# Properties

 Label 169.4.a.c.1.1 Level $169$ Weight $4$ Character 169.1 Self dual yes Analytic conductor $9.971$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [169,4,Mod(1,169)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(169, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("169.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$169 = 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 169.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: $$9.97132279097$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 13) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 169.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -9.00000 q^{5} -3.00000 q^{6} +15.0000 q^{7} -21.0000 q^{8} -26.0000 q^{9} +O(q^{10})$$ $$q+3.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -9.00000 q^{5} -3.00000 q^{6} +15.0000 q^{7} -21.0000 q^{8} -26.0000 q^{9} -27.0000 q^{10} -48.0000 q^{11} -1.00000 q^{12} +45.0000 q^{14} +9.00000 q^{15} -71.0000 q^{16} +45.0000 q^{17} -78.0000 q^{18} +6.00000 q^{19} -9.00000 q^{20} -15.0000 q^{21} -144.000 q^{22} -162.000 q^{23} +21.0000 q^{24} -44.0000 q^{25} +53.0000 q^{27} +15.0000 q^{28} -144.000 q^{29} +27.0000 q^{30} +264.000 q^{31} -45.0000 q^{32} +48.0000 q^{33} +135.000 q^{34} -135.000 q^{35} -26.0000 q^{36} +303.000 q^{37} +18.0000 q^{38} +189.000 q^{40} -192.000 q^{41} -45.0000 q^{42} +97.0000 q^{43} -48.0000 q^{44} +234.000 q^{45} -486.000 q^{46} +111.000 q^{47} +71.0000 q^{48} -118.000 q^{49} -132.000 q^{50} -45.0000 q^{51} -414.000 q^{53} +159.000 q^{54} +432.000 q^{55} -315.000 q^{56} -6.00000 q^{57} -432.000 q^{58} +522.000 q^{59} +9.00000 q^{60} +376.000 q^{61} +792.000 q^{62} -390.000 q^{63} +433.000 q^{64} +144.000 q^{66} -36.0000 q^{67} +45.0000 q^{68} +162.000 q^{69} -405.000 q^{70} +357.000 q^{71} +546.000 q^{72} -1098.00 q^{73} +909.000 q^{74} +44.0000 q^{75} +6.00000 q^{76} -720.000 q^{77} -830.000 q^{79} +639.000 q^{80} +649.000 q^{81} -576.000 q^{82} -438.000 q^{83} -15.0000 q^{84} -405.000 q^{85} +291.000 q^{86} +144.000 q^{87} +1008.00 q^{88} -438.000 q^{89} +702.000 q^{90} -162.000 q^{92} -264.000 q^{93} +333.000 q^{94} -54.0000 q^{95} +45.0000 q^{96} -852.000 q^{97} -354.000 q^{98} +1248.00 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 3.00000 1.06066 0.530330 0.847791i $$-0.322068\pi$$
0.530330 + 0.847791i $$0.322068\pi$$
$$3$$ −1.00000 −0.192450 −0.0962250 0.995360i $$-0.530677\pi$$
−0.0962250 + 0.995360i $$0.530677\pi$$
$$4$$ 1.00000 0.125000
$$5$$ −9.00000 −0.804984 −0.402492 0.915423i $$-0.631856\pi$$
−0.402492 + 0.915423i $$0.631856\pi$$
$$6$$ −3.00000 −0.204124
$$7$$ 15.0000 0.809924 0.404962 0.914334i $$-0.367285\pi$$
0.404962 + 0.914334i $$0.367285\pi$$
$$8$$ −21.0000 −0.928078
$$9$$ −26.0000 −0.962963
$$10$$ −27.0000 −0.853815
$$11$$ −48.0000 −1.31569 −0.657843 0.753155i $$-0.728531\pi$$
−0.657843 + 0.753155i $$0.728531\pi$$
$$12$$ −1.00000 −0.0240563
$$13$$ 0 0
$$14$$ 45.0000 0.859054
$$15$$ 9.00000 0.154919
$$16$$ −71.0000 −1.10938
$$17$$ 45.0000 0.642006 0.321003 0.947078i $$-0.395980\pi$$
0.321003 + 0.947078i $$0.395980\pi$$
$$18$$ −78.0000 −1.02138
$$19$$ 6.00000 0.0724471 0.0362235 0.999344i $$-0.488467\pi$$
0.0362235 + 0.999344i $$0.488467\pi$$
$$20$$ −9.00000 −0.100623
$$21$$ −15.0000 −0.155870
$$22$$ −144.000 −1.39550
$$23$$ −162.000 −1.46867 −0.734333 0.678789i $$-0.762505\pi$$
−0.734333 + 0.678789i $$0.762505\pi$$
$$24$$ 21.0000 0.178609
$$25$$ −44.0000 −0.352000
$$26$$ 0 0
$$27$$ 53.0000 0.377772
$$28$$ 15.0000 0.101240
$$29$$ −144.000 −0.922073 −0.461037 0.887381i $$-0.652522\pi$$
−0.461037 + 0.887381i $$0.652522\pi$$
$$30$$ 27.0000 0.164317
$$31$$ 264.000 1.52954 0.764771 0.644302i $$-0.222852\pi$$
0.764771 + 0.644302i $$0.222852\pi$$
$$32$$ −45.0000 −0.248592
$$33$$ 48.0000 0.253204
$$34$$ 135.000 0.680950
$$35$$ −135.000 −0.651976
$$36$$ −26.0000 −0.120370
$$37$$ 303.000 1.34629 0.673147 0.739509i $$-0.264942\pi$$
0.673147 + 0.739509i $$0.264942\pi$$
$$38$$ 18.0000 0.0768417
$$39$$ 0 0
$$40$$ 189.000 0.747088
$$41$$ −192.000 −0.731350 −0.365675 0.930743i $$-0.619162\pi$$
−0.365675 + 0.930743i $$0.619162\pi$$
$$42$$ −45.0000 −0.165325
$$43$$ 97.0000 0.344008 0.172004 0.985096i $$-0.444976\pi$$
0.172004 + 0.985096i $$0.444976\pi$$
$$44$$ −48.0000 −0.164461
$$45$$ 234.000 0.775170
$$46$$ −486.000 −1.55776
$$47$$ 111.000 0.344490 0.172245 0.985054i $$-0.444898\pi$$
0.172245 + 0.985054i $$0.444898\pi$$
$$48$$ 71.0000 0.213499
$$49$$ −118.000 −0.344023
$$50$$ −132.000 −0.373352
$$51$$ −45.0000 −0.123554
$$52$$ 0 0
$$53$$ −414.000 −1.07297 −0.536484 0.843911i $$-0.680248\pi$$
−0.536484 + 0.843911i $$0.680248\pi$$
$$54$$ 159.000 0.400688
$$55$$ 432.000 1.05911
$$56$$ −315.000 −0.751672
$$57$$ −6.00000 −0.0139424
$$58$$ −432.000 −0.978007
$$59$$ 522.000 1.15184 0.575920 0.817506i $$-0.304644\pi$$
0.575920 + 0.817506i $$0.304644\pi$$
$$60$$ 9.00000 0.0193649
$$61$$ 376.000 0.789211 0.394605 0.918851i $$-0.370881\pi$$
0.394605 + 0.918851i $$0.370881\pi$$
$$62$$ 792.000 1.62232
$$63$$ −390.000 −0.779927
$$64$$ 433.000 0.845703
$$65$$ 0 0
$$66$$ 144.000 0.268563
$$67$$ −36.0000 −0.0656433 −0.0328216 0.999461i $$-0.510449\pi$$
−0.0328216 + 0.999461i $$0.510449\pi$$
