# Properties

 Label 931.4.a.a.1.1 Level $931$ Weight $4$ Character 931.1 Self dual yes Analytic conductor $54.931$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [931,4,Mod(1,931)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(931, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("931.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$931 = 7^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 931.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: $$54.9307782153$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 19) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 931.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-3.00000 q^{2} +5.00000 q^{3} +1.00000 q^{4} +12.0000 q^{5} -15.0000 q^{6} +21.0000 q^{8} -2.00000 q^{9} +O(q^{10})$$ $$q-3.00000 q^{2} +5.00000 q^{3} +1.00000 q^{4} +12.0000 q^{5} -15.0000 q^{6} +21.0000 q^{8} -2.00000 q^{9} -36.0000 q^{10} -54.0000 q^{11} +5.00000 q^{12} -11.0000 q^{13} +60.0000 q^{15} -71.0000 q^{16} +93.0000 q^{17} +6.00000 q^{18} -19.0000 q^{19} +12.0000 q^{20} +162.000 q^{22} +183.000 q^{23} +105.000 q^{24} +19.0000 q^{25} +33.0000 q^{26} -145.000 q^{27} -249.000 q^{29} -180.000 q^{30} -56.0000 q^{31} +45.0000 q^{32} -270.000 q^{33} -279.000 q^{34} -2.00000 q^{36} -250.000 q^{37} +57.0000 q^{38} -55.0000 q^{39} +252.000 q^{40} -240.000 q^{41} -196.000 q^{43} -54.0000 q^{44} -24.0000 q^{45} -549.000 q^{46} +168.000 q^{47} -355.000 q^{48} -57.0000 q^{50} +465.000 q^{51} -11.0000 q^{52} +435.000 q^{53} +435.000 q^{54} -648.000 q^{55} -95.0000 q^{57} +747.000 q^{58} -195.000 q^{59} +60.0000 q^{60} +358.000 q^{61} +168.000 q^{62} +433.000 q^{64} -132.000 q^{65} +810.000 q^{66} -961.000 q^{67} +93.0000 q^{68} +915.000 q^{69} -246.000 q^{71} -42.0000 q^{72} -353.000 q^{73} +750.000 q^{74} +95.0000 q^{75} -19.0000 q^{76} +165.000 q^{78} -34.0000 q^{79} -852.000 q^{80} -671.000 q^{81} +720.000 q^{82} -234.000 q^{83} +1116.00 q^{85} +588.000 q^{86} -1245.00 q^{87} -1134.00 q^{88} +168.000 q^{89} +72.0000 q^{90} +183.000 q^{92} -280.000 q^{93} -504.000 q^{94} -228.000 q^{95} +225.000 q^{96} -758.000 q^{97} +108.000 q^{99} +O(q^{100})$$

## 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 $$-0.677932\pi$$
−0.530330 + 0.847791i $$0.677932\pi$$
$$3$$ 5.00000 0.962250 0.481125 0.876652i $$-0.340228\pi$$
0.481125 + 0.876652i $$0.340228\pi$$
$$4$$ 1.00000 0.125000
$$5$$ 12.0000 1.07331 0.536656 0.843801i $$-0.319687\pi$$
0.536656 + 0.843801i $$0.319687\pi$$
$$6$$ −15.0000 −1.02062
$$7$$ 0 0
$$8$$ 21.0000 0.928078
$$9$$ −2.00000 −0.0740741
$$10$$ −36.0000 −1.13842
$$11$$ −54.0000 −1.48015 −0.740073 0.672526i $$-0.765209\pi$$
−0.740073 + 0.672526i $$0.765209\pi$$
$$12$$ 5.00000 0.120281
$$13$$ −11.0000 −0.234681 −0.117340 0.993092i $$-0.537437\pi$$
−0.117340 + 0.993092i $$0.537437\pi$$
$$14$$ 0 0
$$15$$ 60.0000 1.03280
$$16$$ −71.0000 −1.10938
$$17$$ 93.0000 1.32681 0.663406 0.748259i $$-0.269110\pi$$
0.663406 + 0.748259i $$0.269110\pi$$
$$18$$ 6.00000 0.0785674
$$19$$ −19.0000 −0.229416
$$20$$ 12.0000 0.134164
$$21$$ 0 0
$$22$$ 162.000 1.56993
$$23$$ 183.000 1.65905 0.829525 0.558470i $$-0.188611\pi$$
0.829525 + 0.558470i $$0.188611\pi$$
$$24$$ 105.000 0.893043
$$25$$ 19.0000 0.152000
$$26$$ 33.0000 0.248917
$$27$$ −145.000 −1.03353
$$28$$ 0 0
$$29$$ −249.000 −1.59442 −0.797209 0.603703i $$-0.793691\pi$$
−0.797209 + 0.603703i $$0.793691\pi$$
$$30$$ −180.000 −1.09545
$$31$$ −56.0000 −0.324448 −0.162224 0.986754i $$-0.551867\pi$$
−0.162224 + 0.986754i $$0.551867\pi$$
$$32$$ 45.0000 0.248592
$$33$$ −270.000 −1.42427
$$34$$ −279.000 −1.40730
$$35$$ 0 0
$$36$$ −2.00000 −0.00925926
$$37$$ −250.000 −1.11080 −0.555402 0.831582i $$-0.687436\pi$$
−0.555402 + 0.831582i $$0.687436\pi$$
$$38$$ 57.0000 0.243332
$$39$$ −55.0000 −0.225822
$$40$$ 252.000 0.996117
$$41$$ −240.000 −0.914188 −0.457094 0.889418i $$-0.651110\pi$$
−0.457094 + 0.889418i $$0.651110\pi$$
$$42$$ 0 0
$$43$$ −196.000 −0.695110 −0.347555 0.937660i $$-0.612988\pi$$
−0.347555 + 0.937660i $$0.612988\pi$$
$$44$$ −54.0000 −0.185018
$$45$$ −24.0000 −0.0795046
$$46$$ −549.000 −1.75969
$$47$$ 168.000 0.521390 0.260695 0.965421i $$-0.416048\pi$$
0.260695 + 0.965421i $$0.416048\pi$$
$$48$$ −355.000 −1.06750
$$49$$ 0 0
$$50$$ −57.0000 −0.161220
$$51$$ 465.000 1.27673
$$52$$ −11.0000 −0.0293351
$$53$$ 435.000 1.12739 0.563697 0.825982i $$-0.309379\pi$$
0.563697 + 0.825982i $$0.309379\pi$$
$$54$$ 435.000 1.09622
$$55$$ −648.000 −1.58866
$$56$$ 0 0
$$57$$ −95.0000 −0.220755
$$58$$ 747.000 1.69114
$$59$$ −195.000 −0.430285 −0.215143 0.976583i $$-0.569022\pi$$
−0.215143 + 0.976583i $$0.569022\pi$$
$$60$$ 60.0000 0.129099
$$61$$ 358.000 0.751430 0.375715 0.926735i $$-0.377397\pi$$
0.375715 + 0.926735i $$0.377397\pi$$
$$62$$ 168.000 0.344129
$$63$$ 0 0
$$64$$ 433.000 0.845703
$$65$$ −132.000 −0.251886
$$66$$ 810.000 1.51067
$$67$$ −961.000 −1.75231 −0.876155 0.482029i $$-0.839900\pi$$
