# Properties

 Label 3234.2.a.bf.1.3 Level $3234$ Weight $2$ Character 3234.1 Self dual yes Analytic conductor $25.824$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3234,2,Mod(1,3234)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3234, 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("3234.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3234.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: $$25.8236200137$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.2700.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - 15x - 20$$ x^3 - 15*x - 20 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 462) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$4.41883$$ of defining polynomial Character $$\chi$$ $$=$$ 3234.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +4.41883 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +4.41883 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +4.41883 q^{10} -1.00000 q^{11} -1.00000 q^{12} -4.41883 q^{15} +1.00000 q^{16} +2.37683 q^{17} +1.00000 q^{18} -3.68842 q^{19} +4.41883 q^{20} -1.00000 q^{22} +4.73042 q^{23} -1.00000 q^{24} +14.5261 q^{25} -1.00000 q^{27} -6.52608 q^{29} -4.41883 q^{30} -3.37683 q^{31} +1.00000 q^{32} +1.00000 q^{33} +2.37683 q^{34} +1.00000 q^{36} +7.68842 q^{37} -3.68842 q^{38} +4.41883 q^{40} +12.1492 q^{41} +3.06525 q^{43} -1.00000 q^{44} +4.41883 q^{45} +4.73042 q^{46} -4.10725 q^{47} -1.00000 q^{48} +14.5261 q^{50} -2.37683 q^{51} +8.00000 q^{53} -1.00000 q^{54} -4.41883 q^{55} +3.68842 q^{57} -6.52608 q^{58} -12.5261 q^{59} -4.41883 q^{60} +3.79567 q^{61} -3.37683 q^{62} +1.00000 q^{64} +1.00000 q^{66} +10.1492 q^{67} +2.37683 q^{68} -4.73042 q^{69} +0.934749 q^{71} +1.00000 q^{72} -7.46083 q^{73} +7.68842 q^{74} -14.5261 q^{75} -3.68842 q^{76} -0.418833 q^{79} +4.41883 q^{80} +1.00000 q^{81} +12.1492 q^{82} -2.14925 q^{83} +10.5028 q^{85} +3.06525 q^{86} +6.52608 q^{87} -1.00000 q^{88} -12.8377 q^{89} +4.41883 q^{90} +4.73042 q^{92} +3.37683 q^{93} -4.10725 q^{94} -16.2985 q^{95} -1.00000 q^{96} +7.00000 q^{97} -1.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{6} + 3 q^{8} + 3 q^{9}+O(q^{10})$$ 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 - 3 * q^6 + 3 * q^8 + 3 * q^9 $$3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{6} + 3 q^{8} + 3 q^{9} - 3 q^{11} - 3 q^{12} + 3 q^{16} + 3 q^{17} + 3 q^{18} - 9 q^{19} - 3 q^{22} + 3 q^{23} - 3 q^{24} + 15 q^{25} - 3 q^{27} + 9 q^{29} - 6 q^{31} + 3 q^{32} + 3 q^{33} + 3 q^{34} + 3 q^{36} + 21 q^{37} - 9 q^{38} + 12 q^{41} + 3 q^{43} - 3 q^{44} + 3 q^{46} + 3 q^{47} - 3 q^{48} + 15 q^{50} - 3 q^{51} + 24 q^{53} - 3 q^{54} + 9 q^{57} + 9 q^{58} - 9 q^{59} - 6 q^{61} - 6 q^{62} + 3 q^{64} + 3 q^{66} + 6 q^{67} + 3 q^{68} - 3 q^{69} + 9 q^{71} + 3 q^{72} + 21 q^{74} - 15 q^{75} - 9 q^{76} + 12 q^{79} + 3 q^{81} + 12 q^{82} + 18 q^{83} + 3 q^{86} - 9 q^{87} - 3 q^{88} - 12 q^{89} + 3 q^{92} + 6 q^{93} + 3 q^{94} - 3 q^{96} + 21 q^{97} - 3 q^{99}+O(q^{100})$$ 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 - 3 * q^6 + 3 * q^8 + 3 * q^9 - 3 * q^11 - 3 * q^12 + 3 * q^16 + 3 * q^17 + 3 * q^18 - 9 * q^19 - 3 * q^22 + 3 * q^23 - 3 * q^24 + 15 * q^25 - 3 * q^27 + 9 * q^29 - 6 * q^31 + 3 * q^32 + 3 * q^33 + 3 * q^34 + 3 * q^36 + 21 * q^37 - 9 * q^38 + 12 * q^41 + 3 * q^43 - 3 * q^44 + 3 * q^46 + 3 * q^47 - 3 * q^48 + 15 * q^50 - 3 * q^51 + 24 * q^53 - 3 * q^54 + 9 * q^57 + 9 * q^58 - 9 * q^59 - 6 * q^61 - 6 * q^62 + 3 * q^64 + 3 * q^66 + 6 * q^67 + 3 * q^68 - 3 * q^69 + 9 * q^71 + 3 * q^72 + 21 * q^74 - 15 * q^75 - 9 * q^76 + 12 * q^79 + 3 * q^81 + 12 * q^82 + 18 * q^83 + 3 * q^86 - 9 * q^87 - 3 * q^88 - 12 * q^89 + 3 * q^92 + 6 * q^93 + 3 * q^94 - 3 * q^96 + 21 * q^97 - 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ −1.00000 −0.577350
$$4$$ 1.00000 0.500000
$$5$$ 4.41883 1.97616 0.988081 0.153935i $$-0.0491945\pi$$
0.988081 + 0.153935i $$0.0491945\pi$$
$$6$$ −1.00000 −0.408248
$$7$$ 0 0
$$8$$ 1.00000 0.353553
$$9$$ 1.00000 0.333333
$$10$$ 4.41883 1.39736
$$11$$ −1.00000 −0.301511
$$12$$ −1.00000 −0.288675
$$13$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$14$$ 0 0
$$15$$ −4.41883 −1.14094
$$16$$ 1.00000 0.250000
$$17$$ 2.37683 0.576467 0.288233 0.957560i $$-0.406932\pi$$
0.288233 + 0.957560i $$0.406932\pi$$
$$18$$ 1.00000 0.235702
$$19$$ −3.68842 −0.846181 −0.423090 0.906087i $$-0.639055\pi$$
−0.423090 + 0.906087i $$0.639055\pi$$
$$20$$ 4.41883 0.988081
$$21$$ 0 0
$$22$$ −1.00000 −0.213201
$$23$$ 4.73042 0.986360 0.493180 0.869927i $$-0.335834\pi$$
0.493180 + 0.869927i $$0.335834\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ 14.5261 2.90522
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ −6.52608 −1.21186 −0.605932 0.795517i $$-0.707199\pi$$
−0.605932 + 0.795517i $$0.707199\pi$$
$$30$$ −4.41883 −0.806765
$$31$$ −3.37683 −0.606497 −0.303249 0.952911i $$-0.598071\pi$$
−0.303249 + 0.952911i $$0.598071\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 1.00000 0.174078
$$34$$ 2.37683 0.407624
