Properties

 Label 1323.4.a.bf.1.3 Level $1323$ Weight $4$ Character 1323.1 Self dual yes Analytic conductor $78.060$ Analytic rank $1$ Dimension $6$ Inner twists $2$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1323,4,Mod(1,1323)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1323, 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("1323.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1323 = 3^{3} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1323.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: $$78.0595269376$$ Analytic rank: $$1$$ Dimension: $$6$$ Coefficient field: $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 42x^{4} + 369x^{2} - 112$$ x^6 - 42*x^4 + 369*x^2 - 112 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{3}\cdot 3^{4}$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.3 Root $$-0.560992$$ of defining polynomial Character $$\chi$$ $$=$$ 1323.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-0.560992 q^{2} -7.68529 q^{4} +12.9561 q^{5} +8.79932 q^{8} +O(q^{10})$$ $$q-0.560992 q^{2} -7.68529 q^{4} +12.9561 q^{5} +8.79932 q^{8} -7.26827 q^{10} -48.2286 q^{11} -36.3241 q^{13} +56.5460 q^{16} +83.2713 q^{17} +67.4359 q^{19} -99.5714 q^{20} +27.0559 q^{22} +30.6300 q^{23} +42.8607 q^{25} +20.3775 q^{26} -294.244 q^{29} +270.844 q^{31} -102.116 q^{32} -46.7145 q^{34} -204.315 q^{37} -37.8310 q^{38} +114.005 q^{40} +287.529 q^{41} -55.4901 q^{43} +370.651 q^{44} -17.1832 q^{46} -191.301 q^{47} -24.0445 q^{50} +279.161 q^{52} -521.107 q^{53} -624.855 q^{55} +165.069 q^{58} -381.074 q^{59} +155.465 q^{61} -151.941 q^{62} -395.081 q^{64} -470.619 q^{65} -65.1959 q^{67} -639.964 q^{68} +256.370 q^{71} +318.625 q^{73} +114.619 q^{74} -518.264 q^{76} +77.7532 q^{79} +732.615 q^{80} -161.301 q^{82} +836.549 q^{83} +1078.87 q^{85} +31.1295 q^{86} -424.379 q^{88} +1590.06 q^{89} -235.400 q^{92} +107.319 q^{94} +873.706 q^{95} -1189.88 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 36 q^{4}+O(q^{10})$$ 6 * q + 36 * q^4 $$6 q + 36 q^{4} - 180 q^{10} - 108 q^{13} + 420 q^{16} - 198 q^{19} - 84 q^{22} + 420 q^{25} + 90 q^{31} + 648 q^{34} - 402 q^{37} - 2844 q^{40} - 660 q^{43} - 1332 q^{46} - 1224 q^{52} - 846 q^{55} - 1800 q^{58} - 1152 q^{61} + 2964 q^{64} + 924 q^{67} - 1260 q^{73} - 5868 q^{76} - 1500 q^{79} - 4500 q^{82} - 2232 q^{85} - 2460 q^{88} + 4968 q^{94} - 3312 q^{97}+O(q^{100})$$ 6 * q + 36 * q^4 - 180 * q^10 - 108 * q^13 + 420 * q^16 - 198 * q^19 - 84 * q^22 + 420 * q^25 + 90 * q^31 + 648 * q^34 - 402 * q^37 - 2844 * q^40 - 660 * q^43 - 1332 * q^46 - 1224 * q^52 - 846 * q^55 - 1800 * q^58 - 1152 * q^61 + 2964 * q^64 + 924 * q^67 - 1260 * q^73 - 5868 * q^76 - 1500 * q^79 - 4500 * q^82 - 2232 * q^85 - 2460 * q^88 + 4968 * q^94 - 3312 * q^97

Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −0.560992 −0.198341 −0.0991703 0.995070i $$-0.531619\pi$$
−0.0991703 + 0.995070i $$0.531619\pi$$
$$3$$ 0 0
$$4$$ −7.68529 −0.960661
$$5$$ 12.9561 1.15883 0.579415 0.815033i $$-0.303281\pi$$
0.579415 + 0.815033i $$0.303281\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 8.79932 0.388879
$$9$$ 0 0
$$10$$ −7.26827 −0.229843
$$11$$ −48.2286 −1.32195 −0.660976 0.750407i $$-0.729858\pi$$
−0.660976 + 0.750407i $$0.729858\pi$$
$$12$$ 0 0
$$13$$ −36.3241 −0.774962 −0.387481 0.921878i $$-0.626655\pi$$
−0.387481 + 0.921878i $$0.626655\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 56.5460 0.883531
$$17$$ 83.2713 1.18802 0.594008 0.804459i $$-0.297545\pi$$
0.594008 + 0.804459i $$0.297545\pi$$
$$18$$ 0 0
$$19$$ 67.4359 0.814255 0.407128 0.913371i $$-0.366530\pi$$
0.407128 + 0.913371i $$0.366530\pi$$
$$20$$ −99.5714 −1.11324
$$21$$ 0 0
$$22$$ 27.0559 0.262197
$$23$$ 30.6300 0.277687 0.138843 0.990314i $$-0.455662\pi$$
0.138843 + 0.990314i $$0.455662\pi$$
$$24$$ 0 0
$$25$$ 42.8607 0.342885
$$26$$ 20.3775 0.153706
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −294.244 −1.88413 −0.942065 0.335431i $$-0.891118\pi$$
−0.942065 + 0.335431i $$0.891118\pi$$
$$30$$ 0 0
$$31$$ 270.844 1.56919 0.784597 0.620006i $$-0.212870\pi$$
0.784597 + 0.620006i $$0.212870\pi$$
$$32$$ −102.116 −0.564119
$$33$$ 0 0
$$34$$ −46.7145 −0.235632
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −204.315 −0.907817 −0.453909 0.891048i $$-0.649971\pi$$
−0.453909 + 0.891048i $$0.649971\pi$$
$$38$$ −37.8310 −0.161500
$$39$$ 0 0
$$40$$ 114.005 0.450644
$$41$$ 287.529 1.09523 0.547615 0.836730i $$-0.315536\pi$$
0.547615 + 0.836730i $$0.315536\pi$$
$$42$$ 0 0
$$43$$ −55.4901 −0.196794 −0.0983972 0.995147i $$-0.531372\pi$$
−0.0983972 + 0.995147i $$0.531372\pi$$
$$44$$ 370.651 1.26995
$$45$$ 0 0
$$46$$ −17.1832 −0.0550765
