Properties

 Label 507.2.a.f.1.1 Level $507$ Weight $2$ Character 507.1 Self dual yes Analytic conductor $4.048$ Analytic rank $0$ Dimension $2$ Inner twists $2$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [507,2,Mod(1,507)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(507, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("507.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$507 = 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 507.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: $$4.04841538248$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{12})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 39) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.1 Root $$-1.73205$$ of defining polynomial Character $$\chi$$ $$=$$ 507.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.73205 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.73205 q^{6} -3.46410 q^{7} +1.73205 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.73205 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.73205 q^{6} -3.46410 q^{7} +1.73205 q^{8} +1.00000 q^{9} -3.46410 q^{11} -1.00000 q^{12} +6.00000 q^{14} -5.00000 q^{16} +6.00000 q^{17} -1.73205 q^{18} -3.46410 q^{19} +3.46410 q^{21} +6.00000 q^{22} -1.73205 q^{24} -5.00000 q^{25} -1.00000 q^{27} -3.46410 q^{28} +6.00000 q^{29} +3.46410 q^{31} +5.19615 q^{32} +3.46410 q^{33} -10.3923 q^{34} +1.00000 q^{36} +6.92820 q^{37} +6.00000 q^{38} +6.92820 q^{41} -6.00000 q^{42} +4.00000 q^{43} -3.46410 q^{44} +3.46410 q^{47} +5.00000 q^{48} +5.00000 q^{49} +8.66025 q^{50} -6.00000 q^{51} +6.00000 q^{53} +1.73205 q^{54} -6.00000 q^{56} +3.46410 q^{57} -10.3923 q^{58} +10.3923 q^{59} -2.00000 q^{61} -6.00000 q^{62} -3.46410 q^{63} +1.00000 q^{64} -6.00000 q^{66} +10.3923 q^{67} +6.00000 q^{68} -3.46410 q^{71} +1.73205 q^{72} -12.0000 q^{74} +5.00000 q^{75} -3.46410 q^{76} +12.0000 q^{77} -8.00000 q^{79} +1.00000 q^{81} -12.0000 q^{82} +3.46410 q^{83} +3.46410 q^{84} -6.92820 q^{86} -6.00000 q^{87} -6.00000 q^{88} -6.92820 q^{89} -3.46410 q^{93} -6.00000 q^{94} -5.19615 q^{96} +13.8564 q^{97} -8.66025 q^{98} -3.46410 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + 2 q^{4} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 + 2 * q^4 + 2 * q^9 $$2 q - 2 q^{3} + 2 q^{4} + 2 q^{9} - 2 q^{12} + 12 q^{14} - 10 q^{16} + 12 q^{17} + 12 q^{22} - 10 q^{25} - 2 q^{27} + 12 q^{29} + 2 q^{36} + 12 q^{38} - 12 q^{42} + 8 q^{43} + 10 q^{48} + 10 q^{49} - 12 q^{51} + 12 q^{53} - 12 q^{56} - 4 q^{61} - 12 q^{62} + 2 q^{64} - 12 q^{66} + 12 q^{68} - 24 q^{74} + 10 q^{75} + 24 q^{77} - 16 q^{79} + 2 q^{81} - 24 q^{82} - 12 q^{87} - 12 q^{88} - 12 q^{94}+O(q^{100})$$ 2 * q - 2 * q^3 + 2 * q^4 + 2 * q^9 - 2 * q^12 + 12 * q^14 - 10 * q^16 + 12 * q^17 + 12 * q^22 - 10 * q^25 - 2 * q^27 + 12 * q^29 + 2 * q^36 + 12 * q^38 - 12 * q^42 + 8 * q^43 + 10 * q^48 + 10 * q^49 - 12 * q^51 + 12 * q^53 - 12 * q^56 - 4 * q^61 - 12 * q^62 + 2 * q^64 - 12 * q^66 + 12 * q^68 - 24 * q^74 + 10 * q^75 + 24 * q^77 - 16 * q^79 + 2 * q^81 - 24 * q^82 - 12 * q^87 - 12 * q^88 - 12 * q^94

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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.73205 −1.22474 −0.612372 0.790569i $$-0.709785\pi$$
−0.612372 + 0.790569i $$0.709785\pi$$
$$3$$ −1.00000 −0.577350
$$4$$ 1.00000 0.500000
$$5$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$6$$ 1.73205 0.707107
$$7$$ −3.46410 −1.30931 −0.654654 0.755929i $$-0.727186\pi$$
−0.654654 + 0.755929i $$0.727186\pi$$
$$8$$ 1.73205 0.612372
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −3.46410 −1.04447 −0.522233 0.852803i $$-0.674901\pi$$
−0.522233 + 0.852803i $$0.674901\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ 0 0
$$14$$ 6.00000 1.60357
$$15$$ 0 0
$$16$$ −5.00000 −1.25000
$$17$$ 6.00000 1.45521 0.727607 0.685994i $$-0.240633\pi$$
0.727607 + 0.685994i $$0.240633\pi$$
$$18$$ −1.73205 −0.408248
$$19$$ −3.46410 −0.794719 −0.397360 0.917663i $$-0.630073\pi$$
−0.397360 + 0.917663i $$0.630073\pi$$
$$20$$ 0 0
$$21$$ 3.46410 0.755929
$$22$$ 6.00000 1.27920
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ −1.73205 −0.353553
$$25$$ −5.00000 −1.00000
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ −3.46410 −0.654654
$$29$$ 6.00000 1.11417 0.557086 0.830455i $$-0.311919\pi$$
0.557086 + 0.830455i $$0.311919\pi$$
$$30$$ 0 0
$$31$$ 3.46410 0.622171 0.311086 0.950382i $$-0.399307\pi$$
0.311086 + 0.950382i $$0.399307\pi$$
$$32$$ 5.19615 0.918559
$$33$$ 3.46410 0.603023
$$34$$ −10.3923 −1.78227
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 6.92820 1.13899 0.569495 0.821995i $$-0.307139\pi$$
0.569495 + 0.821995i $$0.307139\pi$$
$$38$$ 6.00000 0.973329
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 6.92820 1.08200 0.541002 0.841021i $$-0.318045\pi$$
0.541002 + 0.841021i $$0.318045\pi$$
$$42$$ −6.00000 −0.925820
$$43$$ 4.00000 0.609994 0.304997 0.952353i $$-0.401344\pi$$
0.304997 + 0.952353i $$0.401344\pi$$
$$44$$ −3.46410 −0.522233