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 7.68842 1.26397 0.631984 0.774981i $$-0.282241\pi$$
0.631984 + 0.774981i $$0.282241\pi$$
$$38$$ −3.68842 −0.598340
$$39$$ 0 0
$$40$$ 4.41883 0.698679
$$41$$ 12.1492 1.89739 0.948697 0.316187i $$-0.102403\pi$$
0.948697 + 0.316187i $$0.102403\pi$$
$$42$$ 0 0
$$43$$ 3.06525 0.467446 0.233723 0.972303i $$-0.424909\pi$$
0.233723 + 0.972303i $$0.424909\pi$$
$$44$$ −1.00000 −0.150756
$$45$$ 4.41883 0.658721
$$46$$ 4.73042 0.697462
$$47$$ −4.10725 −0.599104 −0.299552 0.954080i $$-0.596837\pi$$
−0.299552 + 0.954080i $$0.596837\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ 0 0
$$50$$ 14.5261 2.05430
$$51$$ −2.37683 −0.332823
$$52$$ 0 0
$$53$$ 8.00000 1.09888 0.549442 0.835532i $$-0.314840\pi$$
0.549442 + 0.835532i $$0.314840\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ −4.41883 −0.595835
$$56$$ 0 0
$$57$$ 3.68842 0.488543
$$58$$ −6.52608 −0.856917
$$59$$ −12.5261 −1.63076 −0.815379 0.578928i $$-0.803471\pi$$
−0.815379 + 0.578928i $$0.803471\pi$$
$$60$$ −4.41883 −0.570469
$$61$$ 3.79567 0.485985 0.242993 0.970028i $$-0.421871\pi$$
0.242993 + 0.970028i $$0.421871\pi$$
$$62$$ −3.37683 −0.428858
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 1.00000 0.123091
$$67$$ 10.1492 1.23993 0.619964 0.784630i $$-0.287147\pi$$
0.619964 + 0.784630i $$0.287147\pi$$
$$68$$ 2.37683 0.288233
$$69$$ −4.73042 −0.569475
$$70$$ 0 0
$$71$$ 0.934749 0.110934 0.0554672 0.998461i $$-0.482335\pi$$
0.0554672 + 0.998461i $$0.482335\pi$$
$$72$$ 1.00000 0.117851
$$73$$ −7.46083 −0.873224 −0.436612 0.899650i $$-0.643822\pi$$
−0.436612 + 0.899650i $$0.643822\pi$$
$$74$$ 7.68842 0.893760
$$75$$ −14.5261 −1.67733
$$76$$ −3.68842 −0.423090
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −0.418833 −0.0471224 −0.0235612 0.999722i $$-0.507500\pi$$
−0.0235612 + 0.999722i $$0.507500\pi$$
$$80$$ 4.41883 0.494041
$$81$$ 1.00000 0.111111
$$82$$ 12.1492 1.34166
$$83$$ −2.14925 −0.235911 −0.117955 0.993019i $$-0.537634\pi$$
−0.117955 + 0.993019i $$0.537634\pi$$
$$84$$ 0 0
$$85$$ 10.5028 1.13919
$$86$$ 3.06525 0.330534
$$87$$ 6.52608 0.699669
$$88$$ −1.00000 −0.106600
$$89$$ −12.8377 −1.36079 −0.680395 0.732846i $$-0.738192\pi$$
−0.680395 + 0.732846i $$0.738192\pi$$
$$90$$ 4.41883 0.465786
$$91$$ 0 0
$$92$$ 4.73042 0.493180
$$93$$ 3.37683 0.350161
$$94$$ −4.10725 −0.423630
$$95$$ −16.2985 −1.67219
$$96$$ −1.00000 −0.102062
$$97$$ 7.00000 0.710742 0.355371 0.934725i $$-0.384354\pi$$
0.355371 + 0.934725i $$0.384354\pi$$
$$98$$ 0 0
$$99$$ −1.00000 −0.100504
$$100$$ 14.5261 1.45261
$$101$$ −15.3637 −1.52875 −0.764375 0.644772i $$-0.776952\pi$$
−0.764375 + 0.644772i $$0.776952\pi$$
$$102$$ −2.37683 −0.235342
$$103$$ −2.62317 −0.258468 −0.129234 0.991614i $$-0.541252\pi$$
−0.129234 + 0.991614i $$0.541252\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 8.00000 0.777029
$$107$$ 10.9029 1.05402 0.527012 0.849858i $$-0.323312\pi$$
0.527012 + 0.849858i $$0.323312\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ −11.7957 −1.12982 −0.564910 0.825153i $$-0.691089\pi$$
−0.564910 + 0.825153i $$0.691089\pi$$
$$110$$ −4.41883 −0.421319
$$111$$ −7.68842 −0.729752
$$112$$ 0 0
$$113$$ 5.37683 0.505810 0.252905 0.967491i $$-0.418614\pi$$
0.252905 + 0.967491i $$0.418614\pi$$
$$114$$ 3.68842 0.345452
$$115$$ 20.9029 1.94921
$$116$$ −6.52608 −0.605932
$$117$$ 0 0
$$118$$ −12.5261 −1.15312
$$119$$ 0 0
$$120$$ −4.41883 −0.403382
$$121$$ 1.00000 0.0909091
$$122$$ 3.79567 0.343643
$$123$$ −12.1492 −1.09546
$$124$$ −3.37683 −0.303249
$$125$$ 42.0942 3.76502
$$126$$ 0 0
$$127$$ 0.730416 0.0648139 0.0324070 0.999475i $$-0.489683\pi$$
0.0324070 + 0.999475i $$0.489683\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ −3.06525 −0.269880
$$130$$ 0 0
$$131$$ 7.37683 0.644517 0.322258 0.946652i $$-0.395558\pi$$
0.322258 + 0.946652i $$0.395558\pi$$
$$132$$ 1.00000 0.0870388
$$133$$ 0 0
$$134$$ 10.1492 0.876762
$$135$$ −4.41883 −0.380313
$$136$$ 2.37683 0.203812
$$137$$ −8.83767 −0.755053 −0.377526 0.925999i $$-0.623225\pi$$
−0.377526 + 0.925999i $$0.623225\pi$$
$$138$$ −4.73042 −0.402680
$$139$$ 1.47392 0.125016 0.0625080 0.998044i $$-0.480090\pi$$
0.0625080 + 0.998044i $$0.480090\pi$$
$$140$$ 0 0
$$141$$ 4.10725 0.345893
$$142$$ 0.934749 0.0784424
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ −28.8377 −2.39484
$$146$$ −7.46083 −0.617463
$$147$$ 0 0
$$148$$ 7.68842 0.631984
$$149$$ −9.36375 −0.767108 −0.383554 0.923518i $$-0.625300\pi$$
−0.383554 + 0.923518i $$0.625300\pi$$
$$150$$ −14.5261 −1.18605
$$151$$ −13.5681 −1.10415 −0.552077 0.833793i $$-0.686165\pi$$
−0.552077 + 0.833793i $$0.686165\pi$$
$$152$$ −3.68842 −0.299170
$$153$$ 2.37683 0.192156
$$154$$ 0 0
$$155$$ −14.9217 −1.19854
$$156$$ 0 0
$$157$$ −18.7406 −1.49566 −0.747831 0.663890i $$-0.768904\pi$$
−0.747831 + 0.663890i $$0.768904\pi$$
$$158$$ −0.418833 −0.0333205
$$159$$ −8.00000 −0.634441
$$160$$ 4.41883 0.349339
$$161$$ 0 0
$$162$$ 1.00000 0.0785674
$$163$$ 18.7724 1.47037 0.735185 0.677867i $$-0.237096\pi$$
0.735185 + 0.677867i $$0.237096\pi$$
$$164$$ 12.1492 0.948697