−0.876155 + 0.482029i $$0.839900\pi$$
$$68$$ 93.0000 0.165852
$$69$$ 915.000 1.59642
$$70$$ 0 0
$$71$$ −246.000 −0.411195 −0.205597 0.978637i $$-0.565914\pi$$
−0.205597 + 0.978637i $$0.565914\pi$$
$$72$$ −42.0000 −0.0687465
$$73$$ −353.000 −0.565966 −0.282983 0.959125i $$-0.591324\pi$$
−0.282983 + 0.959125i $$0.591324\pi$$
$$74$$ 750.000 1.17819
$$75$$ 95.0000 0.146262
$$76$$ −19.0000 −0.0286770
$$77$$ 0 0
$$78$$ 165.000 0.239520
$$79$$ −34.0000 −0.0484215 −0.0242108 0.999707i $$-0.507707\pi$$
−0.0242108 + 0.999707i $$0.507707\pi$$
$$80$$ −852.000 −1.19071
$$81$$ −671.000 −0.920439
$$82$$ 720.000 0.969643
$$83$$ −234.000 −0.309456 −0.154728 0.987957i $$-0.549450\pi$$
−0.154728 + 0.987957i $$0.549450\pi$$
$$84$$ 0 0
$$85$$ 1116.00 1.42408
$$86$$ 588.000 0.737275
$$87$$ −1245.00 −1.53423
$$88$$ −1134.00 −1.37369
$$89$$ 168.000 0.200089 0.100045 0.994983i $$-0.468101\pi$$
0.100045 + 0.994983i $$0.468101\pi$$
$$90$$ 72.0000 0.0843274
$$91$$ 0 0
$$92$$ 183.000 0.207381
$$93$$ −280.000 −0.312201
$$94$$ −504.000 −0.553017
$$95$$ −228.000 −0.246235
$$96$$ 225.000 0.239208
$$97$$ −758.000 −0.793435 −0.396718 0.917941i $$-0.629851\pi$$
−0.396718 + 0.917941i $$0.629851\pi$$
$$98$$ 0 0
$$99$$ 108.000 0.109640
$$100$$ 19.0000 0.0190000
$$101$$ 726.000 0.715245 0.357622 0.933866i $$-0.383588\pi$$
0.357622 + 0.933866i $$0.383588\pi$$
$$102$$ −1395.00 −1.35417
$$103$$ −2.00000 −0.00191326 −0.000956630 1.00000i $$-0.500305\pi$$
−0.000956630 1.00000i $$0.500305\pi$$
$$104$$ −231.000 −0.217802
$$105$$ 0 0
$$106$$ −1305.00 −1.19578
$$107$$ 1413.00 1.27663 0.638317 0.769773i $$-0.279631\pi$$
0.638317 + 0.769773i $$0.279631\pi$$
$$108$$ −145.000 −0.129191
$$109$$ 389.000 0.341830 0.170915 0.985286i $$-0.445328\pi$$
0.170915 + 0.985286i $$0.445328\pi$$
$$110$$ 1944.00 1.68503
$$111$$ −1250.00 −1.06887
$$112$$ 0 0
$$113$$ 342.000 0.284714 0.142357 0.989815i $$-0.454532\pi$$
0.142357 + 0.989815i $$0.454532\pi$$
$$114$$ 285.000 0.234146
$$115$$ 2196.00 1.78068
$$116$$ −249.000 −0.199302
$$117$$ 22.0000 0.0173838
$$118$$ 585.000 0.456387
$$119$$ 0 0
$$120$$ 1260.00 0.958514
$$121$$ 1585.00 1.19083
$$122$$ −1074.00 −0.797011
$$123$$ −1200.00 −0.879678
$$124$$ −56.0000 −0.0405560
$$125$$ −1272.00 −0.910169
$$126$$ 0 0
$$127$$ −1150.00 −0.803512 −0.401756 0.915747i $$-0.631600\pi$$
−0.401756 + 0.915747i $$0.631600\pi$$
$$128$$ −1659.00 −1.14560
$$129$$ −980.000 −0.668870
$$130$$ 396.000 0.267165
$$131$$ 1452.00 0.968411 0.484205 0.874954i $$-0.339109\pi$$
0.484205 + 0.874954i $$0.339109\pi$$
$$132$$ −270.000 −0.178034
$$133$$ 0 0
$$134$$ 2883.00 1.85861
$$135$$ −1740.00 −1.10930
$$136$$ 1953.00 1.23139
$$137$$ −1689.00 −1.05329 −0.526646 0.850085i $$-0.676551\pi$$
−0.526646 + 0.850085i $$0.676551\pi$$
$$138$$ −2745.00 −1.69326
$$139$$ −2144.00 −1.30829 −0.654143 0.756371i $$-0.726970\pi$$
−0.654143 + 0.756371i $$0.726970\pi$$
$$140$$ 0 0
$$141$$ 840.000 0.501708
$$142$$ 738.000 0.436138
$$143$$ 594.000 0.347362
$$144$$ 142.000 0.0821759
$$145$$ −2988.00 −1.71131
$$146$$ 1059.00 0.600298
$$147$$ 0 0
$$148$$ −250.000 −0.138850
$$149$$ −3000.00 −1.64946 −0.824730 0.565527i $$-0.808673\pi$$
−0.824730 + 0.565527i $$0.808673\pi$$
$$150$$ −285.000 −0.155134
$$151$$ −1006.00 −0.542166 −0.271083 0.962556i $$-0.587382\pi$$
−0.271083 + 0.962556i $$0.587382\pi$$
$$152$$ −399.000 −0.212916
$$153$$ −186.000 −0.0982824
$$154$$ 0 0
$$155$$ −672.000 −0.348234
$$156$$ −55.0000 −0.0282277
$$157$$ −2846.00 −1.44672 −0.723362 0.690469i $$-0.757404\pi$$
−0.723362 + 0.690469i $$0.757404\pi$$
$$158$$ 102.000 0.0513588
$$159$$ 2175.00 1.08483
$$160$$ 540.000 0.266817
$$161$$ 0 0
$$162$$ 2013.00 0.976273
$$163$$ −1600.00 −0.768845 −0.384422 0.923157i $$-0.625599\pi$$
−0.384422 + 0.923157i $$0.625599\pi$$
$$164$$ −240.000 −0.114273
$$165$$ −3240.00 −1.52869
$$166$$ 702.000 0.328228
$$167$$ 2004.00 0.928588 0.464294 0.885681i $$-0.346308\pi$$
0.464294 + 0.885681i $$0.346308\pi$$
$$168$$ 0 0
$$169$$ −2076.00 −0.944925
$$170$$ −3348.00 −1.51047
$$171$$ 38.0000 0.0169938
$$172$$ −196.000 −0.0868887
$$173$$ 462.000 0.203036 0.101518 0.994834i $$-0.467630\pi$$
0.101518 + 0.994834i $$0.467630\pi$$
$$174$$ 3735.00 1.62730
$$175$$ 0 0
$$176$$ 3834.00 1.64204
$$177$$ −975.000 −0.414042
$$178$$ −504.000 −0.212227
$$179$$ 720.000 0.300644 0.150322 0.988637i $$-0.451969\pi$$
0.150322 + 0.988637i $$0.451969\pi$$
$$180$$ −24.0000 −0.00993808
$$181$$ 2338.00 0.960122 0.480061 0.877235i $$-0.340614\pi$$
0.480061 + 0.877235i $$0.340614\pi$$
$$182$$ 0 0
$$183$$ 1790.00 0.723063
$$184$$ 3843.00 1.53973
$$185$$ −3000.00 −1.19224
$$186$$ 840.000 0.331139
$$187$$ −5022.00 −1.96388
$$188$$ 168.000 0.0651737
$$189$$ 0 0
$$190$$ 684.000 0.261171
$$191$$ 2871.00 1.08763 0.543817 0.839204i $$-0.316978\pi$$
0.543817 + 0.839204i $$0.316978\pi$$
$$192$$ 2165.00 0.813778
$$193$$ 1658.00 0.618370 0.309185 0.951002i $$-0.399944\pi$$
0.309185 + 0.951002i $$0.399944\pi$$