$$47$$ −191.301 −0.593706 −0.296853 0.954923i $$-0.595937\pi$$
−0.296853 + 0.954923i $$0.595937\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ −24.0445 −0.0680081
$$51$$ 0 0
$$52$$ 279.161 0.744475
$$53$$ −521.107 −1.35056 −0.675278 0.737563i $$-0.735976\pi$$
−0.675278 + 0.737563i $$0.735976\pi$$
$$54$$ 0 0
$$55$$ −624.855 −1.53192
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 165.069 0.373699
$$59$$ −381.074 −0.840874 −0.420437 0.907322i $$-0.638123\pi$$
−0.420437 + 0.907322i $$0.638123\pi$$
$$60$$ 0 0
$$61$$ 155.465 0.326315 0.163158 0.986600i $$-0.447832\pi$$
0.163158 + 0.986600i $$0.447832\pi$$
$$62$$ −151.941 −0.311235
$$63$$ 0 0
$$64$$ −395.081 −0.771643
$$65$$ −470.619 −0.898048
$$66$$ 0 0
$$67$$ −65.1959 −0.118880 −0.0594399 0.998232i $$-0.518931\pi$$
−0.0594399 + 0.998232i $$0.518931\pi$$
$$68$$ −639.964 −1.14128
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 256.370 0.428529 0.214265 0.976776i $$-0.431265\pi$$
0.214265 + 0.976776i $$0.431265\pi$$
$$72$$ 0 0
$$73$$ 318.625 0.510852 0.255426 0.966829i $$-0.417784\pi$$
0.255426 + 0.966829i $$0.417784\pi$$
$$74$$ 114.619 0.180057
$$75$$ 0 0
$$76$$ −518.264 −0.782223
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 77.7532 0.110733 0.0553666 0.998466i $$-0.482367\pi$$
0.0553666 + 0.998466i $$0.482367\pi$$
$$80$$ 732.615 1.02386
$$81$$ 0 0
$$82$$ −161.301 −0.217229
$$83$$ 836.549 1.10630 0.553152 0.833081i $$-0.313425\pi$$
0.553152 + 0.833081i $$0.313425\pi$$
$$84$$ 0 0
$$85$$ 1078.87 1.37671
$$86$$ 31.1295 0.0390323
$$87$$ 0 0
$$88$$ −424.379 −0.514079
$$89$$ 1590.06 1.89377 0.946887 0.321565i $$-0.104209\pi$$
0.946887 + 0.321565i $$0.104209\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −235.400 −0.266763
$$93$$ 0 0
$$94$$ 107.319 0.117756
$$95$$ 873.706 0.943583
$$96$$ 0 0
$$97$$ −1189.88 −1.24550 −0.622751 0.782420i $$-0.713985\pi$$
−0.622751 + 0.782420i $$0.713985\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ −329.397 −0.329397
$$101$$ 507.845 0.500321 0.250161 0.968204i $$-0.419517\pi$$
0.250161 + 0.968204i $$0.419517\pi$$
$$102$$ 0 0
$$103$$ −1852.78 −1.77242 −0.886212 0.463280i $$-0.846672\pi$$
−0.886212 + 0.463280i $$0.846672\pi$$
$$104$$ −319.628 −0.301366
$$105$$ 0 0
$$106$$ 292.337 0.267870
$$107$$ −614.904 −0.555561 −0.277781 0.960645i $$-0.589599\pi$$
−0.277781 + 0.960645i $$0.589599\pi$$
$$108$$ 0 0
$$109$$ 1017.92 0.894487 0.447243 0.894412i $$-0.352406\pi$$
0.447243 + 0.894412i $$0.352406\pi$$
$$110$$ 350.539 0.303841
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 1204.42 1.00267 0.501337 0.865252i $$-0.332842\pi$$
0.501337 + 0.865252i $$0.332842\pi$$
$$114$$ 0 0
$$115$$ 396.845 0.321792
$$116$$ 2261.35 1.81001
$$117$$ 0 0
$$118$$ 213.779 0.166779
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 995.000 0.747559
$$122$$ −87.2145 −0.0647215
$$123$$ 0 0
$$124$$ −2081.52 −1.50746
$$125$$ −1064.21 −0.761484
$$126$$ 0 0
$$127$$ −2515.55 −1.75763 −0.878813 0.477165i $$-0.841664\pi$$
−0.878813 + 0.477165i $$0.841664\pi$$
$$128$$ 1038.57 0.717167
$$129$$ 0 0
$$130$$ 264.014 0.178119
$$131$$ −2165.37 −1.44419 −0.722096 0.691793i $$-0.756821\pi$$
−0.722096 + 0.691793i $$0.756821\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 36.5743 0.0235787
$$135$$ 0 0
$$136$$ 732.731 0.461994
$$137$$ −795.668 −0.496194 −0.248097 0.968735i $$-0.579805\pi$$
−0.248097 + 0.968735i $$0.579805\pi$$
$$138$$ 0 0
$$139$$ −2400.60 −1.46487 −0.732433 0.680839i $$-0.761615\pi$$
−0.732433 + 0.680839i $$0.761615\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −143.822 −0.0849947
$$143$$ 1751.86 1.02446
$$144$$ 0 0
$$145$$ −3812.26 −2.18338
$$146$$ −178.746 −0.101323
$$147$$ 0 0
$$148$$ 1570.22 0.872105
$$149$$ −1404.26 −0.772091 −0.386046 0.922480i $$-0.626159\pi$$
−0.386046 + 0.922480i $$0.626159\pi$$
$$150$$ 0 0
$$151$$ −1761.48 −0.949322 −0.474661 0.880169i $$-0.657429\pi$$
−0.474661 + 0.880169i $$0.657429\pi$$
$$152$$ 593.390 0.316646
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 3509.09 1.81843
$$156$$ 0 0
$$157$$ −1082.03 −0.550034 −0.275017 0.961439i $$-0.588683\pi$$
−0.275017 + 0.961439i $$0.588683\pi$$
$$158$$ −43.6189 −0.0219629
$$159$$ 0 0
$$160$$ −1323.03 −0.653717
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −2473.27 −1.18847 −0.594237 0.804290i $$-0.702546\pi$$
−0.594237 + 0.804290i $$0.702546\pi$$
$$164$$ −2209.74 −1.05215
$$165$$ 0 0
$$166$$ −469.297 −0.219425
$$167$$ 2240.74 1.03828 0.519142 0.854688i $$-0.326252\pi$$
0.519142 + 0.854688i $$0.326252\pi$$
$$168$$ 0 0
$$169$$ −877.557 −0.399434
$$170$$ −605.238 −0.273057
$$171$$ 0 0
$$172$$ 426.457 0.189053
$$173$$ −2121.28 −0.932242 −0.466121 0.884721i $$-0.654349\pi$$