$$165$$ 4.41883 0.344006
$$166$$ −2.14925 −0.166814
$$167$$ −23.8898 −1.84865 −0.924325 0.381606i $$-0.875371\pi$$
−0.924325 + 0.381606i $$0.875371\pi$$
$$168$$ 0 0
$$169$$ −13.0000 −1.00000
$$170$$ 10.5028 0.805530
$$171$$ −3.68842 −0.282060
$$172$$ 3.06525 0.233723
$$173$$ 11.3768 0.864965 0.432482 0.901642i $$-0.357638\pi$$
0.432482 + 0.901642i $$0.357638\pi$$
$$174$$ 6.52608 0.494741
$$175$$ 0 0
$$176$$ −1.00000 −0.0753778
$$177$$ 12.5261 0.941518
$$178$$ −12.8377 −0.962224
$$179$$ 24.6101 1.83944 0.919722 0.392571i $$-0.128414\pi$$
0.919722 + 0.392571i $$0.128414\pi$$
$$180$$ 4.41883 0.329360
$$181$$ 17.4608 1.29785 0.648927 0.760851i $$-0.275218\pi$$
0.648927 + 0.760851i $$0.275218\pi$$
$$182$$ 0 0
$$183$$ −3.79567 −0.280584
$$184$$ 4.73042 0.348731
$$185$$ 33.9738 2.49781
$$186$$ 3.37683 0.247601
$$187$$ −2.37683 −0.173811
$$188$$ −4.10725 −0.299552
$$189$$ 0 0
$$190$$ −16.2985 −1.18242
$$191$$ −16.8377 −1.21833 −0.609165 0.793043i $$-0.708495\pi$$
−0.609165 + 0.793043i $$0.708495\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ 5.46083 0.393079 0.196540 0.980496i $$-0.437030\pi$$
0.196540 + 0.980496i $$0.437030\pi$$
$$194$$ 7.00000 0.502571
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −9.68842 −0.690271 −0.345136 0.938553i $$-0.612167\pi$$
−0.345136 + 0.938553i $$0.612167\pi$$
$$198$$ −1.00000 −0.0710669
$$199$$ 20.2985 1.43892 0.719461 0.694533i $$-0.244389\pi$$
0.719461 + 0.694533i $$0.244389\pi$$
$$200$$ 14.5261 1.02715
$$201$$ −10.1492 −0.715873
$$202$$ −15.3637 −1.08099
$$203$$ 0 0
$$204$$ −2.37683 −0.166412
$$205$$ 53.6855 3.74956
$$206$$ −2.62317 −0.182765
$$207$$ 4.73042 0.328787
$$208$$ 0 0
$$209$$ 3.68842 0.255133
$$210$$ 0 0
$$211$$ 22.5130 1.54986 0.774929 0.632048i $$-0.217785\pi$$
0.774929 + 0.632048i $$0.217785\pi$$
$$212$$ 8.00000 0.549442
$$213$$ −0.934749 −0.0640480
$$214$$ 10.9029 0.745308
$$215$$ 13.5448 0.923750
$$216$$ −1.00000 −0.0680414
$$217$$ 0 0
$$218$$ −11.7957 −0.798903
$$219$$ 7.46083 0.504156
$$220$$ −4.41883 −0.297918
$$221$$ 0 0
$$222$$ −7.68842 −0.516013
$$223$$ −7.46083 −0.499614 −0.249807 0.968296i $$-0.580367\pi$$
−0.249807 + 0.968296i $$0.580367\pi$$
$$224$$ 0 0
$$225$$ 14.5261 0.968405
$$226$$ 5.37683 0.357662
$$227$$ −2.77241 −0.184012 −0.0920058 0.995758i $$-0.529328\pi$$
−0.0920058 + 0.995758i $$0.529328\pi$$
$$228$$ 3.68842 0.244271
$$229$$ −2.53917 −0.167793 −0.0838965 0.996474i $$-0.526737\pi$$
−0.0838965 + 0.996474i $$0.526737\pi$$
$$230$$ 20.9029 1.37830
$$231$$ 0 0
$$232$$ −6.52608 −0.428458
$$233$$ −24.0522 −1.57571 −0.787855 0.615861i $$-0.788808\pi$$
−0.787855 + 0.615861i $$0.788808\pi$$
$$234$$ 0 0
$$235$$ −18.1492 −1.18393
$$236$$ −12.5261 −0.815379
$$237$$ 0.418833 0.0272061
$$238$$ 0 0
$$239$$ 7.59133 0.491042 0.245521 0.969391i $$-0.421041\pi$$
0.245521 + 0.969391i $$0.421041\pi$$
$$240$$ −4.41883 −0.285234
$$241$$ 16.2985 1.04988 0.524939 0.851140i $$-0.324088\pi$$
0.524939 + 0.851140i $$0.324088\pi$$
$$242$$ 1.00000 0.0642824
$$243$$ −1.00000 −0.0641500
$$244$$ 3.79567 0.242993
$$245$$ 0 0
$$246$$ −12.1492 −0.774608
$$247$$ 0 0
$$248$$ −3.37683 −0.214429
$$249$$ 2.14925 0.136203
$$250$$ 42.0942 2.66227
$$251$$ −8.61008 −0.543463 −0.271732 0.962373i $$-0.587596\pi$$
−0.271732 + 0.962373i $$0.587596\pi$$
$$252$$ 0 0
$$253$$ −4.73042 −0.297399
$$254$$ 0.730416 0.0458304
$$255$$ −10.5028 −0.657713
$$256$$ 1.00000 0.0625000
$$257$$ 0.837665 0.0522521 0.0261261 0.999659i $$-0.491683\pi$$
0.0261261 + 0.999659i $$0.491683\pi$$
$$258$$ −3.06525 −0.190834
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −6.52608 −0.403954
$$262$$ 7.37683 0.455742
$$263$$ 10.8377 0.668279 0.334140 0.942524i $$-0.391554\pi$$
0.334140 + 0.942524i $$0.391554\pi$$
$$264$$ 1.00000 0.0615457
$$265$$ 35.3507 2.17157
$$266$$ 0 0
$$267$$ 12.8377 0.785652
$$268$$ 10.1492 0.619964
$$269$$ 17.8797 1.09014 0.545071 0.838390i $$-0.316503\pi$$
0.545071 + 0.838390i $$0.316503\pi$$
$$270$$ −4.41883 −0.268922
$$271$$ −8.83767 −0.536850 −0.268425 0.963301i $$-0.586503\pi$$
−0.268425 + 0.963301i $$0.586503\pi$$
$$272$$ 2.37683 0.144117
$$273$$ 0 0
$$274$$ −8.83767 −0.533903
$$275$$ −14.5261 −0.875956
$$276$$ −4.73042 −0.284738
$$277$$ −16.0000 −0.961347 −0.480673 0.876900i $$-0.659608\pi$$
−0.480673 + 0.876900i $$0.659608\pi$$
$$278$$ 1.47392 0.0883997
$$279$$ −3.37683 −0.202166
$$280$$ 0 0
$$281$$ −15.9217 −0.949807 −0.474903 0.880038i $$-0.657517\pi$$
−0.474903 + 0.880038i $$0.657517\pi$$
$$282$$ 4.10725 0.244583
$$283$$ −18.0840 −1.07498 −0.537491 0.843269i $$-0.680628\pi$$
−0.537491 + 0.843269i $$0.680628\pi$$
$$284$$ 0.934749 0.0554672
$$285$$ 16.2985 0.965440
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 1.00000 0.0589256
$$289$$ −11.3507 −0.667686
$$290$$ −28.8377 −1.69341
$$291$$ −7.00000 −0.410347
$$292$$ −7.46083 −0.436612
$$293$$ 25.1492 1.46923 0.734617 0.678482i $$-0.237362\pi$$
0.734617 + 0.678482i $$0.237362\pi$$
$$294$$ 0 0
$$295$$ −55.3507 −3.22264
$$296$$ 7.68842 0.446880
$$297$$ 1.00000 0.0580259
$$298$$ −9.36375 −0.542427
$$299$$ 0 0
$$300$$ −14.5261 −0.838664