$$194$$ 2274.00 0.841565
$$195$$ −660.000 −0.242377
$$196$$ 0 0
$$197$$ −4176.00 −1.51029 −0.755146 0.655556i $$-0.772434\pi$$
−0.755146 + 0.655556i $$0.772434\pi$$
$$198$$ −324.000 −0.116291
$$199$$ 241.000 0.0858494 0.0429247 0.999078i $$-0.486332\pi$$
0.0429247 + 0.999078i $$0.486332\pi$$
$$200$$ 399.000 0.141068
$$201$$ −4805.00 −1.68616
$$202$$ −2178.00 −0.758631
$$203$$ 0 0
$$204$$ 465.000 0.159591
$$205$$ −2880.00 −0.981209
$$206$$ 6.00000 0.00202932
$$207$$ −366.000 −0.122893
$$208$$ 781.000 0.260349
$$209$$ 1026.00 0.339569
$$210$$ 0 0
$$211$$ −745.000 −0.243071 −0.121535 0.992587i $$-0.538782\pi$$
−0.121535 + 0.992587i $$0.538782\pi$$
$$212$$ 435.000 0.140924
$$213$$ −1230.00 −0.395672
$$214$$ −4239.00 −1.35408
$$215$$ −2352.00 −0.746070
$$216$$ −3045.00 −0.959194
$$217$$ 0 0
$$218$$ −1167.00 −0.362565
$$219$$ −1765.00 −0.544601
$$220$$ −648.000 −0.198583
$$221$$ −1023.00 −0.311377
$$222$$ 3750.00 1.13371
$$223$$ 1978.00 0.593976 0.296988 0.954881i $$-0.404018\pi$$
0.296988 + 0.954881i $$0.404018\pi$$
$$224$$ 0 0
$$225$$ −38.0000 −0.0112593
$$226$$ −1026.00 −0.301985
$$227$$ −5355.00 −1.56574 −0.782872 0.622183i $$-0.786246\pi$$
−0.782872 + 0.622183i $$0.786246\pi$$
$$228$$ −95.0000 −0.0275944
$$229$$ 6370.00 1.83817 0.919086 0.394057i $$-0.128929\pi$$
0.919086 + 0.394057i $$0.128929\pi$$
$$230$$ −6588.00 −1.88870
$$231$$ 0 0
$$232$$ −5229.00 −1.47974
$$233$$ −2838.00 −0.797955 −0.398978 0.916961i $$-0.630635\pi$$
−0.398978 + 0.916961i $$0.630635\pi$$
$$234$$ −66.0000 −0.0184383
$$235$$ 2016.00 0.559614
$$236$$ −195.000 −0.0537857
$$237$$ −170.000 −0.0465936
$$238$$ 0 0
$$239$$ −369.000 −0.0998687 −0.0499344 0.998753i $$-0.515901\pi$$
−0.0499344 + 0.998753i $$0.515901\pi$$
$$240$$ −4260.00 −1.14576
$$241$$ −6608.00 −1.76622 −0.883109 0.469167i $$-0.844554\pi$$
−0.883109 + 0.469167i $$0.844554\pi$$
$$242$$ −4755.00 −1.26307
$$243$$ 560.000 0.147835
$$244$$ 358.000 0.0939287
$$245$$ 0 0
$$246$$ 3600.00 0.933039
$$247$$ 209.000 0.0538395
$$248$$ −1176.00 −0.301113
$$249$$ −1170.00 −0.297774
$$250$$ 3816.00 0.965380
$$251$$ −4674.00 −1.17538 −0.587690 0.809086i $$-0.699962\pi$$
−0.587690 + 0.809086i $$0.699962\pi$$
$$252$$ 0 0
$$253$$ −9882.00 −2.45564
$$254$$ 3450.00 0.852253
$$255$$ 5580.00 1.37033
$$256$$ 1513.00 0.369385
$$257$$ −4512.00 −1.09514 −0.547570 0.836760i $$-0.684447\pi$$
−0.547570 + 0.836760i $$0.684447\pi$$
$$258$$ 2940.00 0.709443
$$259$$ 0 0
$$260$$ −132.000 −0.0314857
$$261$$ 498.000 0.118105
$$262$$ −4356.00 −1.02715
$$263$$ 3768.00 0.883440 0.441720 0.897153i $$-0.354368\pi$$
0.441720 + 0.897153i $$0.354368\pi$$
$$264$$ −5670.00 −1.32183
$$265$$ 5220.00 1.21005
$$266$$ 0 0
$$267$$ 840.000 0.192536
$$268$$ −961.000 −0.219039
$$269$$ −4758.00 −1.07844 −0.539220 0.842165i $$-0.681281\pi$$
−0.539220 + 0.842165i $$0.681281\pi$$
$$270$$ 5220.00 1.17659
$$271$$ 2041.00 0.457498 0.228749 0.973485i $$-0.426537\pi$$
0.228749 + 0.973485i $$0.426537\pi$$
$$272$$ −6603.00 −1.47193
$$273$$ 0 0
$$274$$ 5067.00 1.11718
$$275$$ −1026.00 −0.224982
$$276$$ 915.000 0.199553
$$277$$ 1964.00 0.426012 0.213006 0.977051i $$-0.431675\pi$$
0.213006 + 0.977051i $$0.431675\pi$$
$$278$$ 6432.00 1.38765
$$279$$ 112.000 0.0240332
$$280$$ 0 0
$$281$$ −5496.00 −1.16678 −0.583388 0.812194i $$-0.698273\pi$$
−0.583388 + 0.812194i $$0.698273\pi$$
$$282$$ −2520.00 −0.532141
$$283$$ −3098.00 −0.650731 −0.325366 0.945588i $$-0.605487\pi$$
−0.325366 + 0.945588i $$0.605487\pi$$
$$284$$ −246.000 −0.0513993
$$285$$ −1140.00 −0.236940
$$286$$ −1782.00 −0.368433
$$287$$ 0 0
$$288$$ −90.0000 −0.0184142
$$289$$ 3736.00 0.760432
$$290$$ 8964.00 1.81512
$$291$$ −3790.00 −0.763484
$$292$$ −353.000 −0.0707458
$$293$$ −117.000 −0.0233284 −0.0116642 0.999932i $$-0.503713\pi$$
−0.0116642 + 0.999932i $$0.503713\pi$$
$$294$$ 0 0
$$295$$ −2340.00 −0.461831
$$296$$ −5250.00 −1.03091
$$297$$ 7830.00 1.52977
$$298$$ 9000.00 1.74952
$$299$$ −2013.00 −0.389347
$$300$$ 95.0000 0.0182828
$$301$$ 0 0
$$302$$ 3018.00 0.575054
$$303$$ 3630.00 0.688244
$$304$$ 1349.00 0.254508
$$305$$ 4296.00 0.806519
$$306$$ 558.000 0.104244
$$307$$ 1420.00 0.263986 0.131993 0.991251i $$-0.457862\pi$$
0.131993 + 0.991251i $$0.457862\pi$$
$$308$$ 0 0
$$309$$ −10.0000 −0.00184104
$$310$$ 2016.00 0.369358
$$311$$ 6561.00 1.19627 0.598135 0.801395i $$-0.295909\pi$$
0.598135 + 0.801395i $$0.295909\pi$$
$$312$$ −1155.00 −0.209580
$$313$$ 1483.00 0.267809 0.133904 0.990994i $$-0.457249\pi$$
0.133904 + 0.990994i $$0.457249\pi$$
$$314$$ 8538.00 1.53448
$$315$$ 0 0
$$316$$ −34.0000 −0.00605269
$$317$$ −1239.00 −0.219524 −0.109762 0.993958i $$-0.535009\pi$$
−0.109762 + 0.993958i $$0.535009\pi$$
$$318$$ −6525.00 −1.15064
$$319$$ 13446.0 2.35997
$$320$$ 5196.00 0.907704
$$321$$ 7065.00 1.22844
$$322$$ 0 0
$$323$$ −1767.00 −0.304392
$$324$$ −671.000 −0.115055
$$325$$ −209.000 −0.0356715
$$326$$ 4800.00 0.815483
$$327$$ 1945.00 0.328926
$$328$$ −5040.00 −0.848437