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 3.46410 0.505291 0.252646 0.967559i $$-0.418699\pi$$
0.252646 + 0.967559i $$0.418699\pi$$
$$48$$ 5.00000 0.721688
$$49$$ 5.00000 0.714286
$$50$$ 8.66025 1.22474
$$51$$ −6.00000 −0.840168
$$52$$ 0 0
$$53$$ 6.00000 0.824163 0.412082 0.911147i $$-0.364802\pi$$
0.412082 + 0.911147i $$0.364802\pi$$
$$54$$ 1.73205 0.235702
$$55$$ 0 0
$$56$$ −6.00000 −0.801784
$$57$$ 3.46410 0.458831
$$58$$ −10.3923 −1.36458
$$59$$ 10.3923 1.35296 0.676481 0.736460i $$-0.263504\pi$$
0.676481 + 0.736460i $$0.263504\pi$$
$$60$$ 0 0
$$61$$ −2.00000 −0.256074 −0.128037 0.991769i $$-0.540868\pi$$
−0.128037 + 0.991769i $$0.540868\pi$$
$$62$$ −6.00000 −0.762001
$$63$$ −3.46410 −0.436436
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ −6.00000 −0.738549
$$67$$ 10.3923 1.26962 0.634811 0.772667i $$-0.281078\pi$$
0.634811 + 0.772667i $$0.281078\pi$$
$$68$$ 6.00000 0.727607
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −3.46410 −0.411113 −0.205557 0.978645i $$-0.565900\pi$$
−0.205557 + 0.978645i $$0.565900\pi$$
$$72$$ 1.73205 0.204124
$$73$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$74$$ −12.0000 −1.39497
$$75$$ 5.00000 0.577350
$$76$$ −3.46410 −0.397360
$$77$$ 12.0000 1.36753
$$78$$ 0 0
$$79$$ −8.00000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ −12.0000 −1.32518
$$83$$ 3.46410 0.380235 0.190117 0.981761i $$-0.439113\pi$$
0.190117 + 0.981761i $$0.439113\pi$$
$$84$$ 3.46410 0.377964
$$85$$ 0 0
$$86$$ −6.92820 −0.747087
$$87$$ −6.00000 −0.643268
$$88$$ −6.00000 −0.639602
$$89$$ −6.92820 −0.734388 −0.367194 0.930144i $$-0.619682\pi$$
−0.367194 + 0.930144i $$0.619682\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ −3.46410 −0.359211
$$94$$ −6.00000 −0.618853
$$95$$ 0 0
$$96$$ −5.19615 −0.530330
$$97$$ 13.8564 1.40690 0.703452 0.710742i $$-0.251641\pi$$
0.703452 + 0.710742i $$0.251641\pi$$
$$98$$ −8.66025 −0.874818
$$99$$ −3.46410 −0.348155
$$100$$ −5.00000 −0.500000
$$101$$ −6.00000 −0.597022 −0.298511 0.954406i $$-0.596490\pi$$
−0.298511 + 0.954406i $$0.596490\pi$$
$$102$$ 10.3923 1.02899
$$103$$ −8.00000 −0.788263 −0.394132 0.919054i $$-0.628955\pi$$
−0.394132 + 0.919054i $$0.628955\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ −10.3923 −1.00939
$$107$$ 12.0000 1.16008 0.580042 0.814587i $$-0.303036\pi$$
0.580042 + 0.814587i $$0.303036\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ −6.92820 −0.663602 −0.331801 0.943349i $$-0.607656\pi$$
−0.331801 + 0.943349i $$0.607656\pi$$
$$110$$ 0 0
$$111$$ −6.92820 −0.657596
$$112$$ 17.3205 1.63663
$$113$$ −6.00000 −0.564433 −0.282216 0.959351i $$-0.591070\pi$$
−0.282216 + 0.959351i $$0.591070\pi$$
$$114$$ −6.00000 −0.561951
$$115$$ 0 0
$$116$$ 6.00000 0.557086
$$117$$ 0 0
$$118$$ −18.0000 −1.65703
$$119$$ −20.7846 −1.90532
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 3.46410 0.313625
$$123$$ −6.92820 −0.624695
$$124$$ 3.46410 0.311086
$$125$$ 0 0
$$126$$ 6.00000 0.534522
$$127$$ 8.00000 0.709885 0.354943 0.934888i $$-0.384500\pi$$
0.354943 + 0.934888i $$0.384500\pi$$
$$128$$ −12.1244 −1.07165
$$129$$ −4.00000 −0.352180
$$130$$ 0 0
$$131$$ −12.0000 −1.04844 −0.524222 0.851581i $$-0.675644\pi$$
−0.524222 + 0.851581i $$0.675644\pi$$
$$132$$ 3.46410 0.301511
$$133$$ 12.0000 1.04053
$$134$$ −18.0000 −1.55496
$$135$$ 0 0
$$136$$ 10.3923 0.891133
$$137$$ −20.7846 −1.77575 −0.887875 0.460086i $$-0.847819\pi$$
−0.887875 + 0.460086i $$0.847819\pi$$
$$138$$ 0 0
$$139$$ −4.00000 −0.339276 −0.169638 0.985506i $$-0.554260\pi$$
−0.169638 + 0.985506i $$0.554260\pi$$
$$140$$ 0 0
$$141$$ −3.46410 −0.291730
$$142$$ 6.00000 0.503509
$$143$$ 0 0
$$144$$ −5.00000 −0.416667
$$145$$ 0 0
$$146$$ 0 0
$$147$$ −5.00000 −0.412393
$$148$$ 6.92820 0.569495
$$149$$ −13.8564 −1.13516 −0.567581 0.823318i $$-0.692120\pi$$
−0.567581 + 0.823318i $$0.692120\pi$$
$$150$$ −8.66025 −0.707107
$$151$$ 10.3923 0.845714 0.422857 0.906196i $$-0.361027\pi$$
0.422857 + 0.906196i $$0.361027\pi$$
$$152$$ −6.00000 −0.486664
$$153$$ 6.00000 0.485071
$$154$$ −20.7846 −1.67487
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 14.0000 1.11732 0.558661 0.829396i $$-0.311315\pi$$
0.558661 + 0.829396i $$0.311315\pi$$
$$158$$ 13.8564 1.10236
$$159$$ −6.00000 −0.475831
$$160$$ 0 0
$$161$$ 0 0
$$162$$ −1.73205 −0.136083
$$163$$ 3.46410 0.271329 0.135665 0.990755i $$-0.456683\pi$$
0.135665 + 0.990755i $$0.456683\pi$$
$$164$$ 6.92820 0.541002
$$165$$ 0 0
$$166$$ −6.00000 −0.465690
$$167$$ 17.3205 1.34030 0.670151 0.742225i $$-0.266230\pi$$
0.670151 + 0.742225i $$0.266230\pi$$
$$168$$ 6.00000 0.462910
$$169$$ 0 0
$$170$$ 0 0
$$171$$ −3.46410 −0.264906
$$172$$ 4.00000 0.304997
$$173$$ 18.0000 1.36851 0.684257 0.729241i $$-0.260127\pi$$
0.684257 + 0.729241i $$0.260127\pi$$
$$174$$ 10.3923 0.787839
$$175$$ 17.3205 1.30931
$$176$$ 17.3205 1.30558
$$177$$ −10.3923 −0.781133
$$178$$ 12.0000 0.899438
$$179$$ 12.0000 0.896922 0.448461 0.893802i $$-0.351972\pi$$
0.448461 + 0.893802i $$0.351972\pi$$
$$180$$ 0 0