$$301$$ 0 0
$$302$$ −13.5681 −0.780755
$$303$$ 15.3637 0.882624
$$304$$ −3.68842 −0.211545
$$305$$ 16.7724 0.960386
$$306$$ 2.37683 0.135875
$$307$$ −3.86950 −0.220844 −0.110422 0.993885i $$-0.535220\pi$$
−0.110422 + 0.993885i $$0.535220\pi$$
$$308$$ 0 0
$$309$$ 2.62317 0.149227
$$310$$ −14.9217 −0.847494
$$311$$ −13.5681 −0.769375 −0.384688 0.923047i $$-0.625691\pi$$
−0.384688 + 0.923047i $$0.625691\pi$$
$$312$$ 0 0
$$313$$ 2.52608 0.142783 0.0713913 0.997448i $$-0.477256\pi$$
0.0713913 + 0.997448i $$0.477256\pi$$
$$314$$ −18.7406 −1.05759
$$315$$ 0 0
$$316$$ −0.418833 −0.0235612
$$317$$ −20.0102 −1.12388 −0.561941 0.827177i $$-0.689945\pi$$
−0.561941 + 0.827177i $$0.689945\pi$$
$$318$$ −8.00000 −0.448618
$$319$$ 6.52608 0.365390
$$320$$ 4.41883 0.247020
$$321$$ −10.9029 −0.608541
$$322$$ 0 0
$$323$$ −8.76675 −0.487795
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ 18.7724 1.03971
$$327$$ 11.7957 0.652302
$$328$$ 12.1492 0.670830
$$329$$ 0 0
$$330$$ 4.41883 0.243249
$$331$$ −16.1492 −0.887643 −0.443821 0.896115i $$-0.646378\pi$$
−0.443821 + 0.896115i $$0.646378\pi$$
$$332$$ −2.14925 −0.117955
$$333$$ 7.68842 0.421323
$$334$$ −23.8898 −1.30719
$$335$$ 44.8478 2.45030
$$336$$ 0 0
$$337$$ 23.1362 1.26031 0.630154 0.776471i $$-0.282992\pi$$
0.630154 + 0.776471i $$0.282992\pi$$
$$338$$ −13.0000 −0.707107
$$339$$ −5.37683 −0.292030
$$340$$ 10.5028 0.569596
$$341$$ 3.37683 0.182866
$$342$$ −3.68842 −0.199447
$$343$$ 0 0
$$344$$ 3.06525 0.165267
$$345$$ −20.9029 −1.12538
$$346$$ 11.3768 0.611622
$$347$$ 15.6566 0.840489 0.420245 0.907411i $$-0.361944\pi$$
0.420245 + 0.907411i $$0.361944\pi$$
$$348$$ 6.52608 0.349835
$$349$$ −20.0102 −1.07112 −0.535560 0.844497i $$-0.679899\pi$$
−0.535560 + 0.844497i $$0.679899\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −1.00000 −0.0533002
$$353$$ 7.46083 0.397100 0.198550 0.980091i $$-0.436377\pi$$
0.198550 + 0.980091i $$0.436377\pi$$
$$354$$ 12.5261 0.665754
$$355$$ 4.13050 0.219224
$$356$$ −12.8377 −0.680395
$$357$$ 0 0
$$358$$ 24.6101 1.30068
$$359$$ −11.2463 −0.593559 −0.296779 0.954946i $$-0.595913\pi$$
−0.296779 + 0.954946i $$0.595913\pi$$
$$360$$ 4.41883 0.232893
$$361$$ −5.39558 −0.283978
$$362$$ 17.4608 0.917721
$$363$$ −1.00000 −0.0524864
$$364$$ 0 0
$$365$$ −32.9682 −1.72563
$$366$$ −3.79567 −0.198403
$$367$$ −5.91600 −0.308813 −0.154406 0.988007i $$-0.549347\pi$$
−0.154406 + 0.988007i $$0.549347\pi$$
$$368$$ 4.73042 0.246590
$$369$$ 12.1492 0.632465
$$370$$ 33.9738 1.76622
$$371$$ 0 0
$$372$$ 3.37683 0.175081
$$373$$ −23.1260 −1.19742 −0.598709 0.800966i $$-0.704320\pi$$
−0.598709 + 0.800966i $$0.704320\pi$$
$$374$$ −2.37683 −0.122903
$$375$$ −42.0942 −2.17373
$$376$$ −4.10725 −0.211815
$$377$$ 0 0
$$378$$ 0 0
$$379$$ −6.77241 −0.347876 −0.173938 0.984757i $$-0.555649\pi$$
−0.173938 + 0.984757i $$0.555649\pi$$
$$380$$ −16.2985 −0.836095
$$381$$ −0.730416 −0.0374203
$$382$$ −16.8377 −0.861490
$$383$$ 25.3637 1.29603 0.648013 0.761629i $$-0.275600\pi$$
0.648013 + 0.761629i $$0.275600\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ 5.46083 0.277949
$$387$$ 3.06525 0.155815
$$388$$ 7.00000 0.355371
$$389$$ 25.2565 1.28056 0.640278 0.768144i $$-0.278819\pi$$
0.640278 + 0.768144i $$0.278819\pi$$
$$390$$ 0 0
$$391$$ 11.2434 0.568604
$$392$$ 0 0
$$393$$ −7.37683 −0.372112
$$394$$ −9.68842 −0.488095
$$395$$ −1.85075 −0.0931214
$$396$$ −1.00000 −0.0502519
$$397$$ −13.4739 −0.676237 −0.338118 0.941104i $$-0.609790\pi$$
−0.338118 + 0.941104i $$0.609790\pi$$
$$398$$ 20.2985 1.01747
$$399$$ 0 0
$$400$$ 14.5261 0.726304
$$401$$ 33.8898 1.69238 0.846189 0.532883i $$-0.178892\pi$$
0.846189 + 0.532883i $$0.178892\pi$$
$$402$$ −10.1492 −0.506199
$$403$$ 0 0
$$404$$ −15.3637 −0.764375
$$405$$ 4.41883 0.219574
$$406$$ 0 0
$$407$$ −7.68842 −0.381101
$$408$$ −2.37683 −0.117671
$$409$$ −4.21450 −0.208394 −0.104197 0.994557i $$-0.533227\pi$$
−0.104197 + 0.994557i $$0.533227\pi$$
$$410$$ 53.6855 2.65134
$$411$$ 8.83767 0.435930
$$412$$ −2.62317 −0.129234
$$413$$ 0 0
$$414$$ 4.73042 0.232487
$$415$$ −9.49717 −0.466198
$$416$$ 0 0
$$417$$ −1.47392 −0.0721781
$$418$$ 3.68842 0.180406
$$419$$ −22.3116 −1.08999 −0.544996 0.838439i $$-0.683469\pi$$
−0.544996 + 0.838439i $$0.683469\pi$$
$$420$$ 0 0
$$421$$ 18.3116 0.892452 0.446226 0.894920i $$-0.352768\pi$$
0.446226 + 0.894920i $$0.352768\pi$$
$$422$$ 22.5130 1.09592
$$423$$ −4.10725 −0.199701
$$424$$ 8.00000 0.388514
$$425$$ 34.5261 1.67476
$$426$$ −0.934749 −0.0452888
$$427$$ 0 0
$$428$$ 10.9029 0.527012
$$429$$ 0 0
$$430$$ 13.5448 0.653190
$$431$$ −28.8377 −1.38906 −0.694531 0.719463i $$-0.744388\pi$$
−0.694531 + 0.719463i $$0.744388\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ 7.62317 0.366346 0.183173 0.983081i $$-0.441363\pi$$
0.183173 + 0.983081i $$0.441363\pi$$
$$434$$ 0 0
$$435$$ 28.8377 1.38266
$$436$$ −11.7957 −0.564910
$$437$$ −17.4477 −0.834639
$$438$$ 7.46083 0.356492
$$439$$ −33.3739 −1.59285 −0.796425 0.604737i $$-0.793278\pi$$
−0.796425 + 0.604737i $$0.793278\pi$$
$$440$$ −4.41883 −0.210660