$$68$$ 45.0000 0.0802508
$$69$$ 162.000 0.282645
$$70$$ −405.000 −0.691525
$$71$$ 357.000 0.596734 0.298367 0.954451i $$-0.403558\pi$$
0.298367 + 0.954451i $$0.403558\pi$$
$$72$$ 546.000 0.893704
$$73$$ −1098.00 −1.76043 −0.880214 0.474578i $$-0.842601\pi$$
−0.880214 + 0.474578i $$0.842601\pi$$
$$74$$ 909.000 1.42796
$$75$$ 44.0000 0.0677424
$$76$$ 6.00000 0.00905588
$$77$$ −720.000 −1.06561
$$78$$ 0 0
$$79$$ −830.000 −1.18205 −0.591027 0.806652i $$-0.701277\pi$$
−0.591027 + 0.806652i $$0.701277\pi$$
$$80$$ 639.000 0.893030
$$81$$ 649.000 0.890261
$$82$$ −576.000 −0.775714
$$83$$ −438.000 −0.579238 −0.289619 0.957142i $$-0.593529\pi$$
−0.289619 + 0.957142i $$0.593529\pi$$
$$84$$ −15.0000 −0.0194837
$$85$$ −405.000 −0.516805
$$86$$ 291.000 0.364876
$$87$$ 144.000 0.177453
$$88$$ 1008.00 1.22106
$$89$$ −438.000 −0.521662 −0.260831 0.965384i $$-0.583997\pi$$
−0.260831 + 0.965384i $$0.583997\pi$$
$$90$$ 702.000 0.822192
$$91$$ 0 0
$$92$$ −162.000 −0.183583
$$93$$ −264.000 −0.294360
$$94$$ 333.000 0.365386
$$95$$ −54.0000 −0.0583188
$$96$$ 45.0000 0.0478416
$$97$$ −852.000 −0.891830 −0.445915 0.895075i $$-0.647122\pi$$
−0.445915 + 0.895075i $$0.647122\pi$$
$$98$$ −354.000 −0.364892
$$99$$ 1248.00 1.26696
$$100$$ −44.0000 −0.0440000
$$101$$ −396.000 −0.390133 −0.195067 0.980790i $$-0.562492\pi$$
−0.195067 + 0.980790i $$0.562492\pi$$
$$102$$ −135.000 −0.131049
$$103$$ −182.000 −0.174107 −0.0870534 0.996204i $$-0.527745\pi$$
−0.0870534 + 0.996204i $$0.527745\pi$$
$$104$$ 0 0
$$105$$ 135.000 0.125473
$$106$$ −1242.00 −1.13805
$$107$$ −612.000 −0.552937 −0.276469 0.961023i $$-0.589164\pi$$
−0.276469 + 0.961023i $$0.589164\pi$$
$$108$$ 53.0000 0.0472215
$$109$$ 1083.00 0.951675 0.475838 0.879533i $$-0.342145\pi$$
0.475838 + 0.879533i $$0.342145\pi$$
$$110$$ 1296.00 1.12335
$$111$$ −303.000 −0.259094
$$112$$ −1065.00 −0.898509
$$113$$ 90.0000 0.0749247 0.0374623 0.999298i $$-0.488073\pi$$
0.0374623 + 0.999298i $$0.488073\pi$$
$$114$$ −18.0000 −0.0147882
$$115$$ 1458.00 1.18225
$$116$$ −144.000 −0.115259
$$117$$ 0 0
$$118$$ 1566.00 1.22171
$$119$$ 675.000 0.519976
$$120$$ −189.000 −0.143777
$$121$$ 973.000 0.731029
$$122$$ 1128.00 0.837085
$$123$$ 192.000 0.140748
$$124$$ 264.000 0.191193
$$125$$ 1521.00 1.08834
$$126$$ −1170.00 −0.827237
$$127$$ −2086.00 −1.45750 −0.728750 0.684780i $$-0.759898\pi$$
−0.728750 + 0.684780i $$0.759898\pi$$
$$128$$ 1659.00 1.14560
$$129$$ −97.0000 −0.0662044
$$130$$ 0 0
$$131$$ −1467.00 −0.978415 −0.489208 0.872167i $$-0.662714\pi$$
−0.489208 + 0.872167i $$0.662714\pi$$
$$132$$ 48.0000 0.0316505
$$133$$ 90.0000 0.0586766
$$134$$ −108.000 −0.0696252
$$135$$ −477.000 −0.304101
$$136$$ −945.000 −0.595831
$$137$$ −414.000 −0.258178 −0.129089 0.991633i $$-0.541205\pi$$
−0.129089 + 0.991633i $$0.541205\pi$$
$$138$$ 486.000 0.299790
$$139$$ −2419.00 −1.47609 −0.738046 0.674750i $$-0.764251\pi$$
−0.738046 + 0.674750i $$0.764251\pi$$
$$140$$ −135.000 −0.0814970
$$141$$ −111.000 −0.0662971
$$142$$ 1071.00 0.632932
$$143$$ 0 0
$$144$$ 1846.00 1.06829
$$145$$ 1296.00 0.742255
$$146$$ −3294.00 −1.86721
$$147$$ 118.000 0.0662073
$$148$$ 303.000 0.168287
$$149$$ −930.000 −0.511333 −0.255666 0.966765i $$-0.582295\pi$$
−0.255666 + 0.966765i $$0.582295\pi$$
$$150$$ 132.000 0.0718517
$$151$$ −1683.00 −0.907024 −0.453512 0.891250i $$-0.649829\pi$$
−0.453512 + 0.891250i $$0.649829\pi$$
$$152$$ −126.000 −0.0672365
$$153$$ −1170.00 −0.618228
$$154$$ −2160.00 −1.13025
$$155$$ −2376.00 −1.23126
$$156$$ 0 0
$$157$$ 1874.00 0.952621 0.476310 0.879277i $$-0.341974\pi$$
0.476310 + 0.879277i $$0.341974\pi$$
$$158$$ −2490.00 −1.25376
$$159$$ 414.000 0.206493
$$160$$ 405.000 0.200113
$$161$$ −2430.00 −1.18951
$$162$$ 1947.00 0.944264
$$163$$ −1194.00 −0.573750 −0.286875 0.957968i $$-0.592616\pi$$
−0.286875 + 0.957968i $$0.592616\pi$$
$$164$$ −192.000 −0.0914188
$$165$$ −432.000 −0.203825
$$166$$ −1314.00 −0.614375
$$167$$ −2388.00 −1.10652 −0.553260 0.833008i $$-0.686617\pi$$
−0.553260 + 0.833008i $$0.686617\pi$$
$$168$$ 315.000 0.144659
$$169$$ 0 0
$$170$$ −1215.00 −0.548154
$$171$$ −156.000 −0.0697638
$$172$$ 97.0000 0.0430011
$$173$$ 1566.00 0.688213 0.344106 0.938931i $$-0.388182\pi$$
0.344106 + 0.938931i $$0.388182\pi$$
$$174$$ 432.000 0.188217
$$175$$ −660.000 −0.285093
$$176$$ 3408.00 1.45959
$$177$$ −522.000 −0.221672
$$178$$ −1314.00 −0.553306
$$179$$ 657.000 0.274338 0.137169 0.990548i $$-0.456200\pi$$
0.137169 + 0.990548i $$0.456200\pi$$
$$180$$ 234.000 0.0968963
$$181$$ 1222.00 0.501826 0.250913 0.968010i $$-0.419269\pi$$
0.250913 + 0.968010i $$0.419269\pi$$
$$182$$ 0 0
$$183$$ −376.000 −0.151884
$$184$$ 3402.00 1.36304
$$185$$ −2727.00 −1.08375
$$186$$ −792.000 −0.312216
$$187$$ −2160.00 −0.844678
$$188$$ 111.000 0.0430612
$$189$$ 795.000 0.305967
$$190$$ −162.000 −0.0618564
$$191$$ 1260.00 0.477332 0.238666 0.971102i $$-0.423290\pi$$
0.238666 + 0.971102i $$0.423290\pi$$
$$192$$ −433.000 −0.162756
$$193$$ 342.000 0.127553 0.0637764 0.997964i $$-0.479686\pi$$
0.0637764 + 0.997964i $$0.479686\pi$$
$$194$$ −2556.00 −0.945928
$$195$$ 0 0
$$196$$ −118.000 −0.0430029
$$197$$ 81.0000 0.0292945 0.0146472 0.999893i $$-0.495337\pi$$
0.0146472 + 0.999893i $$0.495337\pi$$