−0.466121 + 0.884721i $$0.654349\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −2727.13 −1.16799
$$177$$ 0 0
$$178$$ −892.010 −0.375612
$$179$$ −1578.84 −0.659262 −0.329631 0.944110i $$-0.606924\pi$$
−0.329631 + 0.944110i $$0.606924\pi$$
$$180$$ 0 0
$$181$$ −4609.64 −1.89299 −0.946496 0.322716i $$-0.895404\pi$$
−0.946496 + 0.322716i $$0.895404\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 269.523 0.107986
$$185$$ −2647.13 −1.05201
$$186$$ 0 0
$$187$$ −4016.06 −1.57050
$$188$$ 1470.21 0.570350
$$189$$ 0 0
$$190$$ −490.142 −0.187151
$$191$$ −1291.72 −0.489350 −0.244675 0.969605i $$-0.578681\pi$$
−0.244675 + 0.969605i $$0.578681\pi$$
$$192$$ 0 0
$$193$$ 1176.01 0.438605 0.219302 0.975657i $$-0.429622\pi$$
0.219302 + 0.975657i $$0.429622\pi$$
$$194$$ 667.511 0.247033
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −4560.71 −1.64943 −0.824713 0.565552i $$-0.808663\pi$$
−0.824713 + 0.565552i $$0.808663\pi$$
$$198$$ 0 0
$$199$$ 986.014 0.351240 0.175620 0.984458i $$-0.443807\pi$$
0.175620 + 0.984458i $$0.443807\pi$$
$$200$$ 377.145 0.133341
$$201$$ 0 0
$$202$$ −284.897 −0.0992340
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 3725.25 1.26919
$$206$$ 1039.39 0.351543
$$207$$ 0 0
$$208$$ −2053.98 −0.684702
$$209$$ −3252.34 −1.07641
$$210$$ 0 0
$$211$$ 5459.28 1.78120 0.890598 0.454792i $$-0.150286\pi$$
0.890598 + 0.454792i $$0.150286\pi$$
$$212$$ 4004.85 1.29743
$$213$$ 0 0
$$214$$ 344.956 0.110190
$$215$$ −718.936 −0.228051
$$216$$ 0 0
$$217$$ 0 0
$$218$$ −571.045 −0.177413
$$219$$ 0 0
$$220$$ 4802.19 1.47165
$$221$$ −3024.76 −0.920666
$$222$$ 0 0
$$223$$ −4626.41 −1.38927 −0.694636 0.719362i $$-0.744434\pi$$
−0.694636 + 0.719362i $$0.744434\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −675.669 −0.198871
$$227$$ 2437.77 0.712776 0.356388 0.934338i $$-0.384008\pi$$
0.356388 + 0.934338i $$0.384008\pi$$
$$228$$ 0 0
$$229$$ −2946.50 −0.850264 −0.425132 0.905131i $$-0.639772\pi$$
−0.425132 + 0.905131i $$0.639772\pi$$
$$230$$ −222.627 −0.0638243
$$231$$ 0 0
$$232$$ −2589.15 −0.732698
$$233$$ 3441.49 0.967638 0.483819 0.875168i $$-0.339249\pi$$
0.483819 + 0.875168i $$0.339249\pi$$
$$234$$ 0 0
$$235$$ −2478.52 −0.688004
$$236$$ 2928.66 0.807795
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −744.021 −0.201367 −0.100684 0.994918i $$-0.532103\pi$$
−0.100684 + 0.994918i $$0.532103\pi$$
$$240$$ 0 0
$$241$$ 4714.37 1.26008 0.630040 0.776562i $$-0.283038\pi$$
0.630040 + 0.776562i $$0.283038\pi$$
$$242$$ −558.187 −0.148271
$$243$$ 0 0
$$244$$ −1194.79 −0.313478
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −2449.55 −0.631017
$$248$$ 2383.24 0.610226
$$249$$ 0 0
$$250$$ 597.011 0.151033
$$251$$ −2248.41 −0.565411 −0.282706 0.959207i $$-0.591232\pi$$
−0.282706 + 0.959207i $$0.591232\pi$$
$$252$$ 0 0
$$253$$ −1477.24 −0.367089
$$254$$ 1411.20 0.348609
$$255$$ 0 0
$$256$$ 2578.02 0.629400
$$257$$ 6099.67 1.48049 0.740247 0.672335i $$-0.234709\pi$$
0.740247 + 0.672335i $$0.234709\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 3616.85 0.862720
$$261$$ 0 0
$$262$$ 1214.75 0.286442
$$263$$ −3493.26 −0.819026 −0.409513 0.912304i $$-0.634301\pi$$
−0.409513 + 0.912304i $$0.634301\pi$$
$$264$$ 0 0
$$265$$ −6751.51 −1.56506
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 501.049 0.114203
$$269$$ −6393.14 −1.44906 −0.724529 0.689244i $$-0.757943\pi$$
−0.724529 + 0.689244i $$0.757943\pi$$
$$270$$ 0 0
$$271$$ 4222.81 0.946559 0.473279 0.880912i $$-0.343070\pi$$
0.473279 + 0.880912i $$0.343070\pi$$
$$272$$ 4708.66 1.04965
$$273$$ 0 0
$$274$$ 446.363 0.0984153
$$275$$ −2067.11 −0.453278
$$276$$ 0 0
$$277$$ −5240.37 −1.13669 −0.568345 0.822790i $$-0.692416\pi$$
−0.568345 + 0.822790i $$0.692416\pi$$
$$278$$ 1346.72 0.290542
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −5924.79 −1.25781 −0.628903 0.777484i $$-0.716496\pi$$
−0.628903 + 0.777484i $$0.716496\pi$$
$$282$$ 0 0
$$283$$ 1885.61 0.396070 0.198035 0.980195i $$-0.436544\pi$$
0.198035 + 0.980195i $$0.436544\pi$$
$$284$$ −1970.28 −0.411671
$$285$$ 0 0
$$286$$ −982.781 −0.203192
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 2021.11 0.411380
$$290$$ 2138.65 0.433054
$$291$$ 0 0
$$292$$ −2448.72 −0.490756
$$293$$ −3173.12 −0.632680 −0.316340 0.948646i $$-0.602454\pi$$
−0.316340 + 0.948646i $$0.602454\pi$$
$$294$$ 0 0
$$295$$ −4937.23 −0.974430
$$296$$ −1797.84 −0.353031
$$297$$ 0 0
$$298$$ 787.779 0.153137
$$299$$ −1112.61 −0.215197
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 988.179 0.188289
$$303$$ 0 0
$$304$$ 3813.23 0.719419
$$305$$ 2014.22 0.378144
$$306$$ 0 0
$$307$$ −6188.30 −1.15044 −0.575220 0.817999i $$-0.695084\pi$$