$$329$$ 0 0
$$330$$ 9720.00 1.62142
$$331$$ −8899.00 −1.47774 −0.738872 0.673846i $$-0.764641\pi$$
−0.738872 + 0.673846i $$0.764641\pi$$
$$332$$ −234.000 −0.0386820
$$333$$ 500.000 0.0822818
$$334$$ −6012.00 −0.984916
$$335$$ −11532.0 −1.88078
$$336$$ 0 0
$$337$$ 5816.00 0.940112 0.470056 0.882637i $$-0.344234\pi$$
0.470056 + 0.882637i $$0.344234\pi$$
$$338$$ 6228.00 1.00224
$$339$$ 1710.00 0.273966
$$340$$ 1116.00 0.178011
$$341$$ 3024.00 0.480231
$$342$$ −114.000 −0.0180246
$$343$$ 0 0
$$344$$ −4116.00 −0.645116
$$345$$ 10980.0 1.71346
$$346$$ −1386.00 −0.215352
$$347$$ −1578.00 −0.244125 −0.122063 0.992522i $$-0.538951\pi$$
−0.122063 + 0.992522i $$0.538951\pi$$
$$348$$ −1245.00 −0.191779
$$349$$ −1658.00 −0.254300 −0.127150 0.991883i $$-0.540583\pi$$
−0.127150 + 0.991883i $$0.540583\pi$$
$$350$$ 0 0
$$351$$ 1595.00 0.242549
$$352$$ −2430.00 −0.367953
$$353$$ 11367.0 1.71389 0.856947 0.515405i $$-0.172359\pi$$
0.856947 + 0.515405i $$0.172359\pi$$
$$354$$ 2925.00 0.439158
$$355$$ −2952.00 −0.441341
$$356$$ 168.000 0.0250112
$$357$$ 0 0
$$358$$ −2160.00 −0.318881
$$359$$ 2553.00 0.375326 0.187663 0.982233i $$-0.439909\pi$$
0.187663 + 0.982233i $$0.439909\pi$$
$$360$$ −504.000 −0.0737865
$$361$$ 361.000 0.0526316
$$362$$ −7014.00 −1.01836
$$363$$ 7925.00 1.14588
$$364$$ 0 0
$$365$$ −4236.00 −0.607459
$$366$$ −5370.00 −0.766925
$$367$$ 196.000 0.0278777 0.0139389 0.999903i $$-0.495563\pi$$
0.0139389 + 0.999903i $$0.495563\pi$$
$$368$$ −12993.0 −1.84051
$$369$$ 480.000 0.0677176
$$370$$ 9000.00 1.26456
$$371$$ 0 0
$$372$$ −280.000 −0.0390251
$$373$$ 9353.00 1.29834 0.649169 0.760644i $$-0.275117\pi$$
0.649169 + 0.760644i $$0.275117\pi$$
$$374$$ 15066.0 2.08301
$$375$$ −6360.00 −0.875811
$$376$$ 3528.00 0.483890
$$377$$ 2739.00 0.374180
$$378$$ 0 0
$$379$$ 3827.00 0.518680 0.259340 0.965786i $$-0.416495\pi$$
0.259340 + 0.965786i $$0.416495\pi$$
$$380$$ −228.000 −0.0307794
$$381$$ −5750.00 −0.773180
$$382$$ −8613.00 −1.15361
$$383$$ −5694.00 −0.759660 −0.379830 0.925056i $$-0.624018\pi$$
−0.379830 + 0.925056i $$0.624018\pi$$
$$384$$ −8295.00 −1.10235
$$385$$ 0 0
$$386$$ −4974.00 −0.655881
$$387$$ 392.000 0.0514896
$$388$$ −758.000 −0.0991794
$$389$$ 1290.00 0.168138 0.0840689 0.996460i $$-0.473208\pi$$
0.0840689 + 0.996460i $$0.473208\pi$$
$$390$$ 1980.00 0.257080
$$391$$ 17019.0 2.20125
$$392$$ 0 0
$$393$$ 7260.00 0.931854
$$394$$ 12528.0 1.60191
$$395$$ −408.000 −0.0519714
$$396$$ 108.000 0.0137051
$$397$$ −6536.00 −0.826278 −0.413139 0.910668i $$-0.635568\pi$$
−0.413139 + 0.910668i $$0.635568\pi$$
$$398$$ −723.000 −0.0910571
$$399$$ 0 0
$$400$$ −1349.00 −0.168625
$$401$$ 2328.00 0.289912 0.144956 0.989438i $$-0.453696\pi$$
0.144956 + 0.989438i $$0.453696\pi$$
$$402$$ 14415.0 1.78844
$$403$$ 616.000 0.0761418
$$404$$ 726.000 0.0894056
$$405$$ −8052.00 −0.987919
$$406$$ 0 0
$$407$$ 13500.0 1.64415
$$408$$ 9765.00 1.18490
$$409$$ 6676.00 0.807107 0.403554 0.914956i $$-0.367775\pi$$
0.403554 + 0.914956i $$0.367775\pi$$
$$410$$ 8640.00 1.04073
$$411$$ −8445.00 −1.01353
$$412$$ −2.00000 −0.000239158 0
$$413$$ 0 0
$$414$$ 1098.00 0.130347
$$415$$ −2808.00 −0.332143
$$416$$ −495.000 −0.0583398
$$417$$ −10720.0 −1.25890
$$418$$ −3078.00 −0.360167
$$419$$ 8136.00 0.948615 0.474307 0.880359i $$-0.342699\pi$$
0.474307 + 0.880359i $$0.342699\pi$$
$$420$$ 0 0
$$421$$ −8665.00 −1.00310 −0.501551 0.865128i $$-0.667237\pi$$
−0.501551 + 0.865128i $$0.667237\pi$$
$$422$$ 2235.00 0.257815
$$423$$ −336.000 −0.0386215
$$424$$ 9135.00 1.04631
$$425$$ 1767.00 0.201676
$$426$$ 3690.00 0.419674
$$427$$ 0 0
$$428$$ 1413.00 0.159579
$$429$$ 2970.00 0.334249
$$430$$ 7056.00 0.791327
$$431$$ 750.000 0.0838196 0.0419098 0.999121i $$-0.486656\pi$$
0.0419098 + 0.999121i $$0.486656\pi$$
$$432$$ 10295.0 1.14657
$$433$$ 4858.00 0.539170 0.269585 0.962977i $$-0.413113\pi$$
0.269585 + 0.962977i $$0.413113\pi$$
$$434$$ 0 0
$$435$$ −14940.0 −1.64671
$$436$$ 389.000 0.0427287
$$437$$ −3477.00 −0.380612
$$438$$ 5295.00 0.577637
$$439$$ −6500.00 −0.706670 −0.353335 0.935497i $$-0.614952\pi$$
−0.353335 + 0.935497i $$0.614952\pi$$
$$440$$ −13608.0 −1.47440
$$441$$ 0 0
$$442$$ 3069.00 0.330266
$$443$$ 3486.00 0.373871 0.186936 0.982372i $$-0.440144\pi$$
0.186936 + 0.982372i $$0.440144\pi$$
$$444$$ −1250.00 −0.133609
$$445$$ 2016.00 0.214759
$$446$$ −5934.00 −0.630007
$$447$$ −15000.0 −1.58719
$$448$$ 0 0
$$449$$ −15030.0 −1.57975 −0.789877 0.613265i $$-0.789856\pi$$
−0.789877 + 0.613265i $$0.789856\pi$$
$$450$$ 114.000 0.0119422
$$451$$ 12960.0 1.35313
$$452$$ 342.000 0.0355892
$$453$$ −5030.00 −0.521700
$$454$$ 16065.0 1.66072
$$455$$ 0 0
$$456$$ −1995.00 −0.204878
$$457$$ −2959.00 −0.302880 −0.151440 0.988466i $$-0.548391\pi$$
−0.151440 + 0.988466i $$0.548391\pi$$
$$458$$ −19110.0 −1.94968
$$459$$ −13485.0 −1.37130
$$460$$ 2196.00 0.222585
$$461$$ 156.000 0.0157606 0.00788031 0.999969i $$-0.497492\pi$$
0.00788031 + 0.999969i $$0.497492\pi$$