$$198$$ 3744.00 1.34381
$$199$$ −1996.00 −0.711019 −0.355509 0.934673i $$-0.615693\pi$$
−0.355509 + 0.934673i $$0.615693\pi$$
$$200$$ 924.000 0.326683
$$201$$ 36.0000 0.0126331
$$202$$ −1188.00 −0.413799
$$203$$ −2160.00 −0.746809
$$204$$ −45.0000 −0.0154443
$$205$$ 1728.00 0.588726
$$206$$ −546.000 −0.184668
$$207$$ 4212.00 1.41427
$$208$$ 0 0
$$209$$ −288.000 −0.0953176
$$210$$ 405.000 0.133084
$$211$$ 2833.00 0.924321 0.462161 0.886796i $$-0.347074\pi$$
0.462161 + 0.886796i $$0.347074\pi$$
$$212$$ −414.000 −0.134121
$$213$$ −357.000 −0.114841
$$214$$ −1836.00 −0.586478
$$215$$ −873.000 −0.276921
$$216$$ −1113.00 −0.350602
$$217$$ 3960.00 1.23881
$$218$$ 3249.00 1.00940
$$219$$ 1098.00 0.338794
$$220$$ 432.000 0.132388
$$221$$ 0 0
$$222$$ −909.000 −0.274811
$$223$$ −3507.00 −1.05312 −0.526561 0.850138i $$-0.676519\pi$$
−0.526561 + 0.850138i $$0.676519\pi$$
$$224$$ −675.000 −0.201341
$$225$$ 1144.00 0.338963
$$226$$ 270.000 0.0794696
$$227$$ −228.000 −0.0666647 −0.0333324 0.999444i $$-0.510612\pi$$
−0.0333324 + 0.999444i $$0.510612\pi$$
$$228$$ −6.00000 −0.00174281
$$229$$ 5493.00 1.58510 0.792549 0.609808i $$-0.208753\pi$$
0.792549 + 0.609808i $$0.208753\pi$$
$$230$$ 4374.00 1.25397
$$231$$ 720.000 0.205076
$$232$$ 3024.00 0.855756
$$233$$ −3627.00 −1.01980 −0.509898 0.860235i $$-0.670317\pi$$
−0.509898 + 0.860235i $$0.670317\pi$$
$$234$$ 0 0
$$235$$ −999.000 −0.277309
$$236$$ 522.000 0.143980
$$237$$ 830.000 0.227486
$$238$$ 2025.00 0.551518
$$239$$ 6075.00 1.64418 0.822090 0.569357i $$-0.192808\pi$$
0.822090 + 0.569357i $$0.192808\pi$$
$$240$$ −639.000 −0.171864
$$241$$ 210.000 0.0561298 0.0280649 0.999606i $$-0.491065\pi$$
0.0280649 + 0.999606i $$0.491065\pi$$
$$242$$ 2919.00 0.775374
$$243$$ −2080.00 −0.549103
$$244$$ 376.000 0.0986514
$$245$$ 1062.00 0.276933
$$246$$ 576.000 0.149286
$$247$$ 0 0
$$248$$ −5544.00 −1.41953
$$249$$ 438.000 0.111474
$$250$$ 4563.00 1.15436
$$251$$ −7092.00 −1.78344 −0.891719 0.452589i $$-0.850501\pi$$
−0.891719 + 0.452589i $$0.850501\pi$$
$$252$$ −390.000 −0.0974908
$$253$$ 7776.00 1.93230
$$254$$ −6258.00 −1.54591
$$255$$ 405.000 0.0994592
$$256$$ 1513.00 0.369385
$$257$$ 5805.00 1.40897 0.704486 0.709718i $$-0.251177\pi$$
0.704486 + 0.709718i $$0.251177\pi$$
$$258$$ −291.000 −0.0702204
$$259$$ 4545.00 1.09040
$$260$$ 0 0
$$261$$ 3744.00 0.887923
$$262$$ −4401.00 −1.03777
$$263$$ 792.000 0.185691 0.0928457 0.995681i $$-0.470404\pi$$
0.0928457 + 0.995681i $$0.470404\pi$$
$$264$$ −1008.00 −0.234993
$$265$$ 3726.00 0.863722
$$266$$ 270.000 0.0622359
$$267$$ 438.000 0.100394
$$268$$ −36.0000 −0.00820541
$$269$$ 5472.00 1.24027 0.620137 0.784493i $$-0.287077\pi$$
0.620137 + 0.784493i $$0.287077\pi$$
$$270$$ −1431.00 −0.322548
$$271$$ 2331.00 0.522502 0.261251 0.965271i $$-0.415865\pi$$
0.261251 + 0.965271i $$0.415865\pi$$
$$272$$ −3195.00 −0.712225
$$273$$ 0 0
$$274$$ −1242.00 −0.273839
$$275$$ 2112.00 0.463121
$$276$$ 162.000 0.0353306
$$277$$ 1384.00 0.300204 0.150102 0.988671i $$-0.452040\pi$$
0.150102 + 0.988671i $$0.452040\pi$$
$$278$$ −7257.00 −1.56563
$$279$$ −6864.00 −1.47289
$$280$$ 2835.00 0.605084
$$281$$ −4062.00 −0.862344 −0.431172 0.902270i $$-0.641900\pi$$
−0.431172 + 0.902270i $$0.641900\pi$$
$$282$$ −333.000 −0.0703187
$$283$$ 3764.00 0.790624 0.395312 0.918547i $$-0.370636\pi$$
0.395312 + 0.918547i $$0.370636\pi$$
$$284$$ 357.000 0.0745917
$$285$$ 54.0000 0.0112235
$$286$$ 0 0
$$287$$ −2880.00 −0.592338
$$288$$ 1170.00 0.239385
$$289$$ −2888.00 −0.587828
$$290$$ 3888.00 0.787280
$$291$$ 852.000 0.171633
$$292$$ −1098.00 −0.220053
$$293$$ 4227.00 0.842812 0.421406 0.906872i $$-0.361537\pi$$
0.421406 + 0.906872i $$0.361537\pi$$
$$294$$ 354.000 0.0702235
$$295$$ −4698.00 −0.927214
$$296$$ −6363.00 −1.24947
$$297$$ −2544.00 −0.497030
$$298$$ −2790.00 −0.542350
$$299$$ 0 0
$$300$$ 44.0000 0.00846780
$$301$$ 1455.00 0.278621
$$302$$ −5049.00 −0.962044
$$303$$ 396.000 0.0750812
$$304$$ −426.000 −0.0803710
$$305$$ −3384.00 −0.635303
$$306$$ −3510.00 −0.655730
$$307$$ 306.000 0.0568871 0.0284436 0.999595i $$-0.490945\pi$$
0.0284436 + 0.999595i $$0.490945\pi$$
$$308$$ −720.000 −0.133201
$$309$$ 182.000 0.0335069
$$310$$ −7128.00 −1.30595
$$311$$ 2106.00 0.383988 0.191994 0.981396i $$-0.438505\pi$$
0.191994 + 0.981396i $$0.438505\pi$$
$$312$$ 0 0
$$313$$ 10051.0 1.81507 0.907534 0.419979i $$-0.137963\pi$$
0.907534 + 0.419979i $$0.137963\pi$$
$$314$$ 5622.00 1.01041
$$315$$ 3510.00 0.627829
$$316$$ −830.000 −0.147757
$$317$$ −2154.00 −0.381643 −0.190821 0.981625i $$-0.561115\pi$$
−0.190821 + 0.981625i $$0.561115\pi$$
$$318$$ 1242.00 0.219019
$$319$$ 6912.00 1.21316
$$320$$ −3897.00 −0.680778
$$321$$ 612.000 0.106413
$$322$$ −7290.00 −1.26166
$$323$$ 270.000 0.0465115
$$324$$ 649.000 0.111283
$$325$$ 0 0
$$326$$ −3582.00 −0.608554
$$327$$ −1083.00 −0.183150
$$328$$ 4032.00 0.678750
$$329$$ 1665.00 0.279010
$$330$$ −1296.00 −0.216189
$$331$$ 10770.0 1.78844 0.894219 0.447630i $$-0.147732\pi$$
0.894219 + 0.447630i $$0.147732\pi$$
$$332$$ −438.000 −0.0724047
$$333$$ −7878.00 −1.29643
$$334$$ −7164.00 −1.17364
$$335$$ 324.000 0.0528418
$$336$$ 1065.00 0.172918
$$337$$ 2171.00 0.350926 0.175463 0.984486i $$-0.443858\pi$$