−0.575220 + 0.817999i $$0.695084\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ −1968.57 −0.360668
$$311$$ 3750.93 0.683909 0.341954 0.939717i $$-0.388911\pi$$
0.341954 + 0.939717i $$0.388911\pi$$
$$312$$ 0 0
$$313$$ 4172.29 0.753456 0.376728 0.926324i $$-0.377049\pi$$
0.376728 + 0.926324i $$0.377049\pi$$
$$314$$ 607.009 0.109094
$$315$$ 0 0
$$316$$ −597.556 −0.106377
$$317$$ −8410.38 −1.49014 −0.745069 0.666987i $$-0.767584\pi$$
−0.745069 + 0.666987i $$0.767584\pi$$
$$318$$ 0 0
$$319$$ 14191.0 2.49073
$$320$$ −5118.71 −0.894203
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 5615.47 0.967347
$$324$$ 0 0
$$325$$ −1556.88 −0.265723
$$326$$ 1387.48 0.235723
$$327$$ 0 0
$$328$$ 2530.06 0.425912
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −1217.72 −0.202211 −0.101105 0.994876i $$-0.532238\pi$$
−0.101105 + 0.994876i $$0.532238\pi$$
$$332$$ −6429.12 −1.06278
$$333$$ 0 0
$$334$$ −1257.03 −0.205934
$$335$$ −844.684 −0.137761
$$336$$ 0 0
$$337$$ 3880.01 0.627175 0.313587 0.949559i $$-0.398469\pi$$
0.313587 + 0.949559i $$0.398469\pi$$
$$338$$ 492.302 0.0792240
$$339$$ 0 0
$$340$$ −8291.44 −1.32255
$$341$$ −13062.4 −2.07440
$$342$$ 0 0
$$343$$ 0 0
$$344$$ −488.275 −0.0765291
$$345$$ 0 0
$$346$$ 1190.02 0.184901
$$347$$ 9672.45 1.49638 0.748191 0.663484i $$-0.230923\pi$$
0.748191 + 0.663484i $$0.230923\pi$$
$$348$$ 0 0
$$349$$ −3960.64 −0.607473 −0.303737 0.952756i $$-0.598234\pi$$
−0.303737 + 0.952756i $$0.598234\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 4924.93 0.745738
$$353$$ −7639.67 −1.15189 −0.575947 0.817487i $$-0.695367\pi$$
−0.575947 + 0.817487i $$0.695367\pi$$
$$354$$ 0 0
$$355$$ 3321.56 0.496592
$$356$$ −12220.1 −1.81928
$$357$$ 0 0
$$358$$ 885.715 0.130758
$$359$$ 9411.16 1.38357 0.691785 0.722103i $$-0.256824\pi$$
0.691785 + 0.722103i $$0.256824\pi$$
$$360$$ 0 0
$$361$$ −2311.40 −0.336989
$$362$$ 2585.97 0.375457
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 4128.14 0.591991
$$366$$ 0 0
$$367$$ −5707.80 −0.811838 −0.405919 0.913909i $$-0.633048\pi$$
−0.405919 + 0.913909i $$0.633048\pi$$
$$368$$ 1732.00 0.245345
$$369$$ 0 0
$$370$$ 1485.02 0.208655
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −9579.55 −1.32979 −0.664893 0.746938i $$-0.731523\pi$$
−0.664893 + 0.746938i $$0.731523\pi$$
$$374$$ 2252.98 0.311494
$$375$$ 0 0
$$376$$ −1683.32 −0.230880
$$377$$ 10688.2 1.46013
$$378$$ 0 0
$$379$$ 8730.92 1.18332 0.591658 0.806189i $$-0.298473\pi$$
0.591658 + 0.806189i $$0.298473\pi$$
$$380$$ −6714.68 −0.906463
$$381$$ 0 0
$$382$$ 724.646 0.0970579
$$383$$ 9623.21 1.28387 0.641936 0.766758i $$-0.278131\pi$$
0.641936 + 0.766758i $$0.278131\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −659.730 −0.0869931
$$387$$ 0 0
$$388$$ 9144.54 1.19650
$$389$$ 11319.7 1.47541 0.737704 0.675125i $$-0.235910\pi$$
0.737704 + 0.675125i $$0.235910\pi$$
$$390$$ 0 0
$$391$$ 2550.60 0.329896
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 2558.52 0.327148
$$395$$ 1007.38 0.128321
$$396$$ 0 0
$$397$$ −2855.87 −0.361038 −0.180519 0.983572i $$-0.557778\pi$$
−0.180519 + 0.983572i $$0.557778\pi$$
$$398$$ −553.146 −0.0696651
$$399$$ 0 0
$$400$$ 2423.60 0.302950
$$401$$ 1975.97 0.246073 0.123036 0.992402i $$-0.460737\pi$$
0.123036 + 0.992402i $$0.460737\pi$$
$$402$$ 0 0
$$403$$ −9838.18 −1.21607
$$404$$ −3902.94 −0.480639
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 9853.85 1.20009
$$408$$ 0 0
$$409$$ −8244.62 −0.996749 −0.498375 0.866962i $$-0.666070\pi$$
−0.498375 + 0.866962i $$0.666070\pi$$
$$410$$ −2089.84 −0.251731
$$411$$ 0 0
$$412$$ 14239.1 1.70270
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 10838.4 1.28202
$$416$$ 3709.29 0.437170
$$417$$ 0 0
$$418$$ 1824.54 0.213495
$$419$$ 12861.0 1.49952 0.749761 0.661709i $$-0.230169\pi$$
0.749761 + 0.661709i $$0.230169\pi$$
$$420$$ 0 0
$$421$$ −16467.7 −1.90638 −0.953188 0.302377i $$-0.902220\pi$$
−0.953188 + 0.302377i $$0.902220\pi$$
$$422$$ −3062.61 −0.353283
$$423$$ 0 0
$$424$$ −4585.38 −0.525203
$$425$$ 3569.06 0.407353
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 4725.72 0.533706
$$429$$ 0 0
$$430$$ 403.317 0.0452318
$$431$$ −273.346 −0.0305490 −0.0152745 0.999883i $$-0.504862\pi$$
−0.0152745 + 0.999883i $$0.504862\pi$$
$$432$$ 0 0
$$433$$ 13765.3 1.52776 0.763878 0.645361i $$-0.223293\pi$$
0.763878 + 0.645361i $$0.223293\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −7823.01 −0.859298
$$437$$ 2065.56 0.226108
$$438$$ 0 0
$$439$$ −2965.97 −0.322456 −0.161228 0.986917i $$-0.551545\pi$$
−0.161228 + 0.986917i $$0.551545\pi$$
$$440$$ −5498.30 −0.595730
$$441$$ 0 0
$$442$$ 1696.86 0.182605
$$443$$ 14038.4 1.50561 0.752806 0.658242i $$-0.228700\pi$$