$$462$$ 0 0
$$463$$ 4484.00 0.450085 0.225042 0.974349i $$-0.427748\pi$$
0.225042 + 0.974349i $$0.427748\pi$$
$$464$$ 17679.0 1.76881
$$465$$ −3360.00 −0.335089
$$466$$ 8514.00 0.846359
$$467$$ −8766.00 −0.868613 −0.434306 0.900765i $$-0.643006\pi$$
−0.434306 + 0.900765i $$0.643006\pi$$
$$468$$ 22.0000 0.00217297
$$469$$ 0 0
$$470$$ −6048.00 −0.593561
$$471$$ −14230.0 −1.39211
$$472$$ −4095.00 −0.399338
$$473$$ 10584.0 1.02886
$$474$$ 510.000 0.0494200
$$475$$ −361.000 −0.0348712
$$476$$ 0 0
$$477$$ −870.000 −0.0835106
$$478$$ 1107.00 0.105927
$$479$$ 18996.0 1.81200 0.906001 0.423275i $$-0.139119\pi$$
0.906001 + 0.423275i $$0.139119\pi$$
$$480$$ 2700.00 0.256745
$$481$$ 2750.00 0.260684
$$482$$ 19824.0 1.87336
$$483$$ 0 0
$$484$$ 1585.00 0.148854
$$485$$ −9096.00 −0.851604
$$486$$ −1680.00 −0.156803
$$487$$ −7450.00 −0.693207 −0.346603 0.938012i $$-0.612665\pi$$
−0.346603 + 0.938012i $$0.612665\pi$$
$$488$$ 7518.00 0.697385
$$489$$ −8000.00 −0.739821
$$490$$ 0 0
$$491$$ 6180.00 0.568023 0.284012 0.958821i $$-0.408335\pi$$
0.284012 + 0.958821i $$0.408335\pi$$
$$492$$ −1200.00 −0.109960
$$493$$ −23157.0 −2.11549
$$494$$ −627.000 −0.0571054
$$495$$ 1296.00 0.117679
$$496$$ 3976.00 0.359935
$$497$$ 0 0
$$498$$ 3510.00 0.315837
$$499$$ 2576.00 0.231097 0.115549 0.993302i $$-0.463137\pi$$
0.115549 + 0.993302i $$0.463137\pi$$
$$500$$ −1272.00 −0.113771
$$501$$ 10020.0 0.893534
$$502$$ 14022.0 1.24668
$$503$$ 10545.0 0.934748 0.467374 0.884060i $$-0.345200\pi$$
0.467374 + 0.884060i $$0.345200\pi$$
$$504$$ 0 0
$$505$$ 8712.00 0.767681
$$506$$ 29646.0 2.60460
$$507$$ −10380.0 −0.909254
$$508$$ −1150.00 −0.100439
$$509$$ 14694.0 1.27957 0.639784 0.768555i $$-0.279024\pi$$
0.639784 + 0.768555i $$0.279024\pi$$
$$510$$ −16740.0 −1.45345
$$511$$ 0 0
$$512$$ 8733.00 0.753804
$$513$$ 2755.00 0.237108
$$514$$ 13536.0 1.16157
$$515$$ −24.0000 −0.00205353
$$516$$ −980.000 −0.0836087
$$517$$ −9072.00 −0.771733
$$518$$ 0 0
$$519$$ 2310.00 0.195371
$$520$$ −2772.00 −0.233770
$$521$$ −10332.0 −0.868816 −0.434408 0.900716i $$-0.643042\pi$$
−0.434408 + 0.900716i $$0.643042\pi$$
$$522$$ −1494.00 −0.125269
$$523$$ −10937.0 −0.914420 −0.457210 0.889359i $$-0.651151\pi$$
−0.457210 + 0.889359i $$0.651151\pi$$
$$524$$ 1452.00 0.121051
$$525$$ 0 0
$$526$$ −11304.0 −0.937030
$$527$$ −5208.00 −0.430482
$$528$$ 19170.0 1.58005
$$529$$ 21322.0 1.75245
$$530$$ −15660.0 −1.28345
$$531$$ 390.000 0.0318730
$$532$$ 0 0
$$533$$ 2640.00 0.214542
$$534$$ −2520.00 −0.204215
$$535$$ 16956.0 1.37023
$$536$$ −20181.0 −1.62628
$$537$$ 3600.00 0.289295
$$538$$ 14274.0 1.14386
$$539$$ 0 0
$$540$$ −1740.00 −0.138662
$$541$$ 18578.0 1.47640 0.738198 0.674584i $$-0.235677\pi$$
0.738198 + 0.674584i $$0.235677\pi$$
$$542$$ −6123.00 −0.485250
$$543$$ 11690.0 0.923878
$$544$$ 4185.00 0.329835
$$545$$ 4668.00 0.366890
$$546$$ 0 0
$$547$$ 21404.0 1.67307 0.836535 0.547914i $$-0.184578\pi$$
0.836535 + 0.547914i $$0.184578\pi$$
$$548$$ −1689.00 −0.131662
$$549$$ −716.000 −0.0556614
$$550$$ 3078.00 0.238630
$$551$$ 4731.00 0.365785
$$552$$ 19215.0 1.48160
$$553$$ 0 0
$$554$$ −5892.00 −0.451854
$$555$$ −15000.0 −1.14723
$$556$$ −2144.00 −0.163536
$$557$$ −3948.00 −0.300327 −0.150163 0.988661i $$-0.547980\pi$$
−0.150163 + 0.988661i $$0.547980\pi$$
$$558$$ −336.000 −0.0254911
$$559$$ 2156.00 0.163129
$$560$$ 0 0
$$561$$ −25110.0 −1.88974
$$562$$ 16488.0 1.23755
$$563$$ −5724.00 −0.428486 −0.214243 0.976780i $$-0.568729\pi$$
−0.214243 + 0.976780i $$0.568729\pi$$
$$564$$ 840.000 0.0627134
$$565$$ 4104.00 0.305587
$$566$$ 9294.00 0.690205
$$567$$ 0 0
$$568$$ −5166.00 −0.381621
$$569$$ −20592.0 −1.51716 −0.758578 0.651582i $$-0.774105\pi$$
−0.758578 + 0.651582i $$0.774105\pi$$
$$570$$ 3420.00 0.251312
$$571$$ 20684.0 1.51593 0.757967 0.652293i $$-0.226193\pi$$
0.757967 + 0.652293i $$0.226193\pi$$
$$572$$ 594.000 0.0434203
$$573$$ 14355.0 1.04658
$$574$$ 0 0
$$575$$ 3477.00 0.252176
$$576$$ −866.000 −0.0626447
$$577$$ 19573.0 1.41219 0.706096 0.708116i $$-0.250455\pi$$
0.706096 + 0.708116i $$0.250455\pi$$
$$578$$ −11208.0 −0.806559
$$579$$ 8290.00 0.595027
$$580$$ −2988.00 −0.213914
$$581$$ 0 0
$$582$$ 11370.0 0.809797
$$583$$ −23490.0 −1.66871
$$584$$ −7413.00 −0.525260
$$585$$ 264.000 0.0186582
$$586$$ 351.000 0.0247435
$$587$$ −13524.0 −0.950929 −0.475464 0.879735i $$-0.657720\pi$$
−0.475464 + 0.879735i $$0.657720\pi$$
$$588$$ 0 0
$$589$$ 1064.00 0.0744335
$$590$$ 7020.00 0.489845
$$591$$ −20880.0 −1.45328
$$592$$ 17750.0 1.23230
$$593$$ −8994.00 −0.622832 −0.311416 0.950274i $$-0.600803\pi$$
−0.311416 + 0.950274i $$0.600803\pi$$
$$594$$ −23490.0 −1.62257
$$595$$ 0 0
$$596$$ −3000.00 −0.206183
$$597$$ 1205.00 0.0826087
$$598$$ 6039.00 0.412965
$$599$$ 10128.0 0.690850 0.345425 0.938446i $$-0.387735\pi$$
0.345425 + 0.938446i $$0.387735\pi$$
$$600$$ 1995.00 0.135743
$$601$$ 22696.0 1.54041 0.770207 0.637794i $$-0.220153\pi$$