$$181$$ 10.0000 0.743294 0.371647 0.928374i $$-0.378793\pi$$
0.371647 + 0.928374i $$0.378793\pi$$
$$182$$ 0 0
$$183$$ 2.00000 0.147844
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 6.00000 0.439941
$$187$$ −20.7846 −1.51992
$$188$$ 3.46410 0.252646
$$189$$ 3.46410 0.251976
$$190$$ 0 0
$$191$$ 24.0000 1.73658 0.868290 0.496058i $$-0.165220\pi$$
0.868290 + 0.496058i $$0.165220\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$194$$ −24.0000 −1.72310
$$195$$ 0 0
$$196$$ 5.00000 0.357143
$$197$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$198$$ 6.00000 0.426401
$$199$$ −16.0000 −1.13421 −0.567105 0.823646i $$-0.691937\pi$$
−0.567105 + 0.823646i $$0.691937\pi$$
$$200$$ −8.66025 −0.612372
$$201$$ −10.3923 −0.733017
$$202$$ 10.3923 0.731200
$$203$$ −20.7846 −1.45879
$$204$$ −6.00000 −0.420084
$$205$$ 0 0
$$206$$ 13.8564 0.965422
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 12.0000 0.830057
$$210$$ 0 0
$$211$$ −20.0000 −1.37686 −0.688428 0.725304i $$-0.741699\pi$$
−0.688428 + 0.725304i $$0.741699\pi$$
$$212$$ 6.00000 0.412082
$$213$$ 3.46410 0.237356
$$214$$ −20.7846 −1.42081
$$215$$ 0 0
$$216$$ −1.73205 −0.117851
$$217$$ −12.0000 −0.814613
$$218$$ 12.0000 0.812743
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 12.0000 0.805387
$$223$$ 3.46410 0.231973 0.115987 0.993251i $$-0.462997\pi$$
0.115987 + 0.993251i $$0.462997\pi$$
$$224$$ −18.0000 −1.20268
$$225$$ −5.00000 −0.333333
$$226$$ 10.3923 0.691286
$$227$$ 17.3205 1.14960 0.574801 0.818293i $$-0.305079\pi$$
0.574801 + 0.818293i $$0.305079\pi$$
$$228$$ 3.46410 0.229416
$$229$$ −6.92820 −0.457829 −0.228914 0.973447i $$-0.573518\pi$$
−0.228914 + 0.973447i $$0.573518\pi$$
$$230$$ 0 0
$$231$$ −12.0000 −0.789542
$$232$$ 10.3923 0.682288
$$233$$ 6.00000 0.393073 0.196537 0.980497i $$-0.437031\pi$$
0.196537 + 0.980497i $$0.437031\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 10.3923 0.676481
$$237$$ 8.00000 0.519656
$$238$$ 36.0000 2.33353
$$239$$ 10.3923 0.672222 0.336111 0.941822i $$-0.390888\pi$$
0.336111 + 0.941822i $$0.390888\pi$$
$$240$$ 0 0
$$241$$ 13.8564 0.892570 0.446285 0.894891i $$-0.352747\pi$$
0.446285 + 0.894891i $$0.352747\pi$$
$$242$$ −1.73205 −0.111340
$$243$$ −1.00000 −0.0641500
$$244$$ −2.00000 −0.128037
$$245$$ 0 0
$$246$$ 12.0000 0.765092
$$247$$ 0 0
$$248$$ 6.00000 0.381000
$$249$$ −3.46410 −0.219529
$$250$$ 0 0
$$251$$ 12.0000 0.757433 0.378717 0.925513i $$-0.376365\pi$$
0.378717 + 0.925513i $$0.376365\pi$$
$$252$$ −3.46410 −0.218218
$$253$$ 0 0
$$254$$ −13.8564 −0.869428
$$255$$ 0 0
$$256$$ 19.0000 1.18750
$$257$$ −18.0000 −1.12281 −0.561405 0.827541i $$-0.689739\pi$$
−0.561405 + 0.827541i $$0.689739\pi$$
$$258$$ 6.92820 0.431331
$$259$$ −24.0000 −1.49129
$$260$$ 0 0
$$261$$ 6.00000 0.371391
$$262$$ 20.7846 1.28408
$$263$$ −24.0000 −1.47990 −0.739952 0.672660i $$-0.765152\pi$$
−0.739952 + 0.672660i $$0.765152\pi$$
$$264$$ 6.00000 0.369274
$$265$$ 0 0
$$266$$ −20.7846 −1.27439
$$267$$ 6.92820 0.423999
$$268$$ 10.3923 0.634811
$$269$$ 6.00000 0.365826 0.182913 0.983129i $$-0.441447\pi$$
0.182913 + 0.983129i $$0.441447\pi$$
$$270$$ 0 0
$$271$$ 10.3923 0.631288 0.315644 0.948878i $$-0.397780\pi$$
0.315644 + 0.948878i $$0.397780\pi$$
$$272$$ −30.0000 −1.81902
$$273$$ 0 0
$$274$$ 36.0000 2.17484
$$275$$ 17.3205 1.04447
$$276$$ 0 0
$$277$$ 10.0000 0.600842 0.300421 0.953807i $$-0.402873\pi$$
0.300421 + 0.953807i $$0.402873\pi$$
$$278$$ 6.92820 0.415526
$$279$$ 3.46410 0.207390
$$280$$ 0 0
$$281$$ 6.92820 0.413302 0.206651 0.978415i $$-0.433744\pi$$
0.206651 + 0.978415i $$0.433744\pi$$
$$282$$ 6.00000 0.357295
$$283$$ −4.00000 −0.237775 −0.118888 0.992908i $$-0.537933\pi$$
−0.118888 + 0.992908i $$0.537933\pi$$
$$284$$ −3.46410 −0.205557
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −24.0000 −1.41668
$$288$$ 5.19615 0.306186
$$289$$ 19.0000 1.11765
$$290$$ 0 0
$$291$$ −13.8564 −0.812277
$$292$$ 0 0
$$293$$ 27.7128 1.61900 0.809500 0.587120i $$-0.199738\pi$$
0.809500 + 0.587120i $$0.199738\pi$$
$$294$$ 8.66025 0.505076
$$295$$ 0 0
$$296$$ 12.0000 0.697486
$$297$$ 3.46410 0.201008
$$298$$ 24.0000 1.39028
$$299$$ 0 0
$$300$$ 5.00000 0.288675
$$301$$ −13.8564 −0.798670
$$302$$ −18.0000 −1.03578
$$303$$ 6.00000 0.344691
$$304$$ 17.3205 0.993399
$$305$$ 0 0
$$306$$ −10.3923 −0.594089
$$307$$ −10.3923 −0.593120 −0.296560 0.955014i $$-0.595840\pi$$
−0.296560 + 0.955014i $$0.595840\pi$$
$$308$$ 12.0000 0.683763
$$309$$ 8.00000 0.455104
$$310$$ 0 0
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ 0 0
$$313$$ 10.0000 0.565233 0.282617 0.959233i $$-0.408798\pi$$
0.282617 + 0.959233i $$0.408798\pi$$
$$314$$ −24.2487 −1.36843
$$315$$ 0 0
$$316$$ −8.00000 −0.450035
$$317$$ −13.8564 −0.778253 −0.389127 0.921184i $$-0.627223\pi$$
−0.389127 + 0.921184i $$0.627223\pi$$
$$318$$ 10.3923 0.582772
$$319$$ −20.7846 −1.16371
$$320$$ 0 0
$$321$$ −12.0000 −0.669775
$$322$$ 0 0
$$323$$ −20.7846 −1.15649
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ −6.00000 −0.332309