0.752806 + 0.658242i $$0.228700\pi$$
$$444$$ 0 0
$$445$$ 20601.0 2.19456
$$446$$ 2595.38 0.275549
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −6338.13 −0.666180 −0.333090 0.942895i $$-0.608091\pi$$
−0.333090 + 0.942895i $$0.608091\pi$$
$$450$$ 0 0
$$451$$ −13867.1 −1.44784
$$452$$ −9256.30 −0.963230
$$453$$ 0 0
$$454$$ −1367.57 −0.141372
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 9120.26 0.933540 0.466770 0.884379i $$-0.345418\pi$$
0.466770 + 0.884379i $$0.345418\pi$$
$$458$$ 1652.97 0.168642
$$459$$ 0 0
$$460$$ −3049.87 −0.309133
$$461$$ −15109.6 −1.52651 −0.763256 0.646096i $$-0.776400\pi$$
−0.763256 + 0.646096i $$0.776400\pi$$
$$462$$ 0 0
$$463$$ 6418.72 0.644283 0.322142 0.946692i $$-0.395597\pi$$
0.322142 + 0.946692i $$0.395597\pi$$
$$464$$ −16638.3 −1.66469
$$465$$ 0 0
$$466$$ −1930.65 −0.191922
$$467$$ 4304.81 0.426559 0.213279 0.976991i $$-0.431586\pi$$
0.213279 + 0.976991i $$0.431586\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 1390.43 0.136459
$$471$$ 0 0
$$472$$ −3353.19 −0.326998
$$473$$ 2676.21 0.260153
$$474$$ 0 0
$$475$$ 2890.35 0.279196
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 417.390 0.0399393
$$479$$ 5030.65 0.479867 0.239934 0.970789i $$-0.422874\pi$$
0.239934 + 0.970789i $$0.422874\pi$$
$$480$$ 0 0
$$481$$ 7421.58 0.703524
$$482$$ −2644.72 −0.249925
$$483$$ 0 0
$$484$$ −7646.87 −0.718150
$$485$$ −15416.2 −1.44332
$$486$$ 0 0
$$487$$ −13300.6 −1.23759 −0.618794 0.785553i $$-0.712379\pi$$
−0.618794 + 0.785553i $$0.712379\pi$$
$$488$$ 1367.98 0.126897
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −9246.30 −0.849857 −0.424928 0.905227i $$-0.639701\pi$$
−0.424928 + 0.905227i $$0.639701\pi$$
$$492$$ 0 0
$$493$$ −24502.1 −2.23837
$$494$$ 1374.18 0.125156
$$495$$ 0 0
$$496$$ 15315.1 1.38643
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −16233.3 −1.45632 −0.728160 0.685407i $$-0.759624\pi$$
−0.728160 + 0.685407i $$0.759624\pi$$
$$500$$ 8178.73 0.731528
$$501$$ 0 0
$$502$$ 1261.34 0.112144
$$503$$ 1664.13 0.147515 0.0737575 0.997276i $$-0.476501\pi$$
0.0737575 + 0.997276i $$0.476501\pi$$
$$504$$ 0 0
$$505$$ 6579.69 0.579787
$$506$$ 828.721 0.0728086
$$507$$ 0 0
$$508$$ 19332.7 1.68848
$$509$$ 8757.17 0.762583 0.381291 0.924455i $$-0.375479\pi$$
0.381291 + 0.924455i $$0.375479\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −9754.80 −0.842002
$$513$$ 0 0
$$514$$ −3421.86 −0.293642
$$515$$ −24004.8 −2.05394
$$516$$ 0 0
$$517$$ 9226.21 0.784851
$$518$$ 0 0
$$519$$ 0 0
$$520$$ −4141.13 −0.349232
$$521$$ −1641.01 −0.137993 −0.0689963 0.997617i $$-0.521980\pi$$
−0.0689963 + 0.997617i $$0.521980\pi$$
$$522$$ 0 0
$$523$$ 16530.6 1.38209 0.691043 0.722813i $$-0.257151\pi$$
0.691043 + 0.722813i $$0.257151\pi$$
$$524$$ 16641.5 1.38738
$$525$$ 0 0
$$526$$ 1959.69 0.162446
$$527$$ 22553.5 1.86423
$$528$$ 0 0
$$529$$ −11228.8 −0.922890
$$530$$ 3787.54 0.310416
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −10444.2 −0.848762
$$534$$ 0 0
$$535$$ −7966.77 −0.643801
$$536$$ −573.679 −0.0462298
$$537$$ 0 0
$$538$$ 3586.50 0.287407
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −3418.03 −0.271631 −0.135816 0.990734i $$-0.543365\pi$$
−0.135816 + 0.990734i $$0.543365\pi$$
$$542$$ −2368.96 −0.187741
$$543$$ 0 0
$$544$$ −8503.36 −0.670181
$$545$$ 13188.3 1.03656
$$546$$ 0 0
$$547$$ −15983.1 −1.24934 −0.624668 0.780890i $$-0.714766\pi$$
−0.624668 + 0.780890i $$0.714766\pi$$
$$548$$ 6114.94 0.476674
$$549$$ 0 0
$$550$$ 1159.63 0.0899034
$$551$$ −19842.6 −1.53416
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 2939.80 0.225452
$$555$$ 0 0
$$556$$ 18449.3 1.40724
$$557$$ 8736.81 0.664615 0.332307 0.943171i $$-0.392173\pi$$
0.332307 + 0.943171i $$0.392173\pi$$
$$558$$ 0 0
$$559$$ 2015.63 0.152508
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 3323.76 0.249474
$$563$$ −24107.2 −1.80461 −0.902306 0.431095i $$-0.858127\pi$$
−0.902306 + 0.431095i $$0.858127\pi$$
$$564$$ 0 0
$$565$$ 15604.6 1.16193
$$566$$ −1057.81 −0.0785567
$$567$$ 0 0
$$568$$ 2255.89 0.166646
$$569$$ 13291.7 0.979292 0.489646 0.871921i $$-0.337126\pi$$
0.489646 + 0.871921i $$0.337126\pi$$
$$570$$ 0 0
$$571$$ 15208.2 1.11461 0.557305 0.830308i $$-0.311836\pi$$
0.557305 + 0.830308i $$0.311836\pi$$
$$572$$ −13463.6 −0.984161
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 1312.82 0.0952147
$$576$$ 0 0
$$577$$ −10809.5 −0.779904 −0.389952 0.920835i $$-0.627508\pi$$
−0.389952 + 0.920835i $$0.627508\pi$$
$$578$$ −1133.83 −0.0815933
$$579$$ 0 0
$$580$$ 29298.3 2.09749
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 25132.3 1.78537
$$584$$ 2803.68 0.198660
$$585$$ 0 0
$$586$$ 1780.09 0.125486