0.175463 + 0.984486i $$0.443858\pi$$
$$338$$ 0 0
$$339$$ −90.0000 −0.0144193
$$340$$ −405.000 −0.0646006
$$341$$ −12672.0 −2.01240
$$342$$ −468.000 −0.0739957
$$343$$ −6915.00 −1.08856
$$344$$ −2037.00 −0.319267
$$345$$ −1458.00 −0.227525
$$346$$ 4698.00 0.729960
$$347$$ −7047.00 −1.09021 −0.545105 0.838368i $$-0.683510\pi$$
−0.545105 + 0.838368i $$0.683510\pi$$
$$348$$ 144.000 0.0221816
$$349$$ −6873.00 −1.05416 −0.527082 0.849814i $$-0.676714\pi$$
−0.527082 + 0.849814i $$0.676714\pi$$
$$350$$ −1980.00 −0.302387
$$351$$ 0 0
$$352$$ 2160.00 0.327069
$$353$$ −9318.00 −1.40495 −0.702475 0.711709i $$-0.747922\pi$$
−0.702475 + 0.711709i $$0.747922\pi$$
$$354$$ −1566.00 −0.235119
$$355$$ −3213.00 −0.480362
$$356$$ −438.000 −0.0652077
$$357$$ −675.000 −0.100069
$$358$$ 1971.00 0.290979
$$359$$ 4128.00 0.606873 0.303437 0.952852i $$-0.401866\pi$$
0.303437 + 0.952852i $$0.401866\pi$$
$$360$$ −4914.00 −0.719418
$$361$$ −6823.00 −0.994751
$$362$$ 3666.00 0.532267
$$363$$ −973.000 −0.140687
$$364$$ 0 0
$$365$$ 9882.00 1.41712
$$366$$ −1128.00 −0.161097
$$367$$ −2536.00 −0.360703 −0.180352 0.983602i $$-0.557724\pi$$
−0.180352 + 0.983602i $$0.557724\pi$$
$$368$$ 11502.0 1.62930
$$369$$ 4992.00 0.704263
$$370$$ −8181.00 −1.14949
$$371$$ −6210.00 −0.869022
$$372$$ −264.000 −0.0367951
$$373$$ −92.0000 −0.0127710 −0.00638550 0.999980i $$-0.502033\pi$$
−0.00638550 + 0.999980i $$0.502033\pi$$
$$374$$ −6480.00 −0.895917
$$375$$ −1521.00 −0.209451
$$376$$ −2331.00 −0.319713
$$377$$ 0 0
$$378$$ 2385.00 0.324527
$$379$$ 10182.0 1.37998 0.689992 0.723817i $$-0.257614\pi$$
0.689992 + 0.723817i $$0.257614\pi$$
$$380$$ −54.0000 −0.00728985
$$381$$ 2086.00 0.280496
$$382$$ 3780.00 0.506287
$$383$$ −579.000 −0.0772468 −0.0386234 0.999254i $$-0.512297\pi$$
−0.0386234 + 0.999254i $$0.512297\pi$$
$$384$$ −1659.00 −0.220470
$$385$$ 6480.00 0.857796
$$386$$ 1026.00 0.135290
$$387$$ −2522.00 −0.331267
$$388$$ −852.000 −0.111479
$$389$$ −2106.00 −0.274495 −0.137247 0.990537i $$-0.543826\pi$$
−0.137247 + 0.990537i $$0.543826\pi$$
$$390$$ 0 0
$$391$$ −7290.00 −0.942893
$$392$$ 2478.00 0.319280
$$393$$ 1467.00 0.188296
$$394$$ 243.000 0.0310715
$$395$$ 7470.00 0.951535
$$396$$ 1248.00 0.158370
$$397$$ −1974.00 −0.249552 −0.124776 0.992185i $$-0.539821\pi$$
−0.124776 + 0.992185i $$0.539821\pi$$
$$398$$ −5988.00 −0.754149
$$399$$ −90.0000 −0.0112923
$$400$$ 3124.00 0.390500
$$401$$ 11886.0 1.48020 0.740098 0.672499i $$-0.234779\pi$$
0.740098 + 0.672499i $$0.234779\pi$$
$$402$$ 108.000 0.0133994
$$403$$ 0 0
$$404$$ −396.000 −0.0487667
$$405$$ −5841.00 −0.716646
$$406$$ −6480.00 −0.792111
$$407$$ −14544.0 −1.77130
$$408$$ 945.000 0.114668
$$409$$ 1254.00 0.151605 0.0758023 0.997123i $$-0.475848\pi$$
0.0758023 + 0.997123i $$0.475848\pi$$
$$410$$ 5184.00 0.624438
$$411$$ 414.000 0.0496864
$$412$$ −182.000 −0.0217633
$$413$$ 7830.00 0.932903
$$414$$ 12636.0 1.50006
$$415$$ 3942.00 0.466278
$$416$$ 0 0
$$417$$ 2419.00 0.284074
$$418$$ −864.000 −0.101100
$$419$$ 5823.00 0.678931 0.339466 0.940618i $$-0.389754\pi$$
0.339466 + 0.940618i $$0.389754\pi$$
$$420$$ 135.000 0.0156841
$$421$$ −7341.00 −0.849830 −0.424915 0.905233i $$-0.639696\pi$$
−0.424915 + 0.905233i $$0.639696\pi$$
$$422$$ 8499.00 0.980391
$$423$$ −2886.00 −0.331731
$$424$$ 8694.00 0.995797
$$425$$ −1980.00 −0.225986
$$426$$ −1071.00 −0.121808
$$427$$ 5640.00 0.639201
$$428$$ −612.000 −0.0691171
$$429$$ 0 0
$$430$$ −2619.00 −0.293720
$$431$$ −7485.00 −0.836519 −0.418260 0.908328i $$-0.637360\pi$$
−0.418260 + 0.908328i $$0.637360\pi$$
$$432$$ −3763.00 −0.419091
$$433$$ 15203.0 1.68732 0.843660 0.536878i $$-0.180396\pi$$
0.843660 + 0.536878i $$0.180396\pi$$
$$434$$ 11880.0 1.31396
$$435$$ −1296.00 −0.142847
$$436$$ 1083.00 0.118959
$$437$$ −972.000 −0.106401
$$438$$ 3294.00 0.359346
$$439$$ 1762.00 0.191562 0.0957809 0.995402i $$-0.469465\pi$$
0.0957809 + 0.995402i $$0.469465\pi$$
$$440$$ −9072.00 −0.982933
$$441$$ 3068.00 0.331282
$$442$$ 0 0
$$443$$ −7317.00 −0.784743 −0.392372 0.919807i $$-0.628345\pi$$
−0.392372 + 0.919807i $$0.628345\pi$$
$$444$$ −303.000 −0.0323868
$$445$$ 3942.00 0.419930
$$446$$ −10521.0 −1.11700
$$447$$ 930.000 0.0984060
$$448$$ 6495.00 0.684955
$$449$$ −5016.00 −0.527215 −0.263608 0.964630i $$-0.584912\pi$$
−0.263608 + 0.964630i $$0.584912\pi$$
$$450$$ 3432.00 0.359525
$$451$$ 9216.00 0.962227
$$452$$ 90.0000 0.00936558
$$453$$ 1683.00 0.174557
$$454$$ −684.000 −0.0707086
$$455$$ 0 0
$$456$$ 126.000 0.0129397
$$457$$ 9870.00 1.01028 0.505141 0.863037i $$-0.331440\pi$$
0.505141 + 0.863037i $$0.331440\pi$$
$$458$$ 16479.0 1.68125
$$459$$ 2385.00 0.242532
$$460$$ 1458.00 0.147782
$$461$$ −14541.0 −1.46907 −0.734536 0.678570i $$-0.762600\pi$$
−0.734536 + 0.678570i $$0.762600\pi$$
$$462$$ 2160.00 0.217516
$$463$$ −2112.00 −0.211993 −0.105997 0.994366i $$-0.533803\pi$$
−0.105997 + 0.994366i $$0.533803\pi$$
$$464$$ 10224.0 1.02293
$$465$$ 2376.00 0.236956
$$466$$ −10881.0 −1.08166
$$467$$ −3276.00 −0.324615 −0.162307 0.986740i $$-0.551894\pi$$
−0.162307 + 0.986740i $$0.551894\pi$$
$$468$$ 0 0
$$469$$ −540.000 −0.0531661
$$470$$ −2997.00 −0.294130
$$471$$ −1874.00 −0.183332
$$472$$ −10962.0 −1.06900
$$473$$ −4656.00 −0.452607