$$327$$ 6.92820 0.383131
$$328$$ 12.0000 0.662589
$$329$$ −12.0000 −0.661581
$$330$$ 0 0
$$331$$ −3.46410 −0.190404 −0.0952021 0.995458i $$-0.530350\pi$$
−0.0952021 + 0.995458i $$0.530350\pi$$
$$332$$ 3.46410 0.190117
$$333$$ 6.92820 0.379663
$$334$$ −30.0000 −1.64153
$$335$$ 0 0
$$336$$ −17.3205 −0.944911
$$337$$ 14.0000 0.762629 0.381314 0.924445i $$-0.375472\pi$$
0.381314 + 0.924445i $$0.375472\pi$$
$$338$$ 0 0
$$339$$ 6.00000 0.325875
$$340$$ 0 0
$$341$$ −12.0000 −0.649836
$$342$$ 6.00000 0.324443
$$343$$ 6.92820 0.374088
$$344$$ 6.92820 0.373544
$$345$$ 0 0
$$346$$ −31.1769 −1.67608
$$347$$ 36.0000 1.93258 0.966291 0.257454i $$-0.0828835\pi$$
0.966291 + 0.257454i $$0.0828835\pi$$
$$348$$ −6.00000 −0.321634
$$349$$ −6.92820 −0.370858 −0.185429 0.982658i $$-0.559368\pi$$
−0.185429 + 0.982658i $$0.559368\pi$$
$$350$$ −30.0000 −1.60357
$$351$$ 0 0
$$352$$ −18.0000 −0.959403
$$353$$ −34.6410 −1.84376 −0.921878 0.387481i $$-0.873345\pi$$
−0.921878 + 0.387481i $$0.873345\pi$$
$$354$$ 18.0000 0.956689
$$355$$ 0 0
$$356$$ −6.92820 −0.367194
$$357$$ 20.7846 1.10004
$$358$$ −20.7846 −1.09850
$$359$$ 17.3205 0.914141 0.457071 0.889430i $$-0.348899\pi$$
0.457071 + 0.889430i $$0.348899\pi$$
$$360$$ 0 0
$$361$$ −7.00000 −0.368421
$$362$$ −17.3205 −0.910346
$$363$$ −1.00000 −0.0524864
$$364$$ 0 0
$$365$$ 0 0
$$366$$ −3.46410 −0.181071
$$367$$ −16.0000 −0.835193 −0.417597 0.908633i $$-0.637127\pi$$
−0.417597 + 0.908633i $$0.637127\pi$$
$$368$$ 0 0
$$369$$ 6.92820 0.360668
$$370$$ 0 0
$$371$$ −20.7846 −1.07908
$$372$$ −3.46410 −0.179605
$$373$$ 22.0000 1.13912 0.569558 0.821951i $$-0.307114\pi$$
0.569558 + 0.821951i $$0.307114\pi$$
$$374$$ 36.0000 1.86152
$$375$$ 0 0
$$376$$ 6.00000 0.309426
$$377$$ 0 0
$$378$$ −6.00000 −0.308607
$$379$$ −17.3205 −0.889695 −0.444847 0.895606i $$-0.646742\pi$$
−0.444847 + 0.895606i $$0.646742\pi$$
$$380$$ 0 0
$$381$$ −8.00000 −0.409852
$$382$$ −41.5692 −2.12687
$$383$$ −3.46410 −0.177007 −0.0885037 0.996076i $$-0.528208\pi$$
−0.0885037 + 0.996076i $$0.528208\pi$$
$$384$$ 12.1244 0.618718
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 4.00000 0.203331
$$388$$ 13.8564 0.703452
$$389$$ 18.0000 0.912636 0.456318 0.889817i $$-0.349168\pi$$
0.456318 + 0.889817i $$0.349168\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 8.66025 0.437409
$$393$$ 12.0000 0.605320
$$394$$ 0 0
$$395$$ 0 0
$$396$$ −3.46410 −0.174078
$$397$$ −34.6410 −1.73858 −0.869291 0.494300i $$-0.835424\pi$$
−0.869291 + 0.494300i $$0.835424\pi$$
$$398$$ 27.7128 1.38912
$$399$$ −12.0000 −0.600751
$$400$$ 25.0000 1.25000
$$401$$ 6.92820 0.345978 0.172989 0.984924i $$-0.444657\pi$$
0.172989 + 0.984924i $$0.444657\pi$$
$$402$$ 18.0000 0.897758
$$403$$ 0 0
$$404$$ −6.00000 −0.298511
$$405$$ 0 0
$$406$$ 36.0000 1.78665
$$407$$ −24.0000 −1.18964
$$408$$ −10.3923 −0.514496
$$409$$ −27.7128 −1.37031 −0.685155 0.728397i $$-0.740266\pi$$
−0.685155 + 0.728397i $$0.740266\pi$$
$$410$$ 0 0
$$411$$ 20.7846 1.02523
$$412$$ −8.00000 −0.394132
$$413$$ −36.0000 −1.77144
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 4.00000 0.195881
$$418$$ −20.7846 −1.01661
$$419$$ −12.0000 −0.586238 −0.293119 0.956076i $$-0.594693\pi$$
−0.293119 + 0.956076i $$0.594693\pi$$
$$420$$ 0 0
$$421$$ 34.6410 1.68830 0.844150 0.536107i $$-0.180106\pi$$
0.844150 + 0.536107i $$0.180106\pi$$
$$422$$ 34.6410 1.68630
$$423$$ 3.46410 0.168430
$$424$$ 10.3923 0.504695
$$425$$ −30.0000 −1.45521
$$426$$ −6.00000 −0.290701
$$427$$ 6.92820 0.335279
$$428$$ 12.0000 0.580042
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 24.2487 1.16802 0.584010 0.811747i $$-0.301483\pi$$
0.584010 + 0.811747i $$0.301483\pi$$
$$432$$ 5.00000 0.240563
$$433$$ −34.0000 −1.63394 −0.816968 0.576683i $$-0.804347\pi$$
−0.816968 + 0.576683i $$0.804347\pi$$
$$434$$ 20.7846 0.997693
$$435$$ 0 0
$$436$$ −6.92820 −0.331801
$$437$$ 0 0
$$438$$ 0 0
$$439$$ −8.00000 −0.381819 −0.190910 0.981608i $$-0.561144\pi$$
−0.190910 + 0.981608i $$0.561144\pi$$
$$440$$ 0 0
$$441$$ 5.00000 0.238095
$$442$$ 0 0
$$443$$ −36.0000 −1.71041 −0.855206 0.518289i $$-0.826569\pi$$
−0.855206 + 0.518289i $$0.826569\pi$$
$$444$$ −6.92820 −0.328798
$$445$$ 0 0
$$446$$ −6.00000 −0.284108
$$447$$ 13.8564 0.655386
$$448$$ −3.46410 −0.163663
$$449$$ 6.92820 0.326962 0.163481 0.986546i $$-0.447728\pi$$
0.163481 + 0.986546i $$0.447728\pi$$
$$450$$ 8.66025 0.408248
$$451$$ −24.0000 −1.13012
$$452$$ −6.00000 −0.282216
$$453$$ −10.3923 −0.488273
$$454$$ −30.0000 −1.40797
$$455$$ 0 0
$$456$$ 6.00000 0.280976
$$457$$ 27.7128 1.29635 0.648175 0.761491i $$-0.275532\pi$$
0.648175 + 0.761491i $$0.275532\pi$$
$$458$$ 12.0000 0.560723
$$459$$ −6.00000 −0.280056
$$460$$ 0 0
$$461$$ 13.8564 0.645357 0.322679 0.946509i $$-0.395417\pi$$
0.322679 + 0.946509i $$0.395417\pi$$
$$462$$ 20.7846 0.966988
$$463$$ −17.3205 −0.804952 −0.402476 0.915430i $$-0.631850\pi$$
−0.402476 + 0.915430i $$0.631850\pi$$
$$464$$ −30.0000 −1.39272
$$465$$ 0 0
$$466$$ −10.3923 −0.481414
$$467$$ −12.0000 −0.555294 −0.277647 0.960683i $$-0.589555\pi$$