$$587$$ 3843.26 0.270236 0.135118 0.990830i $$-0.456859\pi$$
0.135118 + 0.990830i $$0.456859\pi$$
$$588$$ 0 0
$$589$$ 18264.6 1.27773
$$590$$ 2769.75 0.193269
$$591$$ 0 0
$$592$$ −11553.2 −0.802084
$$593$$ −1402.87 −0.0971484 −0.0485742 0.998820i $$-0.515468\pi$$
−0.0485742 + 0.998820i $$0.515468\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 10792.2 0.741718
$$597$$ 0 0
$$598$$ 624.164 0.0426822
$$599$$ −12738.8 −0.868937 −0.434469 0.900687i $$-0.643064\pi$$
−0.434469 + 0.900687i $$0.643064\pi$$
$$600$$ 0 0
$$601$$ 17292.7 1.17368 0.586841 0.809702i $$-0.300371\pi$$
0.586841 + 0.809702i $$0.300371\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 13537.5 0.911977
$$605$$ 12891.3 0.866293
$$606$$ 0 0
$$607$$ 27165.6 1.81650 0.908251 0.418426i $$-0.137418\pi$$
0.908251 + 0.418426i $$0.137418\pi$$
$$608$$ −6886.31 −0.459336
$$609$$ 0 0
$$610$$ −1129.96 −0.0750012
$$611$$ 6948.86 0.460100
$$612$$ 0 0
$$613$$ −10405.5 −0.685599 −0.342800 0.939409i $$-0.611375\pi$$
−0.342800 + 0.939409i $$0.611375\pi$$
$$614$$ 3471.59 0.228179
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −18326.4 −1.19578 −0.597889 0.801579i $$-0.703994\pi$$
−0.597889 + 0.801579i $$0.703994\pi$$
$$618$$ 0 0
$$619$$ −3575.13 −0.232143 −0.116072 0.993241i $$-0.537030\pi$$
−0.116072 + 0.993241i $$0.537030\pi$$
$$620$$ −26968.3 −1.74689
$$621$$ 0 0
$$622$$ −2104.24 −0.135647
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −19145.5 −1.22531
$$626$$ −2340.62 −0.149441
$$627$$ 0 0
$$628$$ 8315.70 0.528396
$$629$$ −17013.6 −1.07850
$$630$$ 0 0
$$631$$ −11722.1 −0.739538 −0.369769 0.929124i $$-0.620563\pi$$
−0.369769 + 0.929124i $$0.620563\pi$$
$$632$$ 684.175 0.0430617
$$633$$ 0 0
$$634$$ 4718.15 0.295555
$$635$$ −32591.7 −2.03679
$$636$$ 0 0
$$637$$ 0 0
$$638$$ −7961.03 −0.494013
$$639$$ 0 0
$$640$$ 13455.8 0.831074
$$641$$ 24327.5 1.49903 0.749514 0.661988i $$-0.230287\pi$$
0.749514 + 0.661988i $$0.230287\pi$$
$$642$$ 0 0
$$643$$ 10684.8 0.655316 0.327658 0.944796i $$-0.393741\pi$$
0.327658 + 0.944796i $$0.393741\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −3150.23 −0.191864
$$647$$ 21203.0 1.28837 0.644184 0.764870i $$-0.277197\pi$$
0.644184 + 0.764870i $$0.277197\pi$$
$$648$$ 0 0
$$649$$ 18378.7 1.11160
$$650$$ 873.395 0.0527037
$$651$$ 0 0
$$652$$ 19007.8 1.14172
$$653$$ −27562.2 −1.65175 −0.825874 0.563854i $$-0.809318\pi$$
−0.825874 + 0.563854i $$0.809318\pi$$
$$654$$ 0 0
$$655$$ −28054.7 −1.67357
$$656$$ 16258.6 0.967670
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −1389.06 −0.0821091 −0.0410546 0.999157i $$-0.513072\pi$$
−0.0410546 + 0.999157i $$0.513072\pi$$
$$660$$ 0 0
$$661$$ −12490.6 −0.734989 −0.367494 0.930026i $$-0.619784\pi$$
−0.367494 + 0.930026i $$0.619784\pi$$
$$662$$ 683.130 0.0401066
$$663$$ 0 0
$$664$$ 7361.06 0.430218
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −9012.70 −0.523198
$$668$$ −17220.7 −0.997438
$$669$$ 0 0
$$670$$ 473.861 0.0273237
$$671$$ −7497.85 −0.431373
$$672$$ 0 0
$$673$$ −13234.5 −0.758027 −0.379014 0.925391i $$-0.623737\pi$$
−0.379014 + 0.925391i $$0.623737\pi$$
$$674$$ −2176.66 −0.124394
$$675$$ 0 0
$$676$$ 6744.28 0.383721
$$677$$ 4788.89 0.271864 0.135932 0.990718i $$-0.456597\pi$$
0.135932 + 0.990718i $$0.456597\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 9493.34 0.535372
$$681$$ 0 0
$$682$$ 7327.92 0.411438
$$683$$ 7543.01 0.422584 0.211292 0.977423i $$-0.432233\pi$$
0.211292 + 0.977423i $$0.432233\pi$$
$$684$$ 0 0
$$685$$ −10308.8 −0.575004
$$686$$ 0 0
$$687$$ 0 0
$$688$$ −3137.74 −0.173874
$$689$$ 18928.7 1.04663
$$690$$ 0 0
$$691$$ 27685.3 1.52416 0.762082 0.647481i $$-0.224177\pi$$
0.762082 + 0.647481i $$0.224177\pi$$
$$692$$ 16302.6 0.895568
$$693$$ 0 0
$$694$$ −5426.16 −0.296793
$$695$$ −31102.4 −1.69753
$$696$$ 0 0
$$697$$ 23942.9 1.30115
$$698$$ 2221.89 0.120487
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 19866.5 1.07040 0.535198 0.844726i $$-0.320237\pi$$
0.535198 + 0.844726i $$0.320237\pi$$
$$702$$ 0 0
$$703$$ −13778.2 −0.739195
$$704$$ 19054.2 1.02008
$$705$$ 0 0
$$706$$ 4285.79 0.228467
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −3572.12 −0.189216 −0.0946078 0.995515i $$-0.530160\pi$$
−0.0946078 + 0.995515i $$0.530160\pi$$
$$710$$ −1863.37 −0.0984944
$$711$$ 0 0
$$712$$ 13991.4 0.736448
$$713$$ 8295.95 0.435745
$$714$$ 0 0
$$715$$ 22697.3 1.18718
$$716$$ 12133.8 0.633327
$$717$$ 0 0
$$718$$ −5279.58 −0.274418
$$719$$ 32955.5 1.70936 0.854681 0.519154i $$-0.173753\pi$$
0.854681 + 0.519154i $$0.173753\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 1296.68 0.0668385
$$723$$ 0 0
$$724$$ 35426.4 1.81852
$$725$$ −12611.5 −0.646041
$$726$$ 0 0
$$727$$ −6347.31 −0.323808 −0.161904 0.986806i $$-0.551764\pi$$