$$474$$ 2490.00 0.241286
$$475$$ −264.000 −0.0255014
$$476$$ 675.000 0.0649970
$$477$$ 10764.0 1.03323
$$478$$ 18225.0 1.74392
$$479$$ −15453.0 −1.47404 −0.737020 0.675870i $$-0.763768\pi$$
−0.737020 + 0.675870i $$0.763768\pi$$
$$480$$ −405.000 −0.0385117
$$481$$ 0 0
$$482$$ 630.000 0.0595347
$$483$$ 2430.00 0.228921
$$484$$ 973.000 0.0913787
$$485$$ 7668.00 0.717909
$$486$$ −6240.00 −0.582412
$$487$$ −3660.00 −0.340555 −0.170278 0.985396i $$-0.554466\pi$$
−0.170278 + 0.985396i $$0.554466\pi$$
$$488$$ −7896.00 −0.732449
$$489$$ 1194.00 0.110418
$$490$$ 3186.00 0.293732
$$491$$ −747.000 −0.0686591 −0.0343296 0.999411i $$-0.510930\pi$$
−0.0343296 + 0.999411i $$0.510930\pi$$
$$492$$ 192.000 0.0175936
$$493$$ −6480.00 −0.591977
$$494$$ 0 0
$$495$$ −11232.0 −1.01988
$$496$$ −18744.0 −1.69684
$$497$$ 5355.00 0.483309
$$498$$ 1314.00 0.118236
$$499$$ −15804.0 −1.41780 −0.708902 0.705307i $$-0.750809\pi$$
−0.708902 + 0.705307i $$0.750809\pi$$
$$500$$ 1521.00 0.136042
$$501$$ 2388.00 0.212950
$$502$$ −21276.0 −1.89162
$$503$$ −12078.0 −1.07064 −0.535319 0.844650i $$-0.679809\pi$$
−0.535319 + 0.844650i $$0.679809\pi$$
$$504$$ 8190.00 0.723833
$$505$$ 3564.00 0.314051
$$506$$ 23328.0 2.04952
$$507$$ 0 0
$$508$$ −2086.00 −0.182188
$$509$$ 16110.0 1.40287 0.701437 0.712731i $$-0.252542\pi$$
0.701437 + 0.712731i $$0.252542\pi$$
$$510$$ 1215.00 0.105492
$$511$$ −16470.0 −1.42581
$$512$$ −8733.00 −0.753804
$$513$$ 318.000 0.0273685
$$514$$ 17415.0 1.49444
$$515$$ 1638.00 0.140153
$$516$$ −97.0000 −0.00827556
$$517$$ −5328.00 −0.453240
$$518$$ 13635.0 1.15654
$$519$$ −1566.00 −0.132447
$$520$$ 0 0
$$521$$ 3915.00 0.329212 0.164606 0.986359i $$-0.447365\pi$$
0.164606 + 0.986359i $$0.447365\pi$$
$$522$$ 11232.0 0.941784
$$523$$ 16184.0 1.35311 0.676555 0.736392i $$-0.263472\pi$$
0.676555 + 0.736392i $$0.263472\pi$$
$$524$$ −1467.00 −0.122302
$$525$$ 660.000 0.0548662
$$526$$ 2376.00 0.196955
$$527$$ 11880.0 0.981975
$$528$$ −3408.00 −0.280898
$$529$$ 14077.0 1.15698
$$530$$ 11178.0 0.916116
$$531$$ −13572.0 −1.10918
$$532$$ 90.0000 0.00733458
$$533$$ 0 0
$$534$$ 1314.00 0.106484
$$535$$ 5508.00 0.445106
$$536$$ 756.000 0.0609221
$$537$$ −657.000 −0.0527964
$$538$$ 16416.0 1.31551
$$539$$ 5664.00 0.452627
$$540$$ −477.000 −0.0380126
$$541$$ −7923.00 −0.629642 −0.314821 0.949151i $$-0.601945\pi$$
−0.314821 + 0.949151i $$0.601945\pi$$
$$542$$ 6993.00 0.554198
$$543$$ −1222.00 −0.0965765
$$544$$ −2025.00 −0.159598
$$545$$ −9747.00 −0.766084
$$546$$ 0 0
$$547$$ −14389.0 −1.12473 −0.562367 0.826888i $$-0.690109\pi$$
−0.562367 + 0.826888i $$0.690109\pi$$
$$548$$ −414.000 −0.0322723
$$549$$ −9776.00 −0.759981
$$550$$ 6336.00 0.491214
$$551$$ −864.000 −0.0668015
$$552$$ −3402.00 −0.262317
$$553$$ −12450.0 −0.957374
$$554$$ 4152.00 0.318414
$$555$$ 2727.00 0.208567
$$556$$ −2419.00 −0.184512
$$557$$ −10383.0 −0.789842 −0.394921 0.918715i $$-0.629228\pi$$
−0.394921 + 0.918715i $$0.629228\pi$$
$$558$$ −20592.0 −1.56224
$$559$$ 0 0
$$560$$ 9585.00 0.723286
$$561$$ 2160.00 0.162558
$$562$$ −12186.0 −0.914654
$$563$$ 16425.0 1.22954 0.614770 0.788706i $$-0.289249\pi$$
0.614770 + 0.788706i $$0.289249\pi$$
$$564$$ −111.000 −0.00828713
$$565$$ −810.000 −0.0603132
$$566$$ 11292.0 0.838583
$$567$$ 9735.00 0.721043
$$568$$ −7497.00 −0.553815
$$569$$ −12213.0 −0.899817 −0.449908 0.893075i $$-0.648543\pi$$
−0.449908 + 0.893075i $$0.648543\pi$$
$$570$$ 162.000 0.0119043
$$571$$ 6383.00 0.467811 0.233906 0.972259i $$-0.424849\pi$$
0.233906 + 0.972259i $$0.424849\pi$$
$$572$$ 0 0
$$573$$ −1260.00 −0.0918626
$$574$$ −8640.00 −0.628269
$$575$$ 7128.00 0.516971
$$576$$ −11258.0 −0.814381
$$577$$ 6426.00 0.463636 0.231818 0.972759i $$-0.425533\pi$$
0.231818 + 0.972759i $$0.425533\pi$$
$$578$$ −8664.00 −0.623486
$$579$$ −342.000 −0.0245476
$$580$$ 1296.00 0.0927818
$$581$$ −6570.00 −0.469139
$$582$$ 2556.00 0.182044
$$583$$ 19872.0 1.41169
$$584$$ 23058.0 1.63381
$$585$$ 0 0
$$586$$ 12681.0 0.893937
$$587$$ −21330.0 −1.49980 −0.749901 0.661551i $$-0.769899\pi$$
−0.749901 + 0.661551i $$0.769899\pi$$
$$588$$ 118.000 0.00827591
$$589$$ 1584.00 0.110811
$$590$$ −14094.0 −0.983459
$$591$$ −81.0000 −0.00563772
$$592$$ −21513.0 −1.49355
$$593$$ 12084.0 0.836813 0.418407 0.908260i $$-0.362589\pi$$
0.418407 + 0.908260i $$0.362589\pi$$
$$594$$ −7632.00 −0.527180
$$595$$ −6075.00 −0.418573
$$596$$ −930.000 −0.0639166
$$597$$ 1996.00 0.136836
$$598$$ 0 0
$$599$$ 2394.00 0.163299 0.0816496 0.996661i $$-0.473981\pi$$
0.0816496 + 0.996661i $$0.473981\pi$$
$$600$$ −924.000 −0.0628702
$$601$$ −21971.0 −1.49121 −0.745604 0.666389i $$-0.767839\pi$$
−0.745604 + 0.666389i $$0.767839\pi$$
$$602$$ 4365.00 0.295522
$$603$$ 936.000 0.0632121
$$604$$ −1683.00 −0.113378
$$605$$ −8757.00 −0.588467
$$606$$ 1188.00 0.0796356
$$607$$ −15406.0 −1.03017 −0.515083 0.857141i $$-0.672239\pi$$
−0.515083 + 0.857141i $$0.672239\pi$$
$$608$$ −270.000 −0.0180098
$$609$$ 2160.00 0.143724
$$610$$ −10152.0 −0.673840
$$611$$ 0 0
$$612$$ −1170.00 −0.0772785
$$613$$ −9630.00 −0.634506 −0.317253 0.948341i $$-0.602760\pi$$
−0.317253 + 0.948341i $$0.602760\pi$$
$$614$$ 918.000 0.0603379
$$615$$ −1728.00 −0.113300
$$616$$ 15120.0 0.988965