−0.277647 + 0.960683i $$0.589555\pi$$
$$468$$ 0 0
$$469$$ −36.0000 −1.66233
$$470$$ 0 0
$$471$$ −14.0000 −0.645086
$$472$$ 18.0000 0.828517
$$473$$ −13.8564 −0.637118
$$474$$ −13.8564 −0.636446
$$475$$ 17.3205 0.794719
$$476$$ −20.7846 −0.952661
$$477$$ 6.00000 0.274721
$$478$$ −18.0000 −0.823301
$$479$$ −10.3923 −0.474837 −0.237418 0.971408i $$-0.576301\pi$$
−0.237418 + 0.971408i $$0.576301\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ −24.0000 −1.09317
$$483$$ 0 0
$$484$$ 1.00000 0.0454545
$$485$$ 0 0
$$486$$ 1.73205 0.0785674
$$487$$ −38.1051 −1.72671 −0.863354 0.504599i $$-0.831640\pi$$
−0.863354 + 0.504599i $$0.831640\pi$$
$$488$$ −3.46410 −0.156813
$$489$$ −3.46410 −0.156652
$$490$$ 0 0
$$491$$ 12.0000 0.541552 0.270776 0.962642i $$-0.412720\pi$$
0.270776 + 0.962642i $$0.412720\pi$$
$$492$$ −6.92820 −0.312348
$$493$$ 36.0000 1.62136
$$494$$ 0 0
$$495$$ 0 0
$$496$$ −17.3205 −0.777714
$$497$$ 12.0000 0.538274
$$498$$ 6.00000 0.268866
$$499$$ 10.3923 0.465223 0.232612 0.972570i $$-0.425273\pi$$
0.232612 + 0.972570i $$0.425273\pi$$
$$500$$ 0 0
$$501$$ −17.3205 −0.773823
$$502$$ −20.7846 −0.927663
$$503$$ 24.0000 1.07011 0.535054 0.844818i $$-0.320291\pi$$
0.535054 + 0.844818i $$0.320291\pi$$
$$504$$ −6.00000 −0.267261
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 8.00000 0.354943
$$509$$ 41.5692 1.84252 0.921262 0.388943i $$-0.127160\pi$$
0.921262 + 0.388943i $$0.127160\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −8.66025 −0.382733
$$513$$ 3.46410 0.152944
$$514$$ 31.1769 1.37515
$$515$$ 0 0
$$516$$ −4.00000 −0.176090
$$517$$ −12.0000 −0.527759
$$518$$ 41.5692 1.82645
$$519$$ −18.0000 −0.790112
$$520$$ 0 0
$$521$$ 18.0000 0.788594 0.394297 0.918983i $$-0.370988\pi$$
0.394297 + 0.918983i $$0.370988\pi$$
$$522$$ −10.3923 −0.454859
$$523$$ −4.00000 −0.174908 −0.0874539 0.996169i $$-0.527873\pi$$
−0.0874539 + 0.996169i $$0.527873\pi$$
$$524$$ −12.0000 −0.524222
$$525$$ −17.3205 −0.755929
$$526$$ 41.5692 1.81250
$$527$$ 20.7846 0.905392
$$528$$ −17.3205 −0.753778
$$529$$ −23.0000 −1.00000
$$530$$ 0 0
$$531$$ 10.3923 0.450988
$$532$$ 12.0000 0.520266
$$533$$ 0 0
$$534$$ −12.0000 −0.519291
$$535$$ 0 0
$$536$$ 18.0000 0.777482
$$537$$ −12.0000 −0.517838
$$538$$ −10.3923 −0.448044
$$539$$ −17.3205 −0.746047
$$540$$ 0 0
$$541$$ 6.92820 0.297867 0.148933 0.988847i $$-0.452416\pi$$
0.148933 + 0.988847i $$0.452416\pi$$
$$542$$ −18.0000 −0.773166
$$543$$ −10.0000 −0.429141
$$544$$ 31.1769 1.33670
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 20.0000 0.855138 0.427569 0.903983i $$-0.359370\pi$$
0.427569 + 0.903983i $$0.359370\pi$$
$$548$$ −20.7846 −0.887875
$$549$$ −2.00000 −0.0853579
$$550$$ −30.0000 −1.27920
$$551$$ −20.7846 −0.885454
$$552$$ 0 0
$$553$$ 27.7128 1.17847
$$554$$ −17.3205 −0.735878
$$555$$ 0 0
$$556$$ −4.00000 −0.169638
$$557$$ 13.8564 0.587115 0.293557 0.955941i $$-0.405161\pi$$
0.293557 + 0.955941i $$0.405161\pi$$
$$558$$ −6.00000 −0.254000
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 20.7846 0.877527
$$562$$ −12.0000 −0.506189
$$563$$ 12.0000 0.505740 0.252870 0.967500i $$-0.418626\pi$$
0.252870 + 0.967500i $$0.418626\pi$$
$$564$$ −3.46410 −0.145865
$$565$$ 0 0
$$566$$ 6.92820 0.291214
$$567$$ −3.46410 −0.145479
$$568$$ −6.00000 −0.251754
$$569$$ 6.00000 0.251533 0.125767 0.992060i $$-0.459861\pi$$
0.125767 + 0.992060i $$0.459861\pi$$
$$570$$ 0 0
$$571$$ −4.00000 −0.167395 −0.0836974 0.996491i $$-0.526673\pi$$
−0.0836974 + 0.996491i $$0.526673\pi$$
$$572$$ 0 0
$$573$$ −24.0000 −1.00261
$$574$$ 41.5692 1.73507
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$578$$ −32.9090 −1.36883
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −12.0000 −0.497844
$$582$$ 24.0000 0.994832
$$583$$ −20.7846 −0.860811
$$584$$ 0 0
$$585$$ 0 0
$$586$$ −48.0000 −1.98286
$$587$$ −10.3923 −0.428936 −0.214468 0.976731i $$-0.568802\pi$$
−0.214468 + 0.976731i $$0.568802\pi$$
$$588$$ −5.00000 −0.206197
$$589$$ −12.0000 −0.494451
$$590$$ 0 0
$$591$$ 0 0
$$592$$ −34.6410 −1.42374
$$593$$ 6.92820 0.284507 0.142254 0.989830i $$-0.454565\pi$$
0.142254 + 0.989830i $$0.454565\pi$$
$$594$$ −6.00000 −0.246183
$$595$$ 0 0
$$596$$ −13.8564 −0.567581
$$597$$ 16.0000 0.654836
$$598$$ 0 0
$$599$$ 24.0000 0.980613 0.490307 0.871550i $$-0.336885\pi$$
0.490307 + 0.871550i $$0.336885\pi$$
$$600$$ 8.66025 0.353553
$$601$$ 10.0000 0.407909 0.203954 0.978980i $$-0.434621\pi$$
0.203954 + 0.978980i $$0.434621\pi$$
$$602$$ 24.0000 0.978167
$$603$$ 10.3923 0.423207
$$604$$ 10.3923 0.422857
$$605$$ 0 0
$$606$$ −10.3923 −0.422159
$$607$$ 32.0000 1.29884 0.649420 0.760430i $$-0.275012\pi$$
0.649420 + 0.760430i $$0.275012\pi$$
$$608$$ −18.0000 −0.729996
$$609$$ 20.7846 0.842235
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 6.00000 0.242536
$$613$$ 20.7846 0.839482 0.419741 0.907644i $$-0.362121\pi$$
0.419741 + 0.907644i $$0.362121\pi$$
$$614$$ 18.0000 0.726421
$$615$$ 0 0
$$616$$ 20.7846 0.837436