0.770207 + 0.637794i $$0.220153\pi$$
$$602$$ 0 0
$$603$$ 1922.00 0.129801
$$604$$ −1006.00 −0.0677708
$$605$$ 19020.0 1.27814
$$606$$ −10890.0 −0.729993
$$607$$ 5182.00 0.346509 0.173254 0.984877i $$-0.444572\pi$$
0.173254 + 0.984877i $$0.444572\pi$$
$$608$$ −855.000 −0.0570310
$$609$$ 0 0
$$610$$ −12888.0 −0.855442
$$611$$ −1848.00 −0.122360
$$612$$ −186.000 −0.0122853
$$613$$ 10082.0 0.664287 0.332144 0.943229i $$-0.392228\pi$$
0.332144 + 0.943229i $$0.392228\pi$$
$$614$$ −4260.00 −0.279999
$$615$$ −14400.0 −0.944169
$$616$$ 0 0
$$617$$ −12174.0 −0.794338 −0.397169 0.917745i $$-0.630007\pi$$
−0.397169 + 0.917745i $$0.630007\pi$$
$$618$$ 30.0000 0.00195271
$$619$$ −7490.00 −0.486347 −0.243173 0.969983i $$-0.578188\pi$$
−0.243173 + 0.969983i $$0.578188\pi$$
$$620$$ −672.000 −0.0435293
$$621$$ −26535.0 −1.71467
$$622$$ −19683.0 −1.26884
$$623$$ 0 0
$$624$$ 3905.00 0.250521
$$625$$ −17639.0 −1.12890
$$626$$ −4449.00 −0.284054
$$627$$ 5130.00 0.326750
$$628$$ −2846.00 −0.180840
$$629$$ −23250.0 −1.47383
$$630$$ 0 0
$$631$$ 11072.0 0.698525 0.349263 0.937025i $$-0.386432\pi$$
0.349263 + 0.937025i $$0.386432\pi$$
$$632$$ −714.000 −0.0449389
$$633$$ −3725.00 −0.233895
$$634$$ 3717.00 0.232841
$$635$$ −13800.0 −0.862419
$$636$$ 2175.00 0.135604
$$637$$ 0 0
$$638$$ −40338.0 −2.50313
$$639$$ 492.000 0.0304589
$$640$$ −19908.0 −1.22958
$$641$$ −18894.0 −1.16422 −0.582112 0.813108i $$-0.697774\pi$$
−0.582112 + 0.813108i $$0.697774\pi$$
$$642$$ −21195.0 −1.30296
$$643$$ 19834.0 1.21645 0.608224 0.793765i $$-0.291882\pi$$
0.608224 + 0.793765i $$0.291882\pi$$
$$644$$ 0 0
$$645$$ −11760.0 −0.717906
$$646$$ 5301.00 0.322856
$$647$$ −3375.00 −0.205077 −0.102539 0.994729i $$-0.532697\pi$$
−0.102539 + 0.994729i $$0.532697\pi$$
$$648$$ −14091.0 −0.854239
$$649$$ 10530.0 0.636885
$$650$$ 627.000 0.0378353
$$651$$ 0 0
$$652$$ −1600.00 −0.0961056
$$653$$ −24948.0 −1.49509 −0.747543 0.664214i $$-0.768766\pi$$
−0.747543 + 0.664214i $$0.768766\pi$$
$$654$$ −5835.00 −0.348879
$$655$$ 17424.0 1.03941
$$656$$ 17040.0 1.01418
$$657$$ 706.000 0.0419234
$$658$$ 0 0
$$659$$ −9879.00 −0.583962 −0.291981 0.956424i $$-0.594314\pi$$
−0.291981 + 0.956424i $$0.594314\pi$$
$$660$$ −3240.00 −0.191086
$$661$$ 14155.0 0.832928 0.416464 0.909152i $$-0.363269\pi$$
0.416464 + 0.909152i $$0.363269\pi$$
$$662$$ 26697.0 1.56738
$$663$$ −5115.00 −0.299623
$$664$$ −4914.00 −0.287199
$$665$$ 0 0
$$666$$ −1500.00 −0.0872730
$$667$$ −45567.0 −2.64522
$$668$$ 2004.00 0.116073
$$669$$ 9890.00 0.571554
$$670$$ 34596.0 1.99487
$$671$$ −19332.0 −1.11223
$$672$$ 0 0
$$673$$ 8948.00 0.512511 0.256256 0.966609i $$-0.417511\pi$$
0.256256 + 0.966609i $$0.417511\pi$$
$$674$$ −17448.0 −0.997139
$$675$$ −2755.00 −0.157096
$$676$$ −2076.00 −0.118116
$$677$$ 11511.0 0.653477 0.326738 0.945115i $$-0.394050\pi$$
0.326738 + 0.945115i $$0.394050\pi$$
$$678$$ −5130.00 −0.290585
$$679$$ 0 0
$$680$$ 23436.0 1.32166
$$681$$ −26775.0 −1.50664
$$682$$ −9072.00 −0.509362
$$683$$ −10476.0 −0.586900 −0.293450 0.955974i $$-0.594803\pi$$
−0.293450 + 0.955974i $$0.594803\pi$$
$$684$$ 38.0000 0.00212422
$$685$$ −20268.0 −1.13051
$$686$$ 0 0
$$687$$ 31850.0 1.76878
$$688$$ 13916.0 0.771137
$$689$$ −4785.00 −0.264578
$$690$$ −32940.0 −1.81740
$$691$$ −30098.0 −1.65699 −0.828496 0.559995i $$-0.810803\pi$$
−0.828496 + 0.559995i $$0.810803\pi$$
$$692$$ 462.000 0.0253795
$$693$$ 0 0
$$694$$ 4734.00 0.258934
$$695$$ −25728.0 −1.40420
$$696$$ −26145.0 −1.42388
$$697$$ −22320.0 −1.21296
$$698$$ 4974.00 0.269726
$$699$$ −14190.0 −0.767833
$$700$$ 0 0
$$701$$ −14700.0 −0.792028 −0.396014 0.918245i $$-0.629607\pi$$
−0.396014 + 0.918245i $$0.629607\pi$$
$$702$$ −4785.00 −0.257262
$$703$$ 4750.00 0.254836
$$704$$ −23382.0 −1.25176
$$705$$ 10080.0 0.538489
$$706$$ −34101.0 −1.81786
$$707$$ 0 0
$$708$$ −975.000 −0.0517553
$$709$$ 31178.0 1.65150 0.825751 0.564035i $$-0.190752\pi$$
0.825751 + 0.564035i $$0.190752\pi$$
$$710$$ 8856.00 0.468112
$$711$$ 68.0000 0.00358678
$$712$$ 3528.00 0.185699
$$713$$ −10248.0 −0.538276
$$714$$ 0 0
$$715$$ 7128.00 0.372828
$$716$$ 720.000 0.0375805
$$717$$ −1845.00 −0.0960987
$$718$$ −7659.00 −0.398094
$$719$$ 33285.0 1.72645 0.863227 0.504815i $$-0.168439\pi$$
0.863227 + 0.504815i $$0.168439\pi$$
$$720$$ 1704.00 0.0882005
$$721$$ 0 0
$$722$$ −1083.00 −0.0558242
$$723$$ −33040.0 −1.69954
$$724$$ 2338.00 0.120015
$$725$$ −4731.00 −0.242352
$$726$$ −23775.0 −1.21539
$$727$$ 34729.0 1.77170 0.885851 0.463970i $$-0.153575\pi$$
0.885851 + 0.463970i $$0.153575\pi$$
$$728$$ 0 0
$$729$$ 20917.0 1.06269
$$730$$ 12708.0 0.644307
$$731$$ −18228.0 −0.922280
$$732$$ 1790.00 0.0903829
$$733$$ −4196.00 −0.211436 −0.105718 0.994396i $$-0.533714\pi$$
−0.105718 + 0.994396i $$0.533714\pi$$
$$734$$ −588.000 −0.0295688
$$735$$ 0 0
$$736$$ 8235.00 0.412427
$$737$$ 51894.0 2.59368
$$738$$ −1440.00 −0.0718254
$$739$$ −10744.0 −0.534810 −0.267405 0.963584i $$-0.586166\pi$$