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −4.85075 −0.230466 −0.115233 0.993338i $$-0.536761\pi$$
−0.115233 + 0.993338i $$0.536761\pi$$
$$444$$ −7.68842 −0.364876
$$445$$ −56.7275 −2.68914
$$446$$ −7.46083 −0.353281
$$447$$ 9.36375 0.442890
$$448$$ 0 0
$$449$$ 14.3450 0.676982 0.338491 0.940970i $$-0.390083\pi$$
0.338491 + 0.940970i $$0.390083\pi$$
$$450$$ 14.5261 0.684766
$$451$$ −12.1492 −0.572086
$$452$$ 5.37683 0.252905
$$453$$ 13.5681 0.637484
$$454$$ −2.77241 −0.130116
$$455$$ 0 0
$$456$$ 3.68842 0.172726
$$457$$ −25.1827 −1.17800 −0.588998 0.808135i $$-0.700477\pi$$
−0.588998 + 0.808135i $$0.700477\pi$$
$$458$$ −2.53917 −0.118648
$$459$$ −2.37683 −0.110941
$$460$$ 20.9029 0.974603
$$461$$ 13.0187 0.606344 0.303172 0.952936i $$-0.401954\pi$$
0.303172 + 0.952936i $$0.401954\pi$$
$$462$$ 0 0
$$463$$ −14.2145 −0.660604 −0.330302 0.943875i $$-0.607151\pi$$
−0.330302 + 0.943875i $$0.607151\pi$$
$$464$$ −6.52608 −0.302966
$$465$$ 14.9217 0.691976
$$466$$ −24.0522 −1.11420
$$467$$ −2.39558 −0.110854 −0.0554271 0.998463i $$-0.517652\pi$$
−0.0554271 + 0.998463i $$0.517652\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ −18.1492 −0.837162
$$471$$ 18.7406 0.863520
$$472$$ −12.5261 −0.576560
$$473$$ −3.06525 −0.140940
$$474$$ 0.418833 0.0192376
$$475$$ −53.5782 −2.45834
$$476$$ 0 0
$$477$$ 8.00000 0.366295
$$478$$ 7.59133 0.347219
$$479$$ 10.8377 0.495186 0.247593 0.968864i $$-0.420360\pi$$
0.247593 + 0.968864i $$0.420360\pi$$
$$480$$ −4.41883 −0.201691
$$481$$ 0 0
$$482$$ 16.2985 0.742376
$$483$$ 0 0
$$484$$ 1.00000 0.0454545
$$485$$ 30.9318 1.40454
$$486$$ −1.00000 −0.0453609
$$487$$ −34.6435 −1.56985 −0.784923 0.619593i $$-0.787298\pi$$
−0.784923 + 0.619593i $$0.787298\pi$$
$$488$$ 3.79567 0.171822
$$489$$ −18.7724 −0.848918
$$490$$ 0 0
$$491$$ 38.1492 1.72165 0.860826 0.508900i $$-0.169948\pi$$
0.860826 + 0.508900i $$0.169948\pi$$
$$492$$ −12.1492 −0.547730
$$493$$ −15.5114 −0.698599
$$494$$ 0 0
$$495$$ −4.41883 −0.198612
$$496$$ −3.37683 −0.151624
$$497$$ 0 0
$$498$$ 2.14925 0.0963101
$$499$$ −0.623166 −0.0278968 −0.0139484 0.999903i $$-0.504440\pi$$
−0.0139484 + 0.999903i $$0.504440\pi$$
$$500$$ 42.0942 1.88251
$$501$$ 23.8898 1.06732
$$502$$ −8.61008 −0.384287
$$503$$ 31.1362 1.38829 0.694146 0.719834i $$-0.255782\pi$$
0.694146 + 0.719834i $$0.255782\pi$$
$$504$$ 0 0
$$505$$ −67.8898 −3.02106
$$506$$ −4.73042 −0.210293
$$507$$ 13.0000 0.577350
$$508$$ 0.730416 0.0324070
$$509$$ −42.1043 −1.86624 −0.933121 0.359563i $$-0.882926\pi$$
−0.933121 + 0.359563i $$0.882926\pi$$
$$510$$ −10.5028 −0.465073
$$511$$ 0 0
$$512$$ 1.00000 0.0441942
$$513$$ 3.68842 0.162848
$$514$$ 0.837665 0.0369478
$$515$$ −11.5913 −0.510775
$$516$$ −3.06525 −0.134940
$$517$$ 4.10725 0.180637
$$518$$ 0 0
$$519$$ −11.3768 −0.499388
$$520$$ 0 0
$$521$$ −18.0000 −0.788594 −0.394297 0.918983i $$-0.629012\pi$$
−0.394297 + 0.918983i $$0.629012\pi$$
$$522$$ −6.52608 −0.285639
$$523$$ −25.2667 −1.10483 −0.552417 0.833568i $$-0.686294\pi$$
−0.552417 + 0.833568i $$0.686294\pi$$
$$524$$ 7.37683 0.322258
$$525$$ 0 0
$$526$$ 10.8377 0.472545
$$527$$ −8.02617 −0.349626
$$528$$ 1.00000 0.0435194
$$529$$ −0.623166 −0.0270942
$$530$$ 35.3507 1.53553
$$531$$ −12.5261 −0.543586
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 12.8377 0.555540
$$535$$ 48.1782 2.08292
$$536$$ 10.1492 0.438381
$$537$$ −24.6101 −1.06200
$$538$$ 17.8797 0.770847
$$539$$ 0 0
$$540$$ −4.41883 −0.190156
$$541$$ −4.41883 −0.189980 −0.0949902 0.995478i $$-0.530282\pi$$
−0.0949902 + 0.995478i $$0.530282\pi$$
$$542$$ −8.83767 −0.379610
$$543$$ −17.4608 −0.749316
$$544$$ 2.37683 0.101906
$$545$$ −52.1231 −2.23271
$$546$$ 0 0
$$547$$ −27.6622 −1.18275 −0.591376 0.806396i $$-0.701415\pi$$
−0.591376 + 0.806396i $$0.701415\pi$$
$$548$$ −8.83767 −0.377526
$$549$$ 3.79567 0.161995
$$550$$ −14.5261 −0.619394
$$551$$ 24.0709 1.02546
$$552$$ −4.73042 −0.201340
$$553$$ 0 0
$$554$$ −16.0000 −0.679775
$$555$$ −33.9738 −1.44211
$$556$$ 1.47392 0.0625080
$$557$$ −7.98691 −0.338416 −0.169208 0.985580i $$-0.554121\pi$$
−0.169208 + 0.985580i $$0.554121\pi$$
$$558$$ −3.37683 −0.142953
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 2.37683 0.100350
$$562$$ −15.9217 −0.671615
$$563$$ −27.3507 −1.15269 −0.576346 0.817205i $$-0.695522\pi$$
−0.576346 + 0.817205i $$0.695522\pi$$
$$564$$ 4.10725 0.172946
$$565$$ 23.7593 0.999562
$$566$$ −18.0840 −0.760127
$$567$$ 0 0
$$568$$ 0.934749 0.0392212
$$569$$ −29.4477 −1.23451 −0.617257 0.786762i $$-0.711756\pi$$
−0.617257 + 0.786762i $$0.711756\pi$$
$$570$$ 16.2985 0.682669
$$571$$ −14.5261 −0.607898 −0.303949 0.952688i $$-0.598305\pi$$
−0.303949 + 0.952688i $$0.598305\pi$$
$$572$$ 0 0
$$573$$ 16.8377 0.703404
$$574$$ 0 0
$$575$$ 68.7144 2.86559
$$576$$ 1.00000 0.0416667
$$577$$ 13.3956 0.557665 0.278833 0.960340i $$-0.410053\pi$$
0.278833 + 0.960340i $$0.410053\pi$$
$$578$$ −11.3507 −0.472125
$$579$$ −5.46083 −0.226944
$$580$$ −28.8377 −1.19742
$$581$$ 0 0
$$582$$ −7.00000 −0.290159
$$583$$ −8.00000 −0.331326
$$584$$ −7.46083 −0.308731
$$585$$ 0 0