$$617$$ 6.92820 0.278919 0.139459 0.990228i $$-0.455464\pi$$
0.139459 + 0.990228i $$0.455464\pi$$
$$618$$ −13.8564 −0.557386
$$619$$ 31.1769 1.25311 0.626553 0.779379i $$-0.284465\pi$$
0.626553 + 0.779379i $$0.284465\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 24.0000 0.961540
$$624$$ 0 0
$$625$$ 25.0000 1.00000
$$626$$ −17.3205 −0.692267
$$627$$ −12.0000 −0.479234
$$628$$ 14.0000 0.558661
$$629$$ 41.5692 1.65747
$$630$$ 0 0
$$631$$ 38.1051 1.51694 0.758470 0.651707i $$-0.225947\pi$$
0.758470 + 0.651707i $$0.225947\pi$$
$$632$$ −13.8564 −0.551178
$$633$$ 20.0000 0.794929
$$634$$ 24.0000 0.953162
$$635$$ 0 0
$$636$$ −6.00000 −0.237915
$$637$$ 0 0
$$638$$ 36.0000 1.42525
$$639$$ −3.46410 −0.137038
$$640$$ 0 0
$$641$$ 6.00000 0.236986 0.118493 0.992955i $$-0.462194\pi$$
0.118493 + 0.992955i $$0.462194\pi$$
$$642$$ 20.7846 0.820303
$$643$$ 10.3923 0.409832 0.204916 0.978780i $$-0.434308\pi$$
0.204916 + 0.978780i $$0.434308\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 36.0000 1.41640
$$647$$ 24.0000 0.943537 0.471769 0.881722i $$-0.343616\pi$$
0.471769 + 0.881722i $$0.343616\pi$$
$$648$$ 1.73205 0.0680414
$$649$$ −36.0000 −1.41312
$$650$$ 0 0
$$651$$ 12.0000 0.470317
$$652$$ 3.46410 0.135665
$$653$$ 6.00000 0.234798 0.117399 0.993085i $$-0.462544\pi$$
0.117399 + 0.993085i $$0.462544\pi$$
$$654$$ −12.0000 −0.469237
$$655$$ 0 0
$$656$$ −34.6410 −1.35250
$$657$$ 0 0
$$658$$ 20.7846 0.810268
$$659$$ 12.0000 0.467454 0.233727 0.972302i $$-0.424908\pi$$
0.233727 + 0.972302i $$0.424908\pi$$
$$660$$ 0 0
$$661$$ −20.7846 −0.808428 −0.404214 0.914665i $$-0.632455\pi$$
−0.404214 + 0.914665i $$0.632455\pi$$
$$662$$ 6.00000 0.233197
$$663$$ 0 0
$$664$$ 6.00000 0.232845
$$665$$ 0 0
$$666$$ −12.0000 −0.464991
$$667$$ 0 0
$$668$$ 17.3205 0.670151
$$669$$ −3.46410 −0.133930
$$670$$ 0 0
$$671$$ 6.92820 0.267460
$$672$$ 18.0000 0.694365
$$673$$ 46.0000 1.77317 0.886585 0.462566i $$-0.153071\pi$$
0.886585 + 0.462566i $$0.153071\pi$$
$$674$$ −24.2487 −0.934025
$$675$$ 5.00000 0.192450
$$676$$ 0 0
$$677$$ 6.00000 0.230599 0.115299 0.993331i $$-0.463217\pi$$
0.115299 + 0.993331i $$0.463217\pi$$
$$678$$ −10.3923 −0.399114
$$679$$ −48.0000 −1.84207
$$680$$ 0 0
$$681$$ −17.3205 −0.663723
$$682$$ 20.7846 0.795884
$$683$$ −31.1769 −1.19295 −0.596476 0.802631i $$-0.703433\pi$$
−0.596476 + 0.802631i $$0.703433\pi$$
$$684$$ −3.46410 −0.132453
$$685$$ 0 0
$$686$$ −12.0000 −0.458162
$$687$$ 6.92820 0.264327
$$688$$ −20.0000 −0.762493
$$689$$ 0 0
$$690$$ 0 0
$$691$$ −45.0333 −1.71315 −0.856574 0.516024i $$-0.827412\pi$$
−0.856574 + 0.516024i $$0.827412\pi$$
$$692$$ 18.0000 0.684257
$$693$$ 12.0000 0.455842
$$694$$ −62.3538 −2.36692
$$695$$ 0 0
$$696$$ −10.3923 −0.393919
$$697$$ 41.5692 1.57455
$$698$$ 12.0000 0.454207
$$699$$ −6.00000 −0.226941
$$700$$ 17.3205 0.654654
$$701$$ 42.0000 1.58632 0.793159 0.609015i $$-0.208435\pi$$
0.793159 + 0.609015i $$0.208435\pi$$
$$702$$ 0 0
$$703$$ −24.0000 −0.905177
$$704$$ −3.46410 −0.130558
$$705$$ 0 0
$$706$$ 60.0000 2.25813
$$707$$ 20.7846 0.781686
$$708$$ −10.3923 −0.390567
$$709$$ −6.92820 −0.260194 −0.130097 0.991501i $$-0.541529\pi$$
−0.130097 + 0.991501i $$0.541529\pi$$
$$710$$ 0 0
$$711$$ −8.00000 −0.300023
$$712$$ −12.0000 −0.449719
$$713$$ 0 0
$$714$$ −36.0000 −1.34727
$$715$$ 0 0
$$716$$ 12.0000 0.448461
$$717$$ −10.3923 −0.388108
$$718$$ −30.0000 −1.11959
$$719$$ −24.0000 −0.895049 −0.447524 0.894272i $$-0.647694\pi$$
−0.447524 + 0.894272i $$0.647694\pi$$
$$720$$ 0 0
$$721$$ 27.7128 1.03208
$$722$$ 12.1244 0.451222
$$723$$ −13.8564 −0.515325
$$724$$ 10.0000 0.371647
$$725$$ −30.0000 −1.11417
$$726$$ 1.73205 0.0642824
$$727$$ −16.0000 −0.593407 −0.296704 0.954970i $$-0.595887\pi$$
−0.296704 + 0.954970i $$0.595887\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 24.0000 0.887672
$$732$$ 2.00000 0.0739221
$$733$$ −34.6410 −1.27950 −0.639748 0.768585i $$-0.720961\pi$$
−0.639748 + 0.768585i $$0.720961\pi$$
$$734$$ 27.7128 1.02290
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −36.0000 −1.32608
$$738$$ −12.0000 −0.441726
$$739$$ −38.1051 −1.40172 −0.700860 0.713299i $$-0.747200\pi$$
−0.700860 + 0.713299i $$0.747200\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 36.0000 1.32160
$$743$$ −3.46410 −0.127086 −0.0635428 0.997979i $$-0.520240\pi$$
−0.0635428 + 0.997979i $$0.520240\pi$$
$$744$$ −6.00000 −0.219971
$$745$$ 0 0
$$746$$ −38.1051 −1.39513
$$747$$ 3.46410 0.126745
$$748$$ −20.7846 −0.759961
$$749$$ −41.5692 −1.51891
$$750$$ 0 0
$$751$$ −32.0000 −1.16770 −0.583848 0.811863i $$-0.698454\pi$$
−0.583848 + 0.811863i $$0.698454\pi$$
$$752$$ −17.3205 −0.631614
$$753$$ −12.0000 −0.437304
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 3.46410 0.125988
$$757$$ 22.0000 0.799604 0.399802 0.916602i $$-0.369079\pi$$
0.399802 + 0.916602i $$0.369079\pi$$
$$758$$ 30.0000 1.08965
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −48.4974 −1.75803 −0.879015 0.476794i $$-0.841799\pi$$
−0.879015 + 0.476794i $$0.841799\pi$$
$$762$$ 13.8564 0.501965