−0.161904 + 0.986806i $$0.551764\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ −2315.85 −0.117416
$$731$$ −4620.73 −0.233795
$$732$$ 0 0
$$733$$ 22488.7 1.13321 0.566603 0.823991i $$-0.308257\pi$$
0.566603 + 0.823991i $$0.308257\pi$$
$$734$$ 3202.03 0.161020
$$735$$ 0 0
$$736$$ −3127.82 −0.156648
$$737$$ 3144.31 0.157153
$$738$$ 0 0
$$739$$ 19859.5 0.988559 0.494279 0.869303i $$-0.335432\pi$$
0.494279 + 0.869303i $$0.335432\pi$$
$$740$$ 20344.0 1.01062
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 16900.0 0.834455 0.417228 0.908802i $$-0.363002\pi$$
0.417228 + 0.908802i $$0.363002\pi$$
$$744$$ 0 0
$$745$$ −18193.8 −0.894722
$$746$$ 5374.05 0.263751
$$747$$ 0 0
$$748$$ 30864.6 1.50872
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 19441.1 0.944627 0.472314 0.881431i $$-0.343419\pi$$
0.472314 + 0.881431i $$0.343419\pi$$
$$752$$ −10817.3 −0.524558
$$753$$ 0 0
$$754$$ −5995.97 −0.289603
$$755$$ −22822.0 −1.10010
$$756$$ 0 0
$$757$$ −17481.7 −0.839343 −0.419671 0.907676i $$-0.637855\pi$$
−0.419671 + 0.907676i $$0.637855\pi$$
$$758$$ −4897.97 −0.234700
$$759$$ 0 0
$$760$$ 7688.02 0.366939
$$761$$ −4160.98 −0.198207 −0.0991035 0.995077i $$-0.531597\pi$$
−0.0991035 + 0.995077i $$0.531597\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 9927.26 0.470099
$$765$$ 0 0
$$766$$ −5398.54 −0.254644
$$767$$ 13842.2 0.651645
$$768$$ 0 0
$$769$$ −33004.0 −1.54766 −0.773832 0.633391i $$-0.781662\pi$$
−0.773832 + 0.633391i $$0.781662\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −9037.94 −0.421351
$$773$$ 29291.1 1.36291 0.681455 0.731860i $$-0.261348\pi$$
0.681455 + 0.731860i $$0.261348\pi$$
$$774$$ 0 0
$$775$$ 11608.6 0.538054
$$776$$ −10470.1 −0.484349
$$777$$ 0 0
$$778$$ −6350.28 −0.292633
$$779$$ 19389.8 0.891797
$$780$$ 0 0
$$781$$ −12364.4 −0.566495
$$782$$ −1430.87 −0.0654318
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −14018.9 −0.637395
$$786$$ 0 0
$$787$$ −7935.32 −0.359420 −0.179710 0.983720i $$-0.557516\pi$$
−0.179710 + 0.983720i $$0.557516\pi$$
$$788$$ 35050.3 1.58454
$$789$$ 0 0
$$790$$ −565.131 −0.0254512
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −5647.12 −0.252882
$$794$$ 1602.12 0.0716084
$$795$$ 0 0
$$796$$ −7577.80 −0.337422
$$797$$ 20316.5 0.902947 0.451473 0.892285i $$-0.350899\pi$$
0.451473 + 0.892285i $$0.350899\pi$$
$$798$$ 0 0
$$799$$ −15929.9 −0.705332
$$800$$ −4376.78 −0.193428
$$801$$ 0 0
$$802$$ −1108.50 −0.0488062
$$803$$ −15366.8 −0.675323
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 5519.14 0.241195
$$807$$ 0 0
$$808$$ 4468.69 0.194564
$$809$$ 12447.0 0.540931 0.270465 0.962730i $$-0.412822\pi$$
0.270465 + 0.962730i $$0.412822\pi$$
$$810$$ 0 0
$$811$$ 9478.03 0.410380 0.205190 0.978722i $$-0.434219\pi$$
0.205190 + 0.978722i $$0.434219\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ −5527.93 −0.238027
$$815$$ −32043.9 −1.37724
$$816$$ 0 0
$$817$$ −3742.02 −0.160241
$$818$$ 4625.17 0.197696
$$819$$ 0 0
$$820$$ −28629.7 −1.21926
$$821$$ 3763.09 0.159967 0.0799835 0.996796i $$-0.474513\pi$$
0.0799835 + 0.996796i $$0.474513\pi$$
$$822$$ 0 0
$$823$$ 21474.8 0.909554 0.454777 0.890605i $$-0.349719\pi$$
0.454777 + 0.890605i $$0.349719\pi$$
$$824$$ −16303.2 −0.689258
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 45304.9 1.90496 0.952481 0.304597i $$-0.0985218\pi$$
0.952481 + 0.304597i $$0.0985218\pi$$
$$828$$ 0 0
$$829$$ 4614.78 0.193339 0.0966694 0.995317i $$-0.469181\pi$$
0.0966694 + 0.995317i $$0.469181\pi$$
$$830$$ −6080.26 −0.254276
$$831$$ 0 0
$$832$$ 14351.0 0.597994
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 29031.2 1.20319
$$836$$ 24995.2 1.03406
$$837$$ 0 0
$$838$$ −7214.90 −0.297416
$$839$$ −18308.4 −0.753368 −0.376684 0.926342i $$-0.622936\pi$$
−0.376684 + 0.926342i $$0.622936\pi$$
$$840$$ 0 0
$$841$$ 62190.6 2.54995
$$842$$ 9238.22 0.378112
$$843$$ 0 0
$$844$$ −41956.1 −1.71112
$$845$$ −11369.7 −0.462876
$$846$$ 0 0
$$847$$ 0 0
$$848$$ −29466.5 −1.19326
$$849$$ 0 0
$$850$$ −2002.22 −0.0807946
$$851$$ −6258.18 −0.252089
$$852$$ 0 0
$$853$$ 13373.0 0.536790 0.268395 0.963309i $$-0.413507\pi$$
0.268395 + 0.963309i $$0.413507\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −5410.74 −0.216046
$$857$$ −17658.0 −0.703835 −0.351917 0.936031i $$-0.614470\pi$$
−0.351917 + 0.936031i $$0.614470\pi$$
$$858$$ 0 0
$$859$$ −12413.0 −0.493044 −0.246522 0.969137i $$-0.579288\pi$$
−0.246522 + 0.969137i $$0.579288\pi$$
$$860$$ 5525.23 0.219080
$$861$$ 0 0
$$862$$ 153.345 0.00605910
$$863$$ −10219.2 −0.403087 −0.201544 0.979480i $$-0.564596\pi$$
−0.201544 + 0.979480i $$0.564596\pi$$
$$864$$ 0 0
$$865$$ −27483.5 −1.08031