$$617$$ 14748.0 0.962289 0.481144 0.876641i $$-0.340221\pi$$
0.481144 + 0.876641i $$0.340221\pi$$
$$618$$ 546.000 0.0355394
$$619$$ −3672.00 −0.238433 −0.119217 0.992868i $$-0.538038\pi$$
−0.119217 + 0.992868i $$0.538038\pi$$
$$620$$ −2376.00 −0.153907
$$621$$ −8586.00 −0.554822
$$622$$ 6318.00 0.407281
$$623$$ −6570.00 −0.422506
$$624$$ 0 0
$$625$$ −8189.00 −0.524096
$$626$$ 30153.0 1.92517
$$627$$ 288.000 0.0183439
$$628$$ 1874.00 0.119078
$$629$$ 13635.0 0.864329
$$630$$ 10530.0 0.665913
$$631$$ −19875.0 −1.25390 −0.626950 0.779059i $$-0.715697\pi$$
−0.626950 + 0.779059i $$0.715697\pi$$
$$632$$ 17430.0 1.09704
$$633$$ −2833.00 −0.177886
$$634$$ −6462.00 −0.404793
$$635$$ 18774.0 1.17327
$$636$$ 414.000 0.0258116
$$637$$ 0 0
$$638$$ 20736.0 1.28675
$$639$$ −9282.00 −0.574633
$$640$$ −14931.0 −0.922187
$$641$$ −1710.00 −0.105368 −0.0526840 0.998611i $$-0.516778\pi$$
−0.0526840 + 0.998611i $$0.516778\pi$$
$$642$$ 1836.00 0.112868
$$643$$ −16452.0 −1.00903 −0.504513 0.863404i $$-0.668328\pi$$
−0.504513 + 0.863404i $$0.668328\pi$$
$$644$$ −2430.00 −0.148689
$$645$$ 873.000 0.0532936
$$646$$ 810.000 0.0493329
$$647$$ −25902.0 −1.57390 −0.786950 0.617017i $$-0.788341\pi$$
−0.786950 + 0.617017i $$0.788341\pi$$
$$648$$ −13629.0 −0.826231
$$649$$ −25056.0 −1.51546
$$650$$ 0 0
$$651$$ −3960.00 −0.238410
$$652$$ −1194.00 −0.0717188
$$653$$ 18108.0 1.08518 0.542589 0.839999i $$-0.317444\pi$$
0.542589 + 0.839999i $$0.317444\pi$$
$$654$$ −3249.00 −0.194260
$$655$$ 13203.0 0.787609
$$656$$ 13632.0 0.811342
$$657$$ 28548.0 1.69523
$$658$$ 4995.00 0.295935
$$659$$ −32904.0 −1.94500 −0.972502 0.232894i $$-0.925181\pi$$
−0.972502 + 0.232894i $$0.925181\pi$$
$$660$$ −432.000 −0.0254781
$$661$$ 15318.0 0.901363 0.450682 0.892685i $$-0.351181\pi$$
0.450682 + 0.892685i $$0.351181\pi$$
$$662$$ 32310.0 1.89692
$$663$$ 0 0
$$664$$ 9198.00 0.537578
$$665$$ −810.000 −0.0472338
$$666$$ −23634.0 −1.37507
$$667$$ 23328.0 1.35422
$$668$$ −2388.00 −0.138315
$$669$$ 3507.00 0.202673
$$670$$ 972.000 0.0560472
$$671$$ −18048.0 −1.03835
$$672$$ 675.000 0.0387481
$$673$$ 7729.00 0.442691 0.221346 0.975195i $$-0.428955\pi$$
0.221346 + 0.975195i $$0.428955\pi$$
$$674$$ 6513.00 0.372213
$$675$$ −2332.00 −0.132976
$$676$$ 0 0
$$677$$ 19242.0 1.09236 0.546182 0.837667i $$-0.316081\pi$$
0.546182 + 0.837667i $$0.316081\pi$$
$$678$$ −270.000 −0.0152939
$$679$$ −12780.0 −0.722314
$$680$$ 8505.00 0.479635
$$681$$ 228.000 0.0128296
$$682$$ −38016.0 −2.13447
$$683$$ 22518.0 1.26153 0.630767 0.775973i $$-0.282740\pi$$
0.630767 + 0.775973i $$0.282740\pi$$
$$684$$ −156.000 −0.00872048
$$685$$ 3726.00 0.207829
$$686$$ −20745.0 −1.15459
$$687$$ −5493.00 −0.305052
$$688$$ −6887.00 −0.381634
$$689$$ 0 0
$$690$$ −4374.00 −0.241327
$$691$$ 9168.00 0.504728 0.252364 0.967632i $$-0.418792\pi$$
0.252364 + 0.967632i $$0.418792\pi$$
$$692$$ 1566.00 0.0860266
$$693$$ 18720.0 1.02614
$$694$$ −21141.0 −1.15634
$$695$$ 21771.0 1.18823
$$696$$ −3024.00 −0.164690
$$697$$ −8640.00 −0.469531
$$698$$ −20619.0 −1.11811
$$699$$ 3627.00 0.196260
$$700$$ −660.000 −0.0356367
$$701$$ −1170.00 −0.0630389 −0.0315195 0.999503i $$-0.510035\pi$$
−0.0315195 + 0.999503i $$0.510035\pi$$
$$702$$ 0 0
$$703$$ 1818.00 0.0975351
$$704$$ −20784.0 −1.11268
$$705$$ 999.000 0.0533681
$$706$$ −27954.0 −1.49017
$$707$$ −5940.00 −0.315978
$$708$$ −522.000 −0.0277090
$$709$$ −1662.00 −0.0880363 −0.0440181 0.999031i $$-0.514016\pi$$
−0.0440181 + 0.999031i $$0.514016\pi$$
$$710$$ −9639.00 −0.509500
$$711$$ 21580.0 1.13827
$$712$$ 9198.00 0.484143
$$713$$ −42768.0 −2.24639
$$714$$ −2025.00 −0.106140
$$715$$ 0 0
$$716$$ 657.000 0.0342922
$$717$$ −6075.00 −0.316423
$$718$$ 12384.0 0.643686
$$719$$ −30960.0 −1.60586 −0.802930 0.596073i $$-0.796727\pi$$
−0.802930 + 0.596073i $$0.796727\pi$$
$$720$$ −16614.0 −0.859954
$$721$$ −2730.00 −0.141013
$$722$$ −20469.0 −1.05509
$$723$$ −210.000 −0.0108022
$$724$$ 1222.00 0.0627283
$$725$$ 6336.00 0.324570
$$726$$ −2919.00 −0.149221
$$727$$ 8372.00 0.427098 0.213549 0.976932i $$-0.431498\pi$$
0.213549 + 0.976932i $$0.431498\pi$$
$$728$$ 0 0
$$729$$ −15443.0 −0.784586
$$730$$ 29646.0 1.50308
$$731$$ 4365.00 0.220855
$$732$$ −376.000 −0.0189855
$$733$$ −2739.00 −0.138018 −0.0690091 0.997616i $$-0.521984\pi$$
−0.0690091 + 0.997616i $$0.521984\pi$$
$$734$$ −7608.00 −0.382584
$$735$$ −1062.00 −0.0532959
$$736$$ 7290.00 0.365099
$$737$$ 1728.00 0.0863659
$$738$$ 14976.0 0.746984
$$739$$ −6756.00 −0.336297 −0.168148 0.985762i $$-0.553779\pi$$
−0.168148 + 0.985762i $$0.553779\pi$$
$$740$$ −2727.00 −0.135468
$$741$$ 0 0
$$742$$ −18630.0 −0.921737
$$743$$ 29643.0 1.46366 0.731828 0.681490i $$-0.238668\pi$$
0.731828 + 0.681490i $$0.238668\pi$$
$$744$$ 5544.00 0.273189
$$745$$ 8370.00 0.411615
$$746$$ −276.000 −0.0135457
$$747$$ 11388.0 0.557785
$$748$$ −2160.00 −0.105585
$$749$$ −9180.00 −0.447837
$$750$$ −4563.00 −0.222156
$$751$$ −18128.0 −0.880826 −0.440413 0.897795i $$-0.645168\pi$$
−0.440413 + 0.897795i $$0.645168\pi$$
$$752$$ −7881.00 −0.382168
$$753$$ 7092.00 0.343223
$$754$$ 0 0
$$755$$ 15147.0 0.730140
$$756$$ 795.000 0.0382459
$$757$$ −6410.00 −0.307761 −0.153881 0.988089i $$-0.549177\pi$$
−0.153881 + 0.988089i $$0.549177\pi$$
$$758$$ 30546.0 1.46369