$$763$$ 24.0000 0.868858
$$764$$ 24.0000 0.868290
$$765$$ 0 0
$$766$$ 6.00000 0.216789
$$767$$ 0 0
$$768$$ −19.0000 −0.685603
$$769$$ −27.7128 −0.999350 −0.499675 0.866213i $$-0.666547\pi$$
−0.499675 + 0.866213i $$0.666547\pi$$
$$770$$ 0 0
$$771$$ 18.0000 0.648254
$$772$$ 0 0
$$773$$ 13.8564 0.498380 0.249190 0.968455i $$-0.419836\pi$$
0.249190 + 0.968455i $$0.419836\pi$$
$$774$$ −6.92820 −0.249029
$$775$$ −17.3205 −0.622171
$$776$$ 24.0000 0.861550
$$777$$ 24.0000 0.860995
$$778$$ −31.1769 −1.11775
$$779$$ −24.0000 −0.859889
$$780$$ 0 0
$$781$$ 12.0000 0.429394
$$782$$ 0 0
$$783$$ −6.00000 −0.214423
$$784$$ −25.0000 −0.892857
$$785$$ 0 0
$$786$$ −20.7846 −0.741362
$$787$$ −10.3923 −0.370446 −0.185223 0.982697i $$-0.559301\pi$$
−0.185223 + 0.982697i $$0.559301\pi$$
$$788$$ 0 0
$$789$$ 24.0000 0.854423
$$790$$ 0 0
$$791$$ 20.7846 0.739016
$$792$$ −6.00000 −0.213201
$$793$$ 0 0
$$794$$ 60.0000 2.12932
$$795$$ 0 0
$$796$$ −16.0000 −0.567105
$$797$$ 42.0000 1.48772 0.743858 0.668338i $$-0.232994\pi$$
0.743858 + 0.668338i $$0.232994\pi$$
$$798$$ 20.7846 0.735767
$$799$$ 20.7846 0.735307
$$800$$ −25.9808 −0.918559
$$801$$ −6.92820 −0.244796
$$802$$ −12.0000 −0.423735
$$803$$ 0 0
$$804$$ −10.3923 −0.366508
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −6.00000 −0.211210
$$808$$ −10.3923 −0.365600
$$809$$ −30.0000 −1.05474 −0.527372 0.849635i $$-0.676823\pi$$
−0.527372 + 0.849635i $$0.676823\pi$$
$$810$$ 0 0
$$811$$ 38.1051 1.33805 0.669026 0.743239i $$-0.266712\pi$$
0.669026 + 0.743239i $$0.266712\pi$$
$$812$$ −20.7846 −0.729397
$$813$$ −10.3923 −0.364474
$$814$$ 41.5692 1.45700
$$815$$ 0 0
$$816$$ 30.0000 1.05021
$$817$$ −13.8564 −0.484774
$$818$$ 48.0000 1.67828
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −13.8564 −0.483592 −0.241796 0.970327i $$-0.577736\pi$$
−0.241796 + 0.970327i $$0.577736\pi$$
$$822$$ −36.0000 −1.25564
$$823$$ 40.0000 1.39431 0.697156 0.716919i $$-0.254448\pi$$
0.697156 + 0.716919i $$0.254448\pi$$
$$824$$ −13.8564 −0.482711
$$825$$ −17.3205 −0.603023
$$826$$ 62.3538 2.16957
$$827$$ 24.2487 0.843210 0.421605 0.906780i $$-0.361467\pi$$
0.421605 + 0.906780i $$0.361467\pi$$
$$828$$ 0 0
$$829$$ 2.00000 0.0694629 0.0347314 0.999397i $$-0.488942\pi$$
0.0347314 + 0.999397i $$0.488942\pi$$
$$830$$ 0 0
$$831$$ −10.0000 −0.346896
$$832$$ 0 0
$$833$$ 30.0000 1.03944
$$834$$ −6.92820 −0.239904
$$835$$ 0 0
$$836$$ 12.0000 0.415029
$$837$$ −3.46410 −0.119737
$$838$$ 20.7846 0.717992
$$839$$ 3.46410 0.119594 0.0597970 0.998211i $$-0.480955\pi$$
0.0597970 + 0.998211i $$0.480955\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ −60.0000 −2.06774
$$843$$ −6.92820 −0.238620
$$844$$ −20.0000 −0.688428
$$845$$ 0 0
$$846$$ −6.00000 −0.206284
$$847$$ −3.46410 −0.119028
$$848$$ −30.0000 −1.03020
$$849$$ 4.00000 0.137280
$$850$$ 51.9615 1.78227
$$851$$ 0 0
$$852$$ 3.46410 0.118678
$$853$$ −20.7846 −0.711651 −0.355826 0.934552i $$-0.615800\pi$$
−0.355826 + 0.934552i $$0.615800\pi$$
$$854$$ −12.0000 −0.410632
$$855$$ 0 0
$$856$$ 20.7846 0.710403
$$857$$ −42.0000 −1.43469 −0.717346 0.696717i $$-0.754643\pi$$
−0.717346 + 0.696717i $$0.754643\pi$$
$$858$$ 0 0
$$859$$ 4.00000 0.136478 0.0682391 0.997669i $$-0.478262\pi$$
0.0682391 + 0.997669i $$0.478262\pi$$
$$860$$ 0 0
$$861$$ 24.0000 0.817918
$$862$$ −42.0000 −1.43053
$$863$$ −31.1769 −1.06127 −0.530637 0.847599i $$-0.678047\pi$$
−0.530637 + 0.847599i $$0.678047\pi$$
$$864$$ −5.19615 −0.176777
$$865$$ 0 0
$$866$$ 58.8897 2.00115
$$867$$ −19.0000 −0.645274
$$868$$ −12.0000 −0.407307
$$869$$ 27.7128 0.940093
$$870$$ 0 0
$$871$$ 0 0
$$872$$ −12.0000 −0.406371
$$873$$ 13.8564 0.468968
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −48.4974 −1.63764 −0.818821 0.574049i $$-0.805372\pi$$
−0.818821 + 0.574049i $$0.805372\pi$$
$$878$$ 13.8564 0.467631
$$879$$ −27.7128 −0.934730
$$880$$ 0 0
$$881$$ −18.0000 −0.606435 −0.303218 0.952921i $$-0.598061\pi$$
−0.303218 + 0.952921i $$0.598061\pi$$
$$882$$ −8.66025 −0.291606
$$883$$ 20.0000 0.673054 0.336527 0.941674i $$-0.390748\pi$$
0.336527 + 0.941674i $$0.390748\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 62.3538 2.09482
$$887$$ 48.0000 1.61168 0.805841 0.592132i $$-0.201714\pi$$
0.805841 + 0.592132i $$0.201714\pi$$
$$888$$ −12.0000 −0.402694
$$889$$ −27.7128 −0.929458
$$890$$ 0 0
$$891$$ −3.46410 −0.116052
$$892$$ 3.46410 0.115987
$$893$$ −12.0000 −0.401565
$$894$$ −24.0000 −0.802680
$$895$$ 0 0
$$896$$ 42.0000 1.40312
$$897$$ 0 0
$$898$$ −12.0000 −0.400445
$$899$$ 20.7846 0.693206
$$900$$ −5.00000 −0.166667
$$901$$ 36.0000 1.19933
$$902$$ 41.5692 1.38410
$$903$$ 13.8564 0.461112
$$904$$ −10.3923 −0.345643
$$905$$ 0 0
$$906$$ 18.0000 0.598010
$$907$$ 44.0000 1.46100 0.730498 0.682915i $$-0.239288\pi$$
0.730498 + 0.682915i $$0.239288\pi$$
$$908$$ 17.3205 0.574801
$$909$$ −6.00000 −0.199007
$$910$$ 0 0
$$911$$ 24.0000 0.795155 0.397578 0.917568i $$-0.369851\pi$$
0.397578 + 0.917568i $$0.369851\pi$$
$$912$$ −17.3205 −0.573539
$$913$$ −12.0000 −0.397142