−0.267405 + 0.963584i $$0.586166\pi$$
$$740$$ −3000.00 −0.149030
$$741$$ 1045.00 0.0518071
$$742$$ 0 0
$$743$$ −2208.00 −0.109022 −0.0545112 0.998513i $$-0.517360\pi$$
−0.0545112 + 0.998513i $$0.517360\pi$$
$$744$$ −5880.00 −0.289746
$$745$$ −36000.0 −1.77039
$$746$$ −28059.0 −1.37710
$$747$$ 468.000 0.0229227
$$748$$ −5022.00 −0.245485
$$749$$ 0 0
$$750$$ 19080.0 0.928937
$$751$$ 13160.0 0.639434 0.319717 0.947513i $$-0.396412\pi$$
0.319717 + 0.947513i $$0.396412\pi$$
$$752$$ −11928.0 −0.578417
$$753$$ −23370.0 −1.13101
$$754$$ −8217.00 −0.396877
$$755$$ −12072.0 −0.581914
$$756$$ 0 0
$$757$$ 758.000 0.0363936 0.0181968 0.999834i $$-0.494207\pi$$
0.0181968 + 0.999834i $$0.494207\pi$$
$$758$$ −11481.0 −0.550143
$$759$$ −49410.0 −2.36294
$$760$$ −4788.00 −0.228525
$$761$$ −4851.00 −0.231076 −0.115538 0.993303i $$-0.536859\pi$$
−0.115538 + 0.993303i $$0.536859\pi$$
$$762$$ 17250.0 0.820081
$$763$$ 0 0
$$764$$ 2871.00 0.135954
$$765$$ −2232.00 −0.105488
$$766$$ 17082.0 0.805741
$$767$$ 2145.00 0.100980
$$768$$ 7565.00 0.355441
$$769$$ 33091.0 1.55175 0.775873 0.630890i $$-0.217310\pi$$
0.775873 + 0.630890i $$0.217310\pi$$
$$770$$ 0 0
$$771$$ −22560.0 −1.05380
$$772$$ 1658.00 0.0772963
$$773$$ −42357.0 −1.97086 −0.985430 0.170079i $$-0.945598\pi$$
−0.985430 + 0.170079i $$0.945598\pi$$
$$774$$ −1176.00 −0.0546130
$$775$$ −1064.00 −0.0493161
$$776$$ −15918.0 −0.736370
$$777$$ 0 0
$$778$$ −3870.00 −0.178337
$$779$$ 4560.00 0.209729
$$780$$ −660.000 −0.0302972
$$781$$ 13284.0 0.608629
$$782$$ −51057.0 −2.33478
$$783$$ 36105.0 1.64788
$$784$$ 0 0
$$785$$ −34152.0 −1.55279
$$786$$ −21780.0 −0.988380
$$787$$ 39877.0 1.80618 0.903089 0.429454i $$-0.141294\pi$$
0.903089 + 0.429454i $$0.141294\pi$$
$$788$$ −4176.00 −0.188787
$$789$$ 18840.0 0.850091
$$790$$ 1224.00 0.0551240
$$791$$ 0 0
$$792$$ 2268.00 0.101755
$$793$$ −3938.00 −0.176346
$$794$$ 19608.0 0.876400
$$795$$ 26100.0 1.16437
$$796$$ 241.000 0.0107312
$$797$$ 30033.0 1.33478 0.667392 0.744706i $$-0.267410\pi$$
0.667392 + 0.744706i $$0.267410\pi$$
$$798$$ 0 0
$$799$$ 15624.0 0.691786
$$800$$ 855.000 0.0377860
$$801$$ −336.000 −0.0148214
$$802$$ −6984.00 −0.307498
$$803$$ 19062.0 0.837713
$$804$$ −4805.00 −0.210770
$$805$$ 0 0
$$806$$ −1848.00 −0.0807606
$$807$$ −23790.0 −1.03773
$$808$$ 15246.0 0.663802
$$809$$ 585.000 0.0254234 0.0127117 0.999919i $$-0.495954\pi$$
0.0127117 + 0.999919i $$0.495954\pi$$
$$810$$ 24156.0 1.04785
$$811$$ −28361.0 −1.22798 −0.613989 0.789315i $$-0.710436\pi$$
−0.613989 + 0.789315i $$0.710436\pi$$
$$812$$ 0 0
$$813$$ 10205.0 0.440228
$$814$$ −40500.0 −1.74389
$$815$$ −19200.0 −0.825211
$$816$$ −33015.0 −1.41637
$$817$$ 3724.00 0.159469
$$818$$ −20028.0 −0.856067
$$819$$ 0 0
$$820$$ −2880.00 −0.122651
$$821$$ 25068.0 1.06563 0.532813 0.846233i $$-0.321135\pi$$
0.532813 + 0.846233i $$0.321135\pi$$
$$822$$ 25335.0 1.07501
$$823$$ 10901.0 0.461707 0.230854 0.972989i $$-0.425848\pi$$
0.230854 + 0.972989i $$0.425848\pi$$
$$824$$ −42.0000 −0.00177565
$$825$$ −5130.00 −0.216489
$$826$$ 0 0
$$827$$ 12027.0 0.505707 0.252854 0.967505i $$-0.418631\pi$$
0.252854 + 0.967505i $$0.418631\pi$$
$$828$$ −366.000 −0.0153616
$$829$$ 19339.0 0.810219 0.405109 0.914268i $$-0.367233\pi$$
0.405109 + 0.914268i $$0.367233\pi$$
$$830$$ 8424.00 0.352291
$$831$$ 9820.00 0.409930
$$832$$ −4763.00 −0.198470
$$833$$ 0 0
$$834$$ 32160.0 1.33526
$$835$$ 24048.0 0.996665
$$836$$ 1026.00 0.0424461
$$837$$ 8120.00 0.335326
$$838$$ −24408.0 −1.00616
$$839$$ 13188.0 0.542670 0.271335 0.962485i $$-0.412535\pi$$
0.271335 + 0.962485i $$0.412535\pi$$
$$840$$ 0 0
$$841$$ 37612.0 1.54217
$$842$$ 25995.0 1.06395
$$843$$ −27480.0 −1.12273
$$844$$ −745.000 −0.0303838
$$845$$ −24912.0 −1.01420
$$846$$ 1008.00 0.0409642
$$847$$ 0 0
$$848$$ −30885.0 −1.25070
$$849$$ −15490.0 −0.626167
$$850$$ −5301.00 −0.213909
$$851$$ −45750.0 −1.84288
$$852$$ −1230.00 −0.0494590
$$853$$ 4678.00 0.187775 0.0938873 0.995583i $$-0.470071\pi$$
0.0938873 + 0.995583i $$0.470071\pi$$
$$854$$ 0 0
$$855$$ 456.000 0.0182396
$$856$$ 29673.0 1.18482
$$857$$ −15252.0 −0.607933 −0.303966 0.952683i $$-0.598311\pi$$
−0.303966 + 0.952683i $$0.598311\pi$$
$$858$$ −8910.00 −0.354525
$$859$$ 610.000 0.0242293 0.0121146 0.999927i $$-0.496144\pi$$
0.0121146 + 0.999927i $$0.496144\pi$$
$$860$$ −2352.00 −0.0932588
$$861$$ 0 0
$$862$$ −2250.00 −0.0889041
$$863$$ 774.000 0.0305299 0.0152649 0.999883i $$-0.495141\pi$$
0.0152649 + 0.999883i $$0.495141\pi$$
$$864$$ −6525.00 −0.256927
$$865$$ 5544.00 0.217921
$$866$$ −14574.0 −0.571876
$$867$$ 18680.0 0.731726
$$868$$ 0 0
$$869$$ 1836.00 0.0716709
$$870$$ 44820.0 1.74660
$$871$$ 10571.0 0.411234
$$872$$ 8169.00 0.317245
$$873$$ 1516.00 0.0587730
$$874$$ 10431.0 0.403700
$$875$$ 0 0
$$876$$ −1765.00 −0.0680751
$$877$$ −31039.0 −1.19511 −0.597556 0.801827i $$-0.703861\pi$$
−0.597556 + 0.801827i $$0.703861\pi$$
$$878$$ 19500.0 0.749537
$$879$$ −585.000 −0.0224477