$$759$$ −7776.00 −0.371872
$$760$$ 1134.00 0.0541243
$$761$$ 28290.0 1.34758 0.673792 0.738921i $$-0.264664\pi$$
0.673792 + 0.738921i $$0.264664\pi$$
$$762$$ 6258.00 0.297511
$$763$$ 16245.0 0.770784
$$764$$ 1260.00 0.0596665
$$765$$ 10530.0 0.497664
$$766$$ −1737.00 −0.0819326
$$767$$ 0 0
$$768$$ −1513.00 −0.0710881
$$769$$ −27960.0 −1.31114 −0.655568 0.755136i $$-0.727571\pi$$
−0.655568 + 0.755136i $$0.727571\pi$$
$$770$$ 19440.0 0.909830
$$771$$ −5805.00 −0.271157
$$772$$ 342.000 0.0159441
$$773$$ −5649.00 −0.262847 −0.131423 0.991326i $$-0.541955\pi$$
−0.131423 + 0.991326i $$0.541955\pi$$
$$774$$ −7566.00 −0.351362
$$775$$ −11616.0 −0.538399
$$776$$ 17892.0 0.827687
$$777$$ −4545.00 −0.209847
$$778$$ −6318.00 −0.291146
$$779$$ −1152.00 −0.0529842
$$780$$ 0 0
$$781$$ −17136.0 −0.785114
$$782$$ −21870.0 −1.00009
$$783$$ −7632.00 −0.348334
$$784$$ 8378.00 0.381651
$$785$$ −16866.0 −0.766845
$$786$$ 4401.00 0.199718
$$787$$ 756.000 0.0342420 0.0171210 0.999853i $$-0.494550\pi$$
0.0171210 + 0.999853i $$0.494550\pi$$
$$788$$ 81.0000 0.00366181
$$789$$ −792.000 −0.0357363
$$790$$ 22410.0 1.00926
$$791$$ 1350.00 0.0606833
$$792$$ −26208.0 −1.17583
$$793$$ 0 0
$$794$$ −5922.00 −0.264690
$$795$$ −3726.00 −0.166223
$$796$$ −1996.00 −0.0888773
$$797$$ −31194.0 −1.38638 −0.693192 0.720753i $$-0.743796\pi$$
−0.693192 + 0.720753i $$0.743796\pi$$
$$798$$ −270.000 −0.0119773
$$799$$ 4995.00 0.221164
$$800$$ 1980.00 0.0875045
$$801$$ 11388.0 0.502341
$$802$$ 35658.0 1.56998
$$803$$ 52704.0 2.31617
$$804$$ 36.0000 0.00157913
$$805$$ 21870.0 0.957536
$$806$$ 0 0
$$807$$ −5472.00 −0.238691
$$808$$ 8316.00 0.362074
$$809$$ 17055.0 0.741189 0.370594 0.928795i $$-0.379154\pi$$
0.370594 + 0.928795i $$0.379154\pi$$
$$810$$ −17523.0 −0.760118
$$811$$ 35520.0 1.53795 0.768974 0.639280i $$-0.220768\pi$$
0.768974 + 0.639280i $$0.220768\pi$$
$$812$$ −2160.00 −0.0933512
$$813$$ −2331.00 −0.100556
$$814$$ −43632.0 −1.87875
$$815$$ 10746.0 0.461860
$$816$$ 3195.00 0.137068
$$817$$ 582.000 0.0249224
$$818$$ 3762.00 0.160801
$$819$$ 0 0
$$820$$ 1728.00 0.0735907
$$821$$ 1095.00 0.0465478 0.0232739 0.999729i $$-0.492591\pi$$
0.0232739 + 0.999729i $$0.492591\pi$$
$$822$$ 1242.00 0.0527004
$$823$$ 2554.00 0.108174 0.0540868 0.998536i $$-0.482775\pi$$
0.0540868 + 0.998536i $$0.482775\pi$$
$$824$$ 3822.00 0.161585
$$825$$ −2112.00 −0.0891278
$$826$$ 23490.0 0.989494
$$827$$ 21522.0 0.904950 0.452475 0.891777i $$-0.350541\pi$$
0.452475 + 0.891777i $$0.350541\pi$$
$$828$$ 4212.00 0.176784
$$829$$ 13124.0 0.549838 0.274919 0.961467i $$-0.411349\pi$$
0.274919 + 0.961467i $$0.411349\pi$$
$$830$$ 11826.0 0.494562
$$831$$ −1384.00 −0.0577743
$$832$$ 0 0
$$833$$ −5310.00 −0.220865
$$834$$ 7257.00 0.301306
$$835$$ 21492.0 0.890732
$$836$$ −288.000 −0.0119147
$$837$$ 13992.0 0.577819
$$838$$ 17469.0 0.720115
$$839$$ −23424.0 −0.963869 −0.481935 0.876207i $$-0.660066\pi$$
−0.481935 + 0.876207i $$0.660066\pi$$
$$840$$ −2835.00 −0.116449
$$841$$ −3653.00 −0.149781
$$842$$ −22023.0 −0.901381
$$843$$ 4062.00 0.165958
$$844$$ 2833.00 0.115540
$$845$$ 0 0
$$846$$ −8658.00 −0.351854
$$847$$ 14595.0 0.592078
$$848$$ 29394.0 1.19032
$$849$$ −3764.00 −0.152156
$$850$$ −5940.00 −0.239694
$$851$$ −49086.0 −1.97726
$$852$$ −357.000 −0.0143552
$$853$$ 31077.0 1.24743 0.623714 0.781653i $$-0.285623\pi$$
0.623714 + 0.781653i $$0.285623\pi$$
$$854$$ 16920.0 0.677975
$$855$$ 1404.00 0.0561588
$$856$$ 12852.0 0.513169
$$857$$ 19422.0 0.774146 0.387073 0.922049i $$-0.373486\pi$$
0.387073 + 0.922049i $$0.373486\pi$$
$$858$$ 0 0
$$859$$ 1744.00 0.0692718 0.0346359 0.999400i $$-0.488973\pi$$
0.0346359 + 0.999400i $$0.488973\pi$$
$$860$$ −873.000 −0.0346152
$$861$$ 2880.00 0.113996
$$862$$ −22455.0 −0.887263
$$863$$ 19179.0 0.756501 0.378251 0.925703i $$-0.376526\pi$$
0.378251 + 0.925703i $$0.376526\pi$$
$$864$$ −2385.00 −0.0939113
$$865$$ −14094.0 −0.554000
$$866$$ 45609.0 1.78967
$$867$$ 2888.00 0.113128
$$868$$ 3960.00 0.154852
$$869$$ 39840.0 1.55521
$$870$$ −3888.00 −0.151512
$$871$$ 0 0
$$872$$ −22743.0 −0.883228
$$873$$ 22152.0 0.858799
$$874$$ −2916.00 −0.112855
$$875$$ 22815.0 0.881472
$$876$$ 1098.00 0.0423493
$$877$$ 29217.0 1.12496 0.562479 0.826812i $$-0.309848\pi$$
0.562479 + 0.826812i $$0.309848\pi$$
$$878$$ 5286.00 0.203182
$$879$$ −4227.00 −0.162199
$$880$$ −30672.0 −1.17495
$$881$$ 15633.0 0.597831 0.298916 0.954280i $$-0.403375\pi$$
0.298916 + 0.954280i $$0.403375\pi$$
$$882$$ 9204.00 0.351377
$$883$$ −30589.0 −1.16580 −0.582900 0.812544i $$-0.698082\pi$$
−0.582900 + 0.812544i $$0.698082\pi$$
$$884$$ 0 0
$$885$$ 4698.00 0.178442
$$886$$ −21951.0 −0.832346
$$887$$ −25884.0 −0.979819 −0.489910 0.871773i $$-0.662970\pi$$
−0.489910 + 0.871773i $$0.662970\pi$$
$$888$$ 6363.00 0.240460
$$889$$ −31290.0 −1.18046
$$890$$ 11826.0 0.445403
$$891$$ −31152.0 −1.17130
$$892$$ −3507.00 −0.131640
$$893$$ 666.000 0.0249573
$$894$$ 2790.00 0.104375
$$895$$ −5913.00 −0.220838
$$896$$ 24885.0 0.927845
$$897$$ 0 0
$$898$$ −15048.0 −0.559196
$$899$$ −38016.0 −1.41035
$$900$$ 1144.00 0.0423704
$$901$$ −18630.0 −0.688852
$$902$$ 27648.0 1.02060
$$903$$ −1455.00 −0.0536206
$$904$$ −1890.00 −0.0695359
$$905$$ −10998.0 −0.403962