$$586$$ 25.1492 1.03891
$$587$$ −11.2928 −0.466105 −0.233053 0.972464i $$-0.574871\pi$$
−0.233053 + 0.972464i $$0.574871\pi$$
$$588$$ 0 0
$$589$$ 12.4552 0.513206
$$590$$ −55.3507 −2.27875
$$591$$ 9.68842 0.398528
$$592$$ 7.68842 0.315992
$$593$$ −19.5782 −0.803982 −0.401991 0.915644i $$-0.631682\pi$$
−0.401991 + 0.915644i $$0.631682\pi$$
$$594$$ 1.00000 0.0410305
$$595$$ 0 0
$$596$$ −9.36375 −0.383554
$$597$$ −20.2985 −0.830762
$$598$$ 0 0
$$599$$ 10.9318 0.446662 0.223331 0.974743i $$-0.428307\pi$$
0.223331 + 0.974743i $$0.428307\pi$$
$$600$$ −14.5261 −0.593025
$$601$$ −41.2202 −1.68141 −0.840703 0.541497i $$-0.817858\pi$$
−0.840703 + 0.541497i $$0.817858\pi$$
$$602$$ 0 0
$$603$$ 10.1492 0.413309
$$604$$ −13.5681 −0.552077
$$605$$ 4.41883 0.179651
$$606$$ 15.3637 0.624110
$$607$$ −31.3405 −1.27207 −0.636036 0.771660i $$-0.719427\pi$$
−0.636036 + 0.771660i $$0.719427\pi$$
$$608$$ −3.68842 −0.149585
$$609$$ 0 0
$$610$$ 16.7724 0.679095
$$611$$ 0 0
$$612$$ 2.37683 0.0960778
$$613$$ 17.0885 0.690198 0.345099 0.938566i $$-0.387845\pi$$
0.345099 + 0.938566i $$0.387845\pi$$
$$614$$ −3.86950 −0.156160
$$615$$ −53.6855 −2.16481
$$616$$ 0 0
$$617$$ −6.96817 −0.280528 −0.140264 0.990114i $$-0.544795\pi$$
−0.140264 + 0.990114i $$0.544795\pi$$
$$618$$ 2.62317 0.105519
$$619$$ −0.0187473 −0.000753517 0 −0.000376759 1.00000i $$-0.500120\pi$$
−0.000376759 1.00000i $$0.500120\pi$$
$$620$$ −14.9217 −0.599268
$$621$$ −4.73042 −0.189825
$$622$$ −13.5681 −0.544030
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 113.377 4.53507
$$626$$ 2.52608 0.100963
$$627$$ −3.68842 −0.147301
$$628$$ −18.7406 −0.747831
$$629$$ 18.2741 0.728636
$$630$$ 0 0
$$631$$ −5.78550 −0.230317 −0.115159 0.993347i $$-0.536738\pi$$
−0.115159 + 0.993347i $$0.536738\pi$$
$$632$$ −0.418833 −0.0166603
$$633$$ −22.5130 −0.894811
$$634$$ −20.0102 −0.794705
$$635$$ 3.22759 0.128083
$$636$$ −8.00000 −0.317221
$$637$$ 0 0
$$638$$ 6.52608 0.258370
$$639$$ 0.934749 0.0369781
$$640$$ 4.41883 0.174670
$$641$$ 15.3768 0.607348 0.303674 0.952776i $$-0.401787\pi$$
0.303674 + 0.952776i $$0.401787\pi$$
$$642$$ −10.9029 −0.430304
$$643$$ −35.8058 −1.41204 −0.706022 0.708190i $$-0.749512\pi$$
−0.706022 + 0.708190i $$0.749512\pi$$
$$644$$ 0 0
$$645$$ −13.5448 −0.533327
$$646$$ −8.76675 −0.344923
$$647$$ 9.25650 0.363910 0.181955 0.983307i $$-0.441757\pi$$
0.181955 + 0.983307i $$0.441757\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ 12.5261 0.491692
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 18.7724 0.735185
$$653$$ 24.8478 0.972371 0.486185 0.873856i $$-0.338388\pi$$
0.486185 + 0.873856i $$0.338388\pi$$
$$654$$ 11.7957 0.461247
$$655$$ 32.5970 1.27367
$$656$$ 12.1492 0.474348
$$657$$ −7.46083 −0.291075
$$658$$ 0 0
$$659$$ −6.60442 −0.257272 −0.128636 0.991692i $$-0.541060\pi$$
−0.128636 + 0.991692i $$0.541060\pi$$
$$660$$ 4.41883 0.172003
$$661$$ −13.2332 −0.514714 −0.257357 0.966316i $$-0.582852\pi$$
−0.257357 + 0.966316i $$0.582852\pi$$
$$662$$ −16.1492 −0.627658
$$663$$ 0 0
$$664$$ −2.14925 −0.0834070
$$665$$ 0 0
$$666$$ 7.68842 0.297920
$$667$$ −30.8711 −1.19533
$$668$$ −23.8898 −0.924325
$$669$$ 7.46083 0.288452
$$670$$ 44.8478 1.73262
$$671$$ −3.79567 −0.146530
$$672$$ 0 0
$$673$$ 0.130501 0.00503045 0.00251523 0.999997i $$-0.499199\pi$$
0.00251523 + 0.999997i $$0.499199\pi$$
$$674$$ 23.1362 0.891172
$$675$$ −14.5261 −0.559109
$$676$$ −13.0000 −0.500000
$$677$$ 28.1174 1.08064 0.540320 0.841460i $$-0.318303\pi$$
0.540320 + 0.841460i $$0.318303\pi$$
$$678$$ −5.37683 −0.206496
$$679$$ 0 0
$$680$$ 10.5028 0.402765
$$681$$ 2.77241 0.106239
$$682$$ 3.37683 0.129306
$$683$$ 15.9029 0.608508 0.304254 0.952591i $$-0.401593\pi$$
0.304254 + 0.952591i $$0.401593\pi$$
$$684$$ −3.68842 −0.141030
$$685$$ −39.0522 −1.49211
$$686$$ 0 0
$$687$$ 2.53917 0.0968753
$$688$$ 3.06525 0.116862
$$689$$ 0 0
$$690$$ −20.9029 −0.795760
$$691$$ −15.9813 −0.607956 −0.303978 0.952679i $$-0.598315\pi$$
−0.303978 + 0.952679i $$0.598315\pi$$
$$692$$ 11.3768 0.432482
$$693$$ 0 0
$$694$$ 15.6566 0.594316
$$695$$ 6.51300 0.247052
$$696$$ 6.52608 0.247371
$$697$$ 28.8767 1.09378
$$698$$ −20.0102 −0.757396
$$699$$ 24.0522 0.909736
$$700$$ 0 0
$$701$$ 13.9869 0.528278 0.264139 0.964485i $$-0.414912\pi$$
0.264139 + 0.964485i $$0.414912\pi$$
$$702$$ 0 0
$$703$$ −28.3581 −1.06955
$$704$$ −1.00000 −0.0376889
$$705$$ 18.1492 0.683540
$$706$$ 7.46083 0.280792
$$707$$ 0 0
$$708$$ 12.5261 0.470759
$$709$$ 7.64191 0.286998 0.143499 0.989650i $$-0.454165\pi$$
0.143499 + 0.989650i $$0.454165\pi$$
$$710$$ 4.13050 0.155015
$$711$$ −0.418833 −0.0157075
$$712$$ −12.8377 −0.481112
$$713$$ −15.9738 −0.598225
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 24.6101 0.919722
$$717$$ −7.59133 −0.283504
$$718$$ −11.2463 −0.419709
$$719$$ −19.4376 −0.724899 −0.362450 0.932003i $$-0.618060\pi$$
−0.362450 + 0.932003i $$0.618060\pi$$
$$720$$ 4.41883 0.164680
$$721$$ 0 0
$$722$$ −5.39558 −0.200803
$$723$$ −16.2985 −0.606148
$$724$$ 17.4608 0.648927
$$725$$ −94.7984 −3.52072
$$726$$ −1.00000 −0.0371135