$$914$$ −48.0000 −1.58770
$$915$$ 0 0
$$916$$ −6.92820 −0.228914
$$917$$ 41.5692 1.37274
$$918$$ 10.3923 0.342997
$$919$$ −32.0000 −1.05558 −0.527791 0.849374i $$-0.676980\pi$$
−0.527791 + 0.849374i $$0.676980\pi$$
$$920$$ 0 0
$$921$$ 10.3923 0.342438
$$922$$ −24.0000 −0.790398
$$923$$ 0 0
$$924$$ −12.0000 −0.394771
$$925$$ −34.6410 −1.13899
$$926$$ 30.0000 0.985861
$$927$$ −8.00000 −0.262754
$$928$$ 31.1769 1.02343
$$929$$ 20.7846 0.681921 0.340960 0.940078i $$-0.389248\pi$$
0.340960 + 0.940078i $$0.389248\pi$$
$$930$$ 0 0
$$931$$ −17.3205 −0.567657
$$932$$ 6.00000 0.196537
$$933$$ 0 0
$$934$$ 20.7846 0.680093
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −38.0000 −1.24141 −0.620703 0.784046i $$-0.713153\pi$$
−0.620703 + 0.784046i $$0.713153\pi$$
$$938$$ 62.3538 2.03592
$$939$$ −10.0000 −0.326338
$$940$$ 0 0
$$941$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$942$$ 24.2487 0.790066
$$943$$ 0 0
$$944$$ −51.9615 −1.69120
$$945$$ 0 0
$$946$$ 24.0000 0.780307
$$947$$ 51.9615 1.68852 0.844261 0.535932i $$-0.180040\pi$$
0.844261 + 0.535932i $$0.180040\pi$$
$$948$$ 8.00000 0.259828
$$949$$ 0 0
$$950$$ −30.0000 −0.973329
$$951$$ 13.8564 0.449325
$$952$$ −36.0000 −1.16677
$$953$$ −42.0000 −1.36051 −0.680257 0.732974i $$-0.738132\pi$$
−0.680257 + 0.732974i $$0.738132\pi$$
$$954$$ −10.3923 −0.336463
$$955$$ 0 0
$$956$$ 10.3923 0.336111
$$957$$ 20.7846 0.671871
$$958$$ 18.0000 0.581554
$$959$$ 72.0000 2.32500
$$960$$ 0 0
$$961$$ −19.0000 −0.612903
$$962$$ 0 0
$$963$$ 12.0000 0.386695
$$964$$ 13.8564 0.446285
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −10.3923 −0.334194 −0.167097 0.985940i $$-0.553439\pi$$
−0.167097 + 0.985940i $$0.553439\pi$$
$$968$$ 1.73205 0.0556702
$$969$$ 20.7846 0.667698
$$970$$ 0 0
$$971$$ −12.0000 −0.385098 −0.192549 0.981287i $$-0.561675\pi$$
−0.192549 + 0.981287i $$0.561675\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ 13.8564 0.444216
$$974$$ 66.0000 2.11478
$$975$$ 0 0
$$976$$ 10.0000 0.320092
$$977$$ −48.4974 −1.55157 −0.775785 0.630997i $$-0.782646\pi$$
−0.775785 + 0.630997i $$0.782646\pi$$
$$978$$ 6.00000 0.191859
$$979$$ 24.0000 0.767043
$$980$$ 0 0
$$981$$ −6.92820 −0.221201
$$982$$ −20.7846 −0.663264
$$983$$ −51.9615 −1.65732 −0.828658 0.559756i $$-0.810895\pi$$
−0.828658 + 0.559756i $$0.810895\pi$$
$$984$$ −12.0000 −0.382546
$$985$$ 0 0
$$986$$ −62.3538 −1.98575
$$987$$ 12.0000 0.381964
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 32.0000 1.01651 0.508257 0.861206i $$-0.330290\pi$$
0.508257 + 0.861206i $$0.330290\pi$$
$$992$$ 18.0000 0.571501
$$993$$ 3.46410 0.109930
$$994$$ −20.7846 −0.659248
$$995$$ 0 0
$$996$$ −3.46410 −0.109764
$$997$$ 38.0000 1.20347 0.601736 0.798695i $$-0.294476\pi$$
0.601736 + 0.798695i $$0.294476\pi$$
$$998$$ −18.0000 −0.569780
$$999$$ −6.92820 −0.219199
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 507.2.a.f.1.1 2
3.2 odd 2 1521.2.a.l.1.2 2
4.3 odd 2 8112.2.a.bv.1.2 2
13.2 odd 12 507.2.j.a.316.1 2
13.3 even 3 507.2.e.e.22.2 4
13.4 even 6 507.2.e.e.484.1 4
13.5 odd 4 39.2.b.a.25.2 yes 2
13.6 odd 12 507.2.j.c.361.1 2
13.7 odd 12 507.2.j.a.361.1 2
13.8 odd 4 39.2.b.a.25.1 2
13.9 even 3 507.2.e.e.484.2 4
13.10 even 6 507.2.e.e.22.1 4
13.11 odd 12 507.2.j.c.316.1 2
13.12 even 2 inner 507.2.a.f.1.2 2
39.5 even 4 117.2.b.a.64.1 2
39.8 even 4 117.2.b.a.64.2 2
39.38 odd 2 1521.2.a.l.1.1 2
52.31 even 4 624.2.c.e.337.2 2
52.47 even 4 624.2.c.e.337.1 2
52.51 odd 2 8112.2.a.bv.1.1 2
65.8 even 4 975.2.h.f.649.1 4
65.18 even 4 975.2.h.f.649.3 4
65.34 odd 4 975.2.b.d.376.2 2
65.44 odd 4 975.2.b.d.376.1 2
65.47 even 4 975.2.h.f.649.4 4
65.57 even 4 975.2.h.f.649.2 4
91.34 even 4 1911.2.c.d.883.1 2
91.83 even 4 1911.2.c.d.883.2 2
104.5 odd 4 2496.2.c.k.961.1 2
104.21 odd 4 2496.2.c.k.961.2 2
104.83 even 4 2496.2.c.d.961.2 2
104.99 even 4 2496.2.c.d.961.1 2
156.47 odd 4 1872.2.c.e.1585.1 2
156.83 odd 4 1872.2.c.e.1585.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
39.2.b.a.25.1 2 13.8 odd 4
39.2.b.a.25.2 yes 2 13.5 odd 4
117.2.b.a.64.1 2 39.5 even 4
117.2.b.a.64.2 2 39.8 even 4
507.2.a.f.1.1 2 1.1 even 1 trivial
507.2.a.f.1.2 2 13.12 even 2 inner
507.2.e.e.22.1 4 13.10 even 6
507.2.e.e.22.2 4 13.3 even 3
507.2.e.e.484.1 4 13.4 even 6
507.2.e.e.484.2 4 13.9 even 3
507.2.j.a.316.1 2 13.2 odd 12
507.2.j.a.361.1 2 13.7 odd 12
507.2.j.c.316.1 2 13.11 odd 12
507.2.j.c.361.1 2 13.6 odd 12
624.2.c.e.337.1 2 52.47 even 4
624.2.c.e.337.2 2 52.31 even 4
975.2.b.d.376.1 2 65.44 odd 4
975.2.b.d.376.2 2 65.34 odd 4
975.2.h.f.649.1 4 65.8 even 4
975.2.h.f.649.2 4 65.57 even 4
975.2.h.f.649.3 4 65.18 even 4
975.2.h.f.649.4 4 65.47 even 4
1521.2.a.l.1.1 2 39.38 odd 2
1521.2.a.l.1.2 2 3.2 odd 2
1872.2.c.e.1585.1 2 156.47 odd 4
1872.2.c.e.1585.2 2 156.83 odd 4
1911.2.c.d.883.1 2 91.34 even 4
1911.2.c.d.883.2 2 91.83 even 4
2496.2.c.d.961.1 2 104.99 even 4
2496.2.c.d.961.2 2 104.83 even 4
2496.2.c.k.961.1 2 104.5 odd 4
2496.2.c.k.961.2 2 104.21 odd 4
8112.2.a.bv.1.1 2 52.51 odd 2
8112.2.a.bv.1.2 2 4.3 odd 2