 4 2475.2.c.r.199.3 6
15.8 even 4 2475.2.c.r.199.4 6
15.14 odd 2 495.2.a.e.1.2 3
20.19 odd 2 2640.2.a.be.1.1 3
35.34 odd 2 8085.2.a.bk.1.2 3
55.54 odd 2 1815.2.a.m.1.2 3
60.59 even 2 7920.2.a.cj.1.1 3
165.164 even 2 5445.2.a.z.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.a.c.1.2 3 5.4 even 2
495.2.a.e.1.2 3 15.14 odd 2
825.2.a.k.1.2 3 1.1 even 1 trivial
825.2.c.g.199.3 6 5.3 odd 4
825.2.c.g.199.4 6 5.2 odd 4
1815.2.a.m.1.2 3 55.54 odd 2
2475.2.a.bb.1.2 3 3.2 odd 2
2475.2.c.r.199.3 6 15.2 even 4
2475.2.c.r.199.4 6 15.8 even 4
2640.2.a.be.1.1 3 20.19 odd 2
5445.2.a.z.1.2 3 165.164 even 2
7920.2.a.cj.1.1 3 60.59 even 2
8085.2.a.bk.1.2 3 35.34 odd 2
9075.2.a.cf.1.2 3 11.10 odd 2