$$866$$ −7722.22 −0.303016
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −3749.93 −0.146384
$$870$$ 0 0
$$871$$ 2368.18 0.0921272
$$872$$ 8957.00 0.347847
$$873$$ 0 0
$$874$$ −1158.76 −0.0448464
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 952.207 0.0366633 0.0183317 0.999832i $$-0.494165\pi$$
0.0183317 + 0.999832i $$0.494165\pi$$
$$878$$ 1663.89 0.0639561
$$879$$ 0 0
$$880$$ −35333.0 −1.35350
$$881$$ 25182.5 0.963021 0.481510 0.876440i $$-0.340088\pi$$
0.481510 + 0.876440i $$0.340088\pi$$
$$882$$ 0 0
$$883$$ −3941.95 −0.150234 −0.0751172 0.997175i $$-0.523933\pi$$
−0.0751172 + 0.997175i $$0.523933\pi$$
$$884$$ 23246.1 0.884448
$$885$$ 0 0
$$886$$ −7875.45 −0.298624
$$887$$ 41630.5 1.57589 0.787946 0.615744i $$-0.211144\pi$$
0.787946 + 0.615744i $$0.211144\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ −11557.0 −0.435271
$$891$$ 0 0
$$892$$ 35555.3 1.33462
$$893$$ −12900.6 −0.483428
$$894$$ 0 0
$$895$$ −20455.6 −0.763972
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 3555.64 0.132130
$$899$$ −79694.3 −2.95657
$$900$$ 0 0
$$901$$ −43393.2 −1.60448
$$902$$ 7779.34 0.287166
$$903$$ 0 0
$$904$$ 10598.1 0.389918
$$905$$ −59722.9 −2.19365
$$906$$ 0 0
$$907$$ 34713.1 1.27081 0.635407 0.772177i $$-0.280832\pi$$
0.635407 + 0.772177i $$0.280832\pi$$
$$908$$ −18734.9 −0.684737
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 22126.6 0.804705 0.402352 0.915485i $$-0.368193\pi$$
0.402352 + 0.915485i $$0.368193\pi$$
$$912$$ 0 0
$$913$$ −40345.6 −1.46248
$$914$$ −5116.39 −0.185159
$$915$$ 0 0
$$916$$ 22644.7 0.816816
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −38198.8 −1.37112 −0.685561 0.728015i $$-0.740443\pi$$
−0.685561 + 0.728015i $$0.740443\pi$$
$$920$$ 3491.97 0.125138
$$921$$ 0 0
$$922$$ 8476.33 0.302769
$$923$$ −9312.43 −0.332094
$$924$$ 0 0
$$925$$ −8757.09 −0.311277
$$926$$ −3600.85 −0.127787
$$927$$ 0 0
$$928$$ 30047.1 1.06287
$$929$$ −13387.9 −0.472813 −0.236407 0.971654i $$-0.575970\pi$$
−0.236407 + 0.971654i $$0.575970\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ −26448.8 −0.929572
$$933$$ 0 0
$$934$$ −2414.96 −0.0846039
$$935$$ −52032.5 −1.81994
$$936$$ 0 0
$$937$$ −34574.1 −1.20543 −0.602715 0.797957i $$-0.705914\pi$$
−0.602715 + 0.797957i $$0.705914\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 19048.2 0.660939
$$941$$ −2922.81 −0.101255 −0.0506274 0.998718i $$-0.516122\pi$$
−0.0506274 + 0.998718i $$0.516122\pi$$
$$942$$ 0 0
$$943$$ 8807.01 0.304131
$$944$$ −21548.2 −0.742938
$$945$$ 0 0
$$946$$ −1501.33 −0.0515989
$$947$$ 9913.84 0.340186 0.170093 0.985428i $$-0.445593\pi$$
0.170093 + 0.985428i $$0.445593\pi$$
$$948$$ 0 0
$$949$$ −11573.8 −0.395891
$$950$$ −1621.46 −0.0553759
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −51173.4 −1.73942 −0.869710 0.493562i $$-0.835694\pi$$
−0.869710 + 0.493562i $$0.835694\pi$$
$$954$$ 0 0
$$955$$ −16735.7 −0.567073
$$956$$ 5718.02 0.193446
$$957$$ 0 0
$$958$$ −2822.16 −0.0951772
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 43565.5 1.46237
$$962$$ −4163.45 −0.139537
$$963$$ 0 0
$$964$$ −36231.3 −1.21051
$$965$$ 15236.5 0.508268
$$966$$ 0 0
$$967$$ 8758.72 0.291273 0.145637 0.989338i $$-0.453477\pi$$
0.145637 + 0.989338i $$0.453477\pi$$
$$968$$ 8755.33 0.290710
$$969$$ 0 0
$$970$$ 8648.34 0.286270
$$971$$ 8293.18 0.274089 0.137045 0.990565i $$-0.456240\pi$$
0.137045 + 0.990565i $$0.456240\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 7461.50 0.245464
$$975$$ 0 0
$$976$$ 8790.91 0.288309
$$977$$ −18969.7 −0.621181 −0.310590 0.950544i $$-0.600527\pi$$
−0.310590 + 0.950544i $$0.600527\pi$$
$$978$$ 0 0
$$979$$ −76686.4 −2.50348
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 5187.10 0.168561
$$983$$ 56313.2 1.82718 0.913588 0.406642i $$-0.133300\pi$$
0.913588 + 0.406642i $$0.133300\pi$$
$$984$$ 0 0
$$985$$ −59089.0 −1.91140
$$986$$ 13745.5 0.443960
$$987$$ 0 0
$$988$$ 18825.5 0.606193
$$989$$ −1699.66 −0.0546472
$$990$$ 0 0
$$991$$ −8558.30 −0.274332 −0.137166 0.990548i $$-0.543799\pi$$
−0.137166 + 0.990548i $$0.543799\pi$$
$$992$$ −27657.6 −0.885212
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 12774.9 0.407027
$$996$$ 0 0
$$997$$ −36163.8 −1.14876 −0.574382 0.818587i $$-0.694758\pi$$
−0.574382 + 0.818587i $$0.694758\pi$$
$$998$$ 9106.77 0.288847
$$999$$ 0 0
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 1323.4.a.bf.1.3 6
3.2 odd 2 inner 1323.4.a.bf.1.4 yes 6
7.6 odd 2 1323.4.a.bg.1.3 yes 6
21.20 even 2 1323.4.a.bg.1.4 yes 6

By twisted newform
Twist Min Dim Char Parity Ord Type
1323.4.a.bf.1.3 6 1.1 even 1 trivial
1323.4.a.bf.1.4 yes 6 3.2 odd 2 inner
1323.4.a.bg.1.3 yes 6 7.6 odd 2
1323.4.a.bg.1.4 yes 6 21.20 even 2