$$727$$ −44.9217 −1.66605 −0.833026 0.553234i $$-0.813394\pi$$
−0.833026 + 0.553234i $$0.813394\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ −32.9682 −1.22021
$$731$$ 7.28559 0.269467
$$732$$ −3.79567 −0.140292
$$733$$ 25.0420 0.924947 0.462474 0.886633i $$-0.346962\pi$$
0.462474 + 0.886633i $$0.346962\pi$$
$$734$$ −5.91600 −0.218364
$$735$$ 0 0
$$736$$ 4.73042 0.174365
$$737$$ −10.1492 −0.373852
$$738$$ 12.1492 0.447220
$$739$$ −6.62317 −0.243637 −0.121819 0.992552i $$-0.538873\pi$$
−0.121819 + 0.992552i $$0.538873\pi$$
$$740$$ 33.9738 1.24890
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −39.3972 −1.44534 −0.722671 0.691192i $$-0.757086\pi$$
−0.722671 + 0.691192i $$0.757086\pi$$
$$744$$ 3.37683 0.123801
$$745$$ −41.3768 −1.51593
$$746$$ −23.1260 −0.846703
$$747$$ −2.14925 −0.0786369
$$748$$ −2.37683 −0.0869056
$$749$$ 0 0
$$750$$ −42.0942 −1.53706
$$751$$ −10.7072 −0.390710 −0.195355 0.980733i $$-0.562586\pi$$
−0.195355 + 0.980733i $$0.562586\pi$$
$$752$$ −4.10725 −0.149776
$$753$$ 8.61008 0.313769
$$754$$ 0 0
$$755$$ −59.9551 −2.18199
$$756$$ 0 0
$$757$$ −17.2797 −0.628043 −0.314022 0.949416i $$-0.601676\pi$$
−0.314022 + 0.949416i $$0.601676\pi$$
$$758$$ −6.77241 −0.245985
$$759$$ 4.73042 0.171703
$$760$$ −16.2985 −0.591209
$$761$$ −32.7087 −1.18569 −0.592846 0.805316i $$-0.701996\pi$$
−0.592846 + 0.805316i $$0.701996\pi$$
$$762$$ −0.730416 −0.0264602
$$763$$ 0 0
$$764$$ −16.8377 −0.609165
$$765$$ 10.5028 0.379731
$$766$$ 25.3637 0.916429
$$767$$ 0 0
$$768$$ −1.00000 −0.0360844
$$769$$ 12.2145 0.440466 0.220233 0.975447i $$-0.429318\pi$$
0.220233 + 0.975447i $$0.429318\pi$$
$$770$$ 0 0
$$771$$ −0.837665 −0.0301678
$$772$$ 5.46083 0.196540
$$773$$ 0.204334 0.00734937 0.00367468 0.999993i $$-0.498830\pi$$
0.00367468 + 0.999993i $$0.498830\pi$$
$$774$$ 3.06525 0.110178
$$775$$ −49.0522 −1.76201
$$776$$ 7.00000 0.251285
$$777$$ 0 0
$$778$$ 25.2565 0.905489
$$779$$ −44.8115 −1.60554
$$780$$ 0 0
$$781$$ −0.934749 −0.0334480
$$782$$ 11.2434 0.402064
$$783$$ 6.52608 0.233223
$$784$$ 0 0
$$785$$ −82.8115 −2.95567
$$786$$ −7.37683 −0.263123
$$787$$ 25.5579 0.911041 0.455521 0.890225i $$-0.349453\pi$$
0.455521 + 0.890225i $$0.349453\pi$$
$$788$$ −9.68842 −0.345136
$$789$$ −10.8377 −0.385831
$$790$$ −1.85075 −0.0658468
$$791$$ 0 0
$$792$$ −1.00000 −0.0355335
$$793$$ 0 0
$$794$$ −13.4739 −0.478171
$$795$$ −35.3507 −1.25376
$$796$$ 20.2985 0.719461
$$797$$ 30.0477 1.06434 0.532171 0.846637i $$-0.321376\pi$$
0.532171 + 0.846637i $$0.321376\pi$$
$$798$$ 0 0
$$799$$ −9.76225 −0.345364
$$800$$ 14.5261 0.513575
$$801$$ −12.8377 −0.453597
$$802$$ 33.8898 1.19669
$$803$$ 7.46083 0.263287
$$804$$ −10.1492 −0.357936
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −17.8797 −0.629394
$$808$$ −15.3637 −0.540495
$$809$$ 2.34342 0.0823901 0.0411951 0.999151i $$-0.486883\pi$$
0.0411951 + 0.999151i $$0.486883\pi$$
$$810$$ 4.41883 0.155262
$$811$$ −11.8695 −0.416794 −0.208397 0.978044i $$-0.566825\pi$$
−0.208397 + 0.978044i $$0.566825\pi$$
$$812$$ 0 0
$$813$$ 8.83767 0.309950
$$814$$ −7.68842 −0.269479
$$815$$ 82.9522 2.90569
$$816$$ −2.37683 −0.0832058
$$817$$ −11.3059 −0.395544
$$818$$ −4.21450 −0.147357
$$819$$ 0 0
$$820$$ 53.6855 1.87478
$$821$$ −37.4812 −1.30810 −0.654051 0.756451i $$-0.726932\pi$$
−0.654051 + 0.756451i $$0.726932\pi$$
$$822$$ 8.83767 0.308249
$$823$$ 1.29284 0.0450654 0.0225327 0.999746i $$-0.492827\pi$$
0.0225327 + 0.999746i $$0.492827\pi$$
$$824$$ −2.62317 −0.0913823
$$825$$ 14.5261 0.505733
$$826$$ 0 0
$$827$$ 43.3319 1.50680 0.753399 0.657563i $$-0.228413\pi$$
0.753399 + 0.657563i $$0.228413\pi$$
$$828$$ 4.73042 0.164393
$$829$$ 40.1174 1.39334 0.696668 0.717394i $$-0.254665\pi$$
0.696668 + 0.717394i $$0.254665\pi$$
$$830$$ −9.49717 −0.329652
$$831$$ 16.0000 0.555034
$$832$$ 0 0
$$833$$ 0 0
$$834$$ −1.47392 −0.0510376
$$835$$ −105.565 −3.65323
$$836$$ 3.68842 0.127567
$$837$$ 3.37683 0.116720
$$838$$ −22.3116 −0.770741
$$839$$ 54.6912 1.88815 0.944074 0.329733i $$-0.106959\pi$$
0.944074 + 0.329733i $$0.106959\pi$$
$$840$$ 0 0
$$841$$ 13.5897 0.468612
$$842$$ 18.3116 0.631059
$$843$$ 15.9217 0.548371
$$844$$ 22.5130 0.774929
$$845$$ −57.4448 −1.97616
$$846$$ −4.10725 −0.141210
$$847$$ 0 0
$$848$$ 8.00000 0.274721
$$849$$ 18.0840 0.620641
$$850$$ 34.5261 1.18423
$$851$$ 36.3694 1.24673
$$852$$ −0.934749 −0.0320240
$$853$$ −20.8275 −0.713120 −0.356560 0.934272i $$-0.616050\pi$$
−0.356560 + 0.934272i $$0.616050\pi$$
$$854$$ 0 0
$$855$$ −16.2985 −0.557397
$$856$$ 10.9029 0.372654
$$857$$ −36.8433 −1.25854 −0.629272 0.777185i $$-0.716647\pi$$
−0.629272 + 0.777185i $$0.716647\pi$$
$$858$$ 0 0
$$859$$ 38.5782 1.31627 0.658136 0.752899i $$-0.271345\pi$$
0.658136 + 0.752899i $$0.271345\pi$$
$$860$$ 13.5448 0.461875
$$861$$ 0 0
$$862$$ −28.8377 −0.982215
$$863$$ −17.2565 −0.587418 −0.293709 0.955895i $$-0.594890\pi$$
−0.293709 + 0.955895i $$0.594890\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 50.2723 1.70931
$$866$$ 7.62317 0.259046
$$867$$ 11.3507 0.385489
$$868$$ 0 0
$$869$$ 0.418833 0.0142079