$$880$$ 46008.0 1.76242
$$881$$ −33678.0 −1.28790 −0.643950 0.765067i $$-0.722706\pi$$
−0.643950 + 0.765067i $$0.722706\pi$$
$$882$$ 0 0
$$883$$ −42982.0 −1.63812 −0.819060 0.573708i $$-0.805504\pi$$
−0.819060 + 0.573708i $$0.805504\pi$$
$$884$$ −1023.00 −0.0389222
$$885$$ −11700.0 −0.444397
$$886$$ −10458.0 −0.396550
$$887$$ −4494.00 −0.170117 −0.0850585 0.996376i $$-0.527108\pi$$
−0.0850585 + 0.996376i $$0.527108\pi$$
$$888$$ −26250.0 −0.991996
$$889$$ 0 0
$$890$$ −6048.00 −0.227786
$$891$$ 36234.0 1.36238
$$892$$ 1978.00 0.0742470
$$893$$ −3192.00 −0.119615
$$894$$ 45000.0 1.68347
$$895$$ 8640.00 0.322685
$$896$$ 0 0
$$897$$ −10065.0 −0.374649
$$898$$ 45090.0 1.67558
$$899$$ 13944.0 0.517306
$$900$$ −38.0000 −0.00140741
$$901$$ 40455.0 1.49584
$$902$$ −38880.0 −1.43521
$$903$$ 0 0
$$904$$ 7182.00 0.264236
$$905$$ 28056.0 1.03051
$$906$$ 15090.0 0.553346
$$907$$ −23839.0 −0.872724 −0.436362 0.899771i $$-0.643733\pi$$
−0.436362 + 0.899771i $$0.643733\pi$$
$$908$$ −5355.00 −0.195718
$$909$$ −1452.00 −0.0529811
$$910$$ 0 0
$$911$$ −10332.0 −0.375757 −0.187878 0.982192i $$-0.560161\pi$$
−0.187878 + 0.982192i $$0.560161\pi$$
$$912$$ 6745.00 0.244901
$$913$$ 12636.0 0.458040
$$914$$ 8877.00 0.321253
$$915$$ 21480.0 0.776073
$$916$$ 6370.00 0.229772
$$917$$ 0 0
$$918$$ 40455.0 1.45448
$$919$$ −14371.0 −0.515838 −0.257919 0.966166i $$-0.583037\pi$$
−0.257919 + 0.966166i $$0.583037\pi$$
$$920$$ 46116.0 1.65261
$$921$$ 7100.00 0.254021
$$922$$ −468.000 −0.0167167
$$923$$ 2706.00 0.0964995
$$924$$ 0 0
$$925$$ −4750.00 −0.168842
$$926$$ −13452.0 −0.477387
$$927$$ 4.00000 0.000141723 0
$$928$$ −11205.0 −0.396360
$$929$$ −26889.0 −0.949623 −0.474811 0.880088i $$-0.657484\pi$$
−0.474811 + 0.880088i $$0.657484\pi$$
$$930$$ 10080.0 0.355415
$$931$$ 0 0
$$932$$ −2838.00 −0.0997444
$$933$$ 32805.0 1.15111
$$934$$ 26298.0 0.921303
$$935$$ −60264.0 −2.10785
$$936$$ 462.000 0.0161335
$$937$$ −785.000 −0.0273691 −0.0136845 0.999906i $$-0.504356\pi$$
−0.0136845 + 0.999906i $$0.504356\pi$$
$$938$$ 0 0
$$939$$ 7415.00 0.257699
$$940$$ 2016.00 0.0699518
$$941$$ 18141.0 0.628459 0.314229 0.949347i $$-0.398254\pi$$
0.314229 + 0.949347i $$0.398254\pi$$
$$942$$ 42690.0 1.47656
$$943$$ −43920.0 −1.51668
$$944$$ 13845.0 0.477348
$$945$$ 0 0
$$946$$ −31752.0 −1.09128
$$947$$ 23100.0 0.792660 0.396330 0.918108i $$-0.370284\pi$$
0.396330 + 0.918108i $$0.370284\pi$$
$$948$$ −170.000 −0.00582420
$$949$$ 3883.00 0.132821
$$950$$ 1083.00 0.0369865
$$951$$ −6195.00 −0.211237
$$952$$ 0 0
$$953$$ 45690.0 1.55304 0.776519 0.630094i $$-0.216984\pi$$
0.776519 + 0.630094i $$0.216984\pi$$
$$954$$ 2610.00 0.0885764
$$955$$ 34452.0 1.16737
$$956$$ −369.000 −0.0124836
$$957$$ 67230.0 2.27089
$$958$$ −56988.0 −1.92192
$$959$$ 0 0
$$960$$ 25980.0 0.873438
$$961$$ −26655.0 −0.894733
$$962$$ −8250.00 −0.276498
$$963$$ −2826.00 −0.0945655
$$964$$ −6608.00 −0.220777
$$965$$ 19896.0 0.663705
$$966$$ 0 0
$$967$$ 21584.0 0.717781 0.358891 0.933380i $$-0.383155\pi$$
0.358891 + 0.933380i $$0.383155\pi$$
$$968$$ 33285.0 1.10519
$$969$$ −8835.00 −0.292901
$$970$$ 27288.0 0.903263
$$971$$ 50556.0 1.67087 0.835437 0.549586i $$-0.185214\pi$$
0.835437 + 0.549586i $$0.185214\pi$$
$$972$$ 560.000 0.0184794
$$973$$ 0 0
$$974$$ 22350.0 0.735257
$$975$$ −1045.00 −0.0343249
$$976$$ −25418.0 −0.833617
$$977$$ 8568.00 0.280568 0.140284 0.990111i $$-0.455198\pi$$
0.140284 + 0.990111i $$0.455198\pi$$
$$978$$ 24000.0 0.784699
$$979$$ −9072.00 −0.296162
$$980$$ 0 0
$$981$$ −778.000 −0.0253207
$$982$$ −18540.0 −0.602480
$$983$$ −29706.0 −0.963860 −0.481930 0.876210i $$-0.660064\pi$$
−0.481930 + 0.876210i $$0.660064\pi$$
$$984$$ −25200.0 −0.816409
$$985$$ −50112.0 −1.62102
$$986$$ 69471.0 2.24382
$$987$$ 0 0
$$988$$ 209.000 0.00672993
$$989$$ −35868.0 −1.15322
$$990$$ −3888.00 −0.124817
$$991$$ 30512.0 0.978048 0.489024 0.872270i $$-0.337353\pi$$
0.489024 + 0.872270i $$0.337353\pi$$
$$992$$ −2520.00 −0.0806553
$$993$$ −44495.0 −1.42196
$$994$$ 0 0
$$995$$ 2892.00 0.0921433
$$996$$ −1170.00 −0.0372218
$$997$$ −47756.0 −1.51700 −0.758499 0.651674i $$-0.774067\pi$$
−0.758499 + 0.651674i $$0.774067\pi$$
$$998$$ −7728.00 −0.245116
$$999$$ 36250.0 1.14805
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 931.4.a.a.1.1 1
7.6 odd 2 19.4.a.a.1.1 1
21.20 even 2 171.4.a.d.1.1 1
28.27 even 2 304.4.a.b.1.1 1
35.13 even 4 475.4.b.c.324.2 2
35.27 even 4 475.4.b.c.324.1 2
35.34 odd 2 475.4.a.e.1.1 1
56.13 odd 2 1216.4.a.f.1.1 1
56.27 even 2 1216.4.a.a.1.1 1
77.76 even 2 2299.4.a.b.1.1 1
133.132 even 2 361.4.a.b.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
19.4.a.a.1.1 1 7.6 odd 2
171.4.a.d.1.1 1 21.20 even 2
304.4.a.b.1.1 1 28.27 even 2
361.4.a.b.1.1 1 133.132 even 2
475.4.a.e.1.1 1 35.34 odd 2
475.4.b.c.324.1 2 35.27 even 4
475.4.b.c.324.2 2 35.13 even 4
931.4.a.a.1.1 1 1.1 even 1 trivial
1216.4.a.a.1.1 1 56.27 even 2
1216.4.a.f.1.1 1 56.13 odd 2
2299.4.a.b.1.1 1 77.76 even 2