$$906$$ 5049.00 0.185145
$$907$$ 12305.0 0.450475 0.225237 0.974304i $$-0.427684\pi$$
0.225237 + 0.974304i $$0.427684\pi$$
$$908$$ −228.000 −0.00833309
$$909$$ 10296.0 0.375684
$$910$$ 0 0
$$911$$ 29772.0 1.08276 0.541378 0.840779i $$-0.317903\pi$$
0.541378 + 0.840779i $$0.317903\pi$$
$$912$$ 426.000 0.0154674
$$913$$ 21024.0 0.762095
$$914$$ 29610.0 1.07157
$$915$$ 3384.00 0.122264
$$916$$ 5493.00 0.198137
$$917$$ −22005.0 −0.792442
$$918$$ 7155.00 0.257244
$$919$$ 47644.0 1.71015 0.855076 0.518502i $$-0.173510\pi$$
0.855076 + 0.518502i $$0.173510\pi$$
$$920$$ −30618.0 −1.09722
$$921$$ −306.000 −0.0109479
$$922$$ −43623.0 −1.55819
$$923$$ 0 0
$$924$$ 720.000 0.0256345
$$925$$ −13332.0 −0.473896
$$926$$ −6336.00 −0.224853
$$927$$ 4732.00 0.167658
$$928$$ 6480.00 0.229220
$$929$$ 21924.0 0.774277 0.387138 0.922022i $$-0.373464\pi$$
0.387138 + 0.922022i $$0.373464\pi$$
$$930$$ 7128.00 0.251329
$$931$$ −708.000 −0.0249235
$$932$$ −3627.00 −0.127475
$$933$$ −2106.00 −0.0738985
$$934$$ −9828.00 −0.344306
$$935$$ 19440.0 0.679953
$$936$$ 0 0
$$937$$ 32398.0 1.12956 0.564779 0.825242i $$-0.308961\pi$$
0.564779 + 0.825242i $$0.308961\pi$$
$$938$$ −1620.00 −0.0563911
$$939$$ −10051.0 −0.349310
$$940$$ −999.000 −0.0346636
$$941$$ 2097.00 0.0726464 0.0363232 0.999340i $$-0.488435\pi$$
0.0363232 + 0.999340i $$0.488435\pi$$
$$942$$ −5622.00 −0.194453
$$943$$ 31104.0 1.07411
$$944$$ −37062.0 −1.27782
$$945$$ −7155.00 −0.246299
$$946$$ −13968.0 −0.480062
$$947$$ −20016.0 −0.686835 −0.343417 0.939183i $$-0.611585\pi$$
−0.343417 + 0.939183i $$0.611585\pi$$
$$948$$ 830.000 0.0284358
$$949$$ 0 0
$$950$$ −792.000 −0.0270483
$$951$$ 2154.00 0.0734471
$$952$$ −14175.0 −0.482578
$$953$$ −24993.0 −0.849531 −0.424765 0.905304i $$-0.639643\pi$$
−0.424765 + 0.905304i $$0.639643\pi$$
$$954$$ 32292.0 1.09590
$$955$$ −11340.0 −0.384245
$$956$$ 6075.00 0.205523
$$957$$ −6912.00 −0.233473
$$958$$ −46359.0 −1.56346
$$959$$ −6210.00 −0.209105
$$960$$ 3897.00 0.131016
$$961$$ 39905.0 1.33950
$$962$$ 0 0
$$963$$ 15912.0 0.532458
$$964$$ 210.000 0.00701623
$$965$$ −3078.00 −0.102678
$$966$$ 7290.00 0.242807
$$967$$ −40959.0 −1.36210 −0.681051 0.732236i $$-0.738477\pi$$
−0.681051 + 0.732236i $$0.738477\pi$$
$$968$$ −20433.0 −0.678452
$$969$$ −270.000 −0.00895113
$$970$$ 23004.0 0.761458
$$971$$ −48933.0 −1.61723 −0.808617 0.588335i $$-0.799784\pi$$
−0.808617 + 0.588335i $$0.799784\pi$$
$$972$$ −2080.00 −0.0686379
$$973$$ −36285.0 −1.19552
$$974$$ −10980.0 −0.361213
$$975$$ 0 0
$$976$$ −26696.0 −0.875531
$$977$$ 47388.0 1.55177 0.775884 0.630876i $$-0.217304\pi$$
0.775884 + 0.630876i $$0.217304\pi$$
$$978$$ 3582.00 0.117116
$$979$$ 21024.0 0.686343
$$980$$ 1062.00 0.0346167
$$981$$ −28158.0 −0.916428
$$982$$ −2241.00 −0.0728240
$$983$$ 16803.0 0.545201 0.272600 0.962127i $$-0.412116\pi$$
0.272600 + 0.962127i $$0.412116\pi$$
$$984$$ −4032.00 −0.130625
$$985$$ −729.000 −0.0235816
$$986$$ −19440.0 −0.627886
$$987$$ −1665.00 −0.0536956
$$988$$ 0 0
$$989$$ −15714.0 −0.505234
$$990$$ −33696.0 −1.08175
$$991$$ −57526.0 −1.84397 −0.921985 0.387226i $$-0.873433\pi$$
−0.921985 + 0.387226i $$0.873433\pi$$
$$992$$ −11880.0 −0.380232
$$993$$ −10770.0 −0.344185
$$994$$ 16065.0 0.512627
$$995$$ 17964.0 0.572359
$$996$$ 438.000 0.0139343
$$997$$ −25000.0 −0.794140 −0.397070 0.917788i $$-0.629973\pi$$
−0.397070 + 0.917788i $$0.629973\pi$$
$$998$$ −47412.0 −1.50381
$$999$$ 16059.0 0.508593
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 169.4.a.c.1.1 1
3.2 odd 2 1521.4.a.d.1.1 1
13.2 odd 12 169.4.e.d.147.2 4
13.3 even 3 169.4.c.b.22.1 2
13.4 even 6 169.4.c.c.146.1 2
13.5 odd 4 13.4.b.a.12.1 2
13.6 odd 12 169.4.e.d.23.1 4
13.7 odd 12 169.4.e.d.23.2 4
13.8 odd 4 13.4.b.a.12.2 yes 2
13.9 even 3 169.4.c.b.146.1 2
13.10 even 6 169.4.c.c.22.1 2
13.11 odd 12 169.4.e.d.147.1 4
13.12 even 2 169.4.a.b.1.1 1
39.5 even 4 117.4.b.a.64.2 2
39.8 even 4 117.4.b.a.64.1 2
39.38 odd 2 1521.4.a.i.1.1 1
52.31 even 4 208.4.f.b.129.2 2
52.47 even 4 208.4.f.b.129.1 2
65.8 even 4 325.4.d.b.324.1 2
65.18 even 4 325.4.d.a.324.1 2
65.34 odd 4 325.4.c.b.51.1 2
65.44 odd 4 325.4.c.b.51.2 2
65.47 even 4 325.4.d.a.324.2 2
65.57 even 4 325.4.d.b.324.2 2
104.5 odd 4 832.4.f.e.129.1 2
104.21 odd 4 832.4.f.e.129.2 2
104.83 even 4 832.4.f.c.129.1 2
104.99 even 4 832.4.f.c.129.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
13.4.b.a.12.1 2 13.5 odd 4
13.4.b.a.12.2 yes 2 13.8 odd 4
117.4.b.a.64.1 2 39.8 even 4
117.4.b.a.64.2 2 39.5 even 4
169.4.a.b.1.1 1 13.12 even 2
169.4.a.c.1.1 1 1.1 even 1 trivial
169.4.c.b.22.1 2 13.3 even 3
169.4.c.b.146.1 2 13.9 even 3
169.4.c.c.22.1 2 13.10 even 6
169.4.c.c.146.1 2 13.4 even 6
169.4.e.d.23.1 4 13.6 odd 12
169.4.e.d.23.2 4 13.7 odd 12
169.4.e.d.147.1 4 13.11 odd 12
169.4.e.d.147.2 4 13.2 odd 12
208.4.f.b.129.1 2 52.47 even 4
208.4.f.b.129.2 2 52.31 even 4
325.4.c.b.51.1 2 65.34 odd 4
325.4.c.b.51.2 2 65.44 odd 4
325.4.d.a.324.1 2 65.18 even 4
325.4.d.a.324.2 2 65.47 even 4
325.4.d.b.324.1 2 65.8 even 4
325.4.d.b.324.2 2 65.57 even 4
832.4.f.c.129.1 2 104.83 even 4
832.4.f.c.129.2 2 104.99 even 4
832.4.f.e.129.1 2 104.5 odd 4
832.4.f.e.129.2 2 104.21 odd 4
1521.4.a.d.1.1 1 3.2 odd 2
1521.4.a.i.1.1 1 39.38 odd 2