$$870$$ 28.8377 0.977688
$$871$$ 0 0
$$872$$ −11.7957 −0.399452
$$873$$ 7.00000 0.236914
$$874$$ −17.4477 −0.590179
$$875$$ 0 0
$$876$$ 7.46083 0.252078
$$877$$ 0.250837 0.00847016 0.00423508 0.999991i $$-0.498652\pi$$
0.00423508 + 0.999991i $$0.498652\pi$$
$$878$$ −33.3739 −1.12632
$$879$$ −25.1492 −0.848263
$$880$$ −4.41883 −0.148959
$$881$$ 16.2985 0.549110 0.274555 0.961571i $$-0.411469\pi$$
0.274555 + 0.961571i $$0.411469\pi$$
$$882$$ 0 0
$$883$$ −33.9551 −1.14268 −0.571340 0.820714i $$-0.693576\pi$$
−0.571340 + 0.820714i $$0.693576\pi$$
$$884$$ 0 0
$$885$$ 55.3507 1.86059
$$886$$ −4.85075 −0.162964
$$887$$ 19.4608 0.653431 0.326715 0.945123i $$-0.394058\pi$$
0.326715 + 0.945123i $$0.394058\pi$$
$$888$$ −7.68842 −0.258006
$$889$$ 0 0
$$890$$ −56.7275 −1.90151
$$891$$ −1.00000 −0.0335013
$$892$$ −7.46083 −0.249807
$$893$$ 15.1492 0.506950
$$894$$ 9.36375 0.313171
$$895$$ 108.748 3.63504
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 14.3450 0.478699
$$899$$ 22.0375 0.734992
$$900$$ 14.5261 0.484203
$$901$$ 19.0147 0.633470
$$902$$ −12.1492 −0.404526
$$903$$ 0 0
$$904$$ 5.37683 0.178831
$$905$$ 77.1565 2.56477
$$906$$ 13.5681 0.450769
$$907$$ 0.772415 0.0256476 0.0128238 0.999918i $$-0.495918\pi$$
0.0128238 + 0.999918i $$0.495918\pi$$
$$908$$ −2.77241 −0.0920058
$$909$$ −15.3637 −0.509583
$$910$$ 0 0
$$911$$ −8.94491 −0.296358 −0.148179 0.988961i $$-0.547341\pi$$
−0.148179 + 0.988961i $$0.547341\pi$$
$$912$$ 3.68842 0.122136
$$913$$ 2.14925 0.0711297
$$914$$ −25.1827 −0.832969
$$915$$ −16.7724 −0.554479
$$916$$ −2.53917 −0.0838965
$$917$$ 0 0
$$918$$ −2.37683 −0.0784472
$$919$$ 58.8812 1.94231 0.971157 0.238443i $$-0.0766369\pi$$
0.971157 + 0.238443i $$0.0766369\pi$$
$$920$$ 20.9029 0.689149
$$921$$ 3.86950 0.127504
$$922$$ 13.0187 0.428750
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 111.683 3.67210
$$926$$ −14.2145 −0.467117
$$927$$ −2.62317 −0.0861561
$$928$$ −6.52608 −0.214229
$$929$$ −22.5392 −0.739486 −0.369743 0.929134i $$-0.620554\pi$$
−0.369743 + 0.929134i $$0.620554\pi$$
$$930$$ 14.9217 0.489301
$$931$$ 0 0
$$932$$ −24.0522 −0.787855
$$933$$ 13.5681 0.444199
$$934$$ −2.39558 −0.0783858
$$935$$ −10.5028 −0.343479
$$936$$ 0 0
$$937$$ −12.9478 −0.422987 −0.211494 0.977379i $$-0.567833\pi$$
−0.211494 + 0.977379i $$0.567833\pi$$
$$938$$ 0 0
$$939$$ −2.52608 −0.0824356
$$940$$ −18.1492 −0.591963
$$941$$ 48.8246 1.59164 0.795818 0.605536i $$-0.207041\pi$$
0.795818 + 0.605536i $$0.207041\pi$$
$$942$$ 18.7406 0.610601
$$943$$ 57.4710 1.87151
$$944$$ −12.5261 −0.407689
$$945$$ 0 0
$$946$$ −3.06525 −0.0996599
$$947$$ −31.1231 −1.01136 −0.505682 0.862720i $$-0.668759\pi$$
−0.505682 + 0.862720i $$0.668759\pi$$
$$948$$ 0.418833 0.0136031
$$949$$ 0 0
$$950$$ −53.5782 −1.73831
$$951$$ 20.0102 0.648874
$$952$$ 0 0
$$953$$ 16.2797 0.527353 0.263676 0.964611i $$-0.415065\pi$$
0.263676 + 0.964611i $$0.415065\pi$$
$$954$$ 8.00000 0.259010
$$955$$ −74.4028 −2.40762
$$956$$ 7.59133 0.245521
$$957$$ −6.52608 −0.210958
$$958$$ 10.8377 0.350149
$$959$$ 0 0
$$960$$ −4.41883 −0.142617
$$961$$ −19.5970 −0.632161
$$962$$ 0 0
$$963$$ 10.9029 0.351342
$$964$$ 16.2985 0.524939
$$965$$ 24.1305 0.776788
$$966$$ 0 0
$$967$$ 3.91308 0.125836 0.0629181 0.998019i $$-0.479959\pi$$
0.0629181 + 0.998019i $$0.479959\pi$$
$$968$$ 1.00000 0.0321412
$$969$$ 8.76675 0.281629
$$970$$ 30.9318 0.993161
$$971$$ 39.3132 1.26162 0.630810 0.775938i $$-0.282723\pi$$
0.630810 + 0.775938i $$0.282723\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ 0 0
$$974$$ −34.6435 −1.11005
$$975$$ 0 0
$$976$$ 3.79567 0.121496
$$977$$ −23.3507 −0.747054 −0.373527 0.927619i $$-0.621852\pi$$
−0.373527 + 0.927619i $$0.621852\pi$$
$$978$$ −18.7724 −0.600276
$$979$$ 12.8377 0.410294
$$980$$ 0 0
$$981$$ −11.7957 −0.376607
$$982$$ 38.1492 1.21739
$$983$$ −4.51592 −0.144035 −0.0720177 0.997403i $$-0.522944\pi$$
−0.0720177 + 0.997403i $$0.522944\pi$$
$$984$$ −12.1492 −0.387304
$$985$$ −42.8115 −1.36409
$$986$$ −15.5114 −0.493984
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 14.4999 0.461070
$$990$$ −4.41883 −0.140440
$$991$$ 59.9273 1.90365 0.951827 0.306635i $$-0.0992032\pi$$
0.951827 + 0.306635i $$0.0992032\pi$$
$$992$$ −3.37683 −0.107215
$$993$$ 16.1492 0.512481
$$994$$ 0 0
$$995$$ 89.6957 2.84354
$$996$$ 2.14925 0.0681015
$$997$$ −21.5073 −0.681144 −0.340572 0.940218i $$-0.610621\pi$$
−0.340572 + 0.940218i $$0.610621\pi$$
$$998$$ −0.623166 −0.0197260
$$999$$ −7.68842 −0.243251
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 3234.2.a.bf.1.3 3
3.2 odd 2 9702.2.a.dv.1.1 3
7.2 even 3 462.2.i.g.67.1 6
7.4 even 3 462.2.i.g.331.1 yes 6
7.6 odd 2 3234.2.a.bh.1.1 3
21.2 odd 6 1386.2.k.v.991.3 6
21.11 odd 6 1386.2.k.v.793.3 6
21.20 even 2 9702.2.a.dw.1.3 3

By twisted newform
Twist Min Dim Char Parity Ord Type
462.2.i.g.67.1 6 7.2 even 3
462.2.i.g.331.1 yes 6 7.4 even 3
1386.2.k.v.793.3 6 21.11 odd 6
1386.2.k.v.991.3 6 21.2 odd 6
3234.2.a.bf.1.3 3 1.1 even 1 trivial
3234.2.a.bh.1.1 3 7.6 odd 2
9702.2.a.dv.1.1 3 3.2 odd 2
9702.2.a.dw.1.3 3 21.20